Neglected, semimembranosus osteochondral avulsion fracture of the posteromedial tibial plateau
Rakesh John
2018-06-01
Full Text Available Semimembranosus avulsion fracture is infrequently reported and is easy to miss on plain radiographs; the mechanism of injury is highly controversial. Initial reports linked it to anterior cruciate ligament and medial meniscal tears. We report an osteochondral semimembranosus avulsion fracture of the posteromedial tibial plateau with associated posterior cruciate ligament rupture. Also described is a novel surgical fixation technique for such osteochondral fractures where the surgical exposure is limited due to the obliquity of the fracture line resulting in a greater involvement of the articular cartilage than the small bony component. The fixation technique described may be used for osteochondral fractures where the application of a conventional compression screw may not be feasible. Keywords: Osteochondral fracture, Semimembranosus avulsion fracture, Posteromedial tibial plateau, Neglected, Nonunion
Tractography of the brainstem in major depressive disorder using diffusion tensor imaging.
Yun Ju C Song
Full Text Available BACKGROUND: The brainstem is the main region that innervates neurotransmitter release to the Hypothalamic-Pituitary Adrenal (HPA axis and fronto-limbic circuits, two key brain circuits found to be dysfunctional in Major Depressive Disorder (MDD. However, the brainstem's role in MDD has only been evaluated in limited reports. Using Diffusion Tensor Imaging (DTI, we investigated whether major brainstem white matter tracts that relate to these two circuits differ in MDD patients compared to healthy controls. METHODS: MDD patients (n = 95 and age- and gender-matched controls (n = 34 were assessed using probabilistic tractography of DTI to delineate three distinct brainstem tracts: the nigrostriatal tract (connecting brainstem to striatum, solitary tract (connecting brainstem to amygdala and corticospinal tract (connecting brainstem to precentral cortex. Fractional anisotropy (FA was used to measure the white matter integrity of these tracts, and measures were compared between MDD and control participants. RESULTS: MDD participants were characterized by a significant and specific decrease in white matter integrity of the right solitary tract (p<0.009 using independent t-test, which is a "bottom up" afferent pathway that connects the brainstem to the amygdala. This decrease was not related to symptom severity. CONCLUSIONS: The results provide new evidence to suggest that structural connectivity between the brainstem and the amygdala is altered in MDD. These results are interesting in light of predominant theories regarding amygdala-mediated emotional reactivity observed in functional imaging studies of MDD. The characterization of altered white matter integrity in the solitary tract in MDD supports the possibility of dysfunctional brainstem-amygdala connectivity impacting vulnerable circuits in MDD.
TensorLy: Tensor Learning in Python
Kossaifi, Jean; Panagakis, Yannis; Pantic, Maja
2016-01-01
Tensor methods are gaining increasing traction in machine learning. However, there are scant to no resources available to perform tensor learning and decomposition in Python. To answer this need we developed TensorLy. TensorLy is a state of the art general purpose library for tensor learning.
Peng Fang
Full Text Available Magnetic resonance imaging studies have reported significant functional and structural differences between depressed patients and controls. Little attention has been given, however, to the abnormalities in anatomical connectivity in depressed patients. In the present study, we aim to investigate the alterations in connectivity of whole-brain anatomical networks in those suffering from major depression by using machine learning approaches. Brain anatomical networks were extracted from diffusion magnetic resonance images obtained from both 22 first-episode, treatment-naive adults with major depressive disorder and 26 matched healthy controls. Using machine learning approaches, we differentiated depressed patients from healthy controls based on their whole-brain anatomical connectivity patterns and identified the most discriminating features that represent between-group differences. Classification results showed that 91.7% (patients=86.4%, controls=96.2%; permutation test, p<0.0001 of subjects were correctly classified via leave-one-out cross-validation. Moreover, the strengths of all the most discriminating connections were increased in depressed patients relative to the controls, and these connections were primarily located within the cortical-limbic network, especially the frontal-limbic network. These results not only provide initial steps toward the development of neurobiological diagnostic markers for major depressive disorder, but also suggest that abnormal cortical-limbic anatomical networks may contribute to the anatomical basis of emotional dysregulation and cognitive impairments associated with this disease.
Ota, Miho; Noda, Takamasa; Sato, Noriko; Hattori, Kotaro; Hori, Hiroaki; Sasayama, Daimei; Teraishi, Toshiya; Nagashima, Anna; Obu, Satoko; Higuchi, Teruhiko; Kunugi, Hiroshi
2015-06-01
The DSM-IV recognizes some subtypes of major depressive disorder (MDD). It is known that the effectiveness of antidepressants differs among the MDD subtypes, and thus the differentiation of the subtypes is important. However, little is known as to structural brain changes in MDD with atypical features (aMDD) in comparison with MDD with melancholic features (mMDD), which prompted us to examine possible differences in white matter integrity assessed with diffusion tensor imaging (DTI) between these two subtypes. Subjects were 21 patients with mMDD, 24 with aMDD, and 37 age- and sex-matched healthy volunteers whose DTI data were obtained by 1.5 tesla magnetic resonance imaging. We compared fractional anisotropy and mean diffusivity value derived from DTI data on a voxel-by-voxel basis among the two diagnostic groups and healthy subjects. There were significant decreases of fractional anisotropy and increases of mean diffusivity in patients with MDD compared with healthy subjects in the corpus callosum, inferior fronto-occipital fasciculus, and left superior longitudinal fasciculus. However, we detected no significant difference in any brain region between mMDD and aMDD. Our results suggest that patients with MDD had reduced white matter integrity in some regions; however, there was no major difference between aMDD and mMDD. © 2014 The Authors. Psychiatry and Clinical Neurosciences © 2014 Japanese Society of Psychiatry and Neurology.
TensorLy: Tensor Learning in Python
Kossaifi, Jean; Panagakis, Yannis; Pantic, Maja
2016-01-01
Tensors are higher-order extensions of matrices. While matrix methods form the cornerstone of machine learning and data analysis, tensor methods have been gaining increasing traction. However, software support for tensor operations is not on the same footing. In order to bridge this gap, we have developed \\emph{TensorLy}, a high-level API for tensor methods and deep tensorized neural networks in Python. TensorLy aims to follow the same standards adopted by the main projects of the Python scie...
Distal insertions of the semimembranosus tendon: MR imaging with anatomic correlation
Maeseneer, Michel de [Universitair Ziekenhuis Brussel, Department of Radiology, Jette, Brussels (Belgium); Vrije Universiteit Brussel, Department of Experimental Anatomy, Brussels (Belgium); Shahabpour, Maryam; Milants, Annemieke; Ridder, Filip de; Mey, Johan de [Universitair Ziekenhuis Brussel, Department of Radiology, Jette, Brussels (Belgium); Lenchik, Leon [Wake Forest University, Department of Radiology, Winston-Salem, NC (United States); Cattrysse, Erik [Vrije Universiteit Brussel, Department of Experimental Anatomy, Brussels (Belgium)
2014-06-15
The purpose of this study is to investigate the distal insertions of the semimembranosus tendon with MR imaging, correlated with findings in cadavers. Four fresh cadaveric specimens were studied with 3-T MR imaging. Sequences included proton density (PD) sequences (TE, 13; TR, 4957; FOV, 170 x 170; matrix, 424 x 413; NA, 2; slice thickness, 2.5 mm) in the axial, coronal, and sagittal planes and 3D fast field echo (FFE) sequences (TR 9.4; TE 6.9; FOV, 159 x 105; matrix, 200 x 211; NA, 2; slice thickness, 0.57 mm). One specimen was dissected and three specimens were sectioned with a bandsaw in the axial, coronal, and sagittal plane. The sections were photographed and correlated with MR images. To standardize the analysis, the semimembranosus muscle and tendon were assessed at seven levels for the axial sections, and at three levels for the coronal and sagittal sections. Anatomic dissection revealed six insertions of the distal semimembranosus tendon: direct arm, anterior arm, posterior oblique ligament extension, oblique popliteal ligament extension, distal tibial expansion (popliteus aponeurosis), and meniscal arm. Axial MR images showed five of six insertions: direct arm, anterior arm, oblique popliteal ligament extension, posterior oblique ligament extension, and distal tibial expansion. Sagittal MR images showed four of six insertions: direct arm, anterior arm, oblique popliteal ligament arm, and distal tibial expansion. Sagittal MR images were ideal for showing the direct arm insertion, but were less optimal than the axial images for showing the other insertions. The anterior arm was seen but volume averaging was present with the gracilis tendon. Coronal MR images optimally revealed the anterior arm, although magic angle artifact was present at its posterior aspect. The common semimembranosus tendon and meniscal arm were also well depicted. The division in anterior arm, direct arm, and oblique popliteal ligament arm was poorly seen on coronal images due to
Tensor surgery and tensor rank
M. Christandl (Matthias); J. Zuiddam (Jeroen)
2018-01-01
textabstractWe introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new vertices
Tensor surgery and tensor rank
M. Christandl (Matthias); J. Zuiddam (Jeroen)
2016-01-01
textabstractWe introduce a method for transforming low-order tensors into higher-order tensors and apply it to tensors defined by graphs and hypergraphs. The transformation proceeds according to a surgery-like procedure that splits vertices, creates and absorbs virtual edges and inserts new
The distal semimembranosus complex: normal MR anatomy, variants, biomechanics and pathology
Beltran, Javier; Jbara, Marlena; Maimon, Ron; Matityahu, Amir; Hwang, Ki; Padron, Mario; Mota, Javier; Beltran, Luis; Sundaram, Murali
2003-01-01
To describe the normal MR anatomy and variations of the distal semimembranosus tendinous arms and the posterior oblique ligament as seen in the three orthogonal planes, to review the biomechanics of this complex and to illustrate pathologic examples. The distal semimembranosus tendon divides into five tendinous arms named the anterior, direct, capsular, inferior and the oblique popliteal ligament. These arms intertwine with the branches of the posterior oblique ligament in the posterior medial aspect of the knee, providing stability. This tendon-ligamentous complex also acts synergistically with the popliteus muscle and actively pulls the posterior horn of the medial meniscus during knee flexion. Pathologic conditions involving this complex include complete and partial tears, insertional tendinosis, avulsion fractures and bursitis. (orig.)
The distal semimembranosus complex: normal MR anatomy, variants, biomechanics and pathology
Beltran, Javier; Jbara, Marlena; Maimon, Ron [Department of Radiology, Maimonides Medical Center, 4802 Tenth Avenue, NY 11219, Brooklyn (United States); Matityahu, Amir; Hwang, Ki [Department of Orthopedic Surgery, Maimonides Medical Center, Brooklyn, NY (United States); Padron, Mario [Department of Radiology, Clinica CEMTRO, Madrid (Spain); Mota, Javier [Department of Radiology, Instituto Clinica Corachan, Barcelona (Spain); Beltran, Luis [New York Medical College, Valhalla, NY (United States); Sundaram, Murali [Department of Radiology, Mayo Clinic, Rochester, MN (United States)
2003-08-01
To describe the normal MR anatomy and variations of the distal semimembranosus tendinous arms and the posterior oblique ligament as seen in the three orthogonal planes, to review the biomechanics of this complex and to illustrate pathologic examples. The distal semimembranosus tendon divides into five tendinous arms named the anterior, direct, capsular, inferior and the oblique popliteal ligament. These arms intertwine with the branches of the posterior oblique ligament in the posterior medial aspect of the knee, providing stability. This tendon-ligamentous complex also acts synergistically with the popliteus muscle and actively pulls the posterior horn of the medial meniscus during knee flexion. Pathologic conditions involving this complex include complete and partial tears, insertional tendinosis, avulsion fractures and bursitis. (orig.)
Liu, Xiaodan; Watanabe, Keita; Kakeda, Shingo; Yoshimura, Reiji; Abe, Osamu; Ide, Satoru; Hayashi, Kenji; Katsuki, Asuka; Umene-Nakano, Wakako; Watanabe, Rieko; Ueda, Issei; Nakamura, Jun; Korogi, Yukunori
2016-06-01
Higher daytime cortisol levels because of a hyperactive hypothalamic-pituitary-adrenal axis have been reported in patients with major depressive disorder (MDD). The elevated glucocorticoids inhibit the proliferation of the oligodendrocytes that are responsible for myelinating the axons of white matter fibre tracts. To evaluate the relationship between white matter integrity and serum cortisol levels during a first depressive episode in drug-naive patients with MDD (MDD group) using a tract-based spatial statistics (TBSS) method. The MDD group (n = 29) and a healthy control group (n = 47) underwent diffusion tensor imaging (DTI) scans and an analysis was conducted using TBSS. Morning blood samples were obtained from both groups for cortisol measurement. Compared with the controls, the MDD group had significantly reduced fractional anisotropy values (Plevels in the MDD group (Plevels in the MDD group may injure the white matter integrity in the frontal-subcortical and frontal-limbic circuits. © The Royal College of Psychiatrists 2016.
Zhou Yan, E-mail: clare1475@hotmail.com [Department of Radiology, Ren-Ji Hospital, Jiao Tong University Medical School, Shanghai 200127 (China); Qin Lingdi, E-mail: flyfool318@hotmail.com [Department of Radiology, Ren-Ji Hospital, Jiao Tong University Medical School, Shanghai 200127 (China); Chen Jun, E-mail: doctor_cj@msn.com [Shanghai Mental Health Center, Jiao Tong University Medical School, Shanghai, 200030 (China); Qian Lijun, E-mail: dearqlj@hotmail.com [Department of Radiology, Ren-Ji Hospital, Jiao Tong University Medical School, Shanghai 200127 (China); Tao Jing, E-mail: jing318@hotmail.com [Department of Radiology, Ren-Ji Hospital, Jiao Tong University Medical School, Shanghai 200127 (China); Fang Yiru, E-mail: fangyr@sina.com [Shanghai Mental Health Center, Jiao Tong University Medical School, Shanghai, 200030 (China); Xu Jianrong, E-mail: xujianr@hotmail.com [Department of Radiology, Ren-Ji Hospital, Jiao Tong University Medical School, Shanghai 200127 (China)
2011-11-15
Treatment-resistant depression (TRD) is a therapeutic challenge for clinicians. Despite a growing interest in this area, an understanding of the pathophysiology of depression, particularly TRD, remains lacking. This study aims to detect the white matter abnormalities of whole brain fractional anisotropy (FA) in patients with TRD compared with major depressive disorder (MDD) before treatment by voxel-based analysis using diffusion tensor imaging. A total of 100 patients first diagnosed with untreated MDD underwent diffusion tensor imaging scans. 8 weeks after the first treatment, 54 patients showed response to the medication, whereas 46 did not. Finally, 20 patients were diagnosed with TRD after undergoing another treatment. A total of 20 patients with TRD and another 20 with MDD before treatment matched in gender, age, and education was enrolled in the research. For every subject, an FA map was generated and analyzed using SPM5. Subsequently, t-test was conducted to compare the FA values voxel to voxel between the two groups (p < 0.001 [FDR corrected], t > 7.57, voxel size > 30). Voxel-based morphometric (VBM) analysis was performed using T1W images. Significant reductions in FA were found in the white matter located in the bilateral of the hippocampus (left hippocampus: t = 7.63, voxel size = 50; right hippocampus: t = 7.82, voxel size = 48). VBM analysis revealed no morphological abnormalities between the two groups. Investigation of brain anisotropy revealed significantly decreased FA in both sides of the hippocampus. Although preliminary, our findings suggest that microstructural abnormalities in the hippocampus indicate vulnerability to treatment resistance.
Zhou Yan; Qin Lingdi; Chen Jun; Qian Lijun; Tao Jing; Fang Yiru; Xu Jianrong
2011-01-01
Treatment-resistant depression (TRD) is a therapeutic challenge for clinicians. Despite a growing interest in this area, an understanding of the pathophysiology of depression, particularly TRD, remains lacking. This study aims to detect the white matter abnormalities of whole brain fractional anisotropy (FA) in patients with TRD compared with major depressive disorder (MDD) before treatment by voxel-based analysis using diffusion tensor imaging. A total of 100 patients first diagnosed with untreated MDD underwent diffusion tensor imaging scans. 8 weeks after the first treatment, 54 patients showed response to the medication, whereas 46 did not. Finally, 20 patients were diagnosed with TRD after undergoing another treatment. A total of 20 patients with TRD and another 20 with MDD before treatment matched in gender, age, and education was enrolled in the research. For every subject, an FA map was generated and analyzed using SPM5. Subsequently, t-test was conducted to compare the FA values voxel to voxel between the two groups (p 7.57, voxel size > 30). Voxel-based morphometric (VBM) analysis was performed using T1W images. Significant reductions in FA were found in the white matter located in the bilateral of the hippocampus (left hippocampus: t = 7.63, voxel size = 50; right hippocampus: t = 7.82, voxel size = 48). VBM analysis revealed no morphological abnormalities between the two groups. Investigation of brain anisotropy revealed significantly decreased FA in both sides of the hippocampus. Although preliminary, our findings suggest that microstructural abnormalities in the hippocampus indicate vulnerability to treatment resistance.
Abe, Osamu; Yamasue, Hidenori; Kasai, Kiyoto
2008-01-01
Previous neuroimaging studies have revealed that frontolimbic dysfunction may contribute to the pathophysiology of major depressive disorder. We used voxel-based analysis to simultaneously elucidate regional changes in gray/white matter volume, mean diffusivity (MD), and fractional anisotropy (FA) in the central nervous system of patients with unipolar major depression. We studied 21 right-handed patients and 42 age- and gender-matched right-handed normal subjects without central nervous system disorders. All image processing and statistical analyses were performed using SPM5 software. Local areas showing significant gray matter volume reduction in depressive patients compared with normal controls were observed in the right parahippocampal gyrus, hippocampus, bilateral middle frontal gyri, bilateral anterior cingulate cortices, left parietal and occipital lobes, and right superior temporal gyrus. Local areas showing increased mean diffusivity in depressive patients were observed in the bilateral parahippocampal gyri, hippocampus, pons, cerebellum, left frontal and temporal lobes, and right frontal lobe. There was no significant difference between the 2 groups for fractional anisotropy and white matter volume in the entire brain. Although there was no local area in which FA and MD were significantly correlated with disease severity, FA tended to correlate negatively with depression days (total accumulated days in depressive state) in the right anterior cingulate and the left frontal white matter (FDR-corrected P=0.055 for both areas). These results suggest that the frontolimbic neural circuit may play an important role in the neuropathology of patients with major depression. (author)
Delgado-Suárez, E J; Rubio-Lozano, M S; Toledo-López, V M; Torrescano-Urrutia, G R; Ponce-Alquicira, E; Huerta-Leidenz, N
2016-12-01
The study included fresh pork semimembranosus (SM, n=289) and triceps brachii (TB, n=283) muscles sourced from meat packers of Mexico and the USA. Samples were analyzed for moisture, protein, and fat content, pH, shear force (WBSF), cook loss, water holding capacity (WHC), instrumental color, emulsion capacity (EC) and stability (ES), and consumer sensory ratings. SM from the USA had lower WBSF (P0.05) across countries. TB from Mexico had higher (Ppork exhibits better technological properties, while country of origin has less effect on consumer acceptability. Copyright © 2016 The Authors. Published by Elsevier Ltd.. All rights reserved.
Semimembranosus muscle herniation: a rare case with emphasis on muscle biomechanics
Naffaa, Lena [American University of Beirut, Department of Diagnostic Radiology, P.O. Box 11-0236, Riad El-Solh, Beirut (Lebanon); Moukaddam, Hicham [Saint Rita Medical Center, Lima, OH (United States); Samim, Mohammad [New York University, Department of Radiology, Hospital for Joint Disease, New York, NY (United States); Lemieux, Aaron [University of California, San Diego School of Medicine, La Jolla, CA (United States); Smitaman, Edward [University of California, San Diego, Teleradiology and Education Center, San Diego, CA (United States)
2017-03-15
Muscle herniations are rare and most reported cases involve muscles of the lower leg. We use a case of muscle herniation involving the semimembranosus muscle, presenting as a painful mass in an adolescent male after an unspecified American football injury, to highlight a simple concept of muscle biomechanics as it pertains to muscle hernia(s): decreased traction upon muscle fibers can increase conspicuity of muscle herniation(s) - this allows a better understanding of the apt provocative maneuvers to employ, during dynamic ultrasound or magnetic resonance imaging, in order to maximize diagnostic yield and, thereby, limit patient morbidity related to any muscle herniation. Our patient subsequently underwent successful decompressive fasciotomy and has since returned to his normal daily activities. (orig.)
Semimembranosus muscle herniation: a rare case with emphasis on muscle biomechanics
Naffaa, Lena; Moukaddam, Hicham; Samim, Mohammad; Lemieux, Aaron; Smitaman, Edward
2017-01-01
Muscle herniations are rare and most reported cases involve muscles of the lower leg. We use a case of muscle herniation involving the semimembranosus muscle, presenting as a painful mass in an adolescent male after an unspecified American football injury, to highlight a simple concept of muscle biomechanics as it pertains to muscle hernia(s): decreased traction upon muscle fibers can increase conspicuity of muscle herniation(s) - this allows a better understanding of the apt provocative maneuvers to employ, during dynamic ultrasound or magnetic resonance imaging, in order to maximize diagnostic yield and, thereby, limit patient morbidity related to any muscle herniation. Our patient subsequently underwent successful decompressive fasciotomy and has since returned to his normal daily activities. (orig.)
Comparison of Magnetic Susceptibility Tensor and Diffusion Tensor of the Brain.
Li, Wei; Liu, Chunlei
2013-10-01
Susceptibility tensor imaging (STI) provides a novel approach for noninvasive assessment of the white matter pathways of the brain. Using mouse brain ex vivo , we compared STI with diffusion tensor imaging (DTI), in terms of tensor values, principal tensor values, anisotropy values, and tensor orientations. Despite the completely different biophysical underpinnings, magnetic susceptibility tensors and diffusion tensors show many similarities in the tensor and principal tensor images, for example, the tensors perpendicular to the fiber direction have the highest gray-white matter contrast, and the largest principal tensor is along the fiber direction. Comparison to DTI fractional anisotropy, the susceptibility anisotropy provides much higher sensitivity to the chemical composition of the white matter, especially myelin. The high sensitivity can be further enhanced with the perfusion of ProHance, a gadolinium-based contrast agent. Regarding the tensor orientations, the direction of the largest principal susceptibility tensor agrees with that of diffusion tensors in major white matter fiber bundles. The STI fiber tractography can reconstruct the fiber pathways for the whole corpus callosum and for white matter fiber bundles that are in close contact but in different orientations. There are some differences between susceptibility and diffusion tensor orientations, which are likely due to the limitations in the current STI reconstruction. With the development of more accurate reconstruction methods, STI holds the promise for probing the white matter micro-architectures with more anatomical details and higher chemical sensitivity.
Tensor calculus for physics a concise guide
Neuenschwander, Dwight E
2015-01-01
Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts—these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry. Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the "inertia tensor," and in electricity and magnetism...
Huijing, P A; van Lookeren Campagne, A A; Koper, J F
1989-01-01
Rat gastrocnemius medialis (GM) and semimembranosus (SM) muscles have a very different morphology. GM is a very pennate muscle, combining relatively short muscle fibre length with sizable fibre angles and long muscle and aponeurosis lengths. SM is a more parallel-fibred muscle, combining a relatively long fibre length with a small fibre angle and short aponeurosis length. The mechanisms of fibre shortening as well as angle increase are operational in GM as well as SM. However, as a consequence of isometric contraction, changes of fibre length and angle are greater for GM than for SM at any relative muscle length. These differences are particularly notable at short muscle lengths: at 80% of optimum muscle length, fibre length changes of approximately 30% are coupled to fibre angle changes of 15 degrees in GM, while for SM these changes are 4% and 0.6 degrees, respectively. A considerable difference was found for normalized active slack muscle length (GM approximately 80 and SM approximately 45%). This is explained by differences of degree of pennation as well as factors related to differences found for estimated fibre length-force characteristics. Estimated normalized active fibre slack length was considerably smaller for SM than for GM (approximately 40 and 60%, respectively). The most likely explanation of these findings are differences of distribution of optimum fibre lengths, possibly in combination with differences of myofilament lengths and/or fibre length distributions.
Subramaniyan, Sivakumar Allur; Hwang, Inho
2017-01-01
Variations in physical toughness between muscles and animals are a function of growth rate and extend of collagen type I and III. The current study was designed to investigate the ability of growth rate, collagen concentration, collagen synthesizing and degrading genes on two different fibroblast cells derived from Hanwoo m. longissimus dorsi (LD) and semimembranosus (SM) muscles. Fibroblast cell survival time was determined for understanding about the characteristics of proliferation rate between the two fibroblasts. We examined the collagen concentration and protein expression of collagen type I and III between the two fibroblasts. The mRNA expression of collagen synthesis and collagen degrading genes to elucidate the molecular mechanisms on toughness and tenderness through collagen production between the two fibroblast cells. From our results the growth rate, collagen content and protein expression of collagen type I and III were significantly higher in SM than LD muscle fibroblast. The mRNA expressions of collagen synthesized genes were increased whereas the collagen degrading genes were decreased in SM than LD muscle. Results from confocal microscopical investigation showed increased fluorescence of collagen type I and III appearing stronger in SM than LD muscle fibroblast. These results implied that the locomotion muscle had higher fibroblast growth rate, leads to produce more collagen, and cause tougher than positional muscle. This in vitro study mirrored that background toughness of various muscles in live animal is likely associated with fibroblast growth pattern, collagen synthesis and its gene expression.
Tensor Transpose and Its Properties
Pan, Ran
2014-01-01
Tensor transpose is a higher order generalization of matrix transpose. In this paper, we use permutations and symmetry group to define? the tensor transpose. Then we discuss the classification and composition of tensor transposes. Properties of tensor transpose are studied in relation to tensor multiplication, tensor eigenvalues, tensor decompositions and tensor rank.
Hess, Siegfried
2015-01-01
This book presents the science of tensors in a didactic way. The various types and ranks of tensors and the physical basis is presented. Cartesian Tensors are needed for the description of directional phenomena in many branches of physics and for the characterization the anisotropy of material properties. The first sections of the book provide an introduction to the vector and tensor algebra and analysis, with applications to physics, at undergraduate level. Second rank tensors, in particular their symmetries, are discussed in detail. Differentiation and integration of fields, including generalizations of the Stokes law and the Gauss theorem, are treated. The physics relevant for the applications in mechanics, quantum mechanics, electrodynamics and hydrodynamics is presented. The second part of the book is devoted to tensors of any rank, at graduate level. Special topics are irreducible, i.e. symmetric traceless tensors, isotropic tensors, multipole potential tensors, spin tensors, integration and spin-...
Gurau, Razvan
2017-01-01
Written by the creator of the modern theory of random tensors, this book is the first self-contained introductory text to this rapidly developing theory. Starting from notions familiar to the average researcher or PhD student in mathematical or theoretical physics, the book presents in detail the theory and its applications to physics. The recent detections of the Higgs boson at the LHC and gravitational waves at LIGO mark new milestones in Physics confirming long standing predictions of Quantum Field Theory and General Relativity. These two experimental results only reinforce today the need to find an underlying common framework of the two: the elusive theory of Quantum Gravity. Over the past thirty years, several alternatives have been proposed as theories of Quantum Gravity, chief among them String Theory. While these theories are yet to be tested experimentally, key lessons have already been learned. Whatever the theory of Quantum Gravity may be, it must incorporate random geometry in one form or another....
Tensor rank is not multiplicative under the tensor product
Christandl, Matthias; Jensen, Asger Kjærulff; Zuiddam, Jeroen
2018-01-01
The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an ℓ-tensor. The tensor product of s and t is a (k+ℓ)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection b...
Tensor rank is not multiplicative under the tensor product
M. Christandl (Matthias); A. K. Jensen (Asger Kjærulff); J. Zuiddam (Jeroen)
2018-01-01
textabstractThe tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an ℓ-tensor. The tensor product of s and t is a (k+ℓ)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the
Tensor rank is not multiplicative under the tensor product
M. Christandl (Matthias); A. K. Jensen (Asger Kjærulff); J. Zuiddam (Jeroen)
2017-01-01
textabstractThe tensor rank of a tensor is the smallest number r such that the tensor can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor (not to be confused with the "tensor Kronecker product" used in
Vladimir Milo Tomovic
2014-02-01
Full Text Available The effects of pig breed (Large White and Landrace in combination with muscle type (M. semimembranosus and M. longissimus dorsi on T45min, T24h, pH45min, pH24h, colour (CIEL*a*b* values, water-holding capacity (filter paper press method: ratio of the area of pressed meat film – M and the wet area on the filter paper – T; M/T value and moisture, protein, total fat and total ash content were investigated. Interaction effect between breed and muscle was not found (p>0.05 for any parameter. The T45min, T24h, pH45min, and M/T value were influenced by the muscle, whereas T24h was also influenced by the breed. The pH45min was higher (p<0.01 and water-holding capacity was better (p<0.001 in M. semimembranosus muscle than in M. longissimus dorsi muscle. Based on the criteria for CIEL* and M/T values, pork meat was classified into seven technological quality classes. The percentages of pale and exudative, reddish-pink and exudative, and reddish-pink and non-exudative pork were 23.5, 26.5, and 27.7%, respectively. Composition was in the characteristic range for modern lean pigs.
Tensor rank is not multiplicative under the tensor product
Christandl, Matthias; Jensen, Asger Kjærulff; Zuiddam, Jeroen
2017-01-01
The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specif...
Tensor gauge condition and tensor field decomposition
Zhu, Ben-Chao; Chen, Xiang-Song
2015-10-01
We discuss various proposals of separating a tensor field into pure-gauge and gauge-invariant components. Such tensor field decomposition is intimately related to the effort of identifying the real gravitational degrees of freedom out of the metric tensor in Einstein’s general relativity. We show that as for a vector field, the tensor field decomposition has exact correspondence to and can be derived from the gauge-fixing approach. The complication for the tensor field, however, is that there are infinitely many complete gauge conditions in contrast to the uniqueness of Coulomb gauge for a vector field. The cause of such complication, as we reveal, is the emergence of a peculiar gauge-invariant pure-gauge construction for any gauge field of spin ≥ 2. We make an extensive exploration of the complete tensor gauge conditions and their corresponding tensor field decompositions, regarding mathematical structures, equations of motion for the fields and nonlinear properties. Apparently, no single choice is superior in all aspects, due to an awkward fact that no gauge-fixing can reduce a tensor field to be purely dynamical (i.e. transverse and traceless), as can the Coulomb gauge in a vector case.
Tensor structure for Nori motives
Barbieri-Viale, Luca; Huber, Annette; Prest, Mike
2018-01-01
We construct a tensor product on Freyd's universal abelian category attached to an additive tensor category or a tensor quiver and establish a universal property. This is used to give an alternative construction for the tensor product on Nori motives.
Tensor eigenvalues and their applications
Qi, Liqun; Chen, Yannan
2018-01-01
This book offers an introduction to applications prompted by tensor analysis, especially by the spectral tensor theory developed in recent years. It covers applications of tensor eigenvalues in multilinear systems, exponential data fitting, tensor complementarity problems, and tensor eigenvalue complementarity problems. It also addresses higher-order diffusion tensor imaging, third-order symmetric and traceless tensors in liquid crystals, piezoelectric tensors, strong ellipticity for elasticity tensors, and higher-order tensors in quantum physics. This book is a valuable reference resource for researchers and graduate students who are interested in applications of tensor eigenvalues.
Hohmann, Manuel [Physikalisches Institut, Universitaet Tartu (Estonia)
2016-07-01
Tensor harmonics are a useful mathematical tool for finding solutions to differential equations which transform under a particular representation of the rotation group SO(3). In order to make use of this tool also in the setting of Finsler geometry, where the objects of relevance are d-tensors instead of tensors, we construct a set of d-tensor harmonics for both SO(3) and SO(4) symmetries and show how these can be used for calculations in Finsler geometry and gravity.
Sirlin, Samuel W.
1993-01-01
Eight-page report describes systems of notation used most commonly to represent tensors of various ranks, with emphasis on tensors in Cartesian coordinate systems. Serves as introductory or refresher text for scientists, engineers, and others familiar with basic concepts of coordinate systems, vectors, and partial derivatives. Indicial tensor, vector, dyadic, and matrix notations, and relationships among them described.
Beig, Robert; Krammer, Werner
2004-01-01
For a conformally flat 3-space, we derive a family of linear second-order partial differential operators which sends vectors into trace-free, symmetric 2-tensors. These maps, which are parametrized by conformal Killing vectors on the 3-space, are such that the divergence of the resulting tensor field depends only on the divergence of the original vector field. In particular, these maps send source-free electric fields into TT tensors. Moreover, if the original vector field is the Coulomb field on R 3 {0}, the resulting tensor fields on R 3 {0} are nothing but the family of TT tensors originally written by Bowen and York
Monitoring of color and pH in muscles of pork leg (m. adductor and m. semimembranosus
Martina Bednářová
2014-02-01
Full Text Available In order to identify PSE pork meat, pH and color testing was performed directly in a cutting plant (72 hours post mortem in this research. Specifically pork leg muscles musculi adductor (AD and semimembranosus (SM from five selected suppliers (A, B, C, D, E were examined. Twenty samples of meat for each muscle were examined from each supplier. The measured pH values ranged from 5.43 to 5.63, and the L* values from 46.13 to 57.18. No statistically significant differences in pH values and color were detected among the various suppliers with the exception of the a* and b* parameters for two suppliers, namely A and B (p<0.01. On the contrary, a statistically significant difference (p<0.5 was recorded between individual muscles (AD/SM across all the suppliers (A, B, C, D, E with the exception of a* parameter from suppliers B, C, D, E, and pH values for the E supplier. Our results revealed that individual muscles differ in values of pH and color. In comparison with literature, pH and lightness L* values in musculus adductor point to PSE (pale, soft and exudative meat, while the values of musculus semimebranosus to RFN (red, firm and non-exudative. Use of PSE meat in production of meat products can cause several problems. In particular, it causes light color, low water-holding capacity, poor fat emulsifying ability, lower yield, granular or crumbly texture and poor consistency of the finished product. Therefore classification of the meat directly cutting plant may be possible solution for this problem. The finished product pruduces from muscles of musculi semimembranosus can obtain better quality than the finished product from musculi adductor.
Schnyer, David M; Clasen, Peter C; Gonzalez, Christopher; Beevers, Christopher G
2017-06-30
Using MRI to diagnose mental disorders has been a long-term goal. Despite this, the vast majority of prior neuroimaging work has been descriptive rather than predictive. The current study applies support vector machine (SVM) learning to MRI measures of brain white matter to classify adults with Major Depressive Disorder (MDD) and healthy controls. In a precisely matched group of individuals with MDD (n =25) and healthy controls (n =25), SVM learning accurately (74%) classified patients and controls across a brain map of white matter fractional anisotropy values (FA). The study revealed three main findings: 1) SVM applied to DTI derived FA maps can accurately classify MDD vs. healthy controls; 2) prediction is strongest when only right hemisphere white matter is examined; and 3) removing FA values from a region identified by univariate contrast as significantly different between MDD and healthy controls does not change the SVM accuracy. These results indicate that SVM learning applied to neuroimaging data can classify the presence versus absence of MDD and that predictive information is distributed across brain networks rather than being highly localized. Finally, MDD group differences revealed through typical univariate contrasts do not necessarily reveal patterns that provide accurate predictive information. Copyright © 2017 Elsevier Ireland Ltd. All rights reserved.
Categorical Tensor Network States
Jacob D. Biamonte
2011-12-01
Full Text Available We examine the use of string diagrams and the mathematics of category theory in the description of quantum states by tensor networks. This approach lead to a unification of several ideas, as well as several results and methods that have not previously appeared in either side of the literature. Our approach enabled the development of a tensor network framework allowing a solution to the quantum decomposition problem which has several appealing features. Specifically, given an n-body quantum state |ψ〉, we present a new and general method to factor |ψ〉 into a tensor network of clearly defined building blocks. We use the solution to expose a previously unknown and large class of quantum states which we prove can be sampled efficiently and exactly. This general framework of categorical tensor network states, where a combination of generic and algebraically defined tensors appear, enhances the theory of tensor network states.
Efficient MATLAB computations with sparse and factored tensors.
Bader, Brett William; Kolda, Tamara Gibson (Sandia National Lab, Livermore, CA)
2006-12-01
In this paper, the term tensor refers simply to a multidimensional or N-way array, and we consider how specially structured tensors allow for efficient storage and computation. First, we study sparse tensors, which have the property that the vast majority of the elements are zero. We propose storing sparse tensors using coordinate format and describe the computational efficiency of this scheme for various mathematical operations, including those typical to tensor decomposition algorithms. Second, we study factored tensors, which have the property that they can be assembled from more basic components. We consider two specific types: a Tucker tensor can be expressed as the product of a core tensor (which itself may be dense, sparse, or factored) and a matrix along each mode, and a Kruskal tensor can be expressed as the sum of rank-1 tensors. We are interested in the case where the storage of the components is less than the storage of the full tensor, and we demonstrate that many elementary operations can be computed using only the components. All of the efficiencies described in this paper are implemented in the Tensor Toolbox for MATLAB.
Cartesian tensors an introduction
Temple, G
2004-01-01
This undergraduate text provides an introduction to the theory of Cartesian tensors, defining tensors as multilinear functions of direction, and simplifying many theorems in a manner that lends unity to the subject. The author notes the importance of the analysis of the structure of tensors in terms of spectral sets of projection operators as part of the very substance of quantum theory. He therefore provides an elementary discussion of the subject, in addition to a view of isotropic tensors and spinor analysis within the confines of Euclidean space. The text concludes with an examination of t
Linear Invariant Tensor Interpolation Applied to Cardiac Diffusion Tensor MRI
Gahm, Jin Kyu; Wisniewski, Nicholas; Kindlmann, Gordon; Kung, Geoffrey L.; Klug, William S.; Garfinkel, Alan; Ennis, Daniel B.
2015-01-01
Purpose Various methods exist for interpolating diffusion tensor fields, but none of them linearly interpolate tensor shape attributes. Linear interpolation is expected not to introduce spurious changes in tensor shape. Methods Herein we define a new linear invariant (LI) tensor interpolation method that linearly interpolates components of tensor shape (tensor invariants) and recapitulates the interpolated tensor from the linearly interpolated tensor invariants and the eigenvectors of a linearly interpolated tensor. The LI tensor interpolation method is compared to the Euclidean (EU), affine-invariant Riemannian (AI), log-Euclidean (LE) and geodesic-loxodrome (GL) interpolation methods using both a synthetic tensor field and three experimentally measured cardiac DT-MRI datasets. Results EU, AI, and LE introduce significant microstructural bias, which can be avoided through the use of GL or LI. Conclusion GL introduces the least microstructural bias, but LI tensor interpolation performs very similarly and at substantially reduced computational cost. PMID:23286085
Wit, B. de; Rocek, M.
1982-01-01
We construct a conformally invariant theory of the N = 1 supersymmetric tensor gauge multiplet and discuss the situation in N = 2. We show that our results give rise to the recently proposed variant of Poincare supergravity, and provide the complete tensor calculus for the theory. Finally, we argue that this theory cannot be quantized sensibly. (orig.)
Time integration of tensor trains
Lubich, Christian; Oseledets, Ivan; Vandereycken, Bart
2014-01-01
A robust and efficient time integrator for dynamical tensor approximation in the tensor train or matrix product state format is presented. The method is based on splitting the projector onto the tangent space of the tensor manifold. The algorithm can be used for updating time-dependent tensors in the given data-sparse tensor train / matrix product state format and for computing an approximate solution to high-dimensional tensor differential equations within this data-sparse format. The formul...
Suwandy, Via; Carne, Alan; van de Ven, Remy; Bekhit, Alaa El-Din A; Hopkins, David L
2015-02-01
The effects of pulsed electric field (PEF) and ageing (3, 7, 14 and 21 days) on the shear force, protein profile, and post-mortem proteolysis of beef loins (M. Longissimus lumborum, LL) and topsides (M. Semimembranosus, SM) were investigated using a range of pulsed electric field treatments [voltages (5 and 10 kV) and frequencies (20, 50, and 90 Hz)]. PEF treatment decreased the shear force of beef LL and SM muscles by up to 19%. The reduction in the shear force in the LL was not affected by the treatment intensity whereas the reduction in the SM was dependent on PEF frequency. PEF treated beef loins showed increased proteolysis, both early post-mortem and during subsequent post-mortem storage reflected by increased degradation of troponin-T and desmin. The most prominent troponin-T degradation was found in samples treated with 5 kV-90 Hz, 10 kV-20 Hz at day 3 and day 7 post-treatment in addition to 10 kV-50 Hz in subsequent post-treatment times. The degradation of desmin in PEF treated beef loins increased with ageing time.
Pietrasik, Z; Shand, P J
2011-05-01
The individual and combined effects of moisture enhancement with a salt/phosphate solution (ME), blade tenderization (BT), and enzyme injection with proteinases derived from Aspergillus oryzae or Bacillus subtilis on cooking properties, Warner-Bratzler shear force (WBSF), and sensory characteristics of beef semimembranosus were investigated. ME significantly (P < 0.01) reduced WBSF and increased (P < 0.05) sensory scores for juiciness and tenderness. BT increased (P < 0.05) initial and overall tenderness scores and made connective tissue less perceptible. BT combined with ME resulted in the highest initial and overall tenderness scores, however, combining ME with either proteinase was as effective for reducing WBSF and increasing tenderness, particularly at 20 (vs. 10) ppm enzyme inclusion. Tenderness of enzyme-injected steaks was increased without compromising other palatability attributes. All treatments increased the frequency of steaks rated slightly tender or higher, with the ME+BT combination, or ME with inclusion of 20 ppm of either proteinase, being most effective. Copyright © 2010 The American Meat Science Association. Published by Elsevier Ltd. All rights reserved.
Tensor spherical harmonics and tensor multipoles. II. Minkowski space
Daumens, M.; Minnaert, P.
1976-01-01
The bases of tensor spherical harmonics and of tensor multipoles discussed in the preceding paper are generalized in the Hilbert space of Minkowski tensor fields. The transformation properties of the tensor multipoles under Lorentz transformation lead to the notion of irreducible tensor multipoles. We show that the usual 4-vector multipoles are themselves irreducible, and we build the irreducible tensor multipoles of the second order. We also give their relations with the symmetric tensor multipoles defined by Zerilli for application to the gravitational radiation
Tensors and their applications
Islam, Nazrul
2006-01-01
About the Book: The book is written is in easy-to-read style with corresponding examples. The main aim of this book is to precisely explain the fundamentals of Tensors and their applications to Mechanics, Elasticity, Theory of Relativity, Electromagnetic, Riemannian Geometry and many other disciplines of science and engineering, in a lucid manner. The text has been explained section wise, every concept has been narrated in the form of definition, examples and questions related to the concept taught. The overall package of the book is highly useful and interesting for the people associated with the field. Contents: Preliminaries Tensor Algebra Metric Tensor and Riemannian Metric Christoffel`s Symbols and Covariant Differentiation Riemann-Christoffel Tensor The e-Systems and the Generalized Krönecker Deltas Geometry Analytical Mechanics Curvature of a Curve, Geodesic Parallelism of Vectors Ricci`s Coefficients of Rotation and Congruence Hyper Surfaces
Symmetric Tensor Decomposition
Brachat, Jerome; Comon, Pierre; Mourrain, Bernard
2010-01-01
We present an algorithm for decomposing a symmetric tensor, of dimension n and order d, as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables...... of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation...... of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions and for detecting the rank....
Scheunert, M.
1982-10-01
We develop a graded tensor calculus corresponding to arbitrary Abelian groups of degrees and arbitrary commutation factors. The standard basic constructions and definitions like tensor products, spaces of multilinear mappings, contractions, symmetrization, symmetric algebra, as well as the transpose, adjoint, and trace of a linear mapping, are generalized to the graded case and a multitude of canonical isomorphisms is presented. Moreover, the graded versions of the classical Lie algebras are introduced and some of their basic properties are described. (orig.)
Lazzeretti, Paolo
2018-04-01
It is shown that nonsymmetric second-rank current density tensors, related to the current densities induced by magnetic fields and nuclear magnetic dipole moments, are fundamental properties of a molecule. Together with magnetizability, nuclear magnetic shielding, and nuclear spin-spin coupling, they completely characterize its response to magnetic perturbations. Gauge invariance, resolution into isotropic, deviatoric, and antisymmetric parts, and contributions of current density tensors to magnetic properties are discussed. The components of the second-rank tensor properties are rationalized via relationships explicitly connecting them to the direction of the induced current density vectors and to the components of the current density tensors. The contribution of the deviatoric part to the average value of magnetizability, nuclear shielding, and nuclear spin-spin coupling, uniquely determined by the antisymmetric part of current density tensors, vanishes identically. The physical meaning of isotropic and anisotropic invariants of current density tensors has been investigated, and the connection between anisotropy magnitude and electron delocalization has been discussed.
A Review of Tensors and Tensor Signal Processing
Cammoun, L.; Castaño-Moraga, C. A.; Muñoz-Moreno, E.; Sosa-Cabrera, D.; Acar, B.; Rodriguez-Florido, M. A.; Brun, A.; Knutsson, H.; Thiran, J. P.
Tensors have been broadly used in mathematics and physics, since they are a generalization of scalars or vectors and allow to represent more complex properties. In this chapter we present an overview of some tensor applications, especially those focused on the image processing field. From a mathematical point of view, a lot of work has been developed about tensor calculus, which obviously is more complex than scalar or vectorial calculus. Moreover, tensors can represent the metric of a vector space, which is very useful in the field of differential geometry. In physics, tensors have been used to describe several magnitudes, such as the strain or stress of materials. In solid mechanics, tensors are used to define the generalized Hooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dynamics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics and mathematics. They have been used, for instance, in medical imaging, where we can highlight two applications: the diffusion tensor image, which represents how molecules diffuse inside the tissues and is broadly used for brain imaging; and the tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide information about the local structure or to define anisotropic image filters.
Tensor spaces and exterior algebra
Yokonuma, Takeo
1992-01-01
This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. To facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. In particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.
Tensor analysis for physicists
Schouten, J A
1989-01-01
This brilliant study by a famed mathematical scholar and former professor of mathematics at the University of Amsterdam integrates a concise exposition of the mathematical basis of tensor analysis with admirably chosen physical examples of the theory. The first five chapters incisively set out the mathematical theory underlying the use of tensors. The tensor algebra in EN and RN is developed in Chapters I and II. Chapter II introduces a sub-group of the affine group, then deals with the identification of quantities in EN. The tensor analysis in XN is developed in Chapter IV. In chapters VI through IX, Professor Schouten presents applications of the theory that are both intrinsically interesting and good examples of the use and advantages of the calculus. Chapter VI, intimately connected with Chapter III, shows that the dimensions of physical quantities depend upon the choice of the underlying group, and that tensor calculus is the best instrument for dealing with the properties of anisotropic media. In Chapte...
Generalized dielectric permittivity tensor
Borzdov, G.N.; Barkovskii, L.M.; Fedorov, F.I.
1986-01-01
The authors deal with the question of what is to be done with the formalism of the electrodynamics of dispersive media based on the introduction of dielectric-permittivity tensors for purely harmonic fields when Voigt waves and waves of more general form exist. An attempt is made to broaden and generalize the formalism to take into account dispersion of waves of the given type. In dispersive media, the polarization, magnetization, and conduction current-density vectors of point and time are determined by the values of the electromagnetic field vectors in the vicinity of this point (spatial dispersion) in the preceding instants of time (time dispersion). The dielectric-permittivity tensor and other tensors of electrodynamic parameters of the medium are introduced in terms of a set of evolution operators and not the set of harmonic function. It is noted that a magnetic-permeability tensor and an elastic-modulus tensor may be introduced for an acoustic field in dispersive anisotropic media with coupling equations of general form
Hayashi K
2014-06-01
Full Text Available Kenji Hayashi,1 Reiji Yoshimura,1 Shingo Kakeda,2 Taro Kishi,3 Osamu Abe,4 Wakako Umene-Nakano,1 Asuka Katsuki,1 Hikaru Hori,1 Atsuko Ikenouchi-Sugita,1 Keita Watanabe,2 Satoru Ide,2 Issei Ueda,2 Junji Moriya,2 Nakao Iwata,3 Yukunori Korogi,2 Marek Kubicki,5 Jun Nakamura1 1Department of Psychiatry, 2Department of Radiology, University of Occupational and Environmental Health, Kitakyushu, Japan; 3Department of Psychiatry, Fujita Health University, Toyoake, Japan; 4Department of Radiology, Nihon University School of Medicine, Tokyo, Japan; 5Psychiatry Neuroimaging Laboratory, Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA Abstract: We investigated the association between the Val158Met polymorphism of the catechol-O-methyltransferase (COMT gene, the Val66Met polymorphism of the brain-derived neurotrophic factor (BDNF gene, and white matter changes in patients with major depressive disorder (MDD and healthy subjects using diffusion tensor imaging (DTI. We studied 30 patients with MDD (17 males and 13 females, with mean age ± standard deviation [SD] =44±12 years and 30 sex- and age-matched healthy controls (17 males and 13 females, aged 44±13 years. Using DTI analysis with a tract-based spatial statistics (TBSS approach, we investigated the differences in fractional anisotropy, radial diffusivity, and axial diffusivity distribution among the three groups (patients with the COMT gene Val158Met, those with the BDNF gene Val66Met, and the healthy subjects. In a voxel-wise-based group comparison, we found significant decreases in fractional anisotropy and axial diffusivity within the temporal lobe white matter in the Met-carriers with MDD compared with the controls (P<0.05. No correlations in fractional anisotropy, axial diffusivity, or radial diffusivity were observed between the MDD patients and the controls, either among those with the BDNF Val/Val genotype or among the BDNF Met-carriers. These results suggest an association
Killing tensors and conformal Killing tensors from conformal Killing vectors
Rani, Raffaele; Edgar, S Brian; Barnes, Alan
2003-01-01
Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition, we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate that it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors, and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors
Chatzistavrakidis, Athanasios [Van Swinderen Institute for Particle Physics and Gravity, University of Groningen,Nijenborgh 4, 9747 AG Groningen (Netherlands); Khoo, Fech Scen [Department of Physics and Earth Sciences, Jacobs University Bremen,Campus Ring 1, 28759 Bremen (Germany); Roest, Diederik [Van Swinderen Institute for Particle Physics and Gravity, University of Groningen,Nijenborgh 4, 9747 AG Groningen (Netherlands); Schupp, Peter [Department of Physics and Earth Sciences, Jacobs University Bremen,Campus Ring 1, 28759 Bremen (Germany)
2017-03-13
The particular structure of Galileon interactions allows for higher-derivative terms while retaining second order field equations for scalar fields and Abelian p-forms. In this work we introduce an index-free formulation of these interactions in terms of two sets of Grassmannian variables. We employ this to construct Galileon interactions for mixed-symmetry tensor fields and coupled systems thereof. We argue that these tensors are the natural generalization of scalars with Galileon symmetry, similar to p-forms and scalars with a shift-symmetry. The simplest case corresponds to linearised gravity with Lovelock invariants, relating the Galileon symmetry to diffeomorphisms. Finally, we examine the coupling of a mixed-symmetry tensor to gravity, and demonstrate in an explicit example that the inclusion of appropriate counterterms retains second order field equations.
Tensors, relativity, and cosmology
Dalarsson, Mirjana
2015-01-01
Tensors, Relativity, and Cosmology, Second Edition, combines relativity, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject that is followed by detailed mathematical derivations. The book includes a section on general relativity that gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes and Penrose processes), and considers the energy-momentum tensor for various solutions. In addition, a section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects, with a final section on cosmology discussing cosmological models, observational tests, and scenarios for the early universe. This fully revised and updated second edition includes new material on relativistic effects, such as the behavior of clocks and measuring rods in m...
Alsing, Paul M; McDonald, Jonathan R; Miller, Warner A
2011-01-01
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The three-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The four-dimensional Ric is the Einstein tensor for such spacetimes. More recently, the Ric was used by Hamilton to define a nonlinear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincare conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area-an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimensions.
Alsing, Paul M.; McDonald, Jonathan R.; Miller, Warner A.
2011-08-01
The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The three-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The four-dimensional Ric is the Einstein tensor for such spacetimes. More recently, the Ric was used by Hamilton to define a nonlinear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincarè conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area—an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimensions.
Ziegel, Johanna; Nyengaard, Jens Randel; Jensen, Eva B. Vedel
In the present paper, statistical procedures for estimating shape and orientation of arbitrary three-dimensional particles are developed. The focus of this work is on the case where the particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle s...
The evolution of tensor polarization
Huang, H.; Lee, S.Y.; Ratner, L.
1993-01-01
By using the equation of motion for the vector polarization, the spin transfer matrix for spin tensor polarization, the spin transfer matrix for spin tensor polarization is derived. The evolution equation for the tensor polarization is studied in the presence of an isolate spin resonance and in the presence of a spin rotor, or snake
Tensor Calculus: Unlearning Vector Calculus
Lee, Wha-Suck; Engelbrecht, Johann; Moller, Rita
2018-01-01
Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can…
Diffusion tensor image registration using hybrid connectivity and tensor features.
Wang, Qian; Yap, Pew-Thian; Wu, Guorong; Shen, Dinggang
2014-07-01
Most existing diffusion tensor imaging (DTI) registration methods estimate structural correspondences based on voxelwise matching of tensors. The rich connectivity information that is given by DTI, however, is often neglected. In this article, we propose to integrate complementary information given by connectivity features and tensor features for improved registration accuracy. To utilize connectivity information, we place multiple anchors representing different brain anatomies in the image space, and define the connectivity features for each voxel as the geodesic distances from all anchors to the voxel under consideration. The geodesic distance, which is computed in relation to the tensor field, encapsulates information of brain connectivity. We also extract tensor features for every voxel to reflect the local statistics of tensors in its neighborhood. We then combine both connectivity features and tensor features for registration of tensor images. From the images, landmarks are selected automatically and their correspondences are determined based on their connectivity and tensor feature vectors. The deformation field that deforms one tensor image to the other is iteratively estimated and optimized according to the landmarks and their associated correspondences. Experimental results show that, by using connectivity features and tensor features simultaneously, registration accuracy is increased substantially compared with the cases using either type of features alone. Copyright © 2013 Wiley Periodicals, Inc.
Gogny interactions with tensor terms
Anguiano, M.; Lallena, A.M.; Bernard, R.N. [Universidad de Granada, Departamento de Fisica Atomica, Molecular y Nuclear, Granada (Spain); Co' , G. [INFN, Lecce (Italy); De Donno, V. [Universita del Salento, Dipartimento di Matematica e Fisica ' ' E. De Giorgi' ' , Lecce (Italy); Grasso, M. [Universite Paris-Sud, Institut de Physique Nucleaire, IN2P3-CNRS, Orsay (France)
2016-07-15
We present a perturbative approach to include tensor terms in the Gogny interaction. We do not change the values of the usual parameterisations, with the only exception of the spin-orbit term, and we add tensor terms whose only free parameters are the strengths of the interactions. We identify observables sensitive to the presence of the tensor force in Hartree-Fock, Hartree-Fock-Bogoliubov and random phase approximation calculations. We show the need of including two tensor contributions, at least: a pure tensor term and a tensor-isospin term. We show results relevant for the inclusion of the tensor term for single-particle energies, charge-conserving magnetic excitations and Gamow-Teller excitations. (orig.)
The geomagnetic field gradient tensor
Kotsiaros, Stavros; Olsen, Nils
2012-01-01
We develop the general mathematical basis for space magnetic gradiometry in spherical coordinates. The magnetic gradient tensor is a second rank tensor consisting of 3 × 3 = 9 spatial derivatives. Since the geomagnetic field vector B is always solenoidal (∇ · B = 0) there are only eight independent...... tensor elements. Furthermore, in current free regions the magnetic gradient tensor becomes symmetric, further reducing the number of independent elements to five. In that case B is a Laplacian potential field and the gradient tensor can be expressed in series of spherical harmonics. We present properties...... of the magnetic gradient tensor and provide explicit expressions of its elements in terms of spherical harmonics. Finally we discuss the benefit of using gradient measurements for exploring the Earth’s magnetic field from space, in particular the advantage of the various tensor elements for a better determination...
Littlejohn, R.G.
1982-01-01
The Hamiltonian structures discovered by Morrison and Greene for various fluid equations were obtained by guessing a Hamiltonian and a suitable Poisson bracket formula, expressed in terms of noncanonical (but physical) coordinates. In general, such a procedure for obtaining a Hamiltonian system does not produce a Hamiltonian phase space in the usual sense (a symplectic manifold), but rather a family of symplectic manifolds. To state the matter in terms of a system with a finite number of degrees of freedom, the family of symplectic manifolds is parametrized by a set of Casimir functions, which are characterized by having vanishing Poisson brackets with all other functions. The number of independent Casimir functions is the corank of the Poisson tensor J/sup ij/, the components of which are the Poisson brackets of the coordinates among themselves. Thus, these Casimir functions exist only when the Poisson tensor is singular
Dillon, Joshua V.; Langmore, Ian; Tran, Dustin; Brevdo, Eugene; Vasudevan, Srinivas; Moore, Dave; Patton, Brian; Alemi, Alex; Hoffman, Matt; Saurous, Rif A.
2017-01-01
The TensorFlow Distributions library implements a vision of probability theory adapted to the modern deep-learning paradigm of end-to-end differentiable computation. Building on two basic abstractions, it offers flexible building blocks for probabilistic computation. Distributions provide fast, numerically stable methods for generating samples and computing statistics, e.g., log density. Bijectors provide composable volume-tracking transformations with automatic caching. Together these enable...
The tensor distribution function.
Leow, A D; Zhu, S; Zhan, L; McMahon, K; de Zubicaray, G I; Meredith, M; Wright, M J; Toga, A W; Thompson, P M
2009-01-01
Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of six directions, second-order tensors (represented by three-by-three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high-angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues.
Tensor Permutation Matrices in Finite Dimensions
Christian, Rakotonirina
2005-01-01
We have generalised the properties with the tensor product, of one 4x4 matrix which is a permutation matrix, and we call a tensor commutation matrix. Tensor commutation matrices can be constructed with or without calculus. A formula allows us to construct a tensor permutation matrix, which is a generalisation of tensor commutation matrix, has been established. The expression of an element of a tensor commutation matrix has been generalised in the case of any element of a tensor permutation ma...
Tensor Factorization for Low-Rank Tensor Completion.
Zhou, Pan; Lu, Canyi; Lin, Zhouchen; Zhang, Chao
2018-03-01
Recently, a tensor nuclear norm (TNN) based method was proposed to solve the tensor completion problem, which has achieved state-of-the-art performance on image and video inpainting tasks. However, it requires computing tensor singular value decomposition (t-SVD), which costs much computation and thus cannot efficiently handle tensor data, due to its natural large scale. Motivated by TNN, we propose a novel low-rank tensor factorization method for efficiently solving the 3-way tensor completion problem. Our method preserves the low-rank structure of a tensor by factorizing it into the product of two tensors of smaller sizes. In the optimization process, our method only needs to update two smaller tensors, which can be more efficiently conducted than computing t-SVD. Furthermore, we prove that the proposed alternating minimization algorithm can converge to a Karush-Kuhn-Tucker point. Experimental results on the synthetic data recovery, image and video inpainting tasks clearly demonstrate the superior performance and efficiency of our developed method over state-of-the-arts including the TNN and matricization methods.
Tensor Train Neighborhood Preserving Embedding
Wang, Wenqi; Aggarwal, Vaneet; Aeron, Shuchin
2018-05-01
In this paper, we propose a Tensor Train Neighborhood Preserving Embedding (TTNPE) to embed multi-dimensional tensor data into low dimensional tensor subspace. Novel approaches to solve the optimization problem in TTNPE are proposed. For this embedding, we evaluate novel trade-off gain among classification, computation, and dimensionality reduction (storage) for supervised learning. It is shown that compared to the state-of-the-arts tensor embedding methods, TTNPE achieves superior trade-off in classification, computation, and dimensionality reduction in MNIST handwritten digits and Weizmann face datasets.
Notes on super Killing tensors
Howe, P.S. [Department of Mathematics, King’s College London,The Strand, London WC2R 2LS (United Kingdom); Lindström, University [Department of Physics and Astronomy, Theoretical Physics, Uppsala University,SE-751 20 Uppsala (Sweden); Theoretical Physics, Imperial College London,Prince Consort Road, London SW7 2AZ (United Kingdom)
2016-03-14
The notion of a Killing tensor is generalised to a superspace setting. Conserved quantities associated with these are defined for superparticles and Poisson brackets are used to define a supersymmetric version of the even Schouten-Nijenhuis bracket. Superconformal Killing tensors in flat superspaces are studied for spacetime dimensions 3,4,5,6 and 10. These tensors are also presented in analytic superspaces and super-twistor spaces for 3,4 and 6 dimensions. Algebraic structures associated with superconformal Killing tensors are also briefly discussed.
Tensor norms and operator ideals
Defant, A; Floret, K
1992-01-01
The three chapters of this book are entitled Basic Concepts, Tensor Norms, and Special Topics. The first may serve as part of an introductory course in Functional Analysis since it shows the powerful use of the projective and injective tensor norms, as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the Resumé and deals with the relation between tensor norms and operator ideals. The last chapter deals with special questions. Each section is accompanied by a series of exer
Typesafe Abstractions for Tensor Operations
Chen, Tongfei
2017-01-01
We propose a typesafe abstraction to tensors (i.e. multidimensional arrays) exploiting the type-level programming capabilities of Scala through heterogeneous lists (HList), and showcase typesafe abstractions of common tensor operations and various neural layers such as convolution or recurrent neural networks. This abstraction could lay the foundation of future typesafe deep learning frameworks that runs on Scala/JVM.
Indicial tensor manipulation on MACSYMA
Bogen, R.A.; Pavelle, R.
1977-01-01
A new computational tool for physical calculations is described. It is the first computer system capable of performing indicial tensor calculus (as opposed to component tensor calculus). It is now operational on the symbolic manipulation system MACSYMA. The authors outline the capabilities of the system and describe some of the physical problems considered as well as others being examined at this time. (Auth.)
Killing-Yano tensors and Nambu mechanics
Baleanu, D.
1998-01-01
Killing-Yano tensors were introduced in 1952 by Kentaro-Yano from mathematical point of view. The physical interpretation of Killing-Yano tensors of rank higher than two was unclear. We found that all Killing-Yano tensors η i 1 i 2 . .. i n with covariant derivative zero are Nambu tensors. We found that in the case of flat space case all Killing-Yano tensors are Nambu tensors. In the case of Taub-NUT and Kerr-Newmann metric Killing-Yano tensors of order two generate Nambu tensors of rank 3
A General Sparse Tensor Framework for Electronic Structure Theory.
Manzer, Samuel; Epifanovsky, Evgeny; Krylov, Anna I; Head-Gordon, Martin
2017-03-14
Linear-scaling algorithms must be developed in order to extend the domain of applicability of electronic structure theory to molecules of any desired size. However, the increasing complexity of modern linear-scaling methods makes code development and maintenance a significant challenge. A major contributor to this difficulty is the lack of robust software abstractions for handling block-sparse tensor operations. We therefore report the development of a highly efficient symbolic block-sparse tensor library in order to provide access to high-level software constructs to treat such problems. Our implementation supports arbitrary multi-dimensional sparsity in all input and output tensors. We avoid cumbersome machine-generated code by implementing all functionality as a high-level symbolic C++ language library and demonstrate that our implementation attains very high performance for linear-scaling sparse tensor contractions.
MATLAB tensor classes for fast algorithm prototyping.
Bader, Brett William; Kolda, Tamara Gibson (Sandia National Laboratories, Livermore, CA)
2004-10-01
Tensors (also known as mutidimensional arrays or N-way arrays) are used in a variety of applications ranging from chemometrics to psychometrics. We describe four MATLAB classes for tensor manipulations that can be used for fast algorithm prototyping. The tensor class extends the functionality of MATLAB's multidimensional arrays by supporting additional operations such as tensor multiplication. The tensor as matrix class supports the 'matricization' of a tensor, i.e., the conversion of a tensor to a matrix (and vice versa), a commonly used operation in many algorithms. Two additional classes represent tensors stored in decomposed formats: cp tensor and tucker tensor. We descibe all of these classes and then demonstrate their use by showing how to implement several tensor algorithms that have appeared in the literature.
Efficient tensor completion for color image and video recovery: Low-rank tensor train
Bengua, Johann A.; Phien, Ho N.; Tuan, Hoang D.; Do, Minh N.
2016-01-01
This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks to its definition from a well-balanced matricization scheme. Accordingly, new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution. The first one called simple low-rank tensor completion via tensor tra...
Random SU(2) invariant tensors
Li, Youning; Han, Muxin; Ruan, Dong; Zeng, Bei
2018-04-01
SU(2) invariant tensors are states in the (local) SU(2) tensor product representation but invariant under the global group action. They are of importance in the study of loop quantum gravity. A random tensor is an ensemble of tensor states. An average over the ensemble is carried out when computing any physical quantities. The random tensor exhibits a phenomenon known as ‘concentration of measure’, which states that for any bipartition the average value of entanglement entropy of its reduced density matrix is asymptotically the maximal possible as the local dimensions go to infinity. We show that this phenomenon is also true when the average is over the SU(2) invariant subspace instead of the entire space for rank-n tensors in general. It is shown in our earlier work Li et al (2017 New J. Phys. 19 063029) that the subleading correction of the entanglement entropy has a mild logarithmic divergence when n = 4. In this paper, we show that for n > 4 the subleading correction is not divergent but a finite number. In some special situation, the number could be even smaller than 1/2, which is the subleading correction of random state over the entire Hilbert space of tensors.
Tensor Product of Polygonal Cell Complexes
Chien, Yu-Yen
2017-01-01
We introduce the tensor product of polygonal cell complexes, which interacts nicely with the tensor product of link graphs of complexes. We also develop the unique factorization property of polygonal cell complexes with respect to the tensor product, and study the symmetries of tensor products of polygonal cell complexes.
The Einstein tensor characterizing some Riemann spaces
Rahman, M.S.
1993-07-01
A formal definition of the Einstein tensor is given. Mention is made of how this tensor plays a role of expressing certain conditions in a precise form. The cases of reducing the Einstein tensor to a zero tensor are studied on its merit. A lucid account of results, formulated as theorems, on Einstein symmetric and Einstein recurrent spaces is then presented. (author). 5 refs
Colored Tensor Models - a Review
Razvan Gurau
2012-04-01
Full Text Available Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a 1/N expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions, non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.
Thermodynamical inequivalence of quantum stress-energy and spin tensors
Becattini, F.; Tinti, L.
2011-01-01
It is shown that different couples of stress-energy and spin tensors of quantum-relativistic fields, which would be otherwise equivalent, are in fact inequivalent if the second law of thermodynamics is taken into account. The proof of the inequivalence is based on the analysis of a macroscopic system at full thermodynamical equilibrium with a macroscopic total angular momentum and a specific instance is given for the free Dirac field, for which we show that the canonical and Belinfante stress-energy tensors are not equivalent. For this particular case, we show that the difference between the predicted angular momentum densities for a rotating system at full thermodynamical equilibrium is a quantum effect, persisting in the nonrelativistic limit, corresponding to a polarization of particles of the order of (ℎ/2π)ω/KT (ω being the angular velocity) and could in principle be measured experimentally. This result implies that specific stress-energy and spin tensors are physically meaningful even in the absence of gravitational coupling and raises the issue of finding the thermodynamically right (or the right class of) tensors. We argue that the maximization of the thermodynamic potential theoretically allows us to discriminate between two different couples, yet for the present we are unable to provide a theoretical method to single out the best couple of tensors in a given quantum field theory. The existence of a nonvanishing spin tensor would have major consequences in hydrodynamics, gravity and cosmology.
Tensor-GMRES method for large sparse systems of nonlinear equations
Feng, Dan; Pulliam, Thomas H.
1994-01-01
This paper introduces a tensor-Krylov method, the tensor-GMRES method, for large sparse systems of nonlinear equations. This method is a coupling of tensor model formation and solution techniques for nonlinear equations with Krylov subspace projection techniques for unsymmetric systems of linear equations. Traditional tensor methods for nonlinear equations are based on a quadratic model of the nonlinear function, a standard linear model augmented by a simple second order term. These methods are shown to be significantly more efficient than standard methods both on nonsingular problems and on problems where the Jacobian matrix at the solution is singular. A major disadvantage of the traditional tensor methods is that the solution of the tensor model requires the factorization of the Jacobian matrix, which may not be suitable for problems where the Jacobian matrix is large and has a 'bad' sparsity structure for an efficient factorization. We overcome this difficulty by forming and solving the tensor model using an extension of a Newton-GMRES scheme. Like traditional tensor methods, we show that the new tensor method has significant computational advantages over the analogous Newton counterpart. Consistent with Krylov subspace based methods, the new tensor method does not depend on the factorization of the Jacobian matrix. As a matter of fact, the Jacobian matrix is never needed explicitly.
Tensor Completion Algorithms in Big Data Analytics
Song, Qingquan; Ge, Hancheng; Caverlee, James; Hu, Xia
2017-01-01
Tensor completion is a problem of filling the missing or unobserved entries of partially observed tensors. Due to the multidimensional character of tensors in describing complex datasets, tensor completion algorithms and their applications have received wide attention and achievement in areas like data mining, computer vision, signal processing, and neuroscience. In this survey, we provide a modern overview of recent advances in tensor completion algorithms from the perspective of big data an...
Development of the Tensoral Computer Language
Ferziger, Joel; Dresselhaus, Eliot
1996-01-01
The research scientist or engineer wishing to perform large scale simulations or to extract useful information from existing databases is required to have expertise in the details of the particular database, the numerical methods and the computer architecture to be used. This poses a significant practical barrier to the use of simulation data. The goal of this research was to develop a high-level computer language called Tensoral, designed to remove this barrier. The Tensoral language provides a framework in which efficient generic data manipulations can be easily coded and implemented. First of all, Tensoral is general. The fundamental objects in Tensoral represent tensor fields and the operators that act on them. The numerical implementation of these tensors and operators is completely and flexibly programmable. New mathematical constructs and operators can be easily added to the Tensoral system. Tensoral is compatible with existing languages. Tensoral tensor operations co-exist in a natural way with a host language, which may be any sufficiently powerful computer language such as Fortran, C, or Vectoral. Tensoral is very-high-level. Tensor operations in Tensoral typically act on entire databases (i.e., arrays) at one time and may, therefore, correspond to many lines of code in a conventional language. Tensoral is efficient. Tensoral is a compiled language. Database manipulations are simplified optimized and scheduled by the compiler eventually resulting in efficient machine code to implement them.
Fowler, Stephanie M; Schmidt, Heinar; van de Ven, Remy; Wynn, Peter; Hopkins, David L
2015-10-01
Complementary studies were conducted to determine the potential for a Raman spectroscopic hand held device to predict meat quality traits of fresh lamb m. semimembranosus (topside) after ageing and freezing/thawing. Spectra were collected from 80 fresh muscles at 24h and 5d PM, another 80 muscles were measured at 24h, 5d and following freezing/thawing. Shear force, cooking loss, sarcomere length, colour, particle size, collagen content, pH24, pHu, purge and thaw loss were also measured. Results indicated a potential to predict pHu (R(2)cv=0.59), pH24 (R(2)cv=0.48) and purge (R(2)cv=0.42) using spectra collected 24h PM. L* could be predicted using spectra collected 24h (R(2)cv=0.33) or 5d PM (R(2)cv=0.33). This suggests that Raman spectroscopy is suited to identifying carcases which deviate from the normal metabolic processes and related meat quality traits. Copyright © 2015. Published by Elsevier Ltd.
Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature
Loveridge, Lee C.
2004-01-01
Various interpretations of the Riemann Curvature Tensor, Ricci Tensor, and Scalar Curvature are described. Also, the physical meanings of the Einstein Tensor and Einstein's Equations are discussed. Finally a derivation of Newtonian Gravity from Einstein's Equations is given.
The tensor rank of tensor product of two three-qubit W states is eight
Chen, Lin; Friedland, Shmuel
2017-01-01
We show that the tensor rank of tensor product of two three-qubit W states is not less than eight. Combining this result with the recent result of M. Christandl, A. K. Jensen, and J. Zuiddam that the tensor rank of tensor product of two three-qubit W states is at most eight, we deduce that the tensor rank of tensor product of two three-qubit W states is eight. We also construct the upper bound of the tensor rank of tensor product of many three-qubit W states.
Link prediction via generalized coupled tensor factorisation
Ermiş, Beyza; Evrim, Acar Ataman; Taylan Cemgil, A.
2012-01-01
and higher-order tensors. We propose to use an approach based on probabilistic interpretation of tensor factorisation models, i.e., Generalised Coupled Tensor Factorisation, which can simultaneously fit a large class of tensor models to higher-order tensors/matrices with com- mon latent factors using...... different loss functions. Numerical experiments demonstrate that joint analysis of data from multiple sources via coupled factorisation improves the link prediction performance and the selection of right loss function and tensor model is crucial for accurately predicting missing links....
Spectral Tensor-Train Decomposition
Bigoni, Daniele; Engsig-Karup, Allan Peter; Marzouk, Youssef M.
2016-01-01
The accurate approximation of high-dimensional functions is an essential task in uncertainty quantification and many other fields. We propose a new function approximation scheme based on a spectral extension of the tensor-train (TT) decomposition. We first define a functional version of the TT...... adaptive Smolyak approach. The method is also used to approximate the solution of an elliptic PDE with random input data. The open source software and examples presented in this work are available online (http://pypi.python.org/pypi/TensorToolbox/)....
Confinement through tensor gauge fields
Salam, A.; Strathdee, J.
1977-12-01
Using the 0(3,2)-symmetric de Sitter solution of Einstein's equation describing a strongly interacting tensor field it is shown that hadronic bags confining quarks can be represented as de Sitter ''micro-universes'' with radii given 1/R 2 =lambdak 2 /6. Here k 2 and lambda are the strong coupling and the ''cosmological'' constant which apear in the Einstein equation used. Surprisingly the energy spectrum for the two-body hadronic states is the same as that for a harmonic oscillator potential, though the wave functions are completely different. The Einstein equation can be extended to include colour for the tensor fields
Tensor product of quantum logics
Pulmannová, Sylvia
1985-01-01
A quantum logic is the couple (L,M) where L is an orthomodular σ-lattice and M is a strong set of states on L. The Jauch-Piron property in the σ-form is also supposed for any state of M. A ``tensor product'' of quantum logics is defined. This definition is compared with the definition of a free orthodistributive product of orthomodular σ-lattices. The existence and uniqueness of the tensor product in special cases of Hilbert space quantum logics and one quantum and one classical logic are studied.
The 'gravitating' tensor in the dualistic theory
Mahanta, M.N.
1989-01-01
The exact microscopic system of Einstein-type field equations of the dualistic gravitation theory is investigated as well as an analysis of the modified energy-momentum tensor or so called 'gravitating' tensor is presented
Mean template for tensor-based morphometry using deformation tensors.
Leporé, Natasha; Brun, Caroline; Pennec, Xavier; Chou, Yi-Yu; Lopez, Oscar L; Aizenstein, Howard J; Becker, James T; Toga, Arthur W; Thompson, Paul M
2007-01-01
Tensor-based morphometry (TBM) studies anatomical differences between brain images statistically, to identify regions that differ between groups, over time, or correlate with cognitive or clinical measures. Using a nonlinear registration algorithm, all images are mapped to a common space, and statistics are most commonly performed on the Jacobian determinant (local expansion factor) of the deformation fields. In, it was shown that the detection sensitivity of the standard TBM approach could be increased by using the full deformation tensors in a multivariate statistical analysis. Here we set out to improve the common space itself, by choosing the shape that minimizes a natural metric on the deformation tensors from that space to the population of control subjects. This method avoids statistical bias and should ease nonlinear registration of new subjects data to a template that is 'closest' to all subjects' anatomies. As deformation tensors are symmetric positive-definite matrices and do not form a vector space, all computations are performed in the log-Euclidean framework. The control brain B that is already the closest to 'average' is found. A gradient descent algorithm is then used to perform the minimization that iteratively deforms this template and obtains the mean shape. We apply our method to map the profile of anatomical differences in a dataset of 26 HIV/AIDS patients and 14 controls, via a log-Euclidean Hotelling's T2 test on the deformation tensors. These results are compared to the ones found using the 'best' control, B. Statistics on both shapes are evaluated using cumulative distribution functions of the p-values in maps of inter-group differences.
Reciprocal mass tensor : a general form
Roy, C.L.
1978-01-01
Using the results of earlier treatment of wave packets, a general form of reciprocal mass tensor has been obtained. The elements of this tensor are seen to be dependent on momentum as well as space coordinates of the particle under consideration. The conditions under which the tensor would reduce to the usual space-independent form, are discussed and the impact of the space-dependence of this tensor on the motion of Bloch electrons, is examined. (author)
A new deteriorated energy-momentum tensor
Duff, M.J.
1982-01-01
The stress-tensor of a scalar field theory is not unique because of the possibility of adding an 'improvement term'. In supersymmetric field theories the stress-tensor will appear in a super-current multiplet along with the sypersymmetry current. The general question of the supercurrent multiplet for arbitrary deteriorated stress tensors and their relationship to supercurrent multiplets for models with gauge antisymmetric tensors is answered for various models of N = 1, 2 and 4 supersymmetry. (U.K.)
Tensor-based spatiotemporal saliency detection
Dou, Hao; Li, Bin; Deng, Qianqian; Zhang, LiRui; Pan, Zhihong; Tian, Jinwen
2018-03-01
This paper proposes an effective tensor-based spatiotemporal saliency computation model for saliency detection in videos. First, we construct the tensor representation of video frames. Then, the spatiotemporal saliency can be directly computed by the tensor distance between different tensors, which can preserve the complete temporal and spatial structure information of object in the spatiotemporal domain. Experimental results demonstrate that our method can achieve encouraging performance in comparison with the state-of-the-art methods.
Akkerman, Erik M.
2010-01-01
Both in diffusion tensor imaging (DTI) and in generalized diffusion tensor imaging (GDTI) the relation between the diffusion tensor and the measured apparent diffusion coefficients is given by a tensorial equation, which needs to be inverted in order to solve the diffusion tensor. The traditional
Efficient Tensor Strategy for Recommendation
Aboagye Emelia Opoku
2017-07-01
Full Text Available The era of big data has witnessed the explosion of tensor datasets, and large scale Probabilistic Tensor Factorization (PTF analysis is important to accommodate such increasing trend of data. Sparsity, and Cold-Start are some of the inherent problems of recommender systems in the era of big data. This paper proposes a novel Sentiment-Based Probabilistic Tensor Analysis technique senti-PTF to address the problems. The propose framework first applies a Natural Language Processing technique to perform sentiment analysis taking advantage of the huge sums of textual data generated available from the social media which are predominantly left untouched. Although some current studies do employ review texts, many of them do not consider how sentiments in reviews influence recommendation algorithm for prediction. There is therefore this big data text analytics gap whose modeling is computationally expensive. From our experiments, our novel machine learning sentiment-based tensor analysis is computationally less expensive, and addresses the cold-start problem, for optimal recommendation prediction.
Weyl tensors for asymmetric complex curvatures
Oliveira, C.G.
Considering a second rank Hermitian field tensor and a general Hermitian connection the associated complex curvature tensor is constructed. The Weyl tensor that corresponds to this complex curvature is determined. The formalism is applied to the Weyl unitary field theory and to the Moffat gravitational theory. (Author) [pt
Spherical Tensor Calculus for Local Adaptive Filtering
Reisert, Marco; Burkhardt, Hans
In 3D image processing tensors play an important role. While rank-1 and rank-2 tensors are well understood and commonly used, higher rank tensors are rare. This is probably due to their cumbersome rotation behavior which prevents a computationally efficient use. In this chapter we want to introduce the notion of a spherical tensor which is based on the irreducible representations of the 3D rotation group. In fact, any ordinary cartesian tensor can be decomposed into a sum of spherical tensors, while each spherical tensor has a quite simple rotation behavior. We introduce so called tensorial harmonics that provide an orthogonal basis for spherical tensor fields of any rank. It is just a generalization of the well known spherical harmonics. Additionally we propose a spherical derivative which connects spherical tensor fields of different degree by differentiation. Based on the proposed theory we present two applications. We propose an efficient algorithm for dense tensor voting in 3D, which makes use of tensorial harmonics decomposition of the tensor-valued voting field. In this way it is possible to perform tensor voting by linear-combinations of convolutions in an efficient way. Secondly, we propose an anisotropic smoothing filter that uses a local shape and orientation adaptive filter kernel which can be computed efficiently by the use spherical derivatives.
A new Weyl-like tensor of geometric origin
Vishwakarma, Ram Gopal
2018-04-01
A set of new tensors of purely geometric origin have been investigated, which form a hierarchy. A tensor of a lower rank plays the role of the potential for the tensor of one rank higher. The tensors have interesting mathematical and physical properties. The highest rank tensor of the hierarchy possesses all the geometrical properties of the Weyl tensor.
Tensor voting for robust color edge detection
Moreno, Rodrigo; García, Miguel Ángel; Puig, Domenec
2014-01-01
The final publication is available at Springer via http://dx.doi.org/10.1007/978-94-007-7584-8_9 This chapter proposes two robust color edge detection methods based on tensor voting. The first method is a direct adaptation of the classical tensor voting to color images where tensors are initialized with either the gradient or the local color structure tensor. The second method is based on an extension of tensor voting in which the encoding and voting processes are specifically tailored to ...
The Physical Interpretation of the Lanczos Tensor
Roberts, Mark D.
1999-01-01
The field equations of general relativity can be written as first order differential equations in the Weyl tensor, the Weyl tensor in turn can be written as a first order differential equation in a three index tensor called the Lanczos tensor. The Lanczos tensor plays a similar role in general relativity to that of the vector potential in electro-magnetic theory. The Aharonov-Bohm effect shows that when quantum mechanics is applied to electro-magnetic theory the vector potential is dynamicall...
Robust estimation of adaptive tensors of curvature by tensor voting.
Tong, Wai-Shun; Tang, Chi-Keung
2005-03-01
Although curvature estimation from a given mesh or regularly sampled point set is a well-studied problem, it is still challenging when the input consists of a cloud of unstructured points corrupted by misalignment error and outlier noise. Such input is ubiquitous in computer vision. In this paper, we propose a three-pass tensor voting algorithm to robustly estimate curvature tensors, from which accurate principal curvatures and directions can be calculated. Our quantitative estimation is an improvement over the previous two-pass algorithm, where only qualitative curvature estimation (sign of Gaussian curvature) is performed. To overcome misalignment errors, our improved method automatically corrects input point locations at subvoxel precision, which also rejects outliers that are uncorrectable. To adapt to different scales locally, we define the RadiusHit of a curvature tensor to quantify estimation accuracy and applicability. Our curvature estimation algorithm has been proven with detailed quantitative experiments, performing better in a variety of standard error metrics (percentage error in curvature magnitudes, absolute angle difference in curvature direction) in the presence of a large amount of misalignment noise.
Antisymmetric tensor generalizations of affine vector fields.
Houri, Tsuyoshi; Morisawa, Yoshiyuki; Tomoda, Kentaro
2016-02-01
Tensor generalizations of affine vector fields called symmetric and antisymmetric affine tensor fields are discussed as symmetry of spacetimes. We review the properties of the symmetric ones, which have been studied in earlier works, and investigate the properties of the antisymmetric ones, which are the main theme in this paper. It is shown that antisymmetric affine tensor fields are closely related to one-lower-rank antisymmetric tensor fields which are parallelly transported along geodesics. It is also shown that the number of linear independent rank- p antisymmetric affine tensor fields in n -dimensions is bounded by ( n + 1)!/ p !( n - p )!. We also derive the integrability conditions for antisymmetric affine tensor fields. Using the integrability conditions, we discuss the existence of antisymmetric affine tensor fields on various spacetimes.
Extended vector-tensor theories
Kimura, Rampei; Naruko, Atsushi; Yoshida, Daisuke, E-mail: rampei@th.phys.titech.ac.jp, E-mail: naruko@th.phys.titech.ac.jp, E-mail: yoshida@th.phys.titech.ac.jp [Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551 (Japan)
2017-01-01
Recently, several extensions of massive vector theory in curved space-time have been proposed in many literatures. In this paper, we consider the most general vector-tensor theories that contain up to two derivatives with respect to metric and vector field. By imposing a degeneracy condition of the Lagrangian in the context of ADM decomposition of space-time to eliminate an unwanted mode, we construct a new class of massive vector theories where five degrees of freedom can propagate, corresponding to three for massive vector modes and two for massless tensor modes. We find that the generalized Proca and the beyond generalized Proca theories up to the quartic Lagrangian, which should be included in this formulation, are degenerate theories even in curved space-time. Finally, introducing new metric and vector field transformations, we investigate the properties of thus obtained theories under such transformations.
Scalar-tensor linear inflation
Artymowski, Michał [Institute of Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków (Poland); Racioppi, Antonio, E-mail: Michal.Artymowski@uj.edu.pl, E-mail: Antonio.Racioppi@kbfi.ee [National Institute of Chemical Physics and Biophysics, Rävala 10, 10143 Tallinn (Estonia)
2017-04-01
We investigate two approaches to non-minimally coupled gravity theories which present linear inflation as attractor solution: a) the scalar-tensor theory approach, where we look for a scalar-tensor theory that would restore results of linear inflation in the strong coupling limit for a non-minimal coupling to gravity of the form of f (φ) R /2; b) the particle physics approach, where we motivate the form of the Jordan frame potential by loop corrections to the inflaton field. In both cases the Jordan frame potentials are modifications of the induced gravity inflationary scenario, but instead of the Starobinsky attractor they lead to linear inflation in the strong coupling limit.
Diffusion tensor optical coherence tomography
Marks, Daniel L.; Blackmon, Richard L.; Oldenburg, Amy L.
2018-01-01
In situ measurements of diffusive particle transport provide insight into tissue architecture, drug delivery, and cellular function. Analogous to diffusion-tensor magnetic resonance imaging (DT-MRI), where the anisotropic diffusion of water molecules is mapped on the millimeter scale to elucidate the fibrous structure of tissue, here we propose diffusion-tensor optical coherence tomography (DT-OCT) for measuring directional diffusivity and flow of optically scattering particles within tissue. Because DT-OCT is sensitive to the sub-resolution motion of Brownian particles as they are constrained by tissue macromolecules, it has the potential to quantify nanoporous anisotropic tissue structure at micrometer resolution as relevant to extracellular matrices, neurons, and capillaries. Here we derive the principles of DT-OCT, relating the detected optical signal from a minimum of six probe beams with the six unique diffusion tensor and three flow vector components. The optimal geometry of the probe beams is determined given a finite numerical aperture, and a high-speed hardware implementation is proposed. Finally, Monte Carlo simulations are employed to assess the ability of the proposed DT-OCT system to quantify anisotropic diffusion of nanoparticles in a collagen matrix, an extracellular constituent that is known to become highly aligned during tumor development.
Sparse alignment for robust tensor learning.
Lai, Zhihui; Wong, Wai Keung; Xu, Yong; Zhao, Cairong; Sun, Mingming
2014-10-01
Multilinear/tensor extensions of manifold learning based algorithms have been widely used in computer vision and pattern recognition. This paper first provides a systematic analysis of the multilinear extensions for the most popular methods by using alignment techniques, thereby obtaining a general tensor alignment framework. From this framework, it is easy to show that the manifold learning based tensor learning methods are intrinsically different from the alignment techniques. Based on the alignment framework, a robust tensor learning method called sparse tensor alignment (STA) is then proposed for unsupervised tensor feature extraction. Different from the existing tensor learning methods, L1- and L2-norms are introduced to enhance the robustness in the alignment step of the STA. The advantage of the proposed technique is that the difficulty in selecting the size of the local neighborhood can be avoided in the manifold learning based tensor feature extraction algorithms. Although STA is an unsupervised learning method, the sparsity encodes the discriminative information in the alignment step and provides the robustness of STA. Extensive experiments on the well-known image databases as well as action and hand gesture databases by encoding object images as tensors demonstrate that the proposed STA algorithm gives the most competitive performance when compared with the tensor-based unsupervised learning methods.
Shape anisotropy: tensor distance to anisotropy measure
Weldeselassie, Yonas T.; El-Hilo, Saba; Atkins, M. S.
2011-03-01
Fractional anisotropy, defined as the distance of a diffusion tensor from its closest isotropic tensor, has been extensively studied as quantitative anisotropy measure for diffusion tensor magnetic resonance images (DT-MRI). It has been used to reveal the white matter profile of brain images, as guiding feature for seeding and stopping in fiber tractography and for the diagnosis and assessment of degenerative brain diseases. Despite its extensive use in DT-MRI community, however, not much attention has been given to the mathematical correctness of its derivation from diffusion tensors which is achieved using Euclidean dot product in 9D space. But, recent progress in DT-MRI has shown that the space of diffusion tensors does not form a Euclidean vector space and thus Euclidean dot product is not appropriate for tensors. In this paper, we propose a novel and robust rotationally invariant diffusion anisotropy measure derived using the recently proposed Log-Euclidean and J-divergence tensor distance measures. An interesting finding of our work is that given a diffusion tensor, its closest isotropic tensor is different for different tensor distance metrics used. We demonstrate qualitatively that our new anisotropy measure reveals superior white matter profile of DT-MR brain images and analytically show that it has a higher signal to noise ratio than fractional anisotropy.
Transposes, L-Eigenvalues and Invariants of Third Order Tensors
Qi, Liqun
2017-01-01
Third order tensors have wide applications in mechanics, physics and engineering. The most famous and useful third order tensor is the piezoelectric tensor, which plays a key role in the piezoelectric effect, first discovered by Curie brothers. On the other hand, the Levi-Civita tensor is famous in tensor calculus. In this paper, we study third order tensors and (third order) hypermatrices systematically, by regarding a third order tensor as a linear operator which transforms a second order t...
Tensor SOM and tensor GTM: Nonlinear tensor analysis by topographic mappings.
Iwasaki, Tohru; Furukawa, Tetsuo
2016-05-01
In this paper, we propose nonlinear tensor analysis methods: the tensor self-organizing map (TSOM) and the tensor generative topographic mapping (TGTM). TSOM is a straightforward extension of the self-organizing map from high-dimensional data to tensorial data, and TGTM is an extension of the generative topographic map, which provides a theoretical background for TSOM using a probabilistic generative model. These methods are useful tools for analyzing and visualizing tensorial data, especially multimodal relational data. For given n-mode relational data, TSOM and TGTM can simultaneously organize a set of n-topographic maps. Furthermore, they can be used to explore the tensorial data space by interactively visualizing the relationships between modes. We present the TSOM algorithm and a theoretical description from the viewpoint of TGTM. Various TSOM variations and visualization techniques are also described, along with some applications to real relational datasets. Additionally, we attempt to build a comprehensive description of the TSOM family by adapting various data structures. Copyright © 2016 Elsevier Ltd. All rights reserved.
Applications of tensor functions in creep mechanics
Betten, J.
1991-01-01
Within this contribution a short survey is given of some recent advances in the mathematical modelling of materials behaviour under creep conditions. The mechanical behaviour of anisotropic solids requires a suitable mathematical modelling. The properties of tensor functions with several argument tensors constitute a rational basis for a consistent mathematical modelling of complex material behaviour. This paper presents certain principles, methods, and recent successfull applications of tensor functions in solid mechanics. The rules for specifying irreducible sets of tensor invariants and tensor generators for material tensors of rank two and four are also discussed. Furthermore, it is very important that the scalar coefficients in constitutive and evolutional equations are determined as functions of the integrity basis and experimental data. It is explained in detail that these coefficients can be determined by using tensorial interpolation methods. Some examples for practical use are discussed. (orig./RHM)
Seamless warping of diffusion tensor fields
Xu, Dongrong; Hao, Xuejun; Bansal, Ravi
2008-01-01
To warp diffusion tensor fields accurately, tensors must be reoriented in the space to which the tensors are warped based on both the local deformation field and the orientation of the underlying fibers in the original image. Existing algorithms for warping tensors typically use forward mapping...... of seams, including voxels in which the deformation is extensive. Backward mapping, however, cannot reorient tensors in the template space because information about the directional orientation of fiber tracts is contained in the original, unwarped imaging space only, and backward mapping alone cannot...... transfer that information to the template space. To combine the advantages of forward and backward mapping, we propose a novel method for the spatial normalization of diffusion tensor (DT) fields that uses a bijection (a bidirectional mapping with one-to-one correspondences between image spaces) to warp DT...
The Topology of Symmetric Tensor Fields
Levin, Yingmei; Batra, Rajesh; Hesselink, Lambertus; Levy, Yuval
1997-01-01
Combinatorial topology, also known as "rubber sheet geometry", has extensive applications in geometry and analysis, many of which result from connections with the theory of differential equations. A link between topology and differential equations is vector fields. Recent developments in scientific visualization have shown that vector fields also play an important role in the analysis of second-order tensor fields. A second-order tensor field can be transformed into its eigensystem, namely, eigenvalues and their associated eigenvectors without loss of information content. Eigenvectors behave in a similar fashion to ordinary vectors with even simpler topological structures due to their sign indeterminacy. Incorporating information about eigenvectors and eigenvalues in a display technique known as hyperstreamlines reveals the structure of a tensor field. The simplify and often complex tensor field and to capture its important features, the tensor is decomposed into an isotopic tensor and a deviator. A tensor field and its deviator share the same set of eigenvectors, and therefore they have a similar topological structure. A a deviator determines the properties of a tensor field, while the isotopic part provides a uniform bias. Degenerate points are basic constituents of tensor fields. In 2-D tensor fields, there are only two types of degenerate points; while in 3-D, the degenerate points can be characterized in a Q'-R' plane. Compressible and incompressible flows share similar topological feature due to the similarity of their deviators. In the case of the deformation tensor, the singularities of its deviator represent the area of vortex core in the field. In turbulent flows, the similarities and differences of the topology of the deformation and the Reynolds stress tensors reveal that the basic addie-viscosity assuptions have their validity in turbulence modeling under certain conditions.
On improving the efficiency of tensor voting
Moreno, Rodrigo; Garcia, Miguel Angel; Puig, Domenec; Pizarro, Luis; Burgeth, Bernhard; Weickert, Joachim
2011-01-01
This paper proposes two alternative formulations to reduce the high computational complexity of tensor voting, a robust perceptual grouping technique used to extract salient information from noisy data. The first scheme consists of numerical approximations of the votes, which have been derived from an in-depth analysis of the plate and ball voting processes. The second scheme simplifies the formulation while keeping the same perceptual meaning of the original tensor voting: The stick tensor v...
Schrimpf, Martin
2016-01-01
Google's Machine Learning framework TensorFlow was open-sourced in November 2015 [1] and has since built a growing community around it. TensorFlow is supposed to be flexible for research purposes while also allowing its models to be deployed productively. This work is aimed towards people with experience in Machine Learning considering whether they should use TensorFlow in their environment. Several aspects of the framework important for such a decision are examined, such as the heterogenity,...
Efficient Low Rank Tensor Ring Completion
Wang, Wenqi; Aggarwal, Vaneet; Aeron, Shuchin
2017-01-01
Using the matrix product state (MPS) representation of the recently proposed tensor ring decompositions, in this paper we propose a tensor completion algorithm, which is an alternating minimization algorithm that alternates over the factors in the MPS representation. This development is motivated in part by the success of matrix completion algorithms that alternate over the (low-rank) factors. In this paper, we propose a spectral initialization for the tensor ring completion algorithm and ana...
The Riemann-Lovelock Curvature Tensor
Kastor, David
2012-01-01
In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth-order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k \\le D
The 1/ N Expansion of Tensor Models with Two Symmetric Tensors
Gurau, Razvan
2018-06-01
It is well known that tensor models for a tensor with no symmetry admit a 1/ N expansion dominated by melonic graphs. This result relies crucially on identifying jackets, which are globally defined ribbon graphs embedded in the tensor graph. In contrast, no result of this kind has so far been established for symmetric tensors because global jackets do not exist. In this paper we introduce a new approach to the 1/ N expansion in tensor models adapted to symmetric tensors. In particular we do not use any global structure like the jackets. We prove that, for any rank D, a tensor model with two symmetric tensors and interactions the complete graph K D+1 admits a 1/ N expansion dominated by melonic graphs.
Dictionary-Based Tensor Canonical Polyadic Decomposition
Cohen, Jeremy Emile; Gillis, Nicolas
2018-04-01
To ensure interpretability of extracted sources in tensor decomposition, we introduce in this paper a dictionary-based tensor canonical polyadic decomposition which enforces one factor to belong exactly to a known dictionary. A new formulation of sparse coding is proposed which enables high dimensional tensors dictionary-based canonical polyadic decomposition. The benefits of using a dictionary in tensor decomposition models are explored both in terms of parameter identifiability and estimation accuracy. Performances of the proposed algorithms are evaluated on the decomposition of simulated data and the unmixing of hyperspectral images.
Bayesian regularization of diffusion tensor images
Frandsen, Jesper; Hobolth, Asger; Østergaard, Leif
2007-01-01
Diffusion tensor imaging (DTI) is a powerful tool in the study of the course of nerve fibre bundles in the human brain. Using DTI, the local fibre orientation in each image voxel can be described by a diffusion tensor which is constructed from local measurements of diffusion coefficients along...... several directions. The measured diffusion coefficients and thereby the diffusion tensors are subject to noise, leading to possibly flawed representations of the three dimensional fibre bundles. In this paper we develop a Bayesian procedure for regularizing the diffusion tensor field, fully utilizing...
A RENORMALIZATION PROCEDURE FOR TENSOR MODELS AND SCALAR-TENSOR THEORIES OF GRAVITY
SASAKURA, NAOKI
2010-01-01
Tensor models are more-index generalizations of the so-called matrix models, and provide models of quantum gravity with the idea that spaces and general relativity are emergent phenomena. In this paper, a renormalization procedure for the tensor models whose dynamical variable is a totally symmetric real three-tensor is discussed. It is proven that configurations with certain Gaussian forms are the attractors of the three-tensor under the renormalization procedure. Since these Gaussian config...
The tensor network theory library
Al-Assam, S.; Clark, S. R.; Jaksch, D.
2017-09-01
In this technical paper we introduce the tensor network theory (TNT) library—an open-source software project aimed at providing a platform for rapidly developing robust, easy to use and highly optimised code for TNT calculations. The objectives of this paper are (i) to give an overview of the structure of TNT library, and (ii) to help scientists decide whether to use the TNT library in their research. We show how to employ the TNT routines by giving examples of ground-state and dynamical calculations of one-dimensional bosonic lattice system. We also discuss different options for gaining access to the software available at www.tensornetworktheory.org.
Dirac tensor with heavy photon
Bytev, V.V.; Kuraev, E.A. [Joint Institute of Nuclear Research, Moscow (Russian Federation). Bogoliubov Lab. of Theoretical Physics; Scherbakova, E.S. [Hamburg Univ. (Germany). 1. Inst. fuer Theoretische Physik
2012-01-15
For the large-angles hard photon emission by initial leptons in process of high energy annihilation of e{sup +}e{sup -} {yields} to hadrons the Dirac tensor is obtained, taking into account the lowest order radiative corrections. The case of large-angles emission of two hard photons by initial leptons is considered. This result is being completed by the kinematics case of collinear hard photons emission as well as soft virtual and real photons and can be used for construction of Monte-Carlo generators. (orig.)
3D reconstruction of tensors and vectors
Defrise, Michel; Gullberg, Grant T.
2005-01-01
Here we have developed formulations for the reconstruction of 3D tensor fields from planar (Radon) and line-integral (X-ray) projections of 3D vector and tensor fields. Much of the motivation for this work is the potential application of MRI to perform diffusion tensor tomography. The goal is to develop a theory for the reconstruction of both Radon planar and X-ray or line-integral projections because of the flexibility of MRI to obtain both of these type of projections in 3D. The development presented here for the linear tensor tomography problem provides insight into the structure of the nonlinear MRI diffusion tensor inverse problem. A particular application of tensor imaging in MRI is the potential application of cardiac diffusion tensor tomography for determining in vivo cardiac fiber structure. One difficulty in the cardiac application is the motion of the heart. This presents a need for developing future theory for tensor tomography in a motion field. This means developing a better understanding of the MRI signal for diffusion processes in a deforming media. The techniques developed may allow the application of MRI tensor tomography for the study of structure of fiber tracts in the brain, atherosclerotic plaque, and spine in addition to fiber structure in the heart. However, the relations presented are also applicable to other fields in medical imaging such as diffraction tomography using ultrasound. The mathematics presented can also be extended to exponential Radon transform of tensor fields and to other geometric acquisitions such as cone beam tomography of tensor fields
Raman scattering tensors of tyrosine.
Tsuboi, M; Ezaki, Y; Aida, M; Suzuki, M; Yimit, A; Ushizawa, K; Ueda, T
1998-01-01
Polarized Raman scattering measurements have been made of a single crystal of L-tyrosine by the use of a Raman microscope with the 488.0-nm exciting beam from an argon ion laser. The L-tyrosine crystal belongs to the space group P2(1)2(1)2(1) (orthorhombic), and Raman scattering intensities corresponding to the aa, bb, cc, ab and ac components of the crystal Raman tensor have been determined for each prominent Raman band. A similar set of measurements has been made of L-tyrosine-d4, in which four hydrogen atoms on the benzene ring are replaced by deuterium atoms. The effects of NH3-->ND3 and OH-->OD on the Raman spectrum have also been examined. In addition, depolarization ratios of some bands of L-tyrosine in aqueous solutions of pH 13 and pH 1 were examined. For comparison with these experimental results, on the other hand, ab initio molecular orbital calculations have been made of the normal modes of vibration and their associated polarizability oscillations of the L-tyrosine molecule. On the basis of these experimental data and by referring to the results of the calculations, discussions have been presented on the Raman tensors associated to some Raman bands, including those at 829 cm-1 (benzene ring breathing), 642 cm-1 (benzene ring deformation), and 432 cm-1 (C alpha-C beta-C gamma bending).
The tensor part of the Skyrme energy density functional. I. Spherical nuclei
Lesinski, T.; Meyer, J. [Universite de Lyon, F-69003 Lyon (France)]|[Institut de Physique Nucleaire de Lyon, CNRS/IN2P3, Universite Lyon 1, F-69622 Villeurbanne (France); Bender, M. [DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif-sur-Yvette Cedex (France)]|[Universite Bordeaux, CNRS/IN2P3, Centre d' Etudes Nucleaires de Bordeaux Gradignan, UMR5797, Chemin du Solarium, BP120, F-33175 Gradignan (France); Bennaceur, K. [Universite de Lyon, F-69003 Lyon (France)]|[Institut de Physique Nucleaire de Lyon, CNRS/IN2P3, Universite Lyon 1, F-69622 Villeurbanne (France)]|[DSM/DAPNIA/SPhN, CEA Saclay, F-91191 Gif-sur-Yvette Cedex (France); Duguet, T. [National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824 (United States)
2007-04-15
We perform a systematic study of the impact of the J-vector{sup 2} tensor term in the Skyrme energy functional on properties of spherical nuclei. In the Skyrme energy functional, the tensor terms originate both from zero-range central and tensor forces. We build a set of 36 parameterizations which cover a wide range of the parameter space of the isoscalar and isovector tensor term coupling constants with a fit protocol very similar to that of the successful SLy parameterizations. We analyze the impact of the tensor terms on a large variety of observables in spherical mean-field calculations, such as the spin-orbit splittings and single-particle spectra of doubly-magic nuclei, the evolution of spin-orbit splittings along chains of semi-magic nuclei, mass residuals of spherical nuclei, and known anomalies of radii. The major findings of our study are (i) tensor terms should not be added perturbatively to existing parameterizations, a complete refit of the entire parameter set is imperative. (ii) The free variation of the tensor terms does not lower the {chi}{sup 2} within a standard Skyrme energy functional. (iii) For certain regions of the parameter space of their coupling constants, the tensor terms lead to instabilities of the spherical shell structure, or even the coexistence of two configurations with different spherical shell structure. (iv) The standard spin-orbit interaction does not scale properly with the principal quantum number, such that single-particle states with one or several nodes have too large spin-orbit splittings, while those of node-less intruder levels are tentatively too small. Tensor terms with realistic coupling constants cannot cure this problem. (v) Positive values of the coupling constants of proton-neutron and like-particle tensor terms allow for a qualitative description of the evolution of spin-orbit splittings in chains of Ca, Ni and Sn isotopes. (vi) For the same values of the tensor term coupling constants, however, the overall
Algebraic classification of the Weyl tensor in higher dimensions based on its 'superenergy' tensor
Senovilla, Jose M M
2010-01-01
The algebraic classification of the Weyl tensor in the arbitrary dimension n is recovered by means of the principal directions of its 'superenergy' tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits one to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in the general dimension can be achieved. (fast track communication)
Complete stress tensor determination by microearthquake analysis
Slunga, R.
2010-12-01
Jones 1984 found that half of the shallow strike-slip EQ in California had at least one M>2 foreshock. By the Gutenberg law this means at least 3-20 M>0 (low b-value 0.4-0.8). deformations within the crust. This was confirmed by observations in Iceland after 1990 when anew seismic network in Iceland operated by IMO started. Like the Parkfield project in California the SIL network in Iceland was established in an area predicted (Einarsson et al 1981, Stefansson and Halldorsson 1988) to be struck by major EQs within decades of years. The area of main interest have a detection threshold of M=0. A physical approach was chosen to the earthquake warning problem (Stefansson et al 1993) and therefore all microearthquakes were analyzed for FPS by the spectral amplitude method (Slunga 1981). As the shear slip is caused by the in situ stress it is logical to investigate what bounds the FPS puts on the stress tensor. McKenzie 1969 assumed that the earthquake takes place in a crust containing only one fracture, the fault plane. He found that in s uch a case only very weak constraints could be put on the stress. This was widely accepted t o be valid also for microearthquakes in the real crust and lead to methods (Angelier 1978, G ephart and Forsythe 1984 etc) to put four constraints on the stress tensor by assuming that the same stress tensor is causing the slip on four or more different fractures. Another and more realistic approach is to assume that the crust have frequent fractures with almost all orientations. In such a case one can rely on Coulomb's failure criterion for isotropic mat erial (gives four constraints) instead of the weaker Bolt's criterion (giving only one const raint). One obvious fifth constraint is to require the vertical stress to equal the lithosta tic pressure. A sixth constraint is achieved by requiring that the deviatoric elastic energy is minimized. The water pressure is also needed for the fourth constraint by Coulomb (CFS=0 ). It can be related to
Gravitational Metric Tensor Exterior to Rotating Homogeneous ...
The covariant and contravariant metric tensors exterior to a homogeneous spherical body rotating uniformly about a common φ axis with constant angular velocity ω is constructed. The constructed metric tensors in this gravitational field have seven non-zero distinct components.The Lagrangian for this gravitational field is ...
Tensor Network Quantum Virtual Machine (TNQVM)
2016-11-18
There is a lack of state-of-the-art quantum computing simulation software that scales on heterogeneous systems like Titan. Tensor Network Quantum Virtual Machine (TNQVM) provides a quantum simulator that leverages a distributed network of GPUs to simulate quantum circuits in a manner that leverages recent results from tensor network theory.
Tensor product varieties and crystals. GL case
Malkin, Anton
2001-01-01
The role of Spaltenstein varieties in the tensor product for GL is explained. In particular a direct (non-combinatorial) proof of the fact that the number of irreducible components of a Spaltenstein variety is equal to a Littlewood-Richardson coefficient (i.e. certain tensor product multiplicity) is obtained.
Differential invariants for higher-rank tensors. A progress report
Tapial, V.
2004-07-01
We outline the construction of differential invariants for higher-rank tensors. In section 2 we outline the general method for the construction of differential invariants. A first result is that the simplest tensor differential invariant contains derivatives of the same order as the rank of the tensor. In section 3 we review the construction for the first-rank tensors (vectors) and second-rank tensors (metrics). In section 4 we outline the same construction for higher-rank tensors. (author)
Beyond Low Rank: A Data-Adaptive Tensor Completion Method
Zhang, Lei; Wei, Wei; Shi, Qinfeng; Shen, Chunhua; Hengel, Anton van den; Zhang, Yanning
2017-01-01
Low rank tensor representation underpins much of recent progress in tensor completion. In real applications, however, this approach is confronted with two challenging problems, namely (1) tensor rank determination; (2) handling real tensor data which only approximately fulfils the low-rank requirement. To address these two issues, we develop a data-adaptive tensor completion model which explicitly represents both the low-rank and non-low-rank structures in a latent tensor. Representing the no...
Unique characterization of the Bel-Robinson tensor
Bergqvist, G; Lankinen, P
2004-01-01
We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors
Tensor completion and low-n-rank tensor recovery via convex optimization
Gandy, Silvia; Yamada, Isao; Recht, Benjamin
2011-01-01
In this paper we consider sparsity on a tensor level, as given by the n-rank of a tensor. In an important sparse-vector approximation problem (compressed sensing) and the low-rank matrix recovery problem, using a convex relaxation technique proved to be a valuable solution strategy. Here, we will adapt these techniques to the tensor setting. We use the n-rank of a tensor as a sparsity measure and consider the low-n-rank tensor recovery problem, i.e. the problem of finding the tensor of the lowest n-rank that fulfills some linear constraints. We introduce a tractable convex relaxation of the n-rank and propose efficient algorithms to solve the low-n-rank tensor recovery problem numerically. The algorithms are based on the Douglas–Rachford splitting technique and its dual variant, the alternating direction method of multipliers
Weyl curvature tensor in static spherical sources
Ponce de Leon, J.
1988-01-01
The role of the Weyl curvature tensor in static sources of the Schwarzschild field is studied. It is shown that in general the contribution from the Weyl curvature tensor (the ''purely gravitational field energy'') to the mass-energy inside the body may be positive, negative, or zero. It is proved that a positive (negative) contribution from the Weyl tensor tends to increase (decrease) the effective gravitational mass, the red-shift (from a point in the sphere to infinity), as well as the gravitational force which acts on a constituent matter element of a body. It is also proved that the contribution from the Weyl tensor always is negative in sources with surface gravitational potential larger than (4/9. It is pointed out that large negative contributions from the Weyl tensor could give rise to the phenomenon of gravitational repulsion. A simple example which illustrates the results is discussed
A recursive reduction of tensor Feynman integrals
Diakonidis, T.; Riemann, T.; Tausk, J.B.; Fleischer, J.
2009-07-01
We perform a recursive reduction of one-loop n-point rank R tensor Feynman integrals [in short: (n,R)-integrals] for n≤6 with R≤n by representing (n,R)-integrals in terms of (n,R-1)- and (n-1,R-1)-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, we find the recursive reduction for the tensors. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four-particle production at LHC and ILC, as well as at meson factories. (orig.)
On Lovelock analogs of the Riemann tensor
Camanho, Xián O.; Dadhich, Naresh
2016-03-01
It is possible to define an analog of the Riemann tensor for Nth order Lovelock gravity, its characterizing property being that the trace of its Bianchi derivative yields the corresponding analog of the Einstein tensor. Interestingly there exist two parallel but distinct such analogs and the main purpose of this note is to reconcile both formulations. In addition we will introduce a simple tensor identity and use it to show that any pure Lovelock vacuum in odd d=2N+1 dimensions is Lovelock flat, i.e. any vacuum solution of the theory has vanishing Lovelock-Riemann tensor. Further, in the presence of cosmological constant it is the Lovelock-Weyl tensor that vanishes.
Making tensor factorizations robust to non-gaussian noise.
Chi, Eric C. (Rice University, Houston, TX); Kolda, Tamara Gibson
2011-03-01
Tensors are multi-way arrays, and the CANDECOMP/PARAFAC (CP) tensor factorization has found application in many different domains. The CP model is typically fit using a least squares objective function, which is a maximum likelihood estimate under the assumption of independent and identically distributed (i.i.d.) Gaussian noise. We demonstrate that this loss function can be highly sensitive to non-Gaussian noise. Therefore, we propose a loss function based on the 1-norm because it can accommodate both Gaussian and grossly non-Gaussian perturbations. We also present an alternating majorization-minimization (MM) algorithm for fitting a CP model using our proposed loss function (CPAL1) and compare its performance to the workhorse algorithm for fitting CP models, CP alternating least squares (CPALS).
Diffusion tensor imaging and tractography in clinical neuro sciences
Zarei, M.; Johansen-Berg, H.; Matthews, P.M.
2003-01-01
Rapidly evolving MR technology has allowed better understanding of structure and function of the human brain. Diffusion weighted MRI was developed two decades ago and it is now well established in diagnosis of acute ischaemia in patients with stroke. Diffusion tensor MRI uses the same principles but takes a step further allowing US to measure magnitude of the diffusion along different directions. This lead to the development of diffusion tensor tractography, a technique by which major neural pathways in the living brain can be visualized. There is a growing interest in exploring possible use of these techniques in clinical neurology and psychiatry. This article aims to review the principles of this technique and recent discoveries which may help US to better understand neurological and psychiatric disorders
Efficient Tensor Completion for Color Image and Video Recovery: Low-Rank Tensor Train.
Bengua, Johann A; Phien, Ho N; Tuan, Hoang Duong; Do, Minh N
2017-05-01
This paper proposes a novel approach to tensor completion, which recovers missing entries of data represented by tensors. The approach is based on the tensor train (TT) rank, which is able to capture hidden information from tensors thanks to its definition from a well-balanced matricization scheme. Accordingly, new optimization formulations for tensor completion are proposed as well as two new algorithms for their solution. The first one called simple low-rank tensor completion via TT (SiLRTC-TT) is intimately related to minimizing a nuclear norm based on TT rank. The second one is from a multilinear matrix factorization model to approximate the TT rank of a tensor, and is called tensor completion by parallel matrix factorization via TT (TMac-TT). A tensor augmentation scheme of transforming a low-order tensor to higher orders is also proposed to enhance the effectiveness of SiLRTC-TT and TMac-TT. Simulation results for color image and video recovery show the clear advantage of our method over all other methods.
Tensor network method for reversible classical computation
Yang, Zhi-Cheng; Kourtis, Stefanos; Chamon, Claudio; Mucciolo, Eduardo R.; Ruckenstein, Andrei E.
2018-03-01
We develop a tensor network technique that can solve universal reversible classical computational problems, formulated as vertex models on a square lattice [Nat. Commun. 8, 15303 (2017), 10.1038/ncomms15303]. By encoding the truth table of each vertex constraint in a tensor, the total number of solutions compatible with partial inputs and outputs at the boundary can be represented as the full contraction of a tensor network. We introduce an iterative compression-decimation (ICD) scheme that performs this contraction efficiently. The ICD algorithm first propagates local constraints to longer ranges via repeated contraction-decomposition sweeps over all lattice bonds, thus achieving compression on a given length scale. It then decimates the lattice via coarse-graining tensor contractions. Repeated iterations of these two steps gradually collapse the tensor network and ultimately yield the exact tensor trace for large systems, without the need for manual control of tensor dimensions. Our protocol allows us to obtain the exact number of solutions for computations where a naive enumeration would take astronomically long times.
On improving the efficiency of tensor voting.
Moreno, Rodrigo; Garcia, Miguel Angel; Puig, Domenec; Pizarro, Luis; Burgeth, Bernhard; Weickert, Joachim
2011-11-01
This paper proposes two alternative formulations to reduce the high computational complexity of tensor voting, a robust perceptual grouping technique used to extract salient information from noisy data. The first scheme consists of numerical approximations of the votes, which have been derived from an in-depth analysis of the plate and ball voting processes. The second scheme simplifies the formulation while keeping the same perceptual meaning of the original tensor voting: The stick tensor voting and the stick component of the plate tensor voting must reinforce surfaceness, the plate components of both the plate and ball tensor voting must boost curveness, whereas junctionness must be strengthened by the ball component of the ball tensor voting. Two new parameters have been proposed for the second formulation in order to control the potentially conflictive influence of the stick component of the plate vote and the ball component of the ball vote. Results show that the proposed formulations can be used in applications where efficiency is an issue since they have a complexity of order O(1). Moreover, the second proposed formulation has been shown to be more appropriate than the original tensor voting for estimating saliencies by appropriately setting the two new parameters.
Conformal field theories and tensor categories. Proceedings
Bai, Chengming [Nankai Univ., Tianjin (China). Chern Institute of Mathematics; Fuchs, Juergen [Karlstad Univ. (Sweden). Theoretical Physics; Huang, Yi-Zhi [Rutgers Univ., Piscataway, NJ (United States). Dept. of Mathematics; Kong, Liang [Tsinghua Univ., Beijing (China). Inst. for Advanced Study; Runkel, Ingo; Schweigert, Christoph (eds.) [Hamburg Univ. (Germany). Dept. of Mathematics
2014-08-01
First book devoted completely to the mathematics of conformal field theories, tensor categories and their applications. Contributors include both mathematicians and physicists. Some long expository articles are especially suitable for beginners. The present volume is a collection of seven papers that are either based on the talks presented at the workshop ''Conformal field theories and tensor categories'' held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.
Conformal field theories and tensor categories. Proceedings
Bai, Chengming; Fuchs, Juergen; Huang, Yi-Zhi; Kong, Liang; Runkel, Ingo; Schweigert, Christoph
2014-01-01
First book devoted completely to the mathematics of conformal field theories, tensor categories and their applications. Contributors include both mathematicians and physicists. Some long expository articles are especially suitable for beginners. The present volume is a collection of seven papers that are either based on the talks presented at the workshop ''Conformal field theories and tensor categories'' held June 13 to June 17, 2011 at the Beijing International Center for Mathematical Research, Peking University, or are extensions of the material presented in the talks at the workshop. These papers present new developments beyond rational conformal field theories and modular tensor categories and new applications in mathematics and physics. The topics covered include tensor categories from representation categories of Hopf algebras, applications of conformal field theories and tensor categories to topological phases and gapped systems, logarithmic conformal field theories and the corresponding non-semisimple tensor categories, and new developments in the representation theory of vertex operator algebras. Some of the papers contain detailed introductory material that is helpful for graduate students and researchers looking for an introduction to these research directions. The papers also discuss exciting recent developments in the area of conformal field theories, tensor categories and their applications and will be extremely useful for researchers working in these areas.
Tensor harmonic analysis on homogenous space
Wrobel, G.
1997-01-01
The Hilbert space of tensor functions on a homogenous space with the compact stability group is considered. The functions are decomposed onto a sum of tensor plane waves (defined in the text), components of which are transformed by irreducible representations of the appropriate transformation group. The orthogonality relation and the completeness relation for tensor plane waves are found. The decomposition constitutes a unitary transformation, which allows to obtain the Parseval equality. The Fourier components can be calculated by means of the Fourier transformation, the form of which is given explicitly. (author)
Abelian gauge theories with tensor gauge fields
Kapuscik, E.
1984-01-01
Gauge fields of arbitrary tensor type are introduced. In curved space-time the gravitational field serves as a bridge joining different gauge fields. The theory of second order tensor gauge field is developed on the basis of close analogy to Maxwell electrodynamics. The notion of tensor current is introduced and an experimental test of its detection is proposed. The main result consists in a coupled set of field equations representing a generalization of Maxwell theory in which the Einstein equivalence principle is not satisfied. (author)
Local Tensor Radiation Conditions For Elastic Waves
Krenk, S.; Kirkegaard, Poul Henning
2001-01-01
A local boundary condition is formulated, representing radiation of elastic waves from an arbitrary point source. The boundary condition takes the form of a tensor relation between the stress at a point on an arbitrarily oriented section and the velocity and displacement vectors at the point....... The tensor relation generalizes the traditional normal incidence impedance condition by accounting for the angle between wave propagation and the surface normal and by including a generalized stiffness term due to spreading of the waves. The effectiveness of the local tensor radiation condition...
Scalable Tensor Factorizations with Missing Data
Acar, Evrim; Dunlavy, Daniel M.; Kolda, Tamara G.
2010-01-01
of missing data, many important data sets will be discarded or improperly analyzed. Therefore, we need a robust and scalable approach for factorizing multi-way arrays (i.e., tensors) in the presence of missing data. We focus on one of the most well-known tensor factorizations, CANDECOMP/PARAFAC (CP...... is shown to successfully factor tensors with noise and up to 70% missing data. Moreover, our approach is significantly faster than the leading alternative and scales to larger problems. To show the real-world usefulness of CP-WOPT, we illustrate its applicability on a novel EEG (electroencephalogram...
Surface tensor estimation from linear sections
Kousholt, Astrid; Kiderlen, Markus; Hug, Daniel
From Crofton's formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the n-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In a design based setting we suggest three types of estimators....... These are based on isotropic uniform random lines, vertical sections, and non-isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting....
Surface tensor estimation from linear sections
Kousholt, Astrid; Kiderlen, Markus; Hug, Daniel
2015-01-01
From Crofton’s formula for Minkowski tensors we derive stereological estimators of translation invariant surface tensors of convex bodies in the n-dimensional Euclidean space. The estimators are based on one-dimensional linear sections. In a design based setting we suggest three types of estimators....... These are based on isotropic uniform random lines, vertical sections, and non-isotropic random lines, respectively. Further, we derive estimators of the specific surface tensors associated with a stationary process of convex particles in the model based setting....
Tensor products of higher almost split sequences
Pasquali, Andrea
2015-01-01
We investigate how the higher almost split sequences over a tensor product of algebras are related to those over each factor. Herschend and Iyama gave a precise criterion for when the tensor product of an $n$-representation finite algebra and an $m$-representation finite algebra is $(n+m)$-representation finite. In this case we give a complete description of the higher almost split sequences over the tensor product by expressing every higher almost split sequence as the mapping cone of a suit...
Scalable tensor factorizations for incomplete data
Acar, Evrim; Dunlavy, Daniel M.; KOlda, Tamara G.
2011-01-01
to factorize data sets with missing values with the goal of capturing the underlying latent structure of the data and possibly reconstructing missing values (i.e., tensor completion). We focus on one of the most well-known tensor factorizations that captures multi-linear structure, CANDECOMP/PARAFAC (CP...... experiments, our algorithm is shown to successfully factorize tensors with noise and up to 99% missing data. A unique aspect of our approach is that it scales to sparse large-scale data, e.g., 1000 × 1000 × 1000 with five million known entries (0.5% dense). We further demonstrate the usefulness of CP...
General projective relativity and the vector-tensor gravitational field
Arcidiacono, G.
1986-01-01
In the general projective relativity, the induced 4-dimensional metric is symmetric in three cases, and we obtain the vector-tensor, the scalar-tensor, and the scalar-vector-tensor theories of gravitation. In this work we examine the vector-tensor theory, similar to the Veblen's theory, but with a different physical interpretation
Tucker tensor analysis of Matern functions in spatial statistics
Litvinenko, Alexander
2018-04-20
Low-rank Tucker tensor methods in spatial statistics 1. Motivation: improve statistical models 2. Motivation: disadvantages of matrices 3. Tools: Tucker tensor format 4. Tensor approximation of Matern covariance function via FFT 5. Typical statistical operations in Tucker tensor format 6. Numerical experiments
TensorFlow Agents: Efficient Batched Reinforcement Learning in TensorFlow
Hafner, Danijar; Davidson, James; Vanhoucke, Vincent
2017-01-01
We introduce TensorFlow Agents, an efficient infrastructure paradigm for building parallel reinforcement learning algorithms in TensorFlow. We simulate multiple environments in parallel, and group them to perform the neural network computation on a batch rather than individual observations. This allows the TensorFlow execution engine to parallelize computation, without the need for manual synchronization. Environments are stepped in separate Python processes to progress them in parallel witho...
Reconstruction of convex bodies from surface tensors
Kousholt, Astrid; Kiderlen, Markus
. The output of the reconstruction algorithm is a polytope P, where the surface tensors of P and K are identical up to rank s. We establish a stability result based on a generalization of Wirtinger’s inequality that shows that for large s, two convex bodies are close in shape when they have identical surface...... that are translates of each other. An algorithm for reconstructing an unknown convex body in R 2 from its surface tensors up to a certain rank is presented. Using the reconstruction algorithm, the shape of an unknown convex body can be approximated when only a finite number s of surface tensors are available...... tensors up to rank s. This is used to establish consistency of the developed reconstruction algorithm....
Reconstruction of convex bodies from surface tensors
Kousholt, Astrid; Kiderlen, Markus
2016-01-01
We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. When only measurements subject to noise...... of surface tensors are available for reconstruction, we recommend to use certain values of the surface tensors, namely harmonic intrinsic volumes instead of the surface tensors evaluated at the standard basis. The second algorithm we present is based on harmonic intrinsic volumes and allows for noisy...... measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based on measurements subject to noise is established under certain assumptions on the noise...
Energy-momentum tensor in scalar QED
Joglekar, S.D.; Misra, A.
1988-01-01
We consider the renormalization of the energy-momentum tensor in scalar quantum electrodynamics. We show the need for adding an improvement term to the conventional energy-momentum tensor. We consider two possible forms for the improvement term: (i) one in which the improvement coefficient is a finite function of bare parameters of the theory (so that the energy-momentum tensor can be obtained from an action that is a finite function of bare quantities); (ii) one in which the improvement coefficient is a finite quantity, i.e., a finite function of renormalized parameters. We establish a negative result; viz., neither form leads to a finite energy-momentum tensor to O(e 2 λ/sup n/). .AE
Unsupervised Tensor Mining for Big Data Practitioners.
Papalexakis, Evangelos E; Faloutsos, Christos
2016-09-01
Multiaspect data are ubiquitous in modern Big Data applications. For instance, different aspects of a social network are the different types of communication between people, the time stamp of each interaction, and the location associated to each individual. How can we jointly model all those aspects and leverage the additional information that they introduce to our analysis? Tensors, which are multidimensional extensions of matrices, are a principled and mathematically sound way of modeling such multiaspect data. In this article, our goal is to popularize tensors and tensor decompositions to Big Data practitioners by demonstrating their effectiveness, outlining challenges that pertain to their application in Big Data scenarios, and presenting our recent work that tackles those challenges. We view this work as a step toward a fully automated, unsupervised tensor mining tool that can be easily and broadly adopted by practitioners in academia and industry.
Calculus of tensors and differential forms
Sinha, Rajnikant
2014-01-01
Calculus of tensors and differential forms is an introductory-level textbook. Through this book, students will familiarize themselves with tools they need in order to use for further study on general relativity and research, such as affine tensors, tensor calculus on manifolds, relative tensors, Lie derivatives, wedge products, differential forms, and Stokes' theorem. The treatment is concrete and in detail, so that abstract concepts do not deter even physics and engineering students. This self contained book requires undergraduate-level calculus of several variables and linear algebra as prerequisite. Fubini's theorem in real analysis, to be used in Stokes' theorem, has been proved earlier than Stokes' theorem so that students don't have to search elsewhere.
Potentials for transverse trace-free tensors
Conboye, Rory; Murchadha, Niall Ó
2014-01-01
In constructing and understanding initial conditions in the 3 + 1 formalism for numerical relativity, the transverse and trace-free (TT) part of the extrinsic curvature plays a key role. We know that TT tensors possess two degrees of freedom per space point. However, finding an expression for a TT tensor depending on only two scalar functions is a non-trivial task. Assuming either axial or translational symmetry, expressions depending on two scalar potentials alone are derived here for all TT tensors in flat 3-space. In a more general spatial slice, only one of these potentials is found, the same potential given in (Baker and Puzio 1999 Phys. Rev. D 59 044030) and (Dain 2001 Phys. Rev. D 64 124002), with the remaining equations reduced to a partial differential equation, depending on boundary conditions for a solution. As an exercise, we also derive the potentials which give the Bowen-York curvature tensor in flat space. (paper)
Correlators in tensor models from character calculus
A. Mironov
2017-11-01
Full Text Available We explain how the calculations of [20], which provided the first evidence for non-trivial structures of Gaussian correlators in tensor models, are efficiently performed with the help of the (Hurwitz character calculus. This emphasizes a close similarity between technical methods in matrix and tensor models and supports a hope to understand the emerging structures in very similar terms. We claim that the 2m-fold Gaussian correlators of rank r tensors are given by r-linear combinations of dimensions with the Young diagrams of size m. The coefficients are made from the characters of the symmetric group Sm and their exact form depends on the choice of the correlator and on the symmetries of the model. As the simplest application of this new knowledge, we provide simple expressions for correlators in the Aristotelian tensor model as tri-linear combinations of dimensions.
Loop optimization for tensor network renormalization
Yang, Shuo; Gu, Zheng-Cheng; Wen, Xiao-Gang
We introduce a tensor renormalization group scheme for coarse-graining a two-dimensional tensor network, which can be successfully applied to both classical and quantum systems on and off criticality. The key idea of our scheme is to deform a 2D tensor network into small loops and then optimize tensors on each loop. In this way we remove short-range entanglement at each iteration step, and significantly improve the accuracy and stability of the renormalization flow. We demonstrate our algorithm in the classical Ising model and a frustrated 2D quantum model. NSF Grant No. DMR-1005541 and NSFC 11274192, BMO Financial Group, John Templeton Foundation, Government of Canada through Industry Canada, Province of Ontario through the Ministry of Economic Development & Innovation.
An introduction to linear algebra and tensors
Akivis, M A; Silverman, Richard A
1978-01-01
Eminently readable, completely elementary treatment begins with linear spaces and ends with analytic geometry, covering multilinear forms, tensors, linear transformation, and more. 250 problems, most with hints and answers. 1972 edition.
Algebraic classification of the conformal tensor
Ares de Parga, Gonzalo; Chavoya, O.; Lopez B, J.L.; Ovando Z, Gerardo
1989-01-01
Starting from the Petrov matrix method, we deduce a new algorithm (adaptable to computers), within the Newman-Penrose formalism, to obtain the algebraic type of the Weyl tensor in general relativity. (author)
Effects of tensor forces in nuclei
Tanihata, Isao
2013-01-01
Recent studies of nuclei far from the stability line have revealed drastic changes in nuclear orbitals and reported the appearance of new magic numbers and the disappearance of magic numbers observed at the stability line. One of the important reasons for such changes is considered to be because of the effect of tensor forces on nuclear structure. Although the role of tensor forces in binding very light nuclei such as deuterons and 4 He has been known, direct experimental evidence for the effect on nuclear structure is scarce. In this paper, I review known effects of tensor forces in nuclei and then discuss the recently raised question of s–p wave mixing in a halo nucleus of 11 Li. Following these reviews, the development of a new experiment to see the high-momentum components due to the tensor forces is discussed and some of the new data are presented. (paper)
The energy–momentum tensor(s in classical gauge theories
Daniel N. Blaschke
2016-11-01
Full Text Available We give an introduction to, and review of, the energy–momentum tensors in classical gauge field theories in Minkowski space, and to some extent also in curved space–time. For the canonical energy–momentum tensor of non-Abelian gauge fields and of matter fields coupled to such fields, we present a new and simple improvement procedure based on gauge invariance for constructing a gauge invariant, symmetric energy–momentum tensor. The relationship with the Einstein–Hilbert tensor following from the coupling to a gravitational field is also discussed.
Geometric decomposition of the conformation tensor in viscoelastic turbulence
Hameduddin, Ismail; Meneveau, Charles; Zaki, Tamer A.; Gayme, Dennice F.
2018-05-01
This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive-definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive-definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive-definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.
Estimation of Uncertainties of Full Moment Tensors
2017-10-06
For our moment tensor inversions, we use the ‘cut-and-paste’ ( CAP ) code of Zhu and Helmberger (1996) and Zhu and Ben-Zion (2013), with some...modifications. For the misfit function we use an L1 norm Silwal and Tape (2016), and we incorporate the number of misfitting polarities into the waveform... norm of the eigenvalue triple provides the magnitude of the moment tensor, leaving two free parameters to define the source type. In the same year
Superconformal tensor calculus in five dimensions
Fujita, Tomoyuki; Ohashi, Keisuke
2001-01-01
We present a full superconformal tensor calculus in five spacetime dimensions in which the Weyl multiplet has 32 Bose plus 32 Fermi degrees of freedom. It is derived using dimensional reduction from the 6D superconformal tensor calculus. We present two types of 32+32 Weyl multiplets, a vector multiplet, linear multiplet, hypermultiplet and nonlinear multiplet. Their superconformal transformation laws and the embedding and invariant action formulas are given. (author)
Goldsborough, Peter
2016-01-01
Deep learning is a branch of artificial intelligence employing deep neural network architectures that has significantly advanced the state-of-the-art in computer vision, speech recognition, natural language processing and other domains. In November 2015, Google released $\\textit{TensorFlow}$, an open source deep learning software library for defining, training and deploying machine learning models. In this paper, we review TensorFlow and put it in context of modern deep learning concepts and ...
Geometrical foundations of tensor calculus and relativity
Schuller , Frédéric; Lorent , Vincent
2006-01-01
Manifolds, particularly space curves: basic notions 1 The first groundform, the covariant metric tensor 11 The second groundform, Meusnier's theorem 19 Transformation groups in the plane 28 Co- and contravariant components for a special affine transformation in the plane 29 Surface vectors 32 Elements of tensor calculus 36 Generalization of the first groundform to the space 46 The covariant (absolute) derivation 57 Examples from elasticity theory 61 Geodesic lines 63 Main curvatur...
Diffusion tensor MRI: clinical applications
Meli, Francisco; Romero, Carlos; Carpintiero, Silvina; Salvatico, Rosana; Lambre, Hector; Vila, Jose
2005-01-01
Purpose: To evaluate the usefulness of diffusion-tensor imaging (DTI) on different neurological diseases, and to know if this technique shows additional information than conventional Magnetic Resonance Imaging (MRI). Materials and method: Eight patients, with neurological diseases (five patients with brain tumors, one with multiple sclerosis (MS), one with variant Creutzfeldt-Jakob disease (vCJD) and the other with delayed CO intoxication were evaluated. A MR scanner of 1.5 T was used and conventional sequences and DTI with twenty-five directions were done. Quantitative maps were gotten, where the fractional anisotropy (FA) through regions of interest (ROIs) in specific anatomic area were quantified (i.e.: internal and external capsules, frontal and temporal bundles, corpus fibers). Results: In the patients with brain tumors, there was a decrease of FA on intra and peritumoral fibers. Some of them had a disruption in their pattern. In patients with MS and CO intoxication, partial interruption along white matter bundles was demonstrated. However, a 'mismatch' between the findings of FLAIR, Diffusion-weighted images (DWI) and DTI, in the case of CO intoxication, was seen. Conclusions: DTI gave more information compared to conventional sequences about ultrastructural brain tissue in almost all the diseases above mentioned. Therefore, there is a work in progress about DTI acquisition, to evaluate a new technique, called tractography. (author)
Measuring Nematic Susceptibilities from the Elastoresistivity Tensor
Hristov, A. T.; Shapiro, M. C.; Hlobil, Patrick; Maharaj, Akash; Chu, Jiun-Haw; Fisher, Ian
The elastoresistivity tensor mijkl relates changes in resistivity to the strain on a material. As a fourth-rank tensor, it contains considerably more information about the material than the simpler (second-rank) resistivity tensor; in particular, certain elastoresistivity coefficients can be related to thermodynamic susceptibilities and serve as a direct probe of symmetry breaking at a phase transition. The aim of this talk is twofold. First, we enumerate how symmetry both constrains the structure of the elastoresistivity tensor into an easy-to-understand form and connects tensor elements to thermodynamic susceptibilities. In the process, we generalize previous studies of elastoresistivity to include the effects of magnetic field. Second, we describe an approach to measuring quantities in the elastoresistivity tensor with a novel transverse measurement, which is immune to relative strain offsets. These techniques are then applied to BaFe2As2 in a proof of principle measurement. This work is supported by the Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515.
On the concircular curvature tensor of Riemannian manifolds
Rahman, M.S.; Lal, S.
1990-06-01
Definition of the concircular curvature tensor, Z hijk , along with Z-tensor, Z ij , is given and some properties of Z hijk are described. Tensors identical with Z hijk are shown. A necessary and sufficient condition that a Riemannian V n has zero Z-tensor is found. A number of theorems on concircular symmetric space, concircular recurrent space (Z n -space) and Z n -space with zero Z-tensor are deduced. (author). 6 refs
(Ln-bar, g)-spaces. Special tensor fields
Manoff, S.; Dimitrov, B.
1998-01-01
The Kronecker tensor field, the contraction tensor field, as well as the multi-Kronecker and multi-contraction tensor fields are determined and the action of the covariant differential operator, the Lie differential operator, the curvature operator, and the deviation operator on these tensor fields is established. The commutation relations between the operators Sym and Asym and the covariant and Lie differential operators are considered acting on symmetric and antisymmetric tensor fields over (L n bar, g)-spaces
Marin Quintero, Maider J.
2013-01-01
The structure tensor for vector valued images is most often defined as the average of the scalar structure tensors in each band. The problem with this definition is the assumption that all bands provide the same amount of edge information giving them the same weights. As a result non-edge pixels can be reinforced and edges can be weakened…
Lepore, N; Brun, C; Chou, Y Y; Chiang, M C; Dutton, R A; Hayashi, K M; Luders, E; Lopez, O L; Aizenstein, H J; Toga, A W; Becker, J T; Thompson, P M
2008-01-01
This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor) of these deformations, as is common, we retain the full deformation tensors and apply a manifold version of Hotelling's $T(2) test to them, in a Log-Euclidean domain. In 2-D and 3-D magnetic resonance imaging (MRI) data from 26 HIV/AIDS patients and 14 matched healthy subjects, we compared multivariate tensor analysis versus univariate tests of simpler tensor-derived indices: the Jacobian determinant, the trace, geodesic anisotropy, and eigenvalues of the deformation tensor, and the angle of rotation of its eigenvectors. We detected consistent, but more extensive patterns of structural abnormalities, with multivariate tests on the full tensor manifold. Their improved power was established by analyzing cumulative p-value plots using false discovery rate (FDR) methods, appropriately controlling for false positives. This increased detection sensitivity may empower drug trials and large-scale studies of disease that use tensor-based morphometry.
Tensor network state correspondence and holography
Singh, Sukhwinder
2018-01-01
In recent years, tensor network states have emerged as a very useful conceptual and simulation framework to study quantum many-body systems at low energies. In this paper, we describe a particular way in which any given tensor network can be viewed as a representation of two different quantum many-body states. The two quantum many-body states are said to correspond to each other by means of the tensor network. We apply this "tensor network state correspondence"—a correspondence between quantum many-body states mediated by tensor networks as we describe—to the multi-scale entanglement renormalization ansatz (MERA) representation of ground states of one dimensional (1D) quantum many-body systems. Since the MERA is a 2D hyperbolic tensor network (the extra dimension is identified as the length scale of the 1D system), the two quantum many-body states obtained from the MERA, via tensor network state correspondence, are seen to live in the bulk and on the boundary of a discrete hyperbolic geometry. The bulk state so obtained from a MERA exhibits interesting features, some of which caricature known features of the holographic correspondence of String theory. We show how (i) the bulk state admits a description in terms of "holographic screens", (ii) the conformal field theory data associated with a critical ground state can be obtained from the corresponding bulk state, in particular, how pointlike boundary operators are identified with extended bulk operators. (iii) We also present numerical results to illustrate that bulk states, dual to ground states of several critical spin chains, have exponentially decaying correlations, and that the bulk correlation length generally decreases with increase in central charge for these spin chains.
Killing-Yano tensors, rank-2 Killing tensors, and conserved quantities in higher dimensions
Krtous, Pavel [Institute of Theoretical Physics, Charles University, V Holesovickach 2, Prague (Czech Republic); Kubiznak, David [Institute of Theoretical Physics, Charles University, V Holesovickach 2, Prague (Czech Republic); Page, Don N. [Theoretical Physics Institute, University of Alberta, Edmonton T6G 2G7, Alberta (Canada); Frolov, Valeri P. [Theoretical Physics Institute, University of Alberta, Edmonton T6G 2G7, Alberta (Canada)
2007-02-15
From the metric and one Killing-Yano tensor of rank D-2 in any D-dimensional spacetime with such a principal Killing-Yano tensor, we show how to generate k = [(D+1)/2] Killing-Yano tensors, of rank D-2j for all 0 {<=} j {<=} k-1, and k rank-2 Killing tensors, giving k constants of geodesic motion that are in involution. For the example of the Kerr-NUT-AdS spacetime (hep-th/0604125) with its principal Killing-Yano tensor (gr-qc/0610144), these constants and the constants from the k Killing vectors give D independent constants in involution, making the geodesic motion completely integrable (hep-th/0611083). The constants of motion are also related to the constants recently obtained in the separation of the Hamilton-Jacobi and Klein-Gordon equations (hep-th/0611245)
Killing-Yano tensors, rank-2 Killing tensors, and conserved quantities in higher dimensions
Krtous, Pavel; Kubiznak, David; Page, Don N.; Frolov, Valeri P.
2007-01-01
From the metric and one Killing-Yano tensor of rank D-2 in any D-dimensional spacetime with such a principal Killing-Yano tensor, we show how to generate k = [(D+1)/2] Killing-Yano tensors, of rank D-2j for all 0 ≤ j ≤ k-1, and k rank-2 Killing tensors, giving k constants of geodesic motion that are in involution. For the example of the Kerr-NUT-AdS spacetime (hep-th/0604125) with its principal Killing-Yano tensor (gr-qc/0610144), these constants and the constants from the k Killing vectors give D independent constants in involution, making the geodesic motion completely integrable (hep-th/0611083). The constants of motion are also related to the constants recently obtained in the separation of the Hamilton-Jacobi and Klein-Gordon equations (hep-th/0611245)
Susceptibility tensor imaging (STI) of the brain.
Li, Wei; Liu, Chunlei; Duong, Timothy Q; van Zijl, Peter C M; Li, Xu
2017-04-01
Susceptibility tensor imaging (STI) is a recently developed MRI technique that allows quantitative determination of orientation-independent magnetic susceptibility parameters from the dependence of gradient echo signal phase on the orientation of biological tissues with respect to the main magnetic field. By modeling the magnetic susceptibility of each voxel as a symmetric rank-2 tensor, individual magnetic susceptibility tensor elements as well as the mean magnetic susceptibility and magnetic susceptibility anisotropy can be determined for brain tissues that would still show orientation dependence after conventional scalar-based quantitative susceptibility mapping to remove such dependence. Similar to diffusion tensor imaging, STI allows mapping of brain white matter fiber orientations and reconstruction of 3D white matter pathways using the principal eigenvectors of the susceptibility tensor. In contrast to diffusion anisotropy, the main determinant factor of the susceptibility anisotropy in brain white matter is myelin. Another unique feature of the susceptibility anisotropy of white matter is its sensitivity to gadolinium-based contrast agents. Mechanistically, MRI-observed susceptibility anisotropy is mainly attributed to the highly ordered lipid molecules in the myelin sheath. STI provides a consistent interpretation of the dependence of phase and susceptibility on orientation at multiple scales. This article reviews the key experimental findings and physical theories that led to the development of STI, its practical implementations, and its applications for brain research. Copyright © 2016 John Wiley & Sons, Ltd. Copyright © 2016 John Wiley & Sons, Ltd.
Susceptibility Tensor Imaging (STI) of the Brain
Li, Wei; Liu, Chunlei; Duong, Timothy Q.; van Zijl, Peter C.M.; Li, Xu
2016-01-01
Susceptibility tensor imaging (STI) is a recently developed MRI technique that allows quantitative determination of orientation-independent magnetic susceptibility parameters from the dependence of gradient echo signal phase on the orientation of biological tissues with respect to the main magnetic field. By modeling the magnetic susceptibility of each voxel as a symmetric rank-2 tensor, individual magnetic susceptibility tensor elements as well as the mean magnetic susceptibility (MMS) and magnetic susceptibility anisotropy (MSA) can be determined for brain tissues that would still show orientation dependence after conventional scalar-based quantitative susceptibility mapping (QSM) to remove such dependence. Similar to diffusion tensor imaging (DTI), STI allows mapping of brain white matter fiber orientations and reconstruction of 3D white matter pathways using the principal eigenvectors of the susceptibility tensor. In contrast to diffusion anisotropy, the main determinant factor of susceptibility anisotropy in brain white matter is myelin. Another unique feature of susceptibility anisotropy of white matter is its sensitivity to gadolinium-based contrast agents. Mechanistically, MRI-observed susceptibility anisotropy is mainly attributed to the highly ordered lipid molecules in myelin sheath. STI provides a consistent interpretation of the dependence of phase and susceptibility on orientation at multiple scales. This article reviews the key experimental findings and physical theories that led to the development of STI, its practical implementations, and its applications for brain research. PMID:27120169
Off-shell N = 2 tensor supermultiplets
Wit, Bernard de; Saueressig, Frank
2006-01-01
A multiplet calculus is presented for an arbitrary number n of N = 2 tensor supermultiplets. For rigid supersymmetry the known couplings are reproduced. In the superconformal case the target spaces parametrized by the scalar fields are cones over (3n-1)-dimensional spaces encoded in homogeneous SU(2) invariant potentials, subject to certain constraints. The coupling to conformal supergravity enables the derivation of a large class of supergravity Lagrangians with vector and tensor multiplets and hypermultiplets. Dualizing the tensor fields into scalars leads to hypermultiplets with hyperkaehler or quaternion-Kaehler target spaces with at least n abelian isometries. It is demonstrated how to use the calculus for the construction of Lagrangians containing higher-derivative couplings of tensor multiplets. For the application of the c-map between vector and tensor supermultiplets to Lagrangians with higher-order derivatives, an off-shell version of this map is proposed. Various other implications of the results are discussed. As an example an elegant derivation of the classification of 4-dimensional quaternion-Kaehler manifolds with two commuting isometries is given
Huf, P A; Carminati, J
2015-01-01
In this paper we: (1) introduce TensorPack, a software package for the algebraic manipulation of tensors in covariant index format in Maple; (2) briefly demonstrate the use of the package with an orthonormal tensor proof of the shearfree conjecture for dust. TensorPack is based on the Riemann and Canon tensor software packages and uses their functions to express tensors in an indexed covariant format. TensorPack uses a string representation as input and provides functions for output in index form. It extends the functionality to basic algebra of tensors, substitution, covariant differentiation, contraction, raising/lowering indices, symmetry functions and other accessory functions. The output can be merged with text in the Maple environment to create a full working document with embedded dynamic functionality. The package offers potential for manipulation of indexed algebraic tensor expressions in a flexible software environment. (paper)
Energy-momentum tensor in the fermion-pairing model
Kawati, S.; Miyata, H.
1980-01-01
The symmetric energy-momentum tensor for the self-interacting fermion theory (psi-barpsi) 2 is expressed in terms of the collective mode within the Hartree approximation. The divergent part of the energy-momentum tensor for the fermion theory induces an effective energy-momentum tensor for the collective mode, and this effective energy-momentum tensor automatically has the Callan-Coleman-Jackiw improved form. The renormalized energy-momentum tensor is structurally equivalent to the Callan-Coleman-Jackiw improved tensor for the Yukawa theory
(Ln-bar, g)-spaces. Ordinary and tensor differentials
Manoff, S.; Dimitrov, B.
1998-01-01
Different types of differentials as special cases of differential operators acting on tensor fields over (L n bar, g)-spaces are considered. The ordinary differential, the covariant differential as a special case of the covariant differential operator, and the Lie differential as a special case of the Lie differential operator are investigated. The tensor differential and its special types (Covariant tensor differential, and Lie tensor differential) are determined and their properties are discussed. Covariant symmetric and antisymmetric (external) tensor differentials, Lie symmetric, and Lie antisymmetric (external) tensor differentials are determined and considered over (L n bar, g)-spaces
Federated Tensor Factorization for Computational Phenotyping
Kim, Yejin; Sun, Jimeng; Yu, Hwanjo; Jiang, Xiaoqian
2017-01-01
Tensor factorization models offer an effective approach to convert massive electronic health records into meaningful clinical concepts (phenotypes) for data analysis. These models need a large amount of diverse samples to avoid population bias. An open challenge is how to derive phenotypes jointly across multiple hospitals, in which direct patient-level data sharing is not possible (e.g., due to institutional policies). In this paper, we developed a novel solution to enable federated tensor factorization for computational phenotyping without sharing patient-level data. We developed secure data harmonization and federated computation procedures based on alternating direction method of multipliers (ADMM). Using this method, the multiple hospitals iteratively update tensors and transfer secure summarized information to a central server, and the server aggregates the information to generate phenotypes. We demonstrated with real medical datasets that our method resembles the centralized training model (based on combined datasets) in terms of accuracy and phenotypes discovery while respecting privacy. PMID:29071165
Tensor calculus for engineers and physicists
de Souza Sánchez Filho, Emil
2016-01-01
This textbook provides a rigorous approach to tensor manifolds in several aspects relevant for Engineers and Physicists working in industry or academia. With a thorough, comprehensive, and unified presentation, this book offers insights into several topics of tensor analysis, which covers all aspects of N dimensional spaces. The main purpose of this book is to give a self-contained yet simple, correct and comprehensive mathematical explanation of tensor calculus for undergraduate and graduate students and for professionals. In addition to many worked problems, this book features a selection of examples, solved step by step. Although no emphasis is placed on special and particular problems of Engineering or Physics, the text covers the fundamentals of these fields of science. The book makes a brief introduction into the basic concept of the tensorial formalism so as to allow the reader to make a quick and easy review of the essential topics that enable having the grounds for the subsequent themes, without need...
Exploring extra dimensions through inflationary tensor modes
Im, Sang Hui; Nilles, Hans Peter; Trautner, Andreas
2018-03-01
Predictions of inflationary schemes can be influenced by the presence of extra dimensions. This could be of particular relevance for the spectrum of gravitational waves in models where the extra dimensions provide a brane-world solution to the hierarchy problem. Apart from models of large as well as exponentially warped extra dimensions, we analyze the size of tensor modes in the Linear Dilaton scheme recently revived in the discussion of the "clockwork mechanism". The results are model dependent, significantly enhanced tensor modes on one side and a suppression on the other. In some cases we are led to a scheme of "remote inflation", where the expansion is driven by energies at a hidden brane. In all cases where tensor modes are enhanced, the requirement of perturbativity of gravity leads to a stringent upper limit on the allowed Hubble rate during inflation.
On an uninterpretated tensor in Dirac's theory
Costa de Beauregard, O.
1989-01-01
Franz, in 1935, deduced systematically from the Dirac equation 10 tensorial equations, 5 with a mechanical interpretation, 5 with an electromagnetic interpretation, which are also consequences of Kemmer's formalism for spins 1 and 0; Durand, in 1944, operating similarly with the second order Dirac equation, obtained, 10 equations, 5 of which expressing the divergences of the Gordon type tensors. Of these equations, together with the tensors they imply, some are easily interpreted by reference to the classical theories, some other remain uniterpreted. Recently (1988) we proposed a theory of the coupling between Einstein's gravity field and the 5 Franz mechanical equations, yielding as a bonus the complete interpretation of the 5 Franz mechanical equations. This is an incitation to reexamine the 5 electromagnetic equations. We show here that two of these, together with one of the Durand equations, implying the same tensor, remain uninterpreted. This is proposed as a challenge to the reader's sagacity [fr
The Riemann-Lovelock curvature tensor
Kastor, David
2012-01-01
In order to study the properties of Lovelock gravity theories in low dimensions, we define the kth-order Riemann-Lovelock tensor as a certain quantity having a total 4k-indices, which is kth order in the Riemann curvature tensor and shares its basic algebraic and differential properties. We show that the kth-order Riemann-Lovelock tensor is determined by its traces in dimensions 2k ≤ D < 4k. In D = 2k + 1 this identity implies that all solutions of pure kth-order Lovelock gravity are 'Riemann-Lovelock' flat. It is verified that the static, spherically symmetric solutions of these theories, which are missing solid angle spacetimes, indeed satisfy this flatness property. This generalizes results from Einstein gravity in D = 3, which corresponds to the k = 1 case. We speculate about some possible further consequences of Riemann-Lovelock curvature. (paper)
Diffusion tensor smoothing through weighted Karcher means
Carmichael, Owen; Chen, Jun; Paul, Debashis; Peng, Jie
2014-01-01
Diffusion tensor magnetic resonance imaging (MRI) quantifies the spatial distribution of water Diffusion at each voxel on a regular grid of locations in a biological specimen by Diffusion tensors– 3 × 3 positive definite matrices. Removal of noise from DTI is an important problem due to the high scientific relevance of DTI and relatively low signal to noise ratio it provides. Leading approaches to this problem amount to estimation of weighted Karcher means of Diffusion tensors within spatial neighborhoods, under various metrics imposed on the space of tensors. However, it is unclear how the behavior of these estimators varies with the magnitude of DTI sensor noise (the noise resulting from the thermal e!ects of MRI scanning) as well as the geometric structure of the underlying Diffusion tensor neighborhoods. In this paper, we combine theoretical analysis, empirical analysis of simulated DTI data, and empirical analysis of real DTI scans to compare the noise removal performance of three kernel-based DTI smoothers that are based on Euclidean, log-Euclidean, and affine-invariant metrics. The results suggest, contrary to conventional wisdom, that imposing a simplistic Euclidean metric may in fact provide comparable or superior noise removal, especially in relatively unstructured regions and/or in the presence of moderate to high levels of sensor noise. On the contrary, log-Euclidean and affine-invariant metrics may lead to better noise removal in highly structured anatomical regions, especially when the sensor noise is of low magnitude. These findings emphasize the importance of considering the interplay of sensor noise magnitude and tensor field geometric structure when assessing Diffusion tensor smoothing options. They also point to the necessity for continued development of smoothing methods that perform well across a large range of scenarios. PMID:25419264
Aspects of the Antisymmetric Tensor Field
Lahiri, Amitabha
1991-02-01
With the possible exception of gravitation, fundamental interactions are generally described by theories of point particles interacting via massless gauge fields. Since the advent of string theories the picture of physical interaction has changed to accommodate one in which extended objects interact with each other. The generalization of the gauge theories to extended objects leads to theories of antisymmetric tensor fields. At scales corresponding to present-day laboratory experiments one expects to see only point particles, their interactions modified by the presence of antisymmetric tensor fields in the theory. Therefore, in order to establish the validity of any theory with antisymmetric tensor fields one needs to look for manifestations of these fields at low energies. The principal problem of gauge theories is the failure to provide a suitable explanation for the generation of masses for the fields in the theory. While there is a known mechanism (spontaneous symmetry breaking) for generating masses for both the matter fields and the gauge fields, the lack of experimental evidence in support of an elementary scalar field suggests that one look for alternative ways of generating masses for the fields. The interaction of gauge fields with an antisymmetric tensor field seems to be an attractive way of doing so, especially since all indications point to the possibility that there will be no remnant degrees of freedom. On the other hand the interaction of such a field with black holes suggest an independent way of verifying the existence of such fields. In this dissertation the origins of the antisymmetric tensor field are discussed in terms of string theory. The interaction of black holes with such a field is discussed next. The last chapter discusses the effects of an antisymmetric tensor field on quantum electrodynamics when the fields are minimally coupled.
Tensor hypercontraction. II. Least-squares renormalization
Parrish, Robert M.; Hohenstein, Edward G.; Martínez, Todd J.; Sherrill, C. David
2012-12-01
The least-squares tensor hypercontraction (LS-THC) representation for the electron repulsion integral (ERI) tensor is presented. Recently, we developed the generic tensor hypercontraction (THC) ansatz, which represents the fourth-order ERI tensor as a product of five second-order tensors [E. G. Hohenstein, R. M. Parrish, and T. J. Martínez, J. Chem. Phys. 137, 044103 (2012)], 10.1063/1.4732310. Our initial algorithm for the generation of the THC factors involved a two-sided invocation of overlap-metric density fitting, followed by a PARAFAC decomposition, and is denoted PARAFAC tensor hypercontraction (PF-THC). LS-THC supersedes PF-THC by producing the THC factors through a least-squares renormalization of a spatial quadrature over the otherwise singular 1/r12 operator. Remarkably, an analytical and simple formula for the LS-THC factors exists. Using this formula, the factors may be generated with O(N^5) effort if exact integrals are decomposed, or O(N^4) effort if the decomposition is applied to density-fitted integrals, using any choice of density fitting metric. The accuracy of LS-THC is explored for a range of systems using both conventional and density-fitted integrals in the context of MP2. The grid fitting error is found to be negligible even for extremely sparse spatial quadrature grids. For the case of density-fitted integrals, the additional error incurred by the grid fitting step is generally markedly smaller than the underlying Coulomb-metric density fitting error. The present results, coupled with our previously published factorizations of MP2 and MP3, provide an efficient, robust O(N^4) approach to both methods. Moreover, LS-THC is generally applicable to many other methods in quantum chemistry.
Diffusion tensor imaging in spinal cord compression
Wang, Wei; Qin, Wen; Hao, Nanxin; Wang, Yibin; Zong, Genlin
2012-01-01
Background Although diffusion tensor imaging has been successfully applied in brain research for decades, several main difficulties have hindered its extended utilization in spinal cord imaging. Purpose To assess the feasibility and clinical value of diffusion tensor imaging and tractography for evaluating chronic spinal cord compression. Material and Methods Single-shot spin-echo echo-planar DT sequences were scanned in 42 spinal cord compression patients and 49 healthy volunteers. The mean values of the apparent diffusion coefficient and fractional anisotropy were measured in region of interest at the cervical and lower thoracic spinal cord. The patients were divided into two groups according to the high signal on T2WI (the SCC-HI group and the SCC-nHI group for with or without high signal). A one-way ANOVA was used. Diffusion tensor tractography was used to visualize the morphological features of normal and impaired white matter. Results There were no statistically significant differences in the apparent diffusion coefficient and fractional anisotropy values between the different spinal cord segments of the normal subjects. All of the patients in the SCC-HI group had increased apparent diffusion coefficient values and decreased fractional anisotropy values at the lesion level compared to the normal controls. However, there were no statistically significant diffusion index differences between the SCC-nHI group and the normal controls. In the diffusion tensor imaging maps, the normal spinal cord sections were depicted as fiber tracts that were color-encoded to a cephalocaudal orientation. The diffusion tensor images were compressed to different degrees in all of the patients. Conclusion Diffusion tensor imaging and tractography are promising methods for visualizing spinal cord tracts and can provide additional information in clinical studies in spinal cord compression
Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery.
Xie, Qi; Zhao, Qian; Meng, Deyu; Xu, Zongben
2017-08-02
It is well known that the sparsity/low-rank of a vector/matrix can be rationally measured by nonzero-entries-number ($l_0$ norm)/nonzero- singular-values-number (rank), respectively. However, data from real applications are often generated by the interaction of multiple factors, which obviously cannot be sufficiently represented by a vector/matrix, while a high order tensor is expected to provide more faithful representation to deliver the intrinsic structure underlying such data ensembles. Unlike the vector/matrix case, constructing a rational high order sparsity measure for tensor is a relatively harder task. To this aim, in this paper we propose a measure for tensor sparsity, called Kronecker-basis-representation based tensor sparsity measure (KBR briefly), which encodes both sparsity insights delivered by Tucker and CANDECOMP/PARAFAC (CP) low-rank decompositions for a general tensor. Then we study the KBR regularization minimization (KBRM) problem, and design an effective ADMM algorithm for solving it, where each involved parameter can be updated with closed-form equations. Such an efficient solver makes it possible to extend KBR to various tasks like tensor completion and tensor robust principal component analysis. A series of experiments, including multispectral image (MSI) denoising, MSI completion and background subtraction, substantiate the superiority of the proposed methods beyond state-of-the-arts.
Fiber crossing in human brain depicted with diffusion tensor MR imaging
Wiegell, M.R.; Larsson, H.B.; Wedeen, V.J.
2000-01-01
Human white matter fiber crossings were investigated with use of the full eigenstructure of the magnetic resonance diffusion tensor. Intravoxel fiber dispersions were characterized by the plane spanned by the major and medium eigenvectors and depicted with three-dimensional graphics. This method...
Reconstruction of convex bodies from surface tensors
Kousholt, Astrid; Kiderlen, Markus
We present two algorithms for reconstruction of the shape of convex bodies in the two-dimensional Euclidean space. The first reconstruction algorithm requires knowledge of the exact surface tensors of a convex body up to rank s for some natural number s. The second algorithm uses harmonic intrinsic...... volumes which are certain values of the surface tensors and allows for noisy measurements. From a generalized version of Wirtinger's inequality, we derive stability results that are utilized to ensure consistency of both reconstruction procedures. Consistency of the reconstruction procedure based...
Improving Tensor Based Recommenders with Clustering
Leginus, Martin; Dolog, Peter; Zemaitis, Valdas
2012-01-01
Social tagging systems (STS) model three types of entities (i.e. tag-user-item) and relationships between them are encoded into a 3-order tensor. Latent relationships and patterns can be discovered by applying tensor factorization techniques like Higher Order Singular Value Decomposition (HOSVD),...... of the recommendations and execution time are improved and memory requirements are decreased. The clustering is motivated by the fact that many tags in a tag space are semantically similar thus the tags can be grouped. Finally, promising experimental results are presented...
Tensor modes in pure natural inflation
Nomura, Yasunori; Yamazaki, Masahito
2018-05-01
We study tensor modes in pure natural inflation [1], a recently-proposed inflationary model in which an axionic inflaton couples to pure Yang-Mills gauge fields. We find that the tensor-to-scalar ratio r is naturally bounded from below. This bound originates from the finiteness of the number of metastable branches of vacua in pure Yang-Mills theories. Details of the model can be probed by future cosmic microwave background experiments and improved lattice gauge theory calculations of the θ-angle dependence of the vacuum energy.
Effective field theory approaches for tensor potentials
Jansen, Maximilian
2016-11-14
Effective field theories are a widely used tool to study physical systems at low energies. We apply them to systematically analyze two and three particles interacting via tensor potentials. Two examples are addressed: pion interactions for anti D{sup 0}D{sup *0} scattering to dynamically generate the X(3872) and dipole interactions for two and three bosons at low energies. For the former, the one-pion exchange and for the latter, the long-range dipole force induce a tensor-like structure of the potential. We apply perturbative as well as non-perturbative methods to determine low-energy observables. The X(3872) is of major interest in modern high-energy physics. Its exotic characteristics require approaches outside the range of the quark model for baryons and mesons. Effective field theories represent such methods and provide access to its peculiar nature. We interpret the X(3872) as a hadronic molecule consisting of neutral D and D{sup *} mesons. It is possible to apply an effective field theory with perturbative pions. Within this framework, we address chiral as well as finite volume extrapolations for low-energy observables, such as the binding energy and the scattering length. We show that the two-point correlation function for the D{sup *0} meson has to be resummed to cure infrared divergences. Moreover, next-to-leading order coupling constants, which were introduced by power counting arguments, appear to be essential to renormalize the scattering amplitude. The binding energy as well as the scattering length display a moderate dependence on the light quark masses. The X(3872) is most likely deeper bound for large light quark masses. In a finite volume on the other hand, the binding energy significantly increases. The dependence on the light quark masses and the volume size can be simultaneously obtained. For bosonic dipoles we apply a non-perturbative, numerical approach. We solve the Lippmann-Schwinger equation for the two-dipole system and the Faddeev
Tucker Tensor analysis of Matern functions in spatial statistics
Litvinenko, Alexander; Keyes, David E.; Khoromskaia, Venera; Khoromskij, Boris N.; Matthies, Hermann G.
2018-01-01
in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence
Diffusion tensor metrics as biomarkers in Alzheimer's disease.
Julio Acosta-Cabronero
Full Text Available Although diffusion tensor imaging has been a major research focus for Alzheimer's disease in recent years, it remains unclear whether it has sufficient stability to have biomarker potential. To date, frequently inconsistent results have been reported, though lack of standardisation in acquisition and analysis make such discrepancies difficult to interpret. There is also, at present, little knowledge of how the biometric properties of diffusion tensor imaging might evolve in the course of Alzheimer's disease.The biomarker question was addressed in this study by adopting a standardised protocol both for the whole brain (tract-based spatial statistics, and for a region of interest: the midline corpus callosum. In order to study the evolution of tensor changes, cross-sectional data from very mild (N = 21 and mild (N = 22 Alzheimer's disease patients were examined as well as a longitudinal cohort (N = 16 that had been rescanned at 12 months.The results revealed that increased axial and mean diffusivity are the first abnormalities to occur and that the first region to develop such significant differences was mesial parietal/splenial white matter; these metrics, however, remained relatively static with advancing disease indicating they are suitable as 'state-specific' markers. In contrast, increased radial diffusivity, and therefore decreased fractional anisotropy-though less detectable early-became increasingly abnormal with disease progression, and, in the splenium of the corpus callosum, correlated significantly with dementia severity; these metrics therefore appear 'stage-specific' and would be ideal for monitoring disease progression. In addition, the cross-sectional and longitudinal analyses showed that the progressive abnormalities in radial diffusivity and fractional anisotropy always occurred in areas that had first shown an increase in axial and mean diffusivity. Given that the former two metrics correlate with dementia severity
Tensor completion for PDEs with uncertain coefficients and Bayesian Update
Litvinenko, Alexander
2017-03-05
In this work, we tried to show connections between Bayesian update and tensor completion techniques. Usually, only a small/sparse vector/tensor of measurements is available. The typical measurement is a function of the solution. The solution of a stochastic PDE is a tensor, the measurement as well. The idea is to use completion techniques to compute all "missing" values of the measurement tensor and only then apply the Bayesian technique.
Tensor completion for PDEs with uncertain coefficients and Bayesian Update
Litvinenko, Alexander
2017-01-01
In this work, we tried to show connections between Bayesian update and tensor completion techniques. Usually, only a small/sparse vector/tensor of measurements is available. The typical measurement is a function of the solution. The solution of a stochastic PDE is a tensor, the measurement as well. The idea is to use completion techniques to compute all "missing" values of the measurement tensor and only then apply the Bayesian technique.
Concatenated image completion via tensor augmentation and completion
Bengua, Johann A.; Tuan, Hoang D.; Phien, Ho N.; Do, Minh N.
2016-01-01
This paper proposes a novel framework called concatenated image completion via tensor augmentation and completion (ICTAC), which recovers missing entries of color images with high accuracy. Typical images are second- or third-order tensors (2D/3D) depending if they are grayscale or color, hence tensor completion algorithms are ideal for their recovery. The proposed framework performs image completion by concatenating copies of a single image that has missing entries into a third-order tensor,...
Norm of the Riemannian Curvature Tensor
We consider the Riemannian functional R p ( g ) = ∫ M | R ( g ) | p d v g defined on the space of Riemannian metrics with unit volume on a closed smooth manifold where R ( g ) and d v g denote the corresponding Riemannian curvature tensor and volume form and p ∈ ( 0 , ∞ ) . First we prove that the Riemannian metrics ...
Abelian tensor models on the lattice
Chaudhuri, Soumyadeep; Giraldo-Rivera, Victor I.; Joseph, Anosh; Loganayagam, R.; Yoon, Junggi
2018-04-01
We consider a chain of Abelian Klebanov-Tarnopolsky fermionic tensor models coupled through quartic nearest-neighbor interactions. We characterize the gauge-singlet spectrum for small chains (L =2 ,3 ,4 ,5 ) and observe that the spectral statistics exhibits strong evidence in favor of quasi-many-body localization.
Tensor squeezed limits and the Higuchi bound
Bordin, Lorenzo [SISSA, via Bonomea 265, 34136, Trieste (Italy); Creminelli, Paolo [Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151, Trieste (Italy); Mirbabayi, Mehrdad [Institute for Advanced Study, Princeton, NJ 08540 (United States); Noreña, Jorge, E-mail: lbordin@sissa.it, E-mail: creminel@ictp.it, E-mail: mehrdadm@ias.edu, E-mail: jorge.norena@pucv.cl [Instituto de Física, Pontificia Universidad Católica de Valparaíso, Avenida Universidad 330, Curauma, Valparaíso (Chile)
2016-09-01
We point out that tensor consistency relations—i.e. the behavior of primordial correlation functions in the limit a tensor mode has a small momentum—are more universal than scalar consistency relations. They hold in the presence of multiple scalar fields and as long as anisotropies are diluted exponentially fast. When de Sitter isometries are approximately respected during inflation this is guaranteed by the Higuchi bound, which forbids the existence of light particles with spin: de Sitter space can support scalar hair but no curly hair. We discuss two indirect ways to look for the violation of tensor consistency relations in observations, as a signature of models in which inflation is not a strong isotropic attractor, such as solid inflation: (a) graviton exchange contribution to the scalar four-point function; (b) quadrupolar anisotropy of the scalar power spectrum due to super-horizon tensor modes. This anisotropy has a well-defined statistics which can be distinguished from cases in which the background has a privileged direction.
Fermionic topological quantum states as tensor networks
Wille, C.; Buerschaper, O.; Eisert, J.
2017-06-01
Tensor network states, and in particular projected entangled pair states, play an important role in the description of strongly correlated quantum lattice systems. They do not only serve as variational states in numerical simulation methods, but also provide a framework for classifying phases of quantum matter and capture notions of topological order in a stringent and rigorous language. The rapid development in this field for spin models and bosonic systems has not yet been mirrored by an analogous development for fermionic models. In this work, we introduce a tensor network formalism capable of capturing notions of topological order for quantum systems with fermionic components. At the heart of the formalism are axioms of fermionic matrix-product operator injectivity, stable under concatenation. Building upon that, we formulate a Grassmann number tensor network ansatz for the ground state of fermionic twisted quantum double models. A specific focus is put on the paradigmatic example of the fermionic toric code. This work shows that the program of describing topologically ordered systems using tensor networks carries over to fermionic models.
Higher-order tensors in diffusion imaging
Schultz, T.; Fuster, A.; Ghosh, A.; Deriche, R.; Florack, L.M.J.; Lim, L.H.; Westin, C.-F.; Vilanova, A.; Burgeth, B.
2014-01-01
Diffusion imaging is a noninvasive tool for probing the microstructure of fibrous nerve and muscle tissue. Higher-order tensors provide a powerful mathematical language to model and analyze the large and complex data that is generated by its modern variants such as High Angular Resolution Diffusion
Visualization and processing of tensor fields
Weickert, Joachim
2007-01-01
Presents information on the visualization and processing of tensor fields. This book serves as an overview for the inquiring scientist, as a basic foundation for developers and practitioners, and as a textbook for specialized classes and seminars for graduate and doctoral students.
Dark energy in scalar-tensor theories
Moeller, J.
2007-12-15
We investigate several aspects of dynamical dark energy in the framework of scalar-tensor theories of gravity. We provide a classification of scalar-tensor coupling functions admitting cosmological scaling solutions. In particular, we recover that Brans-Dicke theory with inverse power-law potential allows for a sequence of background dominated scaling regime and scalar field dominated, accelerated expansion. Furthermore, we compare minimally and non-minimally coupled models, with respect to the small redshift evolution of the dark energy equation of state. We discuss the possibility to discriminate between different models by a reconstruction of the equation-of-state parameter from available observational data. The non-minimal coupling characterizing scalar-tensor models can - in specific cases - alleviate fine tuning problems, which appear if (minimally coupled) quintessence is required to mimic a cosmological constant. Finally, we perform a phase-space analysis of a family of biscalar-tensor models characterized by a specific type of {sigma}-model metric, including two examples from recent literature. In particular, we generalize an axion-dilaton model of Sonner and Townsend, incorporating a perfect fluid background consisting of (dark) matter and radiation. (orig.)
Introduction to vector and tensor analysis
Wrede, Robert C
1972-01-01
A broad introductory treatment, this volume examines general Cartesian coordinates, the cross product, Einstein's special theory of relativity, bases in general coordinate systems, maxima and minima of functions of two variables, line integrals, integral theorems, fundamental notions in n-space, Riemannian geometry, algebraic properties of the curvature tensor, and more. 1963 edition.
Curvature tensor copies in affine geometry
Srivastava, P.P.
1981-01-01
The sets of space-time and spin-connections which give rise to the same curvature tensor are constructed. The corresponding geometries are compared. Results are illustrated by an explicit calculation and comment on the copies in Einstein-Cartan and Weyl-Cartan geometries. (Author) [pt
Positivity of linear maps under tensor powers
Müller-Hermes, Alexander, E-mail: muellerh@ma.tum.de; Wolf, Michael M., E-mail: m.wolf@tum.de [Zentrum Mathematik, Technische Universität München, 85748 Garching (Germany); Reeb, David, E-mail: reeb.qit@gmail.com [Zentrum Mathematik, Technische Universität München, 85748 Garching (Germany); Institute for Theoretical Physics, Leibniz Universität Hannover, 30167 Hannover (Germany)
2016-01-15
We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task.
An introduction to diffusion tensor image analysis.
O'Donnell, Lauren J; Westin, Carl-Fredrik
2011-04-01
Diffusion tensor magnetic resonance imaging (DTI) is a relatively new technology that is popular for imaging the white matter of the brain. This article provides a basic and broad overview of DTI to enable the reader to develop an intuitive understanding of these types of data, and an awareness of their strengths and weaknesses. Copyright © 2011 Elsevier Inc. All rights reserved.
Primordial tensor modes from quantum corrected inflation
Joergensen, Jakob; Sannino, Francesco; Svendsen, Ole
2014-01-01
. Finally we confront these theories with the Planck and BICEP2 data. We demonstrate that the discovery of primordial tensor modes by BICEP2 require the presence of sizable quantum departures from the $\\phi^4$-Inflaton model for the non-minimally coupled scenario which we parametrize and quantify. We...
From stochastic completion fields to tensor voting
Almsick, van M.A.; Duits, R.; Franken, E.M.; Haar Romenij, ter B.M.; Olsen, O.F.; Florack, L.M.J.; Kuijper, A.
2005-01-01
Several image processing algorithms imitate the lateral interaction of neurons in the visual striate cortex V1 to account for the correlations along contours and lines. Here we focus on two methodologies: tensor voting by Guy and Medioni, and stochastic completion fields by Mumford, Williams and
Positivity of linear maps under tensor powers
Müller-Hermes, Alexander; Wolf, Michael M.; Reeb, David
2016-01-01
We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with n copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every n ∈ ℕ, there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all n. For higher dimensions, we reduce the existence question of such non-trivial “tensor-stable positive maps” to a one-parameter family of maps and show that an affirmative answer would imply the existence of non-positive partial transpose bound entanglement. As an application, we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We, furthermore, show that the latter is an upper bound even for the local operations and classical communications-assisted quantum capacity, and that moreover it is a strong converse rate for this task
Dark energy in scalar-tensor theories
Moeller, J.
2007-12-01
We investigate several aspects of dynamical dark energy in the framework of scalar-tensor theories of gravity. We provide a classification of scalar-tensor coupling functions admitting cosmological scaling solutions. In particular, we recover that Brans-Dicke theory with inverse power-law potential allows for a sequence of background dominated scaling regime and scalar field dominated, accelerated expansion. Furthermore, we compare minimally and non-minimally coupled models, with respect to the small redshift evolution of the dark energy equation of state. We discuss the possibility to discriminate between different models by a reconstruction of the equation-of-state parameter from available observational data. The non-minimal coupling characterizing scalar-tensor models can - in specific cases - alleviate fine tuning problems, which appear if (minimally coupled) quintessence is required to mimic a cosmological constant. Finally, we perform a phase-space analysis of a family of biscalar-tensor models characterized by a specific type of σ-model metric, including two examples from recent literature. In particular, we generalize an axion-dilaton model of Sonner and Townsend, incorporating a perfect fluid background consisting of (dark) matter and radiation. (orig.)
Tensors in image processing and computer vision
De Luis García, Rodrigo; Tao, Dacheng; Li, Xuelong
2009-01-01
Tensor signal processing is an emerging field with important applications to computer vision and image processing. This book presents the developments in this branch of signal processing, offering research and discussions by experts in the area. It is suitable for advanced students working in the area of computer vision and image processing.
Tensor B mode and stochastic Faraday mixing
Giovannini, Massimo
2014-01-01
This paper investigates the Faraday effect as a different source of B mode polarization. The E mode polarization is Faraday rotated provided a stochastic large-scale magnetic field is present prior to photon decoupling. In the first part of the paper we discuss the case where the tensor modes of the geometry are absent and we argue that the B mode recently detected by the Bicep2 collaboration cannot be explained by a large-scale magnetic field rotating, through the Faraday effect, the well established E mode polarization. In this case, the observed temperature autocorrelations would be excessively distorted by the magnetic field. In the second part of the paper the formation of Faraday rotation is treated as a stationary, random and Markovian process with the aim of generalizing a set of scaling laws originally derived in the absence of the tensor modes of the geometry. We show that the scalar, vector and tensor modes of the brightness perturbations can all be Faraday rotated even if the vector and tensor par...
Tensor operators in R-matrix approach
Bytsko, A.G.; Rossijskaya Akademiya Nauk, St. Petersburg
1995-12-01
The definitions and some properties (e.g. the Wigner-Eckart theorem, the fusion procedure) of covariant and contravariant q-tensor operators for quasitriangular quantum Lie algebras are formulated in the R-matrix language. The case of U q (sl(n)) (in particular, for n=2) is discussed in more detail. (orig.)
Tensors, differential forms, and variational principles
Lovelock, David
1989-01-01
Incisive, self-contained account of tensor analysis and the calculus of exterior differential forms, interaction between the concept of invariance and the calculus of variations. Emphasis is on analytical techniques, with large number of problems, from routine manipulative exercises to technically difficult assignments.
Tensor algebra and tensor analysis for engineers with applications to continuum mechanics
Itskov, Mikhail
2015-01-01
This is the fourth and revised edition of a well-received book that aims at bridging the gap between the engineering course of tensor algebra on the one side and the mathematical course of classical linear algebra on the other side. In accordance with the contemporary way of scientific publications, a modern absolute tensor notation is preferred throughout. The book provides a comprehensible exposition of the fundamental mathematical concepts of tensor calculus and enriches the presented material with many illustrative examples. In addition, the book also includes advanced chapters dealing with recent developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics. Hence, this monograph addresses graduate students as well as scientists working in this field. In each chapter numerous exercises are included, allowing for self-study and intense practice. Solutions to the exercises are also provided.
Baust, Maximilian; Weinmann, Andreas; Wieczorek, Matthias; Lasser, Tobias; Storath, Martin; Navab, Nassir
2016-08-01
In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup.
The nonabelian tensor square of a bieberbach group with ...
The main objective of this paper is to compute the nonabelian tensor square of one Bieberbach group with elementary abelian 2-group point group of dimension three by using the computational method of the nonabelian tensor square for polycyclic groups. The finding of the computation showed that the nonabelian tensor ...
[An Improved Spectral Quaternion Interpolation Method of Diffusion Tensor Imaging].
Xu, Yonghong; Gao, Shangce; Hao, Xiaofei
2016-04-01
Diffusion tensor imaging(DTI)is a rapid development technology in recent years of magnetic resonance imaging.The diffusion tensor interpolation is a very important procedure in DTI image processing.The traditional spectral quaternion interpolation method revises the direction of the interpolation tensor and can preserve tensors anisotropy,but the method does not revise the size of tensors.The present study puts forward an improved spectral quaternion interpolation method on the basis of traditional spectral quaternion interpolation.Firstly,we decomposed diffusion tensors with the direction of tensors being represented by quaternion.Then we revised the size and direction of the tensor respectively according to different situations.Finally,we acquired the tensor of interpolation point by calculating the weighted average.We compared the improved method with the spectral quaternion method and the Log-Euclidean method by the simulation data and the real data.The results showed that the improved method could not only keep the monotonicity of the fractional anisotropy(FA)and the determinant of tensors,but also preserve the tensor anisotropy at the same time.In conclusion,the improved method provides a kind of important interpolation method for diffusion tensor image processing.
Tensor based structure estimation in multi-channel images
Schou, Jesper; Dierking, Wolfgang; Skriver, Henning
2000-01-01
. In the second part tensors are used for representing the structure information. This approach has the advantage, that tensors can be averaged either spatially or by applying several images, and the resulting tensor provides information of the average strength as well as orientation of the structure...
Relativistic particles with spin and antisymmetric tensor fields
Sandoval Junior, L.
1990-09-01
A study is made on antisymmetric tensor fields particularly on second order tensor field as far as his equivalence to other fields and quantization through the path integral are concerned. Also, a particle model is studied which has been recently proposed and reveals to be equivalent to antisymmetric tensor fields of any order. (L.C.J.A.)
Lepore, Natasha; Brun, Caroline; Chou, Yi-Yu; Chiang, Ming-Chang; Dutton, Rebecca A.; Hayashi, Kiralee M.; Luders, Eileen; Lopez, Oscar L.; Aizenstein, Howard J.; Toga, Arthur W.; Becker, James T.; Thompson, Paul M.
2008-01-01
This paper investigates the performance of a new multivariate method for tensor-based morphometry (TBM). Statistics on Riemannian manifolds are developed that exploit the full information in deformation tensor fields. In TBM, multiple brain images are warped to a common neuroanatomical template via 3-D nonlinear registration; the resulting deformation fields are analyzed statistically to identify group differences in anatomy. Rather than study the Jacobian determinant (volume expansion factor...
A Tensor-Based Structural Damage Identification and Severity Assessment
Anaissi, Ali; Makki Alamdari, Mehrisadat; Rakotoarivelo, Thierry; Khoa, Nguyen Lu Dang
2018-01-01
Early damage detection is critical for a large set of global ageing infrastructure. Structural Health Monitoring systems provide a sensor-based quantitative and objective approach to continuously monitor these structures, as opposed to traditional engineering visual inspection. Analysing these sensed data is one of the major Structural Health Monitoring (SHM) challenges. This paper presents a novel algorithm to detect and assess damage in structures such as bridges. This method applies tensor analysis for data fusion and feature extraction, and further uses one-class support vector machine on this feature to detect anomalies, i.e., structural damage. To evaluate this approach, we collected acceleration data from a sensor-based SHM system, which we deployed on a real bridge and on a laboratory specimen. The results show that our tensor method outperforms a state-of-the-art approach using the wavelet energy spectrum of the measured data. In the specimen case, our approach succeeded in detecting 92.5% of induced damage cases, as opposed to 61.1% for the wavelet-based approach. While our method was applied to bridges, its algorithm and computation can be used on other structures or sensor-data analysis problems, which involve large series of correlated data from multiple sensors. PMID:29301314
Comparison of quality control software tools for diffusion tensor imaging.
Liu, Bilan; Zhu, Tong; Zhong, Jianhui
2015-04-01
Image quality of diffusion tensor imaging (DTI) is critical for image interpretation, diagnostic accuracy and efficiency. However, DTI is susceptible to numerous detrimental artifacts that may impair the reliability and validity of the obtained data. Although many quality control (QC) software tools are being developed and are widely used and each has its different tradeoffs, there is still no general agreement on an image quality control routine for DTIs, and the practical impact of these tradeoffs is not well studied. An objective comparison that identifies the pros and cons of each of the QC tools will be helpful for the users to make the best choice among tools for specific DTI applications. This study aims to quantitatively compare the effectiveness of three popular QC tools including DTI studio (Johns Hopkins University), DTIprep (University of North Carolina at Chapel Hill, University of Iowa and University of Utah) and TORTOISE (National Institute of Health). Both synthetic and in vivo human brain data were used to quantify adverse effects of major DTI artifacts to tensor calculation as well as the effectiveness of different QC tools in identifying and correcting these artifacts. The technical basis of each tool was discussed, and the ways in which particular techniques affect the output of each of the tools were analyzed. The different functions and I/O formats that three QC tools provide for building a general DTI processing pipeline and integration with other popular image processing tools were also discussed. Copyright © 2015 Elsevier Inc. All rights reserved.
Massless and massive quanta resulting from a mediumlike metric tensor
Soln, J.
1985-01-01
A simple model of the ''primordial'' scalar field theory is presented in which the metric tensor is a generalization of the metric tensor from electrodynamics in a medium. The radiation signal corresponding to the scalar field propagates with a velocity that is generally less than c. This signal can be associated simultaneously with imaginary and real effective (momentum-dependent) masses. The requirement that the imaginary effective mass vanishes, which we take to be the prerequisite for the vacuumlike signal propagation, leads to the ''spontaneous'' splitting of the metric tensor into two distinct metric tensors: one metric tensor gives rise to masslesslike radiation and the other to a massive particle. (author)
Auriat, A M; Borich, M R; Snow, N J; Wadden, K P; Boyd, L A
2015-01-01
Diffusion tensor imaging (DTI)-based tractography has been used to demonstrate functionally relevant differences in white matter pathway status after stroke. However, it is now known that the tensor model is insensitive to the complex fiber architectures found in the vast majority of voxels in the human brain. The inability to resolve intra-voxel fiber orientations may have important implications for the utility of standard DTI-based tract reconstruction methods. Intra-voxel fiber orientations can now be identified using novel, tensor-free approaches. Constrained spherical deconvolution (CSD) is one approach to characterize intra-voxel diffusion behavior. In the current study, we performed DTI- and CSD-based tract reconstruction of the corticospinal tract (CST) and corpus callosum (CC) to test the hypothesis that characterization of complex fiber orientations may improve the robustness of fiber tract reconstruction and increase the sensitivity to identify functionally relevant white matter abnormalities in individuals with chronic stroke. Diffusion weighted magnetic resonance imaging was performed in 27 chronic post-stroke participants and 12 healthy controls. Transcallosal pathways and the CST bilaterally were reconstructed using DTI- and CSD-based tractography. Mean fractional anisotropy (FA), apparent diffusion coefficient (ADC), axial diffusivity (AD), and radial diffusivity (RD) were calculated across the tracts of interest. The total number and volume of reconstructed tracts was also determined. Diffusion measures were compared between groups (Stroke, Control) and methods (CSD, DTI). The relationship between post-stroke motor behavior and diffusion measures was evaluated. Overall, CSD methods identified more tracts than the DTI-based approach for both CC and CST pathways. Mean FA, ADC, and RD differed between DTI and CSD for CC-mediated tracts. In these tracts, we discovered a difference in FA for the CC between stroke and healthy control groups using CSD but
A.M. Auriat
2015-01-01
Full Text Available Diffusion tensor imaging (DTI-based tractography has been used to demonstrate functionally relevant differences in white matter pathway status after stroke. However, it is now known that the tensor model is insensitive to the complex fiber architectures found in the vast majority of voxels in the human brain. The inability to resolve intra-voxel fiber orientations may have important implications for the utility of standard DTI-based tract reconstruction methods. Intra-voxel fiber orientations can now be identified using novel, tensor-free approaches. Constrained spherical deconvolution (CSD is one approach to characterize intra-voxel diffusion behavior. In the current study, we performed DTI- and CSD-based tract reconstruction of the corticospinal tract (CST and corpus callosum (CC to test the hypothesis that characterization of complex fiber orientations may improve the robustness of fiber tract reconstruction and increase the sensitivity to identify functionally relevant white matter abnormalities in individuals with chronic stroke. Diffusion weighted magnetic resonance imaging was performed in 27 chronic post-stroke participants and 12 healthy controls. Transcallosal pathways and the CST bilaterally were reconstructed using DTI- and CSD-based tractography. Mean fractional anisotropy (FA, apparent diffusion coefficient (ADC, axial diffusivity (AD, and radial diffusivity (RD were calculated across the tracts of interest. The total number and volume of reconstructed tracts was also determined. Diffusion measures were compared between groups (Stroke, Control and methods (CSD, DTI. The relationship between post-stroke motor behavior and diffusion measures was evaluated. Overall, CSD methods identified more tracts than the DTI-based approach for both CC and CST pathways. Mean FA, ADC, and RD differed between DTI and CSD for CC-mediated tracts. In these tracts, we discovered a difference in FA for the CC between stroke and healthy control groups
Feature Surfaces in Symmetric Tensor Fields Based on Eigenvalue Manifold.
Palacios, Jonathan; Yeh, Harry; Wang, Wenping; Zhang, Yue; Laramee, Robert S; Sharma, Ritesh; Schultz, Thomas; Zhang, Eugene
2016-03-01
Three-dimensional symmetric tensor fields have a wide range of applications in solid and fluid mechanics. Recent advances in the (topological) analysis of 3D symmetric tensor fields focus on degenerate tensors which form curves. In this paper, we introduce a number of feature surfaces, such as neutral surfaces and traceless surfaces, into tensor field analysis, based on the notion of eigenvalue manifold. Neutral surfaces are the boundary between linear tensors and planar tensors, and the traceless surfaces are the boundary between tensors of positive traces and those of negative traces. Degenerate curves, neutral surfaces, and traceless surfaces together form a partition of the eigenvalue manifold, which provides a more complete tensor field analysis than degenerate curves alone. We also extract and visualize the isosurfaces of tensor modes, tensor isotropy, and tensor magnitude, which we have found useful for domain applications in fluid and solid mechanics. Extracting neutral and traceless surfaces using the Marching Tetrahedra method can cause the loss of geometric and topological details, which can lead to false physical interpretation. To robustly extract neutral surfaces and traceless surfaces, we develop a polynomial description of them which enables us to borrow techniques from algebraic surface extraction, a topic well-researched by the computer-aided design (CAD) community as well as the algebraic geometry community. In addition, we adapt the surface extraction technique, called A-patches, to improve the speed of finding degenerate curves. Finally, we apply our analysis to data from solid and fluid mechanics as well as scalar field analysis.
Glyph-Based Comparative Visualization for Diffusion Tensor Fields.
Zhang, Changgong; Schultz, Thomas; Lawonn, Kai; Eisemann, Elmar; Vilanova, Anna
2016-01-01
Diffusion Tensor Imaging (DTI) is a magnetic resonance imaging modality that enables the in-vivo reconstruction and visualization of fibrous structures. To inspect the local and individual diffusion tensors, glyph-based visualizations are commonly used since they are able to effectively convey full aspects of the diffusion tensor. For several applications it is necessary to compare tensor fields, e.g., to study the effects of acquisition parameters, or to investigate the influence of pathologies on white matter structures. This comparison is commonly done by extracting scalar information out of the tensor fields and then comparing these scalar fields, which leads to a loss of information. If the glyph representation is kept, simple juxtaposition or superposition can be used. However, neither facilitates the identification and interpretation of the differences between the tensor fields. Inspired by the checkerboard style visualization and the superquadric tensor glyph, we design a new glyph to locally visualize differences between two diffusion tensors by combining juxtaposition and explicit encoding. Because tensor scale, anisotropy type, and orientation are related to anatomical information relevant for DTI applications, we focus on visualizing tensor differences in these three aspects. As demonstrated in a user study, our new glyph design allows users to efficiently and effectively identify the tensor differences. We also apply our new glyphs to investigate the differences between DTI datasets of the human brain in two different contexts using different b-values, and to compare datasets from a healthy and HIV-infected subject.
Tensoral for post-processing users and simulation authors
Dresselhaus, Eliot
1993-01-01
The CTR post-processing effort aims to make turbulence simulations and data more readily and usefully available to the research and industrial communities. The Tensoral language, which provides the foundation for this effort, is introduced here in the form of a user's guide. The Tensoral user's guide is presented in two main sections. Section one acts as a general introduction and guides database users who wish to post-process simulation databases. Section two gives a brief description of how database authors and other advanced users can make simulation codes and/or the databases they generate available to the user community via Tensoral database back ends. The two-part structure of this document conforms to the two-level design structure of the Tensoral language. Tensoral has been designed to be a general computer language for performing tensor calculus and statistics on numerical data. Tensoral's generality allows it to be used for stand-alone native coding of high-level post-processing tasks (as described in section one of this guide). At the same time, Tensoral's specialization to a minute task (namely, to numerical tensor calculus and statistics) allows it to be easily embedded into applications written partly in Tensoral and partly in other computer languages (here, C and Vectoral). Embedded Tensoral, aimed at advanced users for more general coding (e.g. of efficient simulations, for interfacing with pre-existing software, for visualization, etc.), is described in section two of this guide.
Energy-momentum tensor of the electromagnetic field
Horndeski, G.W.; Wainwright, J.
1977-01-01
In this paper we investigate the energy-momentum tensor of the most general second-order vector-tensor theory of gravitation and electromagnetism which has field equations which are (i) derivable from a variational principle, (ii) consistent with the notion of conservation of charge, and (iii) compatible with Maxwell's equations in a flat space. This energy-momentum tensor turns out to be quadratic in the first partial derivatives of the electromagnetic field tensor and depends upon the curvature tensor. The asymptotic behavior of this energy-momentum tensor is examined for solutions to Maxwell's equations in Minkowski space, and it is demonstrated that this energy-momentum tensor predicts regions of negative energy density in the vicinity of point sources
Quantum mechanics of Yano tensors: Dirac equation in curved spacetime
Cariglia, Marco
2004-01-01
In spacetimes admitting Yano tensors, the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to operators that in the classical limit commute with the Dirac operator and generate conserved quantities. We show that the result is preserved in the full quantum theory, that is, Yano symmetries are not anomalous. This was known for Yano tensors of rank 2, but our main result is to show that it extends to Yano tensors of arbitrary rank. We also describe the conformal Yano equation and show that is invariant under Hodge duality. There is a natural relationship between Yano tensors and supergravity theories. As the simplest possible example, we show that when the spacetime admits a Killing spinor then this generates Yano and conformal Yano tensors. As an application, we construct Yano tensors on maximally symmetric spaces: they are spanned by tensor products of Killing vectors
Algebraic and computational aspects of real tensor ranks
Sakata, Toshio; Miyazaki, Mitsuhiro
2016-01-01
This book provides comprehensive summaries of theoretical (algebraic) and computational aspects of tensor ranks, maximal ranks, and typical ranks, over the real number field. Although tensor ranks have been often argued in the complex number field, it should be emphasized that this book treats real tensor ranks, which have direct applications in statistics. The book provides several interesting ideas, including determinant polynomials, determinantal ideals, absolutely nonsingular tensors, absolutely full column rank tensors, and their connection to bilinear maps and Hurwitz-Radon numbers. In addition to reviews of methods to determine real tensor ranks in details, global theories such as the Jacobian method are also reviewed in details. The book includes as well an accessible and comprehensive introduction of mathematical backgrounds, with basics of positive polynomials and calculations by using the Groebner basis. Furthermore, this book provides insights into numerical methods of finding tensor ranks through...
Wieschhoff, Ged G; Mandell, Jacob C; Czuczman, Gregory J; Nikac, Violeta; Shah, Nehal; Smith, Stacy E
2017-11-01
Hamstring muscle deficiency is increasingly recognized as a risk factor for anterior cruciate ligament (ACL) tears. The purpose of this study is to evaluate the vastus medialis to semimembranosus cross-sectional area (VM:SM CSA) ratio on magnetic resonance imaging (MRI) in patients with ACL tears compared to controls. One hundred knee MRIs of acute ACL tear patients and 100 age-, sex-, and side-matched controls were included. Mechanism of injury, contact versus non-contact, was determined for each ACL tear subject. The VM:SM CSA was measured on individual axial slices with a novel method using image-processing software. One reader measured all 200 knees and the second reader measured 50 knees at random to assess inter-reader variability. The intraclass correlation coefficient (ICC) was calculated to evaluate for correlation between readers. T-tests were performed to evaluate for differences in VM:SM CSA ratios between the ACL tear group and control group. The ICC for agreement between the two readers was 0.991 (95% confidence interval 0.984-0.995). Acute ACL tear patients have an increased VM:SM CSA ratio compared to controls (1.44 vs. 1.28; p = 0.005). Non-contact acute ACL tear patients have an increased VM:SM CSA ratio compared to controls (1.48 vs. 1.20; p = 0.003), whereas contact acute ACL tear patients do not (1.23 vs. 1.26; p = 0.762). Acute non-contact ACL tears are associated with increased VM:SM CSA ratios, which may imply a relative deficiency in hamstring strength. This study also demonstrates a novel method of measuring the relative CSA of muscles on MRI.
Wieschhoff, Ged G.; Mandell, Jacob C.; Czuczman, Gregory J.; Nikac, Violeta; Shah, Nehal; Smith, Stacy E.
2017-01-01
Hamstring muscle deficiency is increasingly recognized as a risk factor for anterior cruciate ligament (ACL) tears. The purpose of this study is to evaluate the vastus medialis to semimembranosus cross-sectional area (VM:SM CSA) ratio on magnetic resonance imaging (MRI) in patients with ACL tears compared to controls. One hundred knee MRIs of acute ACL tear patients and 100 age-, sex-, and side-matched controls were included. Mechanism of injury, contact versus non-contact, was determined for each ACL tear subject. The VM:SM CSA was measured on individual axial slices with a novel method using image-processing software. One reader measured all 200 knees and the second reader measured 50 knees at random to assess inter-reader variability. The intraclass correlation coefficient (ICC) was calculated to evaluate for correlation between readers. T-tests were performed to evaluate for differences in VM:SM CSA ratios between the ACL tear group and control group. The ICC for agreement between the two readers was 0.991 (95% confidence interval 0.984-0.995). Acute ACL tear patients have an increased VM:SM CSA ratio compared to controls (1.44 vs. 1.28; p = 0.005). Non-contact acute ACL tear patients have an increased VM:SM CSA ratio compared to controls (1.48 vs. 1.20; p = 0.003), whereas contact acute ACL tear patients do not (1.23 vs. 1.26; p = 0.762). Acute non-contact ACL tears are associated with increased VM:SM CSA ratios, which may imply a relative deficiency in hamstring strength. This study also demonstrates a novel method of measuring the relative CSA of muscles on MRI. (orig.)
Wieschhoff, Ged G.; Mandell, Jacob C.; Czuczman, Gregory J.; Nikac, Violeta; Shah, Nehal; Smith, Stacy E. [Brigham and Women' s Hospital, Harvard Medical School, Division of Musculoskeletal Imaging and Intervention, Department of Radiology, Boston, MA (United States)
2017-11-15
Hamstring muscle deficiency is increasingly recognized as a risk factor for anterior cruciate ligament (ACL) tears. The purpose of this study is to evaluate the vastus medialis to semimembranosus cross-sectional area (VM:SM CSA) ratio on magnetic resonance imaging (MRI) in patients with ACL tears compared to controls. One hundred knee MRIs of acute ACL tear patients and 100 age-, sex-, and side-matched controls were included. Mechanism of injury, contact versus non-contact, was determined for each ACL tear subject. The VM:SM CSA was measured on individual axial slices with a novel method using image-processing software. One reader measured all 200 knees and the second reader measured 50 knees at random to assess inter-reader variability. The intraclass correlation coefficient (ICC) was calculated to evaluate for correlation between readers. T-tests were performed to evaluate for differences in VM:SM CSA ratios between the ACL tear group and control group. The ICC for agreement between the two readers was 0.991 (95% confidence interval 0.984-0.995). Acute ACL tear patients have an increased VM:SM CSA ratio compared to controls (1.44 vs. 1.28; p = 0.005). Non-contact acute ACL tear patients have an increased VM:SM CSA ratio compared to controls (1.48 vs. 1.20; p = 0.003), whereas contact acute ACL tear patients do not (1.23 vs. 1.26; p = 0.762). Acute non-contact ACL tears are associated with increased VM:SM CSA ratios, which may imply a relative deficiency in hamstring strength. This study also demonstrates a novel method of measuring the relative CSA of muscles on MRI. (orig.)
Relationship between timed 25-foot walk and diffusion tensor imaging in multiple sclerosis.
Klineova, Sylvia; Farber, Rebecca; Saiote, Catarina; Farrell, Colleen; Delman, Bradley N; Tanenbaum, Lawrence N; Friedman, Joshua; Inglese, Matilde; Lublin, Fred D; Krieger, Stephen
2016-01-01
The majority of multiple sclerosis patients experience impaired walking ability, which impacts quality of life. Timed 25-foot walk is commonly used to gauge gait impairment but results can be broadly variable. Objective biological markers that correlate closely with patients' disability are needed. Diffusion tensor imaging, quantifying fiber tract integrity, might provide such information. In this project we analyzed relationships between timed 25-foot walk, conventional and diffusion tensor imaging magnetic resonance imaging markers. A cohort of gait impaired multiple sclerosis patients underwent brain and cervical spinal cord magnetic resonance imaging. Diffusion tensor imaging mean diffusivity and fractional anisotropy were measured on the brain corticospinal tracts and spinal restricted field of vision at C2/3. We analyzed relationships between baseline timed 25-foot walk, conventional and diffusion tensor imaging magnetic resonance imaging markers. Multivariate linear regression analysis showed a statistically significant association between several magnetic resonance imaging and diffusion tensor imaging metrics and timed 25-foot walk: brain mean diffusivity corticospinal tracts (p = 0.004), brain corticospinal tracts axial and radial diffusivity (P = 0.004 and 0.02), grey matter volume (p = 0.05), white matter volume (p = 0.03) and normalized brain volume (P = 0.01). The linear regression model containing mean diffusivity corticospinal tracts and controlled for gait assistance was the best fit model (p = 0.004). Our results suggest an association between diffusion tensor imaging metrics and gait impairment, evidenced by brain mean diffusivity corticospinal tracts and timed 25-foot walk.
Entanglement entropy from the holographic stress tensor
Bhattacharyya, Arpan; Sinha, Aninda
2013-01-01
We consider entanglement entropy in the context of gauge/gravity duality for conformal field theories in even dimensions. The holographic prescription due to Ryu and Takayanagi (RT) leads to an equation describing how the entangling surface extends into the bulk geometry. We show that setting to zero, the time–time component of the Brown–York stress tensor evaluated on the co-dimension 1 entangling surface, leads to the same equation. By considering a spherical entangling surface as an example, we observe that the Euclidean action methods in AdS/CFT will lead to the RT area functional arising as a counterterm needed to regularize the stress tensor. We present arguments leading to a justification for the minimal area prescription. (paper)
Tensor modes on the string theory landscape
Westphal, Alexander
2012-06-01
We attempt an estimate for the distribution of the tensor mode fraction r over the landscape of vacua in string theory. The dynamics of eternal inflation and quantum tunneling lead to a kind of democracy on the landscape, providing no bias towards large-field or small-field inflation regardless of the class of measure. The tensor mode fraction then follows the number frequency distributions of inflationary mechanisms of string theory over the landscape. We show that an estimate of the relative number frequencies for small-field vs large-field inflation, while unattainable on the whole landscape, may be within reach as a regional answer for warped Calabi-Yau flux compactifications of type IIB string theory.
Tensor modes on the string theory landscape
Westphal, Alexander
2012-06-15
We attempt an estimate for the distribution of the tensor mode fraction r over the landscape of vacua in string theory. The dynamics of eternal inflation and quantum tunneling lead to a kind of democracy on the landscape, providing no bias towards large-field or small-field inflation regardless of the class of measure. The tensor mode fraction then follows the number frequency distributions of inflationary mechanisms of string theory over the landscape. We show that an estimate of the relative number frequencies for small-field vs large-field inflation, while unattainable on the whole landscape, may be within reach as a regional answer for warped Calabi-Yau flux compactifications of type IIB string theory.
Kibler, M.; Grenet, G.
1979-07-01
The SU 2 unit tensor operators tsub(k,α) are studied. In the case where the spinor point group G* coincides with U 1 , then tsub(k α) reduces up to a constant to the Wigner-Racah-Schwinger tensor operator tsub(kqα), an operator which produces an angular momentum state. One first investigates those general properties of tsub(kα) which are independent of their realization. The tsub(kα) in terms of two pairs of boson creation and annihilation operators are realized. This leads to look at the Schwinger calculus relative to one angular momentum of two coupled angular momenta. As a by-product, a procedure is given for producing recursion relationships between SU 2 Wigner coefficients. Finally, some of the properties of the Wigner and Racah operators for an arbitrary compact group and the SU 2 coupling coefficients are studied
Tensor Networks and Quantum Error Correction
Ferris, Andrew J.; Poulin, David
2014-07-01
We establish several relations between quantum error correction (QEC) and tensor network (TN) methods of quantum many-body physics. We exhibit correspondences between well-known families of QEC codes and TNs, and demonstrate a formal equivalence between decoding a QEC code and contracting a TN. We build on this equivalence to propose a new family of quantum codes and decoding algorithms that generalize and improve upon quantum polar codes and successive cancellation decoding in a natural way.
Old tensor mesons in QCD sum rules
Aliev, T.M.; Shifman, M.A.
1981-01-01
Tensor mesons f, A 2 and A 3 are analyzed within the framework of QCD sum rules. The effects of gluon and quark condensate is accounted for phenomenologically. Accurate estimates of meson masses and coupling constants of the lowest-lying states are obtained. It is shown that the masses are reproduced within theoretical uncertainty of about 80 MeV. The coupling of f meson to the corresponding quark current is determined. The results are in good aqreement with experimental data [ru
Energy-momentum-tensor in quantumelectrodynamics
Schott, T
1974-01-01
This work deals with the operator properties of the energy-momentum-tensor (ET) in the framework of quantum electrodynamics. The principles of construction of the ET are discussed for quantized fields in the Schwinger variation principle. Dealing with the conserved quantities for quantized fields operator problems are coming up in the Coulomb gauge because Dirac- and Maxwellfield do not commute completely. Further on contemporary commutators of the ET components are investigated mutually. Finally non-canonical methods are developed.
Embryo Cell Membranes Reconstruction by Tensor Voting
Michelin , Gaël; Guignard , Léo; Fiuza , Ulla-Maj; Malandain , Grégoire
2014-01-01
International audience; Image-based studies of developing organs or embryos produce a huge quantity of data. To handle such high-throughput experimental protocols, automated computer-assisted methods are highly desirable. This article aims at designing an efficient cell segmentation method from microscopic images. The proposed approach is twofold: first, cell membranes are enhanced or extracted by the means of structure-based filters, and then perceptual grouping (i.e. tensor voting) allows t...
Sasakian manifolds with purely transversal Bach tensor
Ghosh, Amalendu; Sharma, Ramesh
2017-10-01
We show that a (2n + 1)-dimensional Sasakian manifold (M, g) with a purely transversal Bach tensor has constant scalar curvature ≥2 n (2 n +1 ) , equality holding if and only if (M, g) is Einstein. For dimension 3, M is locally isometric to the unit sphere S3. For dimension 5, if in addition (M, g) is complete, then it has positive Ricci curvature and is compact with finite fundamental group π1(M).
Vector-tensor interaction of gravitation
Zhang Yuan-zhong; Guo han-ying
1982-11-01
In the paper, by using the equation of motion a particle, we show that the antigravity exist in the vector-tensor model of gravitation. Thus the motion of a particle deviates from the geodesic equation. In Newtonian approximation and weak gravitational field, acceleration of a particle in a spherically symmetric and astatic gravitation field is zero. The result is obviously not in agreement with gravitational phenomena.
Anisotropic diffusion tensor applied to temporal mammograms
Karemore, Gopal; Brandt, Sami; Sporring, Jon
2010-01-01
changes related to specific effects like Hormonal Replacement Therapy (HRT) and aging. Given effect-grouped patient data, we demonstrated how anisotropic diffusion tensor and its coherence features computed in an anatomically oriented breast coordinate system followed by statistical learning...
Numerical CP Decomposition of Some Difficult Tensors
Tichavský, Petr; Phan, A. H.; Cichocki, A.
2017-01-01
Roč. 317, č. 1 (2017), s. 362-370 ISSN 0377-0427 R&D Projects: GA ČR(CZ) GA14-13713S Institutional support: RVO:67985556 Keywords : Small matrix multiplication * Canonical polyadic tensor decomposition * Levenberg-Marquardt method Subject RIV: BB - Applied Statistics, Operational Research OBOR OECD: Applied mathematics Impact factor: 1.357, year: 2016 http://library.utia.cas.cz/separaty/2017/SI/tichavsky-0468385. pdf
Experimental status of scalar and tensor mesons
Von Dombrowski, S.
1997-01-01
The recent discoveries of a 0 (1450) and f 0 (1370)/f 0 (1500) in antiproton-proton annihilation at rest shed new light on the interpretation of light scalar mesons. The properties of f 0 (1500) match the expectations of a scalar glueball mixed with nearby qq states. New decay modes of the ξ(2230) are reported in radiative J/Ψ decays, pointing also towards a (tensor) glueball nature of this state. Results from different experiments are discussed and compared. (orig.)
Bayesian approach to magnetotelluric tensor decomposition
Červ, Václav; Pek, Josef; Menvielle, M.
2010-01-01
Roč. 53, č. 2 (2010), s. 21-32 ISSN 1593-5213 R&D Projects: GA AV ČR IAA200120701; GA ČR GA205/04/0746; GA ČR GA205/07/0292 Institutional research plan: CEZ:AV0Z30120515 Keywords : galvanic distortion * telluric distortion * impedance tensor * basic procedure * inversion * noise Subject RIV: DE - Earth Magnetism, Geodesy, Geography Impact factor: 0.336, year: 2010
Monte Carlo Volcano Seismic Moment Tensors
Waite, G. P.; Brill, K. A.; Lanza, F.
2015-12-01
Inverse modeling of volcano seismic sources can provide insight into the geometry and dynamics of volcanic conduits. But given the logistical challenges of working on an active volcano, seismic networks are typically deficient in spatial and temporal coverage; this potentially leads to large errors in source models. In addition, uncertainties in the centroid location and moment-tensor components, including volumetric components, are difficult to constrain from the linear inversion results, which leads to a poor understanding of the model space. In this study, we employ a nonlinear inversion using a Monte Carlo scheme with the objective of defining robustly resolved elements of model space. The model space is randomized by centroid location and moment tensor eigenvectors. Point sources densely sample the summit area and moment tensors are constrained to a randomly chosen geometry within the inversion; Green's functions for the random moment tensors are all calculated from modeled single forces, making the nonlinear inversion computationally reasonable. We apply this method to very-long-period (VLP) seismic events that accompany minor eruptions at Fuego volcano, Guatemala. The library of single force Green's functions is computed with a 3D finite-difference modeling algorithm through a homogeneous velocity-density model that includes topography, for a 3D grid of nodes, spaced 40 m apart, within the summit region. The homogenous velocity and density model is justified by long wavelength of VLP data. The nonlinear inversion reveals well resolved model features and informs the interpretation through a better understanding of the possible models. This approach can also be used to evaluate possible station geometries in order to optimize networks prior to deployment.
Electrical tensor Green functions for cylindrical waveguides
Prijmenko, S.D.; Papkovich, V.G.; Khizhnyak, N.A.
1988-01-01
Formation of electrical tensor Green functions for cylindrical waveguides is considered. Behaviour of these functions in the source region is studied. Cases of electrical tensor Green functions for vector potential G E (r-vector, r'-vector) and electric field G e (r-vector, r'-vector) are analysed. When forming G E (r-vector, r'-vector), its dependence on lateral coordinates is taken into account by means of two-dimensional fundamental vector Hansen functions, several methods are used to take into account the dependence on transverse coordinate. When forming G e (r-vector, r'-vector) we use the fact that G E (r-vector, r'-vector) and G e (r-vector, r'-vector) are the generalized functions. It is shown that G e (r-vector, r'-vector) behaviour in the source region is defined by a singular term, which properties are described by the delta-function. Two variants of solving the problem of defining singular and regular sides of tensor function G E (r-vector, r'-vector) are presented. 23 refs
Tensor integrand reduction via Laurent expansion
Hirschi, Valentin [SLAC, National Accelerator Laboratory,2575 Sand Hill Road, Menlo Park, CA 94025-7090 (United States); Peraro, Tiziano [Higgs Centre for Theoretical Physics, School of Physics and Astronomy,The University of Edinburgh,Edinburgh EH9 3JZ, Scotland (United Kingdom)
2016-06-09
We introduce a new method for the application of one-loop integrand reduction via the Laurent expansion algorithm, as implemented in the public C++ library Ninja. We show how the coefficients of the Laurent expansion can be computed by suitable contractions of the loop numerator tensor with cut-dependent projectors, making it possible to interface Ninja to any one-loop matrix element generator that can provide the components of this tensor. We implemented this technique in the Ninja library and interfaced it to MADLOOP, which is part of the public MADGRAPH5{sub A}MC@NLO framework. We performed a detailed performance study, comparing against other public reduction tools, namely CUTTOOLS, SAMURAI, IREGI, PJFRY++ and GOLEM95. We find that Ninja outperforms traditional integrand reduction in both speed and numerical stability, the latter being on par with that of the tensor integral reduction tool GOLEM95 which is however more limited and slower than Ninja. We considered many benchmark multi-scale processes of increasing complexity, involving QCD and electro-weak corrections as well as effective non-renormalizable couplings, showing that Ninja’s performance scales well with both the rank and multiplicity of the considered process.
Liu, Chunlei; Murphy, Nicole E.; Li, Wei
2012-01-01
Diffusion MRI has become an invaluable tool for studying white matter microstructure and brain connectivity. The emergence of quantitative susceptibility mapping and susceptibility tensor imaging (STI) has provided another unique tool for assessing the structure of white matter. In the highly ordered white matter structure, diffusion MRI measures hindered water mobility induced by various tissue and cell membranes, while susceptibility sensitizes to the molecular composition and axonal arrangement. Integrating these two methods may produce new insights into the complex physiology of white matter. In this study, we investigated the relationship between diffusion and magnetic susceptibility in the white matter. Experiments were conducted on phantoms and human brains in vivo. Diffusion properties were quantified with the diffusion tensor model and also with the higher order tensor model based on the cumulant expansion. Frequency shift and susceptibility tensor were measured with quantitative susceptibility mapping and susceptibility tensor imaging. These diffusion and susceptibility quantities were compared and correlated in regions of single fiber bundles and regions of multiple fiber orientations. Relationships were established with similarities and differences identified. It is believed that diffusion MRI and susceptibility MRI provide complementary information of the microstructure of white matter. Together, they allow a more complete assessment of healthy and diseased brains. PMID:23507987
Non-Newtonian stress tensor and thermal conductivity tensor in granular plane shear flow
Alam, Meheboob; Saha, Saikat
2014-11-01
The non-Newtonian stress tensor and the heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. Closed-form expressions for shear viscosity, pressure, first normal stress difference (N1) and the dissipation rate are given as functions of (i) the density or the area fraction (ν), (ii) the restitution coefficient (e), (iii) the dimensionless shear rate (R), (iv) the temperature anisotropy [ η, the difference between the principal eigenvalues of the second moment tensor] and (v) the angle (ϕ) between the principal directions of the shear tensor and the second moment tensor. Particle simulation data for a sheared hard-disk system is compared with theoretical results, with good agreement for p, μ and N1 over a large range of density. In contrast, the predictions from a Navier-Stokes order constitutive model are found to deviate significantly from both the simulation and the moment theory even at moderate values of e. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic 2nd rank tensor for which explicit expressions are derived.
Depression - major; Depression - clinical; Clinical depression; Unipolar depression; Major depressive disorder ... providers do not know the exact causes of depression. It is believed that chemical changes in the ...
Diffusion Tensor Tractography Reveals Disrupted Structural Connectivity during Brain Aging
Lin, Lan; Tian, Miao; Wang, Qi; Wu, Shuicai
2017-10-01
Brain aging is one of the most crucial biological processes that entail many physical, biological, chemical, and psychological changes, and also a major risk factor for most common neurodegenerative diseases. To improve the quality of life for the elderly, it is important to understand how the brain is changed during the normal aging process. We compared diffusion tensor imaging (DTI)-based brain networks in a cohort of 75 healthy old subjects by using graph theory metrics to describe the anatomical networks and connectivity patterns, and network-based statistic (NBS) analysis was used to identify pairs of regions with altered structural connectivity. The NBS analysis revealed a significant network comprising nine distinct fiber bundles linking 10 different brain regions showed altered white matter structures in young-old group compare with middle-aged group (p < .05, family-wise error-corrected). Our results might guide future studies and help to gain a better understanding of brain aging.
Tensor interaction in heavy-ion scattering. Pt. 1
Nishioka, H.; Johnson, R.C.
1985-01-01
The Heidelberg shape-effect model for heavy-ion tensor interactions is reformulated and generalized using the Hooton-Johnson formulation. The generalized semiclassical model (the turning-point model) predicts that the components of the tensor analysing power anti Tsub(2q) have certain relations with each other for each type of tensor interaction (Tsub(R), Tsub(P) and Tsub(L) types). The predicted relations between the anti Tsub(2q) are very simple and have a direct connection with the properties of the tensor interaction at the turning point. The model predictions are satisfied in quantum-mechanical calculations for 7 Li and 23 Na elastic scattering from 58 Ni in the Fresnel-diffraction energy region. As a consequence of this model, it becomes possible to single out effects from a Tsub(P)- or Tsub(L)-type tensor interaction in polarized heavy-ion scattering. The presence of a Tsub(P)-type tensor interaction is suggested by measured anti T 20 /anti T 22 ratios for 7 Li + 58 Ni scattering. In the turning-point model the three types of tensor operator are not independent, and this is found to be true also in a quantum-mechanical calculation. The model also predicts relations between the components of higher-rank tensor analysing power in the presence of a higher-rank tensor interaction. The rank-3 tensor case is discussed in detail. (orig.)
Tensor network decompositions in the presence of a global symmetry
Singh, Sukhwinder; Pfeifer, Robert N. C.; Vidal, Guifre
2010-01-01
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance in the context of tensor network algorithms as well, thus setting the stage for cross-fertilization between these two areas of research.
Diffusion tensor imaging tensor shape analysis for assessment of regional white matter differences.
Middleton, Dana M; Li, Jonathan Y; Lee, Hui J; Chen, Steven; Dickson, Patricia I; Ellinwood, N Matthew; White, Leonard E; Provenzale, James M
2017-08-01
Purpose The purpose of this study was to investigate a novel tensor shape plot analysis technique of diffusion tensor imaging data as a means to assess microstructural differences in brain tissue. We hypothesized that this technique could distinguish white matter regions with different microstructural compositions. Methods Three normal canines were euthanized at seven weeks old. Their brains were imaged using identical diffusion tensor imaging protocols on a 7T small-animal magnetic resonance imaging system. We examined two white matter regions, the internal capsule and the centrum semiovale, each subdivided into an anterior and posterior region. We placed 100 regions of interest in each of the four brain regions. Eigenvalues for each region of interest triangulated onto tensor shape plots as the weighted average of three shape metrics at the plot's vertices: CS, CL, and CP. Results The distribution of data on the plots for the internal capsule differed markedly from the centrum semiovale data, thus confirming our hypothesis. Furthermore, data for the internal capsule were distributed in a relatively tight cluster, possibly reflecting the compact and parallel nature of its fibers, while data for the centrum semiovale were more widely distributed, consistent with the less compact and often crossing pattern of its fibers. This indicates that the tensor shape plot technique can depict data in similar regions as being alike. Conclusion Tensor shape plots successfully depicted differences in tissue microstructure and reflected the microstructure of individual brain regions. This proof of principle study suggests that if our findings are reproduced in larger samples, including abnormal white matter states, the technique may be useful in assessment of white matter diseases.
Effect of the callipyge phenotype and cooking method on tenderness of several major lamb muscles.
Shackelford, S D; Wheeler, T L; Koohmaraie, M
1997-08-01
We conducted three experiments to determine the effects of the callipyge phenotype on the tenderness of several major lamb muscles and to determine the effect of method of cookery on the tenderness of callipyge lamb at 7 d postmortem. In Exp. 1, chops from normal (n = 23) and callipyge (n = 16) carcasses were open-hearth-broiled. Warner-Bratzler shear force values of longissimus, gluteus medius, semimembranosus, biceps femoris, semitendinosus, adductor, and quadriceps femoris were 123, 44, 28, 26, 19, 16, and 13% greater, respectively, for callipyge (P lamb carcasses (n = 60). Callipyge chops were less tender than normal chops (P cooking method, callipyge samples were less juicy than normal samples (P < .05). These data demonstrate that the callipyge phenotype will likely reduce consumer satisfaction due to reduced tenderness and juiciness; however, reduced tenderness in callipyge leg muscles could be prevented by ovenroasting.
An introduction to tensors and group theory for physicists
Jeevanjee, Nadir
2011-01-01
An Introduction to Tensors and Group Theory for Physicists provides both an intuitive and rigorous approach to tensors and groups and their role in theoretical physics and applied mathematics. A particular aim is to demystify tensors and provide a unified framework for understanding them in the context of classical and quantum physics. Connecting the component formalism prevalent in physics calculations with the abstract but more conceptual formulation found in many mathematical texts, the work will be a welcome addition to the literature on tensors and group theory. Part I of the text begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to classical and quantum physics through the use of tensor products. Part II introduces abstract groups along with matrix Lie groups and Lie algebras, then intertwines this material with that of Part I by introducing representation theory. Exercises and examples are provided throughout for go...
Theoretical study of lithium clusters by electronic stress tensor
Ichikawa, Kazuhide; Nozaki, Hiroo; Komazawa, Naoya; Tachibana, Akitomo
2012-01-01
We study the electronic structure of small lithium clusters Li_n (n = 2 ∼ 8) using the electronic stress tensor. We find that the three eigenvalues of the electronic stress tensor of the Li clusters are negative and degenerate, just like the stress tensor of liquid. This leads us to propose that we may characterize a metallic bond in terms of the electronic stress tensor. Our proposal is that in addition to the negativity of the three eigenvalues of the electronic stress tensor, their degeneracy characterizes some aspects of the metallic nature of chemical bonding. To quantify the degree of degeneracy, we use the differential eigenvalues of the electronic stress tensor. By comparing the Li clusters and hydrocarbon molecules, we show that the sign of the largest eigenvalue and the differential eigenvalues could be useful indices to evaluate the metallicity or covalency of a chemical bond.
Radiative corrections in a vector-tensor model
Chishtie, F.; Gagne-Portelance, M.; Hanif, T.; Homayouni, S.; McKeon, D.G.C.
2006-01-01
In a recently proposed model in which a vector non-Abelian gauge field interacts with an antisymmetric tensor field, it has been shown that the tensor field possesses no physical degrees of freedom. This formal demonstration is tested by computing the one-loop contributions of the tensor field to the self-energy of the vector field. It is shown that despite the large number of Feynman diagrams in which the tensor field contributes, the sum of these diagrams vanishes, confirming that it is not physical. Furthermore, if the tensor field were to couple with a spinor field, it is shown at one-loop order that the spinor self-energy is not renormalizable, and hence this coupling must be excluded. In principle though, this tensor field does couple to the gravitational field
Supergravity tensor calculus in 5D from 6D
Kugo, Taichiro; Ohashi, Keisuke
2000-01-01
Supergravity tensor calculus in five spacetime dimensions is derived by dimensional reduction from the d=6 superconformal tensor calculus. In particular, we obtain an off-shell hypermultiplet in 5D from the on-shell hypermultiplet in 6D. Our tensor calculus retains the dilatation gauge symmetry, so that it is a trivial gauge fixing to make the Einstein term canonical in a general matter-Yang-Mills-supergravity coupled system. (author)
Mesh Denoising based on Normal Voting Tensor and Binary Optimization
Yadav, S. K.; Reitebuch, U.; Polthier, K.
2016-01-01
This paper presents a tensor multiplication based smoothing algorithm that follows a two step denoising method. Unlike other traditional averaging approaches, our approach uses an element based normal voting tensor to compute smooth surfaces. By introducing a binary optimization on the proposed tensor together with a local binary neighborhood concept, our algorithm better retains sharp features and produces smoother umbilical regions than previous approaches. On top of that, we provide a stoc...
Comparison of two global digital algorithms for Minkowski tensor estimation
The geometry of real world objects can be described by Minkowski tensors. Algorithms have been suggested to approximate Minkowski tensors if only a binary image of the object is available. This paper presents implementations of two such algorithms. The theoretical convergence properties...... are confirmed by simulations on test sets, and recommendations for input arguments of the algorithms are given. For increasing resolutions, we obtain more accurate estimators for the Minkowski tensors. Digitisations of more complicated objects are shown to require higher resolutions....
Properties of the tensor correlation in He isotopes
Myo, Takayuki; Sugimoto, Satoru; Kato, Kiyoshi; Toki, Hiroshi; Ikeda, Kiyomi
2006-01-01
We investigate the roles of the tensor correlation on the structures of 4,5 He. For 4 He, we take the high angular momentum states as much as possible with the 2p2h excitations of the shell model type method to describe the tensor correlation. Three specific configurations are found to be favored for the tensor correlation. This correlation is also important to describe the scattering phenomena of the 4 He+nsystem including the higher partial waves consistently
A General Expression for the Quartic Lovelock Tensor
Briggs, C. C.
1997-01-01
A general expression is given for the quartic Lovelock tensor in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general linear connection. In addition, expressions are given (in the appendix) for the coefficient of the quartic Lovelock Lagrangian as well as for lower-order Lovelock tensors and Lovelock Lagrangian coefficients.
Energy-momentum tensor in the quantum field theory
Azakov, S.I.
1977-01-01
An energy-momentum tensor in the scalar field theory is built. The tensor must satisfy the finiteness requirement of the Green function. The Green functions can always be made finite by renormalizations in the S-matrix by introducing counter terms into the Hamiltonian (or Lagrangian) of the interaction. Such a renormalization leads to divergencies in the Green functions. Elimination of these divergencies requires the introduction of new counter terms, which must be taken into account in the energy-momentum tensor
Joint Tensor Feature Analysis For Visual Object Recognition.
Wong, Wai Keung; Lai, Zhihui; Xu, Yong; Wen, Jiajun; Ho, Chu Po
2015-11-01
Tensor-based object recognition has been widely studied in the past several years. This paper focuses on the issue of joint feature selection from the tensor data and proposes a novel method called joint tensor feature analysis (JTFA) for tensor feature extraction and recognition. In order to obtain a set of jointly sparse projections for tensor feature extraction, we define the modified within-class tensor scatter value and the modified between-class tensor scatter value for regression. The k-mode optimization technique and the L(2,1)-norm jointly sparse regression are combined together to compute the optimal solutions. The convergent analysis, computational complexity analysis and the essence of the proposed method/model are also presented. It is interesting to show that the proposed method is very similar to singular value decomposition on the scatter matrix but with sparsity constraint on the right singular value matrix or eigen-decomposition on the scatter matrix with sparse manner. Experimental results on some tensor datasets indicate that JTFA outperforms some well-known tensor feature extraction and selection algorithms.
The Topology of Three-Dimensional Symmetric Tensor Fields
Lavin, Yingmei; Levy, Yuval; Hesselink, Lambertus
1994-01-01
We study the topology of 3-D symmetric tensor fields. The goal is to represent their complex structure by a simple set of carefully chosen points and lines analogous to vector field topology. The basic constituents of tensor topology are the degenerate points, or points where eigenvalues are equal to each other. First, we introduce a new method for locating 3-D degenerate points. We then extract the topological skeletons of the eigenvector fields and use them for a compact, comprehensive description of the tensor field. Finally, we demonstrate the use of tensor field topology for the interpretation of the two-force Boussinesq problem.
QCD vacuum tensor susceptibility and properties of transversely polarized mesons
Bakulev, A.P.; Mikhajlov, S.V.
1999-01-01
We re-estimate the tensor susceptibility of QCD vacuum, χ, and to this end, we re-estimate the leptonic decay constants for transversely polarized ρ-, ρ'- and b 1 -mesons. The origin of the susceptibility is analyzed using duality between ρ- and b 1 -channels in a 2-point correlator of tensor currents and disagree with [2] on both OPE expansion and the value of QCD vacuum tensor susceptibility. Using our value for the latter we determine new estimations of nucleon tensor charges related to the first moment of the transverse structure functions h 1 of a nucleon
3D Inversion of SQUID Magnetic Tensor Data
Zhdanov, Michael; Cai, Hongzhu; Wilson, Glenn
2012-01-01
Developments in SQUID-based technology have enabled direct measurement of magnetic tensor data for geophysical exploration. For quantitative interpretation, we introduce 3D regularized inversion for magnetic tensor data. For mineral exploration-scale targets, our model studies show that magnetic...... tensor data have significantly improved resolution compared to magnetic vector data for the same model. We present a case study for the 3D regularized inversion of magnetic tensor data acquired over a magnetite skarn at Tallawang, Australia. The results obtained from our 3D regularized inversion agree...
Superconformal tensor calculus and matter couplings in six dimensions
Bergshoeff, E.; Sezgin, E.; van Proeyen, A.
1989-01-01
Using superconformal tensor calculus the authors construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. They start from the superconformal algebra which they realize on a 40 + 40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40 + 40 version of the Weyl multiplet. The authors also obtain a 48 + 48 off-shell formulation of Poincare supergravity coupled to a tensor multiplet
Superconformal tensor calculus and matter couplings in six dimensions
Bergshoeff, E.; Sezgin, E.; Proeyen, A. van
1986-01-01
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40 + 40 Weyl multiplet and on several matter multiplets. A special role is played by the tensor multiplet, which cannot be treated as an ordinary matter multiplet, but leads to a second 40 + 40 version of the Weyl multiplet. We also obtain a 48 + 48 off-shell formulation of Poincare supergravity coupled to a tensor multiplet. (orig.)
TENSOR MODELING BASED FOR AIRBORNE LiDAR DATA CLASSIFICATION
N. Li
2016-06-01
Full Text Available Feature selection and description is a key factor in classification of Earth observation data. In this paper a classification method based on tensor decomposition is proposed. First, multiple features are extracted from raw LiDAR point cloud, and raster LiDAR images are derived by accumulating features or the “raw” data attributes. Then, the feature rasters of LiDAR data are stored as a tensor, and tensor decomposition is used to select component features. This tensor representation could keep the initial spatial structure and insure the consideration of the neighborhood. Based on a small number of component features a k nearest neighborhood classification is applied.
p-Norm SDD tensors and eigenvalue localization
Qilong Liu
2016-07-01
Full Text Available Abstract We present a new class of nonsingular tensors (p-norm strictly diagonally dominant tensors, which is a subclass of strong H $\\mathcal{H}$ -tensors. As applications of the results, we give a new eigenvalue inclusion set, which is tighter than those provided by Li et al. (Linear Multilinear Algebra 64:727-736, 2016 in some case. Based on this set, we give a checkable sufficient condition for the positive (semidefiniteness of an even-order symmetric tensor.
Prescribed curvature tensor in locally conformally flat manifolds
Pina, Romildo; Pieterzack, Mauricio
2018-01-01
A global existence theorem for the prescribed curvature tensor problem in locally conformally flat manifolds is proved for a special class of tensors R. Necessary and sufficient conditions for the existence of a metric g ¯ , conformal to Euclidean g, are determined such that R ¯ = R, where R ¯ is the Riemannian curvature tensor of the metric g ¯ . The solution to this problem is given explicitly for special cases of the tensor R, including the case where the metric g ¯ is complete on Rn. Similar problems are considered for locally conformally flat manifolds.
The classification of the Ricci tensor in the general theory of relativity
Cormack, W.J.
1979-10-01
A comprehensive classification of the Ricci tensor in General Relativity using several techniques is given and their connection with existing classification studied under the headings; canonical forms for the Ricci tensor, invariant 2-spaces in the classification of the Ricci tensor, Riemannian curvature and the classification of the Riemann and Ricci tensors, and spinor classifications of the Ricci tensor. (U.K.)
Symmetric Topological Phases and Tensor Network States
Jiang, Shenghan
Classification and simulation of quantum phases are one of main themes in condensed matter physics. Quantum phases can be distinguished by their symmetrical and topological properties. The interplay between symmetry and topology in condensed matter physics often leads to exotic quantum phases and rich phase diagrams. Famous examples include quantum Hall phases, spin liquids and topological insulators. In this thesis, I present our works toward a more systematically understanding of symmetric topological quantum phases in bosonic systems. In the absence of global symmetries, gapped quantum phases are characterized by topological orders. Topological orders in 2+1D are well studied, while a systematically understanding of topological orders in 3+1D is still lacking. By studying a family of exact solvable models, we find at least some topological orders in 3+1D can be distinguished by braiding phases of loop excitations. In the presence of both global symmetries and topological orders, the interplay between them leads to new phases termed as symmetry enriched topological (SET) phases. We develop a framework to classify a large class of SET phases using tensor networks. For each tensor class, we can write down generic variational wavefunctions. We apply our method to study gapped spin liquids on the kagome lattice, which can be viewed as SET phases of on-site symmetries as well as lattice symmetries. In the absence of topological order, symmetry could protect different topological phases, which are often referred to as symmetry protected topological (SPT) phases. We present systematic constructions of tensor network wavefunctions for bosonic symmetry protected topological (SPT) phases respecting both onsite and spatial symmetries.
Holographic duality from random tensor networks
Hayden, Patrick; Nezami, Sepehr; Qi, Xiao-Liang; Thomas, Nathaniel; Walter, Michael; Yang, Zhao [Stanford Institute for Theoretical Physics, Department of Physics, Stanford University,382 Via Pueblo, Stanford, CA 94305 (United States)
2016-11-02
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit toy models realizing many of the interesting structural features of the AdS/CFT correspondence, including the non-uniqueness of bulk operator reconstruction in the boundary theory. In this article, we explore the holographic properties of networks of random tensors. We find that our models naturally incorporate many features that are analogous to those of the AdS/CFT correspondence. When the bond dimension of the tensors is large, we show that the entanglement entropy of all boundary regions, whether connected or not, obey the Ryu-Takayanagi entropy formula, a fact closely related to known properties of the multipartite entanglement of assistance. We also discuss the behavior of Rényi entropies in our models and contrast it with AdS/CFT. Moreover, we find that each boundary region faithfully encodes the physics of the entire bulk entanglement wedge, i.e., the bulk region enclosed by the boundary region and the minimal surface. Our method is to interpret the average over random tensors as the partition function of a classical ferromagnetic Ising model, so that the minimal surfaces of Ryu-Takayanagi appear as domain walls. Upon including the analog of a bulk field, we find that our model reproduces the expected corrections to the Ryu-Takayanagi formula: the bulk minimal surface is displaced and the entropy is augmented by the entanglement of the bulk field. Increasing the entanglement of the bulk field ultimately changes the minimal surface behavior topologically, in a way similar to the effect of creating a black hole. Extrapolating bulk correlation functions to the boundary permits the calculation of the scaling dimensions of boundary operators, which exhibit a large gap between a small number of low-dimension operators and the rest. While we are primarily motivated by the AdS/CFT duality, the main
A supersymmetric SYK-like tensor model
Peng, Cheng; Spradlin, Marcus; Volovich, Anastasia [Department of Physics, Brown University,Providence, RI, 02912 (United States)
2017-05-11
We consider a supersymmetric SYK-like model without quenched disorder that is built by coupling two kinds of fermionic N=1 tensor-valued superfields, “quarks” and “mesons”. We prove that the model has a well-defined large-N limit in which the (s)quark 2-point functions are dominated by mesonic “melon” diagrams. We sum these diagrams to obtain the Schwinger-Dyson equations and show that in the IR, the solution agrees with that of the supersymmetric SYK model.
Tensor glueball-meson mixing phenomenology
Burakovsky, L.; Page, P.R.
2000-01-01
The overpopulated isoscalar tensor states are sifted using Schwinger-type mass relations. Two solutions are found: one where the glueball is the f J (2220), and one where the glueball is more distributed, with f 2 (1820) having the largest component. The f 2 (1565) and f J (1710) cannot be accommodated as glueball-(hybrid) meson mixtures in the absence of significant coupling to decay channels. f 2 '(1525)→ππ is in agreement with experiment. The f J (2220) decays neither flavour democratically nor is narrow. (orig.)
Tensor Network Wavefunctions for Topological Phases
Ware, Brayden Alexander
The combination of quantum effects and interactions in quantum many-body systems can result in exotic phases with fundamentally entangled ground state wavefunctions--topological phases. Topological phases come in two types, both of which will be studied in this thesis. In topologically ordered phases, the pattern of entanglement in the ground state wavefunction encodes the statistics of exotic emergent excitations, a universal indicator of a phase that is robust to all types of perturbations. In symmetry protected topological phases, the entanglement instead encodes a universal response of the system to symmetry defects, an indicator that is robust only to perturbations respecting the protecting symmetry. Finding and creating these phases in physical systems is a motivating challenge that tests all aspects--analytical, numerical, and experimental--of our understanding of the quantum many-body problem. Nearly three decades ago, the creation of simple ansatz wavefunctions--such as the Laughlin fractional quantum hall state, the AKLT state, and the resonating valence bond state--spurred analytical understanding of both the role of entanglement in topological physics and physical mechanisms by which it can arise. However, quantitative understanding of the relevant phase diagrams is still challenging. For this purpose, tensor networks provide a toolbox for systematically improving wavefunction ansatz while still capturing the relevant entanglement properties. In this thesis, we use the tools of entanglement and tensor networks to analyze ansatz states for several proposed new phases. In the first part, we study a featureless phase of bosons on the honeycomb lattice and argue that this phase can be topologically protected under any one of several distinct subsets of the crystalline lattice symmetries. We discuss methods of detecting such phases with entanglement and without. In the second part, we consider the problem of constructing fixed-point wavefunctions for
Scalar-tensor cosmology with cosmological constant
Maslanka, K.
1983-01-01
The equations of scalar-tensor theory of gravitation with cosmological constant in the case of homogeneous and isotropic cosmological model can be reduced to dynamical system of three differential equations with unknown functions H=R/R, THETA=phi/phi, S=e/phi. When new variables are introduced the system becomes more symmetrical and cosmological solutions R(t), phi(t), e(t) are found. It is shown that when cosmological constant is introduced large class of solutions which depend also on Dicke-Brans parameter can be obtained. Investigations of these solutions give general limits for cosmological constant and mean density of matter in plane model. (author)
Tensor Decompositions for Learning Latent Variable Models
2012-12-08
and eigenvectors of tensors is generally significantly more complicated than their matrix counterpart (both algebraically [Qi05, CS11, Lim05] and...The reduction First, let W ∈ Rd×k be a linear transformation such that M2(W,W ) = W M2W = I where I is the k × k identity matrix (i.e., W whitens ...approximate the whitening matrix W ∈ Rd×k from second-moment matrix M2 ∈ Rd×d. To do this, one first multiplies M2 by a random matrix R ∈ Rd×k′ for some k′ ≥ k
Electrical conductivity tensor of an irradiated metal
Corciovei, A.; Dumitru, R.D.
1979-01-01
A method to calculate the electrical conductivity tensor of an irradiated metal is presented. The proposed method relies on the use of the Kubo formula, evaluated by a perturbation method. The one electron Hamiltonian is written as a sum of two terms: the Hamiltonian of the conduction electrons moving in a periodic lattice and the perturbation, namely, the scattering potential due to the irradiation defects of the ideal crystal. Then, the lowest order of the conductivity is determined by the lowest order of the Laplace transform of the current. An integral equation is written for this last quantity. (author)
A supersymmetric SYK-like tensor model
Peng, Cheng; Spradlin, Marcus; Volovich, Anastasia
2017-01-01
We consider a supersymmetric SYK-like model without quenched disorder that is built by coupling two kinds of fermionic N=1 tensor-valued superfields, “quarks” and “mesons”. We prove that the model has a well-defined large-N limit in which the (s)quark 2-point functions are dominated by mesonic “melon” diagrams. We sum these diagrams to obtain the Schwinger-Dyson equations and show that in the IR, the solution agrees with that of the supersymmetric SYK model.
Physical states in the canonical tensor model from the perspective of random tensor networks
Narain, Gaurav [The Institute for Fundamental Study “The Tah Poe Academia Institute”,Naresuan University, Phitsanulok 65000 (Thailand); Sasakura, Naoki [Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502 (Japan); Sato, Yuki [National Institute for Theoretical Physics,School of Physics and Centre for Theoretical Physics,University of the Witwartersrand, WITS 2050 (South Africa)
2015-01-07
Tensor models, generalization of matrix models, are studied aiming for quantum gravity in dimensions larger than two. Among them, the canonical tensor model is formulated as a totally constrained system with first-class constraints, the algebra of which resembles the Dirac algebra of general relativity. When quantized, the physical states are defined to be vanished by the quantized constraints. In explicit representations, the constraint equations are a set of partial differential equations for the physical wave-functions, which do not seem straightforward to be solved due to their non-linear character. In this paper, after providing some explicit solutions for N=2,3, we show that certain scale-free integration of partition functions of statistical systems on random networks (or random tensor networks more generally) provides a series of solutions for general N. Then, by generalizing this form, we also obtain various solutions for general N. Moreover, we show that the solutions for the cases with a cosmological constant can be obtained from those with no cosmological constant for increased N. This would imply the interesting possibility that a cosmological constant can always be absorbed into the dynamics and is not an input parameter in the canonical tensor model. We also observe the possibility of symmetry enhancement in N=3, and comment on an extension of Airy function related to the solutions.
FADTTS: functional analysis of diffusion tensor tract statistics.
Zhu, Hongtu; Kong, Linglong; Li, Runze; Styner, Martin; Gerig, Guido; Lin, Weili; Gilmore, John H
2011-06-01
The aim of this paper is to present a functional analysis of a diffusion tensor tract statistics (FADTTS) pipeline for delineating the association between multiple diffusion properties along major white matter fiber bundles with a set of covariates of interest, such as age, diagnostic status and gender, and the structure of the variability of these white matter tract properties in various diffusion tensor imaging studies. The FADTTS integrates five statistical tools: (i) a multivariate varying coefficient model for allowing the varying coefficient functions in terms of arc length to characterize the varying associations between fiber bundle diffusion properties and a set of covariates, (ii) a weighted least squares estimation of the varying coefficient functions, (iii) a functional principal component analysis to delineate the structure of the variability in fiber bundle diffusion properties, (iv) a global test statistic to test hypotheses of interest, and (v) a simultaneous confidence band to quantify the uncertainty in the estimated coefficient functions. Simulated data are used to evaluate the finite sample performance of FADTTS. We apply FADTTS to investigate the development of white matter diffusivities along the splenium of the corpus callosum tract and the right internal capsule tract in a clinical study of neurodevelopment. FADTTS can be used to facilitate the understanding of normal brain development, the neural bases of neuropsychiatric disorders, and the joint effects of environmental and genetic factors on white matter fiber bundles. The advantages of FADTTS compared with the other existing approaches are that they are capable of modeling the structured inter-subject variability, testing the joint effects, and constructing their simultaneous confidence bands. However, FADTTS is not crucial for estimation and reduces to the functional analysis method for the single measure. Copyright © 2011 Elsevier Inc. All rights reserved.
Interactive Volume Rendering of Diffusion Tensor Data
Hlawitschka, Mario; Weber, Gunther; Anwander, Alfred; Carmichael, Owen; Hamann, Bernd; Scheuermann, Gerik
2007-03-30
As 3D volumetric images of the human body become an increasingly crucial source of information for the diagnosis and treatment of a broad variety of medical conditions, advanced techniques that allow clinicians to efficiently and clearly visualize volumetric images become increasingly important. Interaction has proven to be a key concept in analysis of medical images because static images of 3D data are prone to artifacts and misunderstanding of depth. Furthermore, fading out clinically irrelevant aspects of the image while preserving contextual anatomical landmarks helps medical doctors to focus on important parts of the images without becoming disoriented. Our goal was to develop a tool that unifies interactive manipulation and context preserving visualization of medical images with a special focus on diffusion tensor imaging (DTI) data. At each image voxel, DTI provides a 3 x 3 tensor whose entries represent the 3D statistical properties of water diffusion locally. Water motion that is preferential to specific spatial directions suggests structural organization of the underlying biological tissue; in particular, in the human brain, the naturally occuring diffusion of water in the axon portion of neurons is predominantly anisotropic along the longitudinal direction of the elongated, fiber-like axons [MMM+02]. This property has made DTI an emerging source of information about the structural integrity of axons and axonal connectivity between brain regions, both of which are thought to be disrupted in a broad range of medical disorders including multiple sclerosis, cerebrovascular disease, and autism [Mos02, FCI+01, JLH+99, BGKM+04, BJB+03].
Black holes in vector-tensor theories
Heisenberg, Lavinia [Institute for Theoretical Studies, ETH Zurich, Clausiusstrasse 47, 8092 Zurich (Switzerland); Kase, Ryotaro; Tsujikawa, Shinji [Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan); Minamitsuji, Masato, E-mail: lavinia.heisenberg@eth-its.ethz.ch, E-mail: r.kase@rs.tus.ac.jp, E-mail: masato.minamitsuji@tecnico.ulisboa.pt, E-mail: shinji@rs.kagu.tus.ac.jp [Centro Multidisciplinar de Astrofisica—CENTRA, Departamento de Fisica, Instituto Superior Tecnico—IST, Universidade de Lisboa—UL, Avenida Rovisco Pais 1, 1049-001 Lisboa (Portugal)
2017-08-01
We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordström (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.
Emergent symmetries in the canonical tensor model
Obster, Dennis; Sasakura, Naoki
2018-04-01
The canonical tensor model (CTM) is a tensor model proposing a classically and quantum mechanically consistent description of gravity, formulated as a first-class constraint system with structural similarities to the ADM formalism of general relativity. The classical CTM produces a general relativistic system in a formal continuum limit, the emergence of which should be explained by the quantum CTM. In this paper we study the symmetry properties of a wave function that exactly solves the quantum constraints of the CTM. We have found that it has strong peaks at configurations invariant under some Lie groups, as predicted by a mechanism described in our previous paper. A surprising result is the preference for configurations invariant not only under Lie groups with positive definite signature, but also with Lorentzian signature. Such symmetries could characterize the global structures of spacetimes, and our results are encouraging towards showing spacetime emergence in the CTM. To verify the asymptotic convergence of the wave function we have also analyzed the asymptotic behavior, which for the most part seems to be well under control.
Quantum chaos and holographic tensor models
Krishnan, Chethan [Center for High Energy Physics, Indian Institute of Science,Bangalore 560012 (India); Sanyal, Sambuddha [International Center for Theoretical Sciences, Tata Institute of Fundamental Research,Bangalore 560089 (India); Subramanian, P.N. Bala [Center for High Energy Physics, Indian Institute of Science,Bangalore 560012 (India)
2017-03-10
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp-plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.
Stress tensor fluctuations in de Sitter spacetime
Pérez-Nadal, Guillem; Verdaguer, Enric [Departament de Física Fonamental and Institut de Ciències del Cosmos, Universitat de Barcelona, Av. Diagonal 647, 08028 Barcelona (Spain); Roura, Albert, E-mail: guillem@ffn.ub.es, E-mail: albert.roura@aei.mpg.de, E-mail: enric.verdaguer@ub.edu [Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, 14476 Golm (Germany)
2010-05-01
The two-point function of the stress tensor operator of a quantum field in de Sitter spacetime is calculated for an arbitrary number of dimensions. We assume the field to be in the Bunch-Davies vacuum, and formulate our calculation in terms of de Sitter-invariant bitensors. Explicit results for free minimally coupled scalar fields with arbitrary mass are provided. We find long-range stress tensor correlations for sufficiently light fields (with mass m much smaller than the Hubble scale H), namely, the two-point function decays at large separations like an inverse power of the physical distance with an exponent proportional to m{sup 2}/H{sup 2}. In contrast, we show that for the massless case it decays at large separations like the fourth power of the physical distance. There is thus a discontinuity in the massless limit. As a byproduct of our work, we present a novel and simple geometric interpretation of de Sitter-invariant bitensors for pairs of points which cannot be connected by geodesics.
Quantum chaos and holographic tensor models
Krishnan, Chethan; Sanyal, Sambuddha; Subramanian, P.N. Bala
2017-01-01
A class of tensor models were recently outlined as potentially calculable examples of holography: their perturbative large-N behavior is similar to the Sachdev-Ye-Kitaev (SYK) model, but they are fully quantum mechanical (in the sense that there is no quenched disorder averaging). These facts make them intriguing tentative models for quantum black holes. In this note, we explicitly diagonalize the simplest non-trivial Gurau-Witten tensor model and study its spectral and late-time properties. We find parallels to (a single sample of) SYK where some of these features were recently attributed to random matrix behavior and quantum chaos. In particular, the spectral form factor exhibits a dip-ramp-plateau structure after a running time average, in qualitative agreement with SYK. But we also observe that even though the spectrum has a unique ground state, it has a huge (quasi-?)degeneracy of intermediate energy states, not seen in SYK. If one ignores the delta function due to the degeneracies however, there is level repulsion in the unfolded spacing distribution hinting chaos. Furthermore, there are gaps in the spectrum. The system also has a spectral mirror symmetry which we trace back to the presence of a unitary operator with which the Hamiltonian anticommutes. We use it to argue that to the extent that the model exhibits random matrix behavior, it is controlled not by the Dyson ensembles, but by the BDI (chiral orthogonal) class in the Altland-Zirnbauer classification.
Tensor estimation for double-pulsed diffusional kurtosis imaging.
Shaw, Calvin B; Hui, Edward S; Helpern, Joseph A; Jensen, Jens H
2017-07-01
Double-pulsed diffusional kurtosis imaging (DP-DKI) represents the double diffusion encoding (DDE) MRI signal in terms of six-dimensional (6D) diffusion and kurtosis tensors. Here a method for estimating these tensors from experimental data is described. A standard numerical algorithm for tensor estimation from conventional (i.e. single diffusion encoding) diffusional kurtosis imaging (DKI) data is generalized to DP-DKI. This algorithm is based on a weighted least squares (WLS) fit of the signal model to the data combined with constraints designed to minimize unphysical parameter estimates. The numerical algorithm then takes the form of a quadratic programming problem. The principal change required to adapt the conventional DKI fitting algorithm to DP-DKI is replacing the three-dimensional diffusion and kurtosis tensors with the 6D tensors needed for DP-DKI. In this way, the 6D diffusion and kurtosis tensors for DP-DKI can be conveniently estimated from DDE data by using constrained WLS, providing a practical means for condensing DDE measurements into well-defined mathematical constructs that may be useful for interpreting and applying DDE MRI. Data from healthy volunteers for brain are used to demonstrate the DP-DKI tensor estimation algorithm. In particular, representative parametric maps of selected tensor-derived rotational invariants are presented. Copyright © 2017 John Wiley & Sons, Ltd.
Secoond order parallel tensors on some paracontact manifolds | Liu ...
The object of the present paper is to study the symmetric and skewsymmetric properties of a second order parallel tensor on paracontact metric (k;μ)- spaces and almost β-para-Kenmotsu (k;μ)-spaces. In this paper, we prove that if there exists a second order symmetric parallel tensor on a paracontact metric (k;μ)- space M, ...
Visualizing Tensor Normal Distributions at Multiple Levels of Detail.
Abbasloo, Amin; Wiens, Vitalis; Hermann, Max; Schultz, Thomas
2016-01-01
Despite the widely recognized importance of symmetric second order tensor fields in medicine and engineering, the visualization of data uncertainty in tensor fields is still in its infancy. A recently proposed tensorial normal distribution, involving a fourth order covariance tensor, provides a mathematical description of how different aspects of the tensor field, such as trace, anisotropy, or orientation, vary and covary at each point. However, this wealth of information is far too rich for a human analyst to take in at a single glance, and no suitable visualization tools are available. We propose a novel approach that facilitates visual analysis of tensor covariance at multiple levels of detail. We start with a visual abstraction that uses slice views and direct volume rendering to indicate large-scale changes in the covariance structure, and locations with high overall variance. We then provide tools for interactive exploration, making it possible to drill down into different types of variability, such as in shape or orientation. Finally, we allow the analyst to focus on specific locations of the field, and provide tensor glyph animations and overlays that intuitively depict confidence intervals at those points. Our system is demonstrated by investigating the effects of measurement noise on diffusion tensor MRI, and by analyzing two ensembles of stress tensor fields from solid mechanics.
A tensor approach to the estimation of hydraulic conductivities in ...
Based on the field measurements of the physical properties of fractured rocks, the anisotropic properties of hydraulic conductivity (HC) of the fractured rock aquifer can be assessed and presented using a tensor approach called hydraulic conductivity tensor. Three types of HC values, namely point value, axial value and flow ...
Tensor meson dominance and e+e--physics
Genz, H.; Karlsruhe Univ.; Mallik, S.
1983-01-01
The phenomenological status of tensor meson dominance is reported. Some new results concerning hadronic decays of the 2 ++ -meson chi 2 (3.55) and the heavy lepton tau are also included. Considering experimental errors, tensor meson dominance is in agreement with experiment. (author)
Tensor Excitations in Nambu - Jona-Lasinio Model
Chizhov, M V
1996-01-01
It is shown that in the one-flavour NJL model the vector and axial-vector quasiparticles described by the antisymmetric tensor field are generated. These excitations have tensor interactions with quarks in contrast to usual vector ones. Phenomenological applications are discussed.
Tensor Basis Neural Network v. 1.0 (beta)
2017-03-28
This software package can be used to build, train, and test a neural network machine learning model. The neural network architecture is specifically designed to embed tensor invariance properties by enforcing that the model predictions sit on an invariant tensor basis. This neural network architecture can be used in developing constitutive models for applications such as turbulence modeling, materials science, and electromagnetism.
Energy momentum tensor in local causal perturbation theory
Prange, D.
2001-01-01
We study the energy momentum tensor in the Bogolyubov-Epstein-Glaser approach to perturbation theory. It is found to be locally conserved for a class of theories containing to derivated fields in the interaction. For the massless φ 4 -theory we derive the trace anomaly of the improved tensor. (orig.)
Exploring the tensor networks/AdS correspondence
Bhattacharyya, Arpan [Department of Physics and Center for Field Theory and Particle Physics, Fudan University,220 Handan Road, 200433 Shanghai (China); Centre For High Energy Physics, Indian Institute of Science,560012 Bangalore (India); Gao, Zhe-Shen [Department of Physics and Center for Field Theory and Particle Physics, Fudan University,220 Handan Road, 200433 Shanghai (China); Hung, Ling-Yan [Department of Physics and Center for Field Theory and Particle Physics, Fudan University,220 Handan Road, 200433 Shanghai (China); State Key Laboratory of Surface Physics and Department of Physics, Fudan University,220 Handan Road, 200433 Shanghai (China); Collaborative Innovation Center of Advanced Microstructures, Nanjing University,Nanjing, 210093 (China); Liu, Si-Nong [Department of Physics and Center for Field Theory and Particle Physics, Fudan University,220 Handan Road, 200433 Shanghai (China)
2016-08-11
In this paper we study the recently proposed tensor networks/AdS correspondence. We found that the Coxeter group is a useful tool to describe tensor networks in a negatively curved space. Studying generic tensor network populated by perfect tensors, we find that the physical wave function generically do not admit any connected correlation functions of local operators. To remedy the problem, we assume that wavefunctions admitting such semi-classical gravitational interpretation are composed of tensors close to, but not exactly perfect tensors. Computing corrections to the connected two point correlation functions, we find that the leading contribution is given by structures related to geodesics connecting the operators inserted at the boundary physical dofs. Such considerations admit generalizations at least to three point functions. This is highly suggestive of the emergence of the analogues of Witten diagrams in the tensor network. The perturbations alone however do not give the right entanglement spectrum. Using the Coxeter construction, we also constructed the tensor network counterpart of the BTZ black hole, by orbifolding the discrete lattice on which the network resides. We found that the construction naturally reproduces some of the salient features of the BTZ black hole, such as the appearance of RT surfaces that could wrap the horizon, depending on the size of the entanglement region A.
Black holes with surrounding matter in scalar-tensor theories.
Cardoso, Vitor; Carucci, Isabella P; Pani, Paolo; Sotiriou, Thomas P
2013-09-13
We uncover two mechanisms that can render Kerr black holes unstable in scalar-tensor gravity, both associated with the presence of matter in the vicinity of the black hole and the fact that this introduces an effective mass for the scalar. Our results highlight the importance of understanding the structure of spacetime in realistic, astrophysical black holes in scalar-tensor theories.
A Gradient Based Iterative Solutions for Sylvester Tensor Equations
Zhen Chen
2013-01-01
proposed by Ding and Chen, 2005, and by using tensor arithmetic concepts, an iterative algorithm and its modification are established to solve the Sylvester tensor equation. Convergence analysis indicates that the iterative solutions always converge to the exact solution for arbitrary initial value. Finally, some examples are provided to show that the proposed algorithms are effective.
Relativistic interpretation of the nature of the nuclear tensor force
Zong, Yao-Yao; Sun, Bao-Yuan
2018-02-01
The spin-dependent nature of the nuclear tensor force is studied in detail within the relativistic Hartree-Fock approach. The relativistic formalism for the tensor force is supplemented with an additional Lorentz-invariant tensor formalism in the σ-scalar channel, so as to take into account almost fully the nature of the tensor force brought about by the Fock diagrams in realistic nuclei. Specifically, the tensor sum rules are tested for the spin and pseudo-spin partners with and without nodes, to further understand the nature of the tensor force within the relativistic model. It is shown that the interference between the two components of nucleon spinors causes distinct violations of the tensor sum rules in realistic nuclei, mainly due to the opposite signs on the κ quantities of the upper and lower components, as well as the nodal difference. However, the sum rules can be precisely reproduced if the same radial wave functions are taken for the spin/pseudo-spin partners in addition to neglecting the lower/upper components, revealing clearly the nature of the tensor force. Supported by National Natural Science Foundation of China (11375076, 11675065) and the Fundamental Research Funds for the Central Universities (lzujbky-2016-30)
Parity and isospin in pion condensation and tensor binding
Pace, E.; Palumbo, F.
1978-01-01
In infinite nuclear matter with pion condensates or tensor binding both parity and isospin symmetries are broken. Finite nuclei with pion condensates or tensor binding, however, can have definite parity. They cannot have a definite value of isospin, whose average value is of the order of the number of nucleons. (Auth.)
Superspace actions and duality transformations for N=2 tensor multiplets
Galperin, A.; Ivanov, E.; Ogievetsky, V.
1985-01-01
General actions for self-interacting N=2 tensor multiplets are considered in the harmonic superspace approach. All of them are shown to be equivalent, by superfield duality transformations, to some restricted class of the hypermultiplets actions. In particular, the improved tensor multiplet theory is dual to a free hypermultiplet one. Superspace couplings of these improved matter multiplets against conformal supergravity are also constructed
Multiple M2-branes and the embedding tensor
Bergshoeff, Eric A.; de Roo, Mees; Hohm, Olaf
2008-01-01
We show that the Bagger-Lambert theory of multiple M2-branes fits into the general construction of maximally supersymmetric gauge theories using the embedding tensor technique. We apply the embedding tensor technique in order to systematically obtain the consistent gaugings of N = 8 superconformal
Subtracting a best rank-1 approximation may increase tensor rank
Stegeman, Alwin; Comon, Pierre
2010-01-01
It has been shown that a best rank-R approximation of an order-k tensor may not exist when R >= 2 and k >= 3. This poses a serious problem to data analysts using tensor decompositions it has been observed numerically that, generally, this issue cannot be solved by consecutively computing and
Couplings of self-dual tensor multiplet in six dimensions
Bergshoeff, E.; Sezgin, E.; Sokatchev, E.
1996-01-01
The (1, 0) supersymmetry in six dimensions admits a tensor multiplet which contains a second-rank antisymmetric tensor field with a self-dual field strength and a dilaton. We describe the fully supersymmetric coupling of this multiplet to a Yangâ€“Mills multiplet, in the absence of supergravity. The
Superconformal tensor calculus and matter couplings in six dimensions
Bergshoeff, E.; Sezgin, E.; Proeyen, A. Van
1986-01-01
Using superconformal tensor calculus we construct general interactions of N = 2, d = 6 supergravity with a tensor multiplet and a number of scalar, vector and linear multiplets. We start from the superconformal algebra which we realize on a 40+40 Weyl multiplet and on several matter multiplets. A
Data fusion in metabolomics using coupled matrix and tensor factorizations
Evrim, Acar Ataman; Bro, Rasmus; Smilde, Age Klaas
2015-01-01
of heterogeneous (i.e., in the form of higher order tensors and matrices) data sets with shared/unshared factors. In order to jointly analyze such heterogeneous data sets, we formulate data fusion as a coupled matrix and tensor factorization (CMTF) problem, which has already proved useful in many data mining...
Fast evaluation of nonlinear functionals of tensor product wavelet expansions
Schwab, C.; Stevenson, R.
2011-01-01
Abstract For a nonlinear functional f, and a function u from the span of a set of tensor product interpolets, it is shown how to compute the interpolant of f (u) from the span of this set of tensor product interpolets in linear complexity, assuming that the index set has a certain multiple tree
Gauge theories, duality relations and the tensor hierarchy
Bergshoeff, Eric A.; Hartong, Jelle; Hohm, Olaf; Huebscher, Mechthild; Ortin, Tomas; Hübscher, Mechthild
We compute the complete 3- and 4-dimensional tensor hierarchies, i.e. sets of p-form fields, with 1 We construct gauge-invariant actions that include all the fields in the tensor hierarchies. We elucidate the relation between the gauge transformations of the p-form fields in the action and those of
OPERATOR NORM INEQUALITIES BETWEEN TENSOR UNFOLDINGS ON THE PARTITION LATTICE.
Wang, Miaoyan; Duc, Khanh Dao; Fischer, Jonathan; Song, Yun S
2017-05-01
Interest in higher-order tensors has recently surged in data-intensive fields, with a wide range of applications including image processing, blind source separation, community detection, and feature extraction. A common paradigm in tensor-related algorithms advocates unfolding (or flattening) the tensor into a matrix and applying classical methods developed for matrices. Despite the popularity of such techniques, how the functional properties of a tensor changes upon unfolding is currently not well understood. In contrast to the body of existing work which has focused almost exclusively on matricizations, we here consider all possible unfoldings of an order- k tensor, which are in one-to-one correspondence with the set of partitions of {1, …, k }. We derive general inequalities between the l p -norms of arbitrary unfoldings defined on the partition lattice. In particular, we demonstrate how the spectral norm ( p = 2) of a tensor is bounded by that of its unfoldings, and obtain an improved upper bound on the ratio of the Frobenius norm to the spectral norm of an arbitrary tensor. For specially-structured tensors satisfying a generalized definition of orthogonal decomposability, we prove that the spectral norm remains invariant under specific subsets of unfolding operations.
The Twist Tensor Nuclear Norm for Video Completion.
Hu, Wenrui; Tao, Dacheng; Zhang, Wensheng; Xie, Yuan; Yang, Yehui
2017-12-01
In this paper, we propose a new low-rank tensor model based on the circulant algebra, namely, twist tensor nuclear norm (t-TNN). The twist tensor denotes a three-way tensor representation to laterally store 2-D data slices in order. On one hand, t-TNN convexly relaxes the tensor multirank of the twist tensor in the Fourier domain, which allows an efficient computation using fast Fourier transform. On the other, t-TNN is equal to the nuclear norm of block circulant matricization of the twist tensor in the original domain, which extends the traditional matrix nuclear norm in a block circulant way. We test the t-TNN model on a video completion application that aims to fill missing values and the experiment results validate its effectiveness, especially when dealing with video recorded by a nonstationary panning camera. The block circulant matricization of the twist tensor can be transformed into a circulant block representation with nuclear norm invariance. This representation, after transformation, exploits the horizontal translation relationship between the frames in a video, and endows the t-TNN model with a more powerful ability to reconstruct panning videos than the existing state-of-the-art low-rank models.
The superspace-translation tensor and linearized N = 1 supergravities
Bedding, S.P.; Lang, W.
1982-01-01
The recently proposed superspace-translation tensor is considered as the source of supergravities in the context of N = 1 supersymmetry. It is shown how the structure of this tensor leads to a complete evaluation of the linearized supervielbein in terms of unconstrained prepotentials with derived transformation laws. Connection with formulations using torsion constraints is made. (orig.)
(2, 0) tensor multiplets and conformal supergravity in D = 6
Bergshoeff, Eric; Sezgin, Ergin; Proeyen, Antoine Van
1999-01-01
We construct the supercurrent multiplet that contains the energyâ€“momentum tensor of the (2, 0) tensor multiplet. By coupling this multiplet of currents to the fields of conformal supergravity, we first construct the linearized superconformal transformations rules of the (2, 0) Weyl multiplet.
A General Expression for the Quintic Lovelock Tensor
Briggs, C. C.
1996-01-01
A general expression is given for the quintic Lovelock tensor as well as for the coefficient of the quintic Lovelock Lagrangian in terms of the Riemann-Christoffel and Ricci curvature tensors and the Riemann curvature scalar for n-dimensional differentiable manifolds having a general linear connection.
Aghazadeh, Mustafa; Mirzaei, Mahmoud
2008-01-01
Hydrogen bond (HB) interactions are studied in the real crystalline structure of sulfamerazine by density functional theory (DFT) calculations of the electric field gradient (EFG) tensors at the sites of O-17, N-14, and H-2 nuclei. One-molecule (single) and four-molecule (cluster) models of sulfamerazine are created by available crystal coordinates and the EFG tensors are calculated in both models to indicate the influence of HB interactions on the tensors. Directly relate to the experiments, the calculated EFG tensors are converted to the experimentally measurable nuclear quadrupole resonance (NQR) parameters, quadrupole coupling constant (qcc) and asymmetry parameter (η Q ). The evaluated NQR parameters reveal that due to contribution of the target molecule to N-H...N and N-H...O types of HB interactions, the EFG tensors at the sites of various nuclei are influenced from single model to the target molecule in cluster. Additionally, O2, N4, and H2 nuclei of the target molecule are significantly influenced by HB interactions, consequently, they have the major contributions to HB interactions in cluster model of sulfamerazine. The calculations are performed employing B3LYP method and 6-311++G** basis set using GAUSSIAN 98 suite of program
Decomposition of a symmetric second-order tensor
Heras, José A.
2018-05-01
In the three-dimensional space there are different definitions for the dot and cross products of a vector with a second-order tensor. In this paper we show how these products can uniquely be defined for the case of symmetric tensors. We then decompose a symmetric second-order tensor into its ‘dot’ part, which involves the dot product, and the ‘cross’ part, which involves the cross product. For some physical applications, this decomposition can be interpreted as one in which the dot part identifies with the ‘parallel’ part of the tensor and the cross part identifies with the ‘perpendicular’ part. This decomposition of a symmetric second-order tensor may be suitable for undergraduate courses of vector calculus, mechanics and electrodynamics.
On energy-momentum tensors of gravitational field
Nikishov, A.I.
2001-01-01
The phenomenological approach to gravitation is discussed in which the 3-graviton interaction is reduced to the interaction of each graviton with the energy-momentum tensor of two others. If this is so, (and in general relativity this is not so), then the problem of choosing the correct energy-momentum tensor comes to finding the right 3-graviton vertex. Several energy-momentum tensors od gravitational field are considered and compared in the lowest approximation. Each of them together with the energy-momentum tensor of point-like particles satisfies the conservation laws when equations of motion of particles are the same as in general relativity. It is shown that in Newtonian approximation the considered tensors differ one from other in the way their energy density is distributed between energy density of interaction (nonzero only at locations of particles) and energy density of gravitational field. Stating from Lorentz invariance, the Lagrangians for spin-2, mass-0 field are considered [ru
On the energy-momentum tensor in Moyal space
Balasin, Herbert; Schweda, Manfred; Blaschke, Daniel N.; Gieres, Francois
2015-01-01
We study the properties of the energy-momentum tensor of gauge fields coupled to matter in non-commutative (Moyal) space. In general, the non-commutativity affects the usual conservation law of the tensor as well as its transformation properties (gauge covariance instead of gauge invariance). It is well known that the conservation of the energy-momentum tensor can be achieved by a redefinition involving another star-product. Furthermore, for a pure gauge theory it is always possible to define a gauge invariant energy-momentum tensor by means of a Wilson line. We show that the last two procedures are incompatible with each other if couplings of gauge fields to matter fields (scalars or fermions) are considered: The gauge invariant tensor (constructed via Wilson line) does not allow for a redefinition assuring its conservation, and vice versa the introduction of another star-product does not allow for gauge invariance by means of a Wilson line. (orig.)
Gauge theories, duality relations and the tensor hierarchy
Bergshoeff, Eric A.; Hohm, Olaf; Hartong, Jelle; Huebscher, Mechthild; OrtIn, Tomas
2009-01-01
We compute the complete 3- and 4-dimensional tensor hierarchies, i.e. sets of p-form fields, with 1 ≤ p ≤ D, which realize an off-shell algebra of bosonic gauge transformations. We show how these tensor hierarchies can be put on-shell by introducing a set of duality relations, thereby introducing additional scalars and a metric tensor. These so-called duality hierarchies encode the equations of motion of the bosonic part of the most general gauged supergravity theories in those dimensions, including the (projected) scalar equations of motion. We construct gauge-invariant actions that include all the fields in the tensor hierarchies. We elucidate the relation between the gauge transformations of the p-form fields in the action and those of the same fields in the tensor hierarchy.
Coordinate independent expression for transverse trace-free tensors
Conboye, Rory
2016-01-01
The transverse and trace-free (TT) part of the extrinsic curvature represents half of the dynamical degrees of freedom of the gravitational field in the 3 + 1 formalism. As such, it is part of the freely specifiable initial data for numerical relativity. Though TT tensors in three-space possess only two component degrees of freedom, they cannot ordinarily be given solely by two scalar potentials. Such expressions have been derived, however, in coordinate form, for all TT tensors in flat space which are also translationally or axially symmetric (Conboye and Murchadha 2014 Class. Quantum Grav. 31 085019). Since TT tensors are conformally covariant, these also give TT tensors in conformally flat space. In this article, the work above has been extended by giving a coordinate-independent expression for these TT tensors. The translational and axial symmetry conditions have also been generalized to invariance along any hypersurface orthogonal Killing vector. (paper)
The Scalar-Tensor Theory of Gravitation
Ibanez, J
2003-01-01
Since the scalar-tensor theory of gravitation was proposed almost 50 years ago, it has recently become a robust alternative theory to Einstein's general relativity due to the fact that it appears to represent the lower level of a more fundamental theory and can serve both as a phenomenological theory to explain the recently observed acceleration of the universe, and to solve the cosmological constant problem. To my knowledge The Scalar-Tensor Theory of Gravitation by Y Fujii and K Maeda is the first book to develop a modern view on this topic and is one of the latest titles in the well-presented Cambridge Monographs on Mathematical Physics series. This book is an excellent readable introduction and up-to-date review of the subject. The discussion is well organized; after a comprehensible introduction to the Brans-Dicke theory and the important role played by conformal transformations, the authors review cosmologies with the cosmological constant and how the scalar-tensor theory can serve to explain the accelerating universe, including discussions on dark energy, quintessence and braneworld cosmologies. The book ends with a chapter devoted to quantum effects. To make easy the lectures of the book, each chapter starts with a summary of the subject to be dealt with. As the book proceeds, important issues like conformal frames and the weak equivalence principle are fully discussed. As the authors warn in the preface, the book is not encyclopedic (from my point of view the list of references is fairly short, for example, but this is a minor drawback) and the choice of included topics corresponds to the authors' interests. Nevertheless, the book seems to cover a broad range of the most essential aspects of the subject. Long and 'boring' mathematical derivations are left to appendices so as not to interrupt the flow of the reasoning, allowing the reader to focus on the physical aspects of each subject. These appendices are a valuable help in entering into the mathematical
The effects of noise over the complete space of diffusion tensor shape.
Gahm, Jin Kyu; Kindlmann, Gordon; Ennis, Daniel B
2014-01-01
Diffusion tensor magnetic resonance imaging (DT-MRI) is a technique used to quantify the microstructural organization of biological tissues. Multiple images are necessary to reconstruct the tensor data and each acquisition is subject to complex thermal noise. As such, measures of tensor invariants, which characterize components of tensor shape, derived from the tensor data will be biased from their true values. Previous work has examined this bias, but over a narrow range of tensor shape. Herein, we define the mathematics for constructing a tensor from tensor invariants, which permits an intuitive and principled means for building tensors with a complete range of tensor shape and salient microstructural properties. Thereafter, we use this development to evaluate by simulation the effects of noise on characterizing tensor shape over the complete space of tensor shape for three encoding schemes with different SNR and gradient directions. We also define a new framework for determining the distribution of the true values of tensor invariants given their measures, which provides guidance about the confidence the observer should have in the measures. Finally, we present the statistics of tensor invariant estimates over the complete space of tensor shape to demonstrate how the noise sensitivity of tensor invariants varies across the space of tensor shape as well as how the imaging protocol impacts measures of tensor invariants. Copyright © 2013 Elsevier B.V. All rights reserved.
Endoscopic Anatomy of the Tensor Fold and Anterior Attic.
Li, Bin; Doan, Phi; Gruhl, Robert R; Rubini, Alessia; Marchioni, Daniele; Fina, Manuela
2018-02-01
Objectives The objectives of the study were to (1) study the anatomical variations of the tensor fold and its anatomic relation with transverse crest, supratubal recess, and anterior epitympanic space and (2) explore the most appropriate endoscopic surgical approach to each type of the tensor fold variants. Study Design Cadaver dissection study. Setting Temporal bone dissection laboratory. Subjects and Methods Twenty-eight human temporal bones (26 preserved and 2 fresh) were dissected through an endoscopic transcanal approach between September 2016 and June 2017. The anatomical variations of the tensor fold, transverse crest, supratubal recess, and anterior epitympanic space were studied before and after removing ossicles. Results Three different tensor fold orientations were observed: vertical (type A, 11/28, 39.3%) with attachment to the transverse crest, oblique (type B, 13/28, 46.4%) with attachment to the anterior tegmen tympani, and horizontal (type C, 4/28, 14.3%) with attachment to the tensor tympani canal. The tensor fold was a complete membrane in 20 of 28 (71.4%) specimens, preventing direct ventilation between the supratubal recess and anterior epitympanic space. We identified 3 surgical endoscopic approaches, which allowed visualization of the tensor fold without removing the ossicles. Conclusions The orientation of the tensor fold is the determining structure that dictates the conformation and limits of the epitympanic space. We propose a classification of the tensor fold based on 3 anatomical variants. We also describe 3 different minimally invasive endoscopic approaches to identify the orientation of the tensor fold while maintaining ossicular chain continuity.
Tensor products of quantized tilting modules
Andersen, H.H.
1992-01-01
Let U k denote the quantized enveloping algebra corresponding to a finite dimensional simple complex Lie algebra L. Assume that the quantum parameter is a root of unity in k of order at least the Coxeter number for pound. Also assume that this order is odd and not divisible by 3 if type G 2 occurs. We demonstrate how one can define a reduced tensor product on the family F consisting of those finite dimensional simple U k -modules which are deformations of simple L-modules and which have non-zero quantum dimension. This together with the work of Reshetikhin-Turaev and Turaev-Wenzl prove that (U k , F) is a modular Hopf algebra and hence produces invariants of 3-manifolds. Also by recent work of Duurhus, Jakobsen and Nest it leads to a general topological quantum field theory. The method of proof explores quantized analogues of tilting modules for algebraic groups. (orig.)
Image denoising using non linear diffusion tensors
Benzarti, F.; Amiri, H.
2011-01-01
Image denoising is an important pre-processing step for many image analysis and computer vision system. It refers to the task of recovering a good estimate of the true image from a degraded observation without altering and changing useful structure in the image such as discontinuities and edges. In this paper, we propose a new approach for image denoising based on the combination of two non linear diffusion tensors. One allows diffusion along the orientation of greatest coherences, while the other allows diffusion along orthogonal directions. The idea is to track perfectly the local geometry of the degraded image and applying anisotropic diffusion mainly along the preferred structure direction. To illustrate the effective performance of our model, we present some experimental results on a test and real photographic color images.
Diffusion tensor in electron swarm transport
Makabe, T.; Mori, T.
1983-01-01
Expression for the diffusion tensor of the electron (or light ion) swarm is presented from the higher-order expansion of the velocity distribution in the Boltzmann equation in hydrodynamic stage. Derived diffusion coefficients for the transverse and longitudinal directions include the additional terms representative of the curvature effect under the action of an electric field with the usual-two-term expressions. Numerical analysis is given for the electron swarm in model gases having the momentum transfer cross section Qsub(m)(epsilon)=Q 0 epsilon sup(beta) (β=0, 1/2, 1) using the present theory. As the result, appreciable degree of discrepancy appears between the transverse diffusion coefficient defined here and the conventional expression with increasing of β in Qsub(m). (Author)
Poisson-Jacobi reduction of homogeneous tensors
Grabowski, J; Iglesias, D; Marrero, J C; Padron, E; Urbanski, P
2004-01-01
The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M, homogeneous with respect to a vector field Δ on M, and first-order polydifferential operators on a closed submanifold N of codimension 1 such that Δ is transversal to N. This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on M to the Schouten-Jacobi bracket of first-order polydifferential operators on N and generalizes the Poissonization of Jacobi manifolds. Actually, it can be viewed as a super-Poissonization. This procedure of passing from a homogeneous multivector field to a first-order polydifferential operator can also be understood as a sort of reduction; in the standard case-a half of a Poisson reduction. A dual version of the above correspondence yields in particular the correspondence between Δ-homogeneous symplectic structures on M and contact structures on N
Lectures on tensor categories and modular functors
Bakalov, Bojko
2000-01-01
This book gives an exposition of the relations among the following three topics: monoidal tensor categories (such as a category of representations of a quantum group), 3-dimensional topological quantum field theory, and 2-dimensional modular functors (which naturally arise in 2-dimensional conformal field theory). The following examples are discussed in detail: the category of representations of a quantum group at a root of unity and the Wess-Zumino-Witten modular functor. The idea that these topics are related first appeared in the physics literature in the study of quantum field theory. Pioneering works of Witten and Moore-Seiberg triggered an avalanche of papers, both physical and mathematical, exploring various aspects of these relations. Upon preparing to lecture on the topic at MIT, however, the authors discovered that the existing literature was difficult and that there were gaps to fill. The text is wholly expository and finely succinct. It gathers results, fills existing gaps, and simplifies some pro...
Tensor ghosts in the inflationary cosmology
Clunan, Tim; Sasaki, Misao
2010-01-01
Theories with curvature-squared terms in the action are known to contain ghost modes in general. However, if we regard curvature-squared terms as quantum corrections to the original theory, the emergence of ghosts may be simply due to the perturbative truncation of a full non-perturbative theory. If this is the case, there should be a way to live with ghosts. In this paper, we take the Euclidean path integral approach, in which ghost degrees of freedom can be, and are integrated out in the Euclideanized spacetime. We apply this procedure to Einstein gravity with a Weyl curvature-squared correction in the inflationary background. We find that the amplitude of tensor perturbations is modified by a term of O(α 2 H 2 ) where α 2 is a coupling constant in front of the Weyl-squared term and H is the Hubble parameter during inflation.
Holographic spin networks from tensor network states
Singh, Sukhwinder; McMahon, Nathan A.; Brennen, Gavin K.
2018-01-01
In the holographic correspondence of quantum gravity, a global on-site symmetry at the boundary generally translates to a local gauge symmetry in the bulk. We describe one way how the global boundary on-site symmetries can be gauged within the formalism of the multiscale renormalization ansatz (MERA), in light of the ongoing discussion between tensor networks and holography. We describe how to "lift" the MERA representation of the ground state of a generic one dimensional (1D) local Hamiltonian, which has a global on-site symmetry, to a dual quantum state of a 2D "bulk" lattice on which the symmetry appears gauged. The 2D bulk state decomposes in terms of spin network states, which label a basis in the gauge-invariant sector of the bulk lattice. This decomposition is instrumental to obtain expectation values of gauge-invariant observables in the bulk, and also reveals that the bulk state is generally entangled between the gauge and the remaining ("gravitational") bulk degrees of freedom that are not fixed by the symmetry. We present numerical results for ground states of several 1D critical spin chains to illustrate that the bulk entanglement potentially depends on the central charge of the underlying conformal field theory. We also discuss the possibility of emergent topological order in the bulk using a simple example, and also of emergent symmetries in the nongauge (gravitational) sector in the bulk. More broadly, our holographic model translates the MERA, a tensor network state, to a superposition of spin network states, as they appear in lattice gauge theories in one higher dimension.
Henderson, Tona
1995-01-01
Provides electronic mail addresses for resources and discussion groups related to the following academic majors: art, biology, business, chemistry, computer science, economics, health sciences, history, literature, math, music, philosophy, political science, psychology, sociology, and theater. (AEF)
Minnesota Department of Natural Resources — This data set contains roadway centerlines for major roads (interstates and trunk highways) found on the USGS 1:24,000 mapping series. These roadways are current...
Tensor-Based Dictionary Learning for Spectral CT Reconstruction.
Zhang, Yanbo; Mou, Xuanqin; Wang, Ge; Yu, Hengyong
2017-01-01
Spectral computed tomography (CT) produces an energy-discriminative attenuation map of an object, extending a conventional image volume with a spectral dimension. In spectral CT, an image can be sparsely represented in each of multiple energy channels, and are highly correlated among energy channels. According to this characteristics, we propose a tensor-based dictionary learning method for spectral CT reconstruction. In our method, tensor patches are extracted from an image tensor, which is reconstructed using the filtered backprojection (FBP), to form a training dataset. With the Candecomp/Parafac decomposition, a tensor-based dictionary is trained, in which each atom is a rank-one tensor. Then, the trained dictionary is used to sparsely represent image tensor patches during an iterative reconstruction process, and the alternating minimization scheme is adapted for optimization. The effectiveness of our proposed method is validated with both numerically simulated and real preclinical mouse datasets. The results demonstrate that the proposed tensor-based method generally produces superior image quality, and leads to more accurate material decomposition than the currently popular popular methods.
Tensor Rank Preserving Discriminant Analysis for Facial Recognition.
Tao, Dapeng; Guo, Yanan; Li, Yaotang; Gao, Xinbo
2017-10-12
Facial recognition, one of the basic topics in computer vision and pattern recognition, has received substantial attention in recent years. However, for those traditional facial recognition algorithms, the facial images are reshaped to a long vector, thereby losing part of the original spatial constraints of each pixel. In this paper, a new tensor-based feature extraction algorithm termed tensor rank preserving discriminant analysis (TRPDA) for facial image recognition is proposed; the proposed method involves two stages: in the first stage, the low-dimensional tensor subspace of the original input tensor samples was obtained; in the second stage, discriminative locality alignment was utilized to obtain the ultimate vector feature representation for subsequent facial recognition. On the one hand, the proposed TRPDA algorithm fully utilizes the natural structure of the input samples, and it applies an optimization criterion that can directly handle the tensor spectral analysis problem, thereby decreasing the computation cost compared those traditional tensor-based feature selection algorithms. On the other hand, the proposed TRPDA algorithm extracts feature by finding a tensor subspace that preserves most of the rank order information of the intra-class input samples. Experiments on the three facial databases are performed here to determine the effectiveness of the proposed TRPDA algorithm.
Tensor-based Dictionary Learning for Spectral CT Reconstruction
Zhang, Yanbo; Wang, Ge
2016-01-01
Spectral computed tomography (CT) produces an energy-discriminative attenuation map of an object, extending a conventional image volume with a spectral dimension. In spectral CT, an image can be sparsely represented in each of multiple energy channels, and are highly correlated among energy channels. According to this characteristics, we propose a tensor-based dictionary learning method for spectral CT reconstruction. In our method, tensor patches are extracted from an image tensor, which is reconstructed using the filtered backprojection (FBP), to form a training dataset. With the Candecomp/Parafac decomposition, a tensor-based dictionary is trained, in which each atom is a rank-one tensor. Then, the trained dictionary is used to sparsely represent image tensor patches during an iterative reconstruction process, and the alternating minimization scheme is adapted for optimization. The effectiveness of our proposed method is validated with both numerically simulated and real preclinical mouse datasets. The results demonstrate that the proposed tensor-based method generally produces superior image quality, and leads to more accurate material decomposition than the currently popular popular methods. PMID:27541628
Exact tensor network ansatz for strongly interacting systems
Zaletel, Michael P.
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian. However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems. For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development. Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids. This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.
Reduction schemes for one-loop tensor integrals
Denner, A.; Dittmaier, S.
2006-01-01
We present new methods for the evaluation of one-loop tensor integrals which have been used in the calculation of the complete electroweak one-loop corrections to e + e - ->4 fermions. The described methods for 3-point and 4-point integrals are, in particular, applicable in the case where the conventional Passarino-Veltman reduction breaks down owing to the appearance of Gram determinants in the denominator. One method consists of different variants for expanding tensor coefficients about limits of vanishing Gram determinants or other kinematical determinants, thereby reducing all tensor coefficients to the usual scalar integrals. In a second method a specific tensor coefficient with a logarithmic integrand is evaluated numerically, and the remaining coefficients as well as the standard scalar integral are algebraically derived from this coefficient. For 5-point tensor integrals, we give explicit formulas that reduce the corresponding tensor coefficients to coefficients of 4-point integrals with tensor rank reduced by one. Similar formulas are provided for 6-point functions, and the generalization to functions with more internal propagators is straightforward. All the presented methods are also applicable if infrared (soft or collinear) divergences are treated in dimensional regularization or if mass parameters (for unstable particles) become complex
Algebraic Rainich conditions for the fourth rank tensor V
So, Lau Loi
2011-01-01
Algebraic conditions on the Ricci tensor in the Rainich-Misner-Wheeler unified field theory are known as the Rainich conditions. Penrose and more recently Bergqvist and Lankinen made an analogy from the Ricci tensor to the Bel-Robinson tensor B αβμν , a certain fourth rank tensor quadratic in the Weyl curvature, which also satisfies algebraic Rainich-like conditions. However, we found that not only does the tensor B αβμν fulfill these conditions, but so also does our recently proposed tensor V αβμν , which has many of the desirable properties of B αβμν . For the quasilocal small sphere limit restriction, we found that there are only two fourth rank tensors, B αβμν and V αβμν , which form a basis for good energy expressions. Both of them have the completely trace free and causal properties, these two form necessary and sufficient conditions. Surprisingly either completely traceless or causal is enough to fulfill the algebraic Rainich conditions.
The tensor bi-spectrum in a matter bounce
Chowdhury, Debika; Sreenath, V.; Sriramkumar, L., E-mail: debika@physics.iitm.ac.in, E-mail: sreenath@lsu.edu, E-mail: sriram@physics.iitm.ac.in [Department of Physics, Indian Institute of Technology Madras, Chennai 600036 (India)
2015-11-01
Matter bounces are bouncing scenarios wherein the universe contracts as in a matter dominated phase at early times. Such scenarios are known to lead to a scale invariant spectrum of tensor perturbations, just as de Sitter inflation does. In this work, we examine if the tensor bi-spectrum can discriminate between the inflationary and the bouncing scenarios. Using the Maldacena formalism, we analytically evaluate the tensor bi-spectrum in a matter bounce for an arbitrary triangular configuration of the wavevectors. We show that, over scales of cosmological interest, the non-Gaussianity parameter h{sub NL} that characterizes the amplitude of the tensor bi-spectrum is quite small when compared to the corresponding values in de Sitter inflation. During inflation, the amplitude of the tensor perturbations freeze on super-Hubble scales, a behavior that results in the so-called consistency condition relating the tensor bi-spectrum and the power spectrum in the squeezed limit. In contrast, in the bouncing scenarios, the amplitude of the tensor perturbations grow strongly as one approaches the bounce, which suggests that the consistency condition will not be valid in such situations. We explicitly show that the consistency relation is indeed violated in the matter bounce. We discuss the implications of the results.
Teh, Irvin; McClymont, Darryl; Zdora, Marie-Christine; Whittington, Hannah J; Davidoiu, Valentina; Lee, Jack; Lygate, Craig A; Rau, Christoph; Zanette, Irene; Schneider, Jürgen E
2017-03-10
Diffusion tensor imaging (DTI) is widely used to assess tissue microstructure non-invasively. Cardiac DTI enables inference of cell and sheetlet orientations, which are altered under pathological conditions. However, DTI is affected by many factors, therefore robust validation is critical. Existing histological validation is intrinsically flawed, since it requires further tissue processing leading to sample distortion, is routinely limited in field-of-view and requires reconstruction of three-dimensional volumes from two-dimensional images. In contrast, synchrotron radiation imaging (SRI) data enables imaging of the heart in 3D without further preparation following DTI. The objective of the study was to validate DTI measurements based on structure tensor analysis of SRI data. One isolated, fixed rat heart was imaged ex vivo with DTI and X-ray phase contrast SRI, and reconstructed at 100 μm and 3.6 μm isotropic resolution respectively. Structure tensors were determined from the SRI data and registered to the DTI data. Excellent agreement in helix angles (HA) and transverse angles (TA) was observed between the DTI and structure tensor synchrotron radiation imaging (STSRI) data, where HA DTI-STSRI = -1.4° ± 23.2° and TA DTI-STSRI = -1.4° ± 35.0° (mean ± 1.96 standard deviation across all voxels in the left ventricle). STSRI confirmed that the primary eigenvector of the diffusion tensor corresponds with the cardiomyocyte long-axis across the whole myocardium. We have used STSRI as a novel and high-resolution gold standard for the validation of DTI, allowing like-with-like comparison of three-dimensional tissue structures in the same intact heart free of distortion. This represents a critical step forward in independently verifying the structural basis and informing the interpretation of cardiac DTI data, thereby supporting the further development and adoption of DTI in structure-based electro-mechanical modelling and routine clinical
Tensors and Manifolds With Applications to Physics (2nd edn)
Dray, T
2005-01-01
On the one hand, this is an excellent introduction for mathematicians to the differential geometry underlying general relativity. On the other hand, this is definitely a book for mathematicians. The book's greatest strength is its clear, precise presentation of the basic ideas in differential geometry, combined with equally clear and precise applications to theoretical physics, notably general relativity. But the book's precision is also its greatest weakness; this is not an easy book to read for non-mathematicians, who may not appreciate the notational complexity, some of which is nonstandard. The present edition is very similar to the original, published in 1992. In addition to minor revisions and clarifications of the material, there is now a brief introduction to fibre bundles, and a (very) brief discussion of the gauge theory description of fundamental particles. The index to the symbols used is also a more complete than in the past, but without the descriptive material present in the previous edition. The bulk of the book consists of a careful introduction to tensors and their properties. Tensors are introduced first as linear maps on vector spaces, and only later generalized to tensor fields on manifolds. The differentiation and integration of differential forms is discussed in detail, including Stokes' theorem, Lie differentiation and Hodge duality, and connections, curvature and torsion. To this point, Wasserman's text can be viewed as an expanded version of Bishop and Goldberg's classic text, one major difference being Wasserman's inclusion of the pseudo-Riemannian case from the beginning (in particular, when discussing Hodge duality). Whether one prefers Wasserman's approach to Bishop and Goldberg's is largely a matter of taste: Wasserman's treatment is both more complete and more precise, making it easier to check calculations in detail, but occasionally more difficult to remember what one is calculating. An instructor using this text would be well
Tensors and Manifolds With Applications to Physics (2nd edn)
Dray, T [Oregon State University (United States)
2005-10-21
On the one hand, this is an excellent introduction for mathematicians to the differential geometry underlying general relativity. On the other hand, this is definitely a book for mathematicians. The book's greatest strength is its clear, precise presentation of the basic ideas in differential geometry, combined with equally clear and precise applications to theoretical physics, notably general relativity. But the book's precision is also its greatest weakness; this is not an easy book to read for non-mathematicians, who may not appreciate the notational complexity, some of which is nonstandard. The present edition is very similar to the original, published in 1992. In addition to minor revisions and clarifications of the material, there is now a brief introduction to fibre bundles, and a (very) brief discussion of the gauge theory description of fundamental particles. The index to the symbols used is also a more complete than in the past, but without the descriptive material present in the previous edition. The bulk of the book consists of a careful introduction to tensors and their properties. Tensors are introduced first as linear maps on vector spaces, and only later generalized to tensor fields on manifolds. The differentiation and integration of differential forms is discussed in detail, including Stokes' theorem, Lie differentiation and Hodge duality, and connections, curvature and torsion. To this point, Wasserman's text can be viewed as an expanded version of Bishop and Goldberg's classic text, one major difference being Wasserman's inclusion of the pseudo-Riemannian case from the beginning (in particular, when discussing Hodge duality). Whether one prefers Wasserman's approach to Bishop and Goldberg's is largely a matter of taste: Wasserman's treatment is both more complete and more precise, making it easier to check calculations in detail, but occasionally more difficult to remember what one is calculating. An
Performance of unscented Kalman filter tractography in edema: Analysis of the two-tensor model.
Liao, Ruizhi; Ning, Lipeng; Chen, Zhenrui; Rigolo, Laura; Gong, Shun; Pasternak, Ofer; Golby, Alexandra J; Rathi, Yogesh; O'Donnell, Lauren J
2017-01-01
Diffusion MRI tractography is increasingly used in pre-operative neurosurgical planning to visualize critical fiber tracts. However, a major challenge for conventional tractography, especially in patients with brain tumors, is tracing fiber tracts that are affected by vasogenic edema, which increases water content in the tissue and lowers diffusion anisotropy. One strategy for improving fiber tracking is to use a tractography method that is more sensitive than the traditional single-tensor streamline tractography. We performed experiments to assess the performance of two-tensor unscented Kalman filter (UKF) tractography in edema. UKF tractography fits a diffusion model to the data during fiber tracking, taking advantage of prior information from the previous step along the fiber. We studied UKF performance in a synthetic diffusion MRI digital phantom with simulated edema and in retrospective data from two neurosurgical patients with edema affecting the arcuate fasciculus and corticospinal tracts. We compared the performance of several tractography methods including traditional streamline, UKF single-tensor, and UKF two-tensor. To provide practical guidance on how the UKF method could be employed, we evaluated the impact of using various seed regions both inside and outside the edematous regions, as well as the impact of parameter settings on the tractography sensitivity. We quantified the sensitivity of different methods by measuring the percentage of the patient-specific fMRI activation that was reached by the tractography. We expected that diffusion anisotropy threshold parameters, as well as the inclusion of a free water model, would significantly influence the reconstruction of edematous WM fiber tracts, because edema increases water content in the tissue and lowers anisotropy. Contrary to our initial expectations, varying the fractional anisotropy threshold and including a free water model did not affect the UKF two-tensor tractography output appreciably in
Tensor force and debye screening in quarkonium-type mesons
Kovacs, L.B.; Kovacs, T.G.; Lovas, I.
1990-01-01
We use a non-relativistic quantum-mechanical model to investigate the effect of a screening plasma on two quarkonium-type mesons: the charmonium and bottonium. The stability of these mesons in the plasma is estimated in two cases: including the tensor and spin-orbit term in the potential and without these terms. It turns out that while the bottonium is somewhat stabilized by the tensor force, the charmonium becomes less stabil due to this modification of the potential. Thus the charmonium seems to be a more sensitive probe of the quark-gluon plasma formation than it was thought to be without including the tensor force. (Authors)
Tensor polarized deuteron targets for intermediate energy physics experiments
Meyer, W.; Schilling, E.
1985-03-01
At intermediate energies measurements from a tensor polarized deuteron target are being prepared for the following reactions: the photodisintegration of the deuteron, the elastic pion-deuteron scattering and the elastic electron-deuteron scattering. The experimental situation of the polarization experiments for these reactions is briefly discussed in section 2. In section 3 the definitions of the deuteron polarization and the possibilities to determine the vector and tensor polarization are given. Present tensor polarization values and further improvements in this field are reported in section 4. (orig.)
Extracting the diffusion tensor from molecular dynamics simulation with Milestoning
Mugnai, Mauro L.; Elber, Ron
2015-01-01
We propose an algorithm to extract the diffusion tensor from Molecular Dynamics simulations with Milestoning. A Kramers-Moyal expansion of a discrete master equation, which is the Markovian limit of the Milestoning theory, determines the diffusion tensor. To test the algorithm, we analyze overdamped Langevin trajectories and recover a multidimensional Fokker-Planck equation. The recovery process determines the flux through a mesh and estimates local kinetic parameters. Rate coefficients are converted to the derivatives of the potential of mean force and to coordinate dependent diffusion tensor. We illustrate the computation on simple models and on an atomically detailed system—the diffusion along the backbone torsions of a solvated alanine dipeptide
Some spacetimes with higher rank Killing-Staeckel tensors
Gibbons, G.W.; Houri, T.; Kubiznak, D.; Warnick, C.M.
2011-01-01
By applying the lightlike Eisenhart lift to several known examples of low-dimensional integrable systems admitting integrals of motion of higher-order in momenta, we obtain four- and higher-dimensional Lorentzian spacetimes with irreducible higher-rank Killing tensors. Such metrics, we believe, are first examples of spacetimes admitting higher-rank Killing tensors. Included in our examples is a four-dimensional supersymmetric pp-wave spacetime, whose geodesic flow is superintegrable. The Killing tensors satisfy a non-trivial Poisson-Schouten-Nijenhuis algebra. We discuss the extension to the quantum regime.
Tensor valuations and their applications in stochastic geometry and imaging
Kiderlen, Markus
2017-01-01
The purpose of this volume is to give an up-to-date introduction to tensor valuations and their applications. Starting with classical results concerning scalar-valued valuations on the families of convex bodies and convex polytopes, it proceeds to the modern theory of tensor valuations. Product and Fourier-type transforms are introduced and various integral formulae are derived. New and well-known results are presented, together with generalizations in several directions, including extensions to the non-Euclidean setting and to non-convex sets. A variety of applications of tensor valuations to models in stochastic geometry, to local stereology and to imaging are also discussed.
A Nonlinear GMRES Optimization Algorithm for Canonical Tensor Decomposition
De Sterck, Hans
2011-01-01
A new algorithm is presented for computing a canonical rank-R tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank-R tensor consists of the sum of R rank-one components. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a stand-alone one-step process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear g...
One-loop tensor Feynman integral reduction with signed minors
Fleischer, Jochem; Riemann, Tord; Yundin, Valery
2012-01-01
of the formalism is the immediate evaluation of complete contractions of the tensor integrals with external momenta. This leads to the problem of evaluating sums over products of signed minors with scalar products of chords. Chords are differences of external momenta. These sums may be evaluated analytically......We present an algebraic approach to one-loop tensor integral reduction. The integrals are presented in terms of scalar one- to four-point functions. The reduction is worked out explicitly until five-point functions of rank five. The numerical C++ package PJFry evaluates tensor coefficients in terms...
Scalar-Tensor Black Holes Embedded in an Expanding Universe
Tretyakova, Daria; Latosh, Boris
2018-02-01
In this review we focus our attention on scalar-tensor gravity models and their empirical verification in terms of black hole and wormhole physics. We focus on a black hole, embedded in an expanding universe, describing both cosmological and astrophysical scales. We show that in scalar-tensor gravity it is quite common that the local geometry is isolated from the cosmological expansion, so that it does not backreact on the black hole metric. We try to extract common features of scalar-tensor black holes in an expanding universe and point out the gaps that must be filled.
Scalar-Tensor Black Holes Embedded in an Expanding Universe
Daria Tretyakova
2018-02-01
Full Text Available In this review, we focus our attention on scalar-tensor gravity models and their empirical verification in terms of black hole and wormhole physics. We focus on black holes, embedded in an expanding universe, describing both cosmological and astrophysical scales. We show that in scalar-tensor gravity it is quite common that the local geometry is isolated from the cosmological expansion, so that it does not backreact on the black hole metric. We try to extract common features of scalar-tensor black holes in an expanding universe and point out the issues that are not fully investigated.
Two-perfect fluid interpretation of an energy tensor
Ferrando, J.J.; Morales, J.A.; Portilla, M.
1990-01-01
There are many topics in General Relativity where matter is represented by a mixture of two fluids. In fact, some astrophysical and cosmological situations need to be described by an energy tensor made up of the sum of two or more perfect fluids rather than that with only one. The paper contains the necessary and sufficient conditions for a given energy tensor to be interpreted as a sum of two perfect fluids. Given a tensor of this class, the decomposition in two perfect fluids (which is determined up to a couple of real functions) is obtained
Kamali, Arash; Zhang, Caroline C; Riascos, Roy F; Tandon, Nitin; Bonafante-Mejia, Eliana E; Patel, Rajan; Lincoln, John A; Rabiei, Pejman; Ocasio, Laura; Younes, Kyan; Hasan, Khader M
2018-03-27
The mammillary bodies as part of the hypothalamic nuclei are in the central limbic circuitry of the human brain. The mammillary bodies are shown to be directly or indirectly connected to the amygdala, hippocampus, and thalami as the major gray matter structures of the human limbic system. Although it is not primarily considered as part of the human limbic system, the thalamus is shown to be involved in many limbic functions of the human brain. The major direct connection of the thalami with the hypothalamic nuclei is known to be through the mammillothalamic tract. Given the crucial role of the mammillothalamic tracts in memory functions, diffusion tensor imaging may be helpful in better visualizing the surgical anatomy of this pathway noninvasively. This study aimed to investigate the utility of high spatial resolution diffusion tensor tractography for mapping the trajectory of the mammillothalamic tract in the human brain. Fifteen healthy adults were studied after obtaining written informed consent. We used high spatial resolution diffusion tensor imaging data at 3.0 T. We delineated, for the first time, the detailed trajectory of the mammillothalamic tract of the human brain using deterministic diffusion tensor tractography.
Total magnitude of diffusion tensor imaging as an effective tool for the differentiation of glioma
Smitha, Karavallil A., E-mail: mithamahesh@gmail.com [Department of Imaging Sciences and Interventional Radiology, Sree Chitra Tirunal Institute for Medical Sciences and Technology, Thiruvananthapuram (India); Gupta, Arun kumar, E-mail: gupta209@gmail.com [Department of Neuroimaging and Interventional Radiology, National Institute of Mental Health and Neuro Sciences, Bangalore (India); Jayasree, Ramapurath S., E-mail: jayashreemenon@gmail.com [Biophotonics and Imaging Laboratory, Biomedical Technology Wing, Sree Chitra Tirunal Institute for Medical Sciences and Technology, Thiruvananthapuram (India)
2013-05-15
Objectives: The study aims to evaluate the difference in diffusion properties between high grade glioma and low grade glioma by measuring the total magnitude of diffusion tensor (L), and its isotropic (p) and anisotropic (q) components. Methods: The diffusion tensor parameters p, q, L and FA from the tumor area, adjacent area to the tumor and corresponding contra lateral normal area of 30 high grade glioma and 49 low grade glioma were calculated. Chi square analysis was done to find the changes in age and sex. One Way ANOVA was performed to compare the mean and ROC curve analysis to confirm the discriminative sensitivity. Results: Major variation in the mean values of p, L and FA was observed in different brain areas considered. Variation in the p and L values between low grade and high grade glioma were statistically significant (p < 0.001) and their ROC curve analysis yielded 93.9% and 91.8% sensitivity and 53.3% specificity respectively. Conclusion: Measurement of the isotropic component p and the total value of diffusion tensor L can be effectively correlated with different grades of glioma and can be used to study the diffusion properties of tumor affected brain.
On the skew-symmetric character of the couple-stress tensor
Hadjesfandiari, Ali R.
2013-01-01
In this paper, the skew-symmetric character of the couple-stress tensor is established as the result of arguments from tensor analysis. Consequently, the couple-stress pseudo-tensor has a true vectorial character. The fundamental step in this development is that the isotropic couple-stress tensor cannot exist.
Smirnov, Yu.F.; Tolstoi, V.N.; Kharitonov, Yu.I.
1993-01-01
The tree technique for the quantum algebra su q (2) developed in an earlier study is used to construct the q analog of the algebra of irreducible tensor operators. The adjoint action of the algebra su q (2) on irreducible tensor operators is discussed, and the adjoint R matrix is introduced. A set of expressions is obtained for the matrix elements of various irreducible tensor operators and combinations of them. As an application, the recursion relations for the Clebsch-Gordan and Racah coefficients of the algebra su q (2) are derived. 16 refs
Tensor products of process matrices with indefinite causal structure
Jia, Ding; Sakharwade, Nitica
2018-03-01
Theories with indefinite causal structure have been studied from both the fundamental perspective of quantum gravity and the practical perspective of information processing. In this paper we point out a restriction in forming tensor products of objects with indefinite causal structure in certain models: there exist both classical and quantum objects the tensor products of which violate the normalization condition of probabilities, if all local operations are allowed. We obtain a necessary and sufficient condition for when such unrestricted tensor products of multipartite objects are (in)valid. This poses a challenge to extending communication theory to indefinite causal structures, as the tensor product is the fundamental ingredient in the asymptotic setting of communication theory. We discuss a few options to evade this issue. In particular, we show that the sequential asymptotic setting does not suffer the violation of normalization.
Distance Adaptive Tensor Discriminative Geometry Preserving Projection for Face Recognition
Ziqiang Wang
2012-09-01
Full Text Available There is a growing interest in dimensionality reduction techniques for face recognition, however, the traditional dimensionality reduction algorithms often transform the input face image data into vectors before embedding. Such vectorization often ignores the underlying data structure and leads to higher computational complexity. To effectively cope with these problems, a novel dimensionality reduction algorithm termed distance adaptive tensor discriminative geometry preserving projection (DATDGPP is proposed in this paper. The key idea of DATDGPP is as follows: first, the face image data are directly encoded in high-order tensor structure so that the relationships among the face image data can be preserved; second, the data-adaptive tensor distance is adopted to model the correlation among different coordinates of tensor data; third, the transformation matrix which can preserve discrimination and local geometry information is obtained by an iteration algorithm. Experimental results on three face databases show that the proposed algorithm outperforms other representative dimensionality reduction algorithms.
Vectors, tensors and the basic equations of fluid mechanics
Aris, Rutherford
1962-01-01
Introductory text, geared toward advanced undergraduate and graduate students, applies mathematics of Cartesian and general tensors to physical field theories and demonstrates them in terms of the theory of fluid mechanics. 1962 edition.
Ryu-Takayanagi formula for symmetric random tensor networks
Chirco, Goffredo; Oriti, Daniele; Zhang, Mingyi
2018-06-01
We consider the special case of random tensor networks (RTNs) endowed with gauge symmetry constraints on each tensor. We compute the Rényi entropy for such states and recover the Ryu-Takayanagi (RT) formula in the large-bond regime. The result provides first of all an interesting new extension of the existing derivations of the RT formula for RTNs. Moreover, this extension of the RTN formalism brings it in direct relation with (tensorial) group field theories (and spin networks), and thus provides new tools for realizing the tensor network/geometry duality in the context of background-independent quantum gravity, and for importing quantum gravity tools into tensor network research.
On the projective curvature tensor of generalized Sasakian-space ...
space-forms under some conditions regarding projective curvature tensor. All the results obtained in this paper are in the form of necessary and sufficient conditions. Keywords: Generalized Sasakian-space-forms; projectively flat; ...
An introduction to tensors and group theory for physicists
Jeevanjee, Nadir
2015-01-01
The second edition of this highly praised textbook provides an introduction to tensors, group theory, and their applications in classical and quantum physics. Both intuitive and rigorous, it aims to demystify tensors by giving the slightly more abstract but conceptually much clearer definition found in the math literature, and then connects this formulation to the component formalism of physics calculations. New pedagogical features, such as new illustrations, tables, and boxed sections, as well as additional “invitation” sections that provide accessible introductions to new material, offer increased visual engagement, clarity, and motivation for students. Part I begins with linear algebraic foundations, follows with the modern component-free definition of tensors, and concludes with applications to physics through the use of tensor products. Part II introduces group theory, including abstract groups and Lie groups and their associated Lie algebras, then intertwines this material with that of Part...
Holographic stress tensor for non-relativistic theories
Ross, Simon F.; Saremi, Omid
2009-01-01
We discuss the calculation of the field theory stress tensor from the dual geometry for two recent proposals for gravity duals of non-relativistic conformal field theories. The first of these has a Schroedinger symmetry including Galilean boosts, while the second has just an anisotropic scale invariance (the Lifshitz case). For the Lifshitz case, we construct an appropriate action principle. We propose a definition of the non-relativistic stress tensor complex for the field theory as an appropriate variation of the action in both cases. In the Schroedinger case, we show that this gives physically reasonable results for a simple black hole solution and agrees with an earlier proposal to determine the stress tensor from the familiar AdS prescription. In the Lifshitz case, we solve the linearised equations of motion for a general perturbation around the background, showing that our stress tensor is finite on-shell.
Gauge and non-gauge curvature tensor copies
Srivastava, P.P.
1982-10-01
A procedure for constructing curvature tensor copies is discussed using the anholonomic geometrical framework. The corresponding geometries are compared and the notion of gauge copy is elucidated. An explicit calculation is also made. (author)
Motion Detection in Ultrasound Image-Sequences Using Tensor Voting
Inba, Masafumi; Yanagida, Hirotaka; Tamura, Yasutaka
2008-05-01
Motion detection in ultrasound image sequences using tensor voting is described. We have been developing an ultrasound imaging system adopting a combination of coded excitation and synthetic aperture focusing techniques. In our method, frame rate of the system at distance of 150 mm reaches 5000 frame/s. Sparse array and short duration coded ultrasound signals are used for high-speed data acquisition. However, many artifacts appear in the reconstructed image sequences because of the incompleteness of the transmitted code. To reduce the artifacts, we have examined the application of tensor voting to the imaging method which adopts both coded excitation and synthetic aperture techniques. In this study, the basis of applying tensor voting and the motion detection method to ultrasound images is derived. It was confirmed that velocity detection and feature enhancement are possible using tensor voting in the time and space of simulated ultrasound three-dimensional image sequences.
Renormalization of nonabelian gauge theories with tensor matter fields
Lemes, Vitor; Renan, Ricardo; Sorella, Silvio Paolo
1996-03-01
The renormalizability of a nonabelian model describing the coupling between antisymmetric second rank tensor matter fields and Yang-Mills gauge fields is discussed within the BRS algebraic framework. (author). 12 refs
Tensor analysis and elementary differential geometry for physicists and engineers
Nguyen-Schäfer, Hung
2017-01-01
This book comprehensively presents topics, such as Dirac notation, tensor analysis, elementary differential geometry of moving surfaces, and k-differential forms. Additionally, two new chapters of Cartan differential forms and Dirac and tensor notations in quantum mechanics are added to this second edition. The reader is provided with hands-on calculations and worked-out examples at which he will learn how to handle the bra-ket notation, tensors, differential geometry, and differential forms; and to apply them to the physical and engineering world. Many methods and applications are given in CFD, continuum mechanics, electrodynamics in special relativity, cosmology in the Minkowski four-dimensional spacetime, and relativistic and non-relativistic quantum mechanics. Tensors, differential geometry, differential forms, and Dirac notation are very useful advanced mathematical tools in many fields of modern physics and computational engineering. They are involved in special and general relativity physics, quantum m...
A defect in holographic interpretations of tensor networks
Czech, Bartłomiej [Institute for Advanced Study,Princeton, NJ 08540 (United States); Nguyen, Phuc H.; Swaminathan, Sivaramakrishnan [Theory Group, Department of Physics and Texas Cosmology Center,The University of Texas at Austin,Austin, TX 78712 (United States)
2017-03-16
We initiate the study of how tensor networks reproduce properties of static holographic space-times, which are not locally pure anti-de Sitter. We consider geometries that are holographically dual to ground states of defect, interface and boundary CFTs and compare them to the structure of the requisite MERA networks predicted by the theory of minimal updates. When the CFT is deformed, certain tensors require updating. On the other hand, even identical tensors can contribute differently to estimates of entanglement entropies. We interpret these facts holographically by associating tensor updates to turning on non-normalizable modes in the bulk. In passing, we also clarify and complement existing arguments in support of the theory of minimal updates, propose a novel ansatz called rayed MERA that applies to a class of generalized interface CFTs, and analyze the kinematic spaces of the thin wall and AdS{sub 3}-Janus geometries.
Two new eigenvalue localization sets for tensors and theirs applications
Zhao Jianxing
2017-10-01
Full Text Available A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Qi (J. Symbolic Comput., 2005, 40, 1302-1324 and Li et al. (Numer. Linear Algebra Appl., 2014, 21, 39-50. As an application, a weaker checkable sufficient condition for the positive (semi-definiteness of an even-order real symmetric tensor is obtained. Meanwhile, an S-type E-eigenvalue localization set for tensors is given and proved to be tighter than that presented by Wang et al. (Discrete Cont. Dyn.-B, 2017, 22(1, 187-198. As an application, an S-type upper bound for the Z-spectral radius of weakly symmetric nonnegative tensors is obtained. Finally, numerical examples are given to verify the theoretical results.
An eigenvalue localization set for tensors and its applications
Jianxing Zhao
2017-03-01
Full Text Available Abstract A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al. (Linear Algebra Appl. 481:36-53, 2015 and Huang et al. (J. Inequal. Appl. 2016:254, 2016. As an application of this set, new bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al., the advantage of our results is that, without considering the selection of nonempty proper subsets S of N = { 1 , 2 , … , n } $N=\\{1,2,\\ldots,n\\}$ , we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of M $\\mathcal{M}$ -tensors. Finally, numerical examples are given to verify the theoretical results.
An eigenvalue localization set for tensors and its applications.
Zhao, Jianxing; Sang, Caili
2017-01-01
A new eigenvalue localization set for tensors is given and proved to be tighter than those presented by Li et al . (Linear Algebra Appl. 481:36-53, 2015) and Huang et al . (J. Inequal. Appl. 2016:254, 2016). As an application of this set, new bounds for the minimum eigenvalue of [Formula: see text]-tensors are established and proved to be sharper than some known results. Compared with the results obtained by Huang et al ., the advantage of our results is that, without considering the selection of nonempty proper subsets S of [Formula: see text], we can obtain a tighter eigenvalue localization set for tensors and sharper bounds for the minimum eigenvalue of [Formula: see text]-tensors. Finally, numerical examples are given to verify the theoretical results.
Traffic Speed Data Imputation Method Based on Tensor Completion
Bin Ran
2015-01-01
Full Text Available Traffic speed data plays a key role in Intelligent Transportation Systems (ITS; however, missing traffic data would affect the performance of ITS as well as Advanced Traveler Information Systems (ATIS. In this paper, we handle this issue by a novel tensor-based imputation approach. Specifically, tensor pattern is adopted for modeling traffic speed data and then High accurate Low Rank Tensor Completion (HaLRTC, an efficient tensor completion method, is employed to estimate the missing traffic speed data. This proposed method is able to recover missing entries from given entries, which may be noisy, considering severe fluctuation of traffic speed data compared with traffic volume. The proposed method is evaluated on Performance Measurement System (PeMS database, and the experimental results show the superiority of the proposed approach over state-of-the-art baseline approaches.
Traffic speed data imputation method based on tensor completion.
Ran, Bin; Tan, Huachun; Feng, Jianshuai; Liu, Ying; Wang, Wuhong
2015-01-01
Traffic speed data plays a key role in Intelligent Transportation Systems (ITS); however, missing traffic data would affect the performance of ITS as well as Advanced Traveler Information Systems (ATIS). In this paper, we handle this issue by a novel tensor-based imputation approach. Specifically, tensor pattern is adopted for modeling traffic speed data and then High accurate Low Rank Tensor Completion (HaLRTC), an efficient tensor completion method, is employed to estimate the missing traffic speed data. This proposed method is able to recover missing entries from given entries, which may be noisy, considering severe fluctuation of traffic speed data compared with traffic volume. The proposed method is evaluated on Performance Measurement System (PeMS) database, and the experimental results show the superiority of the proposed approach over state-of-the-art baseline approaches.
A generalization of tensor calculus and its application to physics
Ashtekar, A.
1982-01-01
Penrose's abstract index notation and axiomatic introduction of covariant derivatives in tensor calculus is generalized to fields with internal degrees of freedom. The result provides, in particular, an intrinsic formulation of gauge theories without the use of bundles. (author)
Using Matrix and Tensor Factorizations for the Single-Trial Analysis of Population Spike Trains.
Arno Onken
2016-11-01
Full Text Available Advances in neuronal recording techniques are leading to ever larger numbers of simultaneously monitored neurons. This poses the important analytical challenge of how to capture compactly all sensory information that neural population codes carry in their spatial dimension (differences in stimulus tuning across neurons at different locations, in their temporal dimension (temporal neural response variations, or in their combination (temporally coordinated neural population firing. Here we investigate the utility of tensor factorizations of population spike trains along space and time. These factorizations decompose a dataset of single-trial population spike trains into spatial firing patterns (combinations of neurons firing together, temporal firing patterns (temporal activation of these groups of neurons and trial-dependent activation coefficients (strength of recruitment of such neural patterns on each trial. We validated various factorization methods on simulated data and on populations of ganglion cells simultaneously recorded in the salamander retina. We found that single-trial tensor space-by-time decompositions provided low-dimensional data-robust representations of spike trains that capture efficiently both their spatial and temporal information about sensory stimuli. Tensor decompositions with orthogonality constraints were the most efficient in extracting sensory information, whereas non-negative tensor decompositions worked well even on non-independent and overlapping spike patterns, and retrieved informative firing patterns expressed by the same population in response to novel stimuli. Our method showed that populations of retinal ganglion cells carried information in their spike timing on the ten-milliseconds-scale about spatial details of natural images. This information could not be recovered from the spike counts of these cells. First-spike latencies carried the majority of information provided by the whole spike train about fine
McIntosh, C.B.G.; Foyster, J.M.; Lun, A.W.h.
1981-01-01
A list is given of a canonical set of the Newman--Penrose quantities Phi/sub A/B, the tetrad components of the trace-free Ricci tensor, for each Plebanski class according to Plebanski's classification of this tensor. This comparative list can easily be extended to cover the classification in tetrad language of any second-order, trace-free, symmetric tensor in a space-time. A fourth-order tensor which is the product of two such tensors was defined by Plebanski and used in his classification. This has the same symmetries as the Weyl tensor. The Petrov classification of this tensor, here called the Plebanski tensor, is discussed along with the classification of the Ricci tensor. The use of the Plebanski tensor in a couple of areas of general relativity is also briefly discussed
Structural connectivity via the tensor-based morphometry
Kim, S.; Chung, M.; Hanson, J.; Avants, B.; Gee, J.; Davidson, R.; Pollak, S.
2011-01-01
The tensor-based morphometry (TBM) has been widely used in characterizing tissue volume difference between populations at voxel level. We present a novel computational framework for investigating the white matter connectivity using TBM. Unlike other diffusion tensor imaging (DTI) based white matter connectivity studies, we do not use DTI but only T1-weighted magnetic resonance imaging (MRI). To construct brain network graphs, we have developed a new data-driven approach called the ε-neighbor ...
Review of diffusion tensor imaging and its application in children
Vorona, Gregory A. [Children' s Hospital of Richmond at Virginia Commonwealth University, Department of Radiology, Richmond, VA (United States); Berman, Jeffrey I. [Children' s Hospital of Philadelphia, Department of Radiology, Philadelphia, PA (United States)
2015-09-15
Diffusion MRI is an imaging technique that uses the random motion of water to probe tissue microstructure. Diffusion tensor imaging (DTI) can quantitatively depict the organization and connectivity of white matter. Given the non-invasiveness of the technique, DTI has become a widely used tool for researchers and clinicians to examine the white matter of children. This review covers the basics of diffusion-weighted imaging and diffusion tensor imaging and discusses examples of their clinical application in children. (orig.)
Four dimensional sigma model coupled to the metric tensor field
Ghika, G.; Visinescu, M.
1980-02-01
We discuss the four dimensional nonlinear sigma model with an internal O(n) invariance coupled to the metric tensor field satisfying Einstein equations. We derive a bound on the coupling constant between the sigma field and the metric tensor using the theory of harmonic maps. A special attention is paid to Einstein spaces and some new explicit solutions of the model are constructed. (author)
Optimization via separated representations and the canonical tensor decomposition
Reynolds, Matthew J.; Beylkin, Gregory; Doostan, Alireza
2017-11-01
We introduce a new, quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format. The computational complexity of the algorithm is linear in the dimension of the tensor. We show how to use this algorithm to find global maxima of non-convex multivariate functions in separated form. We demonstrate the performance of the new algorithms on several examples.
Optimization via Separated Representations and the Canonical Tensor Decomposition
Reynolds, Matthew J; Beylkin, Gregory; Doostan, Alireza
2016-01-01
We introduce a new, quadratically convergent algorithm for finding maximum absolute value entries of tensors represented in the canonical format. The computational complexity of the algorithm is linear in the dimension of the tensor. We show how to use this algorithm to find global maxima of non-convex multivariate functions in separated form. We demonstrate the performance of the new algorithms on several examples.
Canonical forms of tensor representations and spontaneous symmetry breaking
Cummins, C.J.
1986-01-01
An algorithm for constructing canonical forms for any tensor representation of the classical compact Lie groups is given. This method is used to find a complete list of the symmetry breaking patterns produced by Higgs fields in the third-rank antisymmetric representations of U(n), SU(n) and SO(n) for n<=7. A simple canonical form is also given for kth-rank symmetric tensor representations. (author)
On deformed tensor potential for inelastic deuteron scattering
Raynal, Jacques.
1980-08-01
Tensor analysing powers for inelastic deuteron scattering have been measured around 12 to 15 MeV. There is no problem to use such a tensor potential for the excited states in coupled channel calculations. However, for transition potentials, form factors are very different. A fit has been done with the first order vibrational model for 64 Ni(d,d') 64 Ni*, 2 + at 1,344 MeV
Smartphone dependence classification using tensor factorization
Kim, Yejin; Yook, In Hye; Yu, Hwanjo; Kim, Dai-Jin
2017-01-01
Excessive smartphone use causes personal and social problems. To address this issue, we sought to derive usage patterns that were directly correlated with smartphone dependence based on usage data. This study attempted to classify smartphone dependence using a data-driven prediction algorithm. We developed a mobile application to collect smartphone usage data. A total of 41,683 logs of 48 smartphone users were collected from March 8, 2015, to January 8, 2016. The participants were classified into the control group (SUC) or the addiction group (SUD) using the Korean Smartphone Addiction Proneness Scale for Adults (S-Scale) and a face-to-face offline interview by a psychiatrist and a clinical psychologist (SUC = 23 and SUD = 25). We derived usage patterns using tensor factorization and found the following six optimal usage patterns: 1) social networking services (SNS) during daytime, 2) web surfing, 3) SNS at night, 4) mobile shopping, 5) entertainment, and 6) gaming at night. The membership vectors of the six patterns obtained a significantly better prediction performance than the raw data. For all patterns, the usage times of the SUD were much longer than those of the SUC. From our findings, we concluded that usage patterns and membership vectors were effective tools to assess and predict smartphone dependence and could provide an intervention guideline to predict and treat smartphone dependence based on usage data. PMID:28636614
Mixed symmetry tensors in the worldline formalism
Corradini, Olindo [Dipartimento di Scienze Fisiche, Informatiche e Matematiche,Università degli Studi di Modena e Reggio Emilia, via Campi 213/A, I-41125 Modena (Italy); INFN - Sezione di Bologna,via Irnerio 46, I-40126 Bologna (Italy); Edwards, James P. [Department of Mathematical Sciences, University of Bath,Claverton Down, Bath BA2 7AY (United Kingdom)
2016-05-10
We consider the first quantised approach to quantum field theory coupled to a non-Abelian gauge field. Representing the colour degrees of freedom with a single family of auxiliary variables the matter field transforms in a reducible representation of the gauge group which — by adding a suitable Chern-Simons term to the particle action — can be projected onto a chosen fully (anti-)symmetric representation. By considering F families of auxiliary variables, we describe how to extend the model to arbitrary tensor products of F reducible representations, which realises a U(F) “flavour” symmetry on the worldline particle model. Gauging this symmetry allows the introduction of constraints on the Hilbert space of the colour fields which can be used to project onto an arbitrary irreducible representation, specified by a certain Young tableau. In particular the occupation numbers of the wavefunction — i.e. the lengths of the columns (rows) of the Young tableau — are fixed through the introduction of Chern-Simons terms. We verify this projection by calculating the number of colour degrees of freedom associated to the matter field. We suggest that, using the worldline approach to quantum field theory, this mechanism will allow the calculation of one-loop scattering amplitudes with the virtual particle in an arbitrary representation of the gauge group.
Diffusion tensor imaging in spinal cord injury
Kamble, Ravindra B; Venkataramana, Neelam K; Naik, Arun L; Rao, Shailesh V
2011-01-01
To assess the feasibility of spinal tractography in patients of spinal cord injury vs a control group and to compare fractional anisotropy (FA) values between the groups. Diffusion tensor imaging (DTI) was performed in the spinal cord of 29 patients (18 patients and 11 controls). DTI was done in the cervical region if the cord injury was at the dorsal or lumbar region and in the conus region if cord injury was in the cervical or dorsal region. FA was calculated for the patients and the controls and the values were compared. The mean FA value was 0.550±0.09 in the control group and 0.367±0.14 in the patients; this difference was statistically significant (P=0.001). Spinal tractography is a feasible technique to assess the extent of spinal cord injury by FA, which is reduced in patients of spinal cord injury, suggesting possible Wallerian degeneration. In future, this technique may become a useful tool for assessing cord injury patients after stem cell therapy, with improvement in FA values indicating axonal regeneration
Diffusion tensor imaging of partial intractable epilepsy
Dumas de la Roque, Anne; Oppenheim, Catherine; Rodrigo, Sebastian; Meder, Jean-Francois; Chassoux, Francine; Devaux, Bertrand; Beuvon, Frederic; Daumas-Duport, Catherine
2005-01-01
Our aim was to assess the value of diffusion tensor imaging (DTI) in patients with partial intractable epilepsy. We used DTI (25 non-collinear directions) in 15 patients with a cortical lesion on conventional MRI. Fractional anisotropy (FA) was measured in the internal capsule, and in the normal-appearing white matter (WM), adjacent tothe lesion, and away from the lesion, at a set distance of 2-3 cm. In each patient, increased or decreased FA measurements were those that varied from mirror values using an arbitrary 10% threshold. Over the whole population, ipsi- and contralateral FA measurements were also compared using a Wilcoxon test (p<0.05). Over the whole population, FA was significantly reduced in the WM adjacent to and away from the lesion, whilst being normal in the internal capsule. FA was reduced by more than 10% in the WM adjacent to and distant from the lesion in 13 and 12 patients respectively. For nine of the ten patients for whom the surgical resection encompassed the limits of the lesion on conventional MRI, histological data showed WM alterations (gliosis, axonal loss, abnormal cells). DTI often reveals WM abnormalities that are undetected on conventional MRI in patients with partial intractable epilepsy. (orig.)
Smartphone dependence classification using tensor factorization.
Jingyun Choi
Full Text Available Excessive smartphone use causes personal and social problems. To address this issue, we sought to derive usage patterns that were directly correlated with smartphone dependence based on usage data. This study attempted to classify smartphone dependence using a data-driven prediction algorithm. We developed a mobile application to collect smartphone usage data. A total of 41,683 logs of 48 smartphone users were collected from March 8, 2015, to January 8, 2016. The participants were classified into the control group (SUC or the addiction group (SUD using the Korean Smartphone Addiction Proneness Scale for Adults (S-Scale and a face-to-face offline interview by a psychiatrist and a clinical psychologist (SUC = 23 and SUD = 25. We derived usage patterns using tensor factorization and found the following six optimal usage patterns: 1 social networking services (SNS during daytime, 2 web surfing, 3 SNS at night, 4 mobile shopping, 5 entertainment, and 6 gaming at night. The membership vectors of the six patterns obtained a significantly better prediction performance than the raw data. For all patterns, the usage times of the SUD were much longer than those of the SUC. From our findings, we concluded that usage patterns and membership vectors were effective tools to assess and predict smartphone dependence and could provide an intervention guideline to predict and treat smartphone dependence based on usage data.
Smartphone dependence classification using tensor factorization.
Choi, Jingyun; Rho, Mi Jung; Kim, Yejin; Yook, In Hye; Yu, Hwanjo; Kim, Dai-Jin; Choi, In Young
2017-01-01
Excessive smartphone use causes personal and social problems. To address this issue, we sought to derive usage patterns that were directly correlated with smartphone dependence based on usage data. This study attempted to classify smartphone dependence using a data-driven prediction algorithm. We developed a mobile application to collect smartphone usage data. A total of 41,683 logs of 48 smartphone users were collected from March 8, 2015, to January 8, 2016. The participants were classified into the control group (SUC) or the addiction group (SUD) using the Korean Smartphone Addiction Proneness Scale for Adults (S-Scale) and a face-to-face offline interview by a psychiatrist and a clinical psychologist (SUC = 23 and SUD = 25). We derived usage patterns using tensor factorization and found the following six optimal usage patterns: 1) social networking services (SNS) during daytime, 2) web surfing, 3) SNS at night, 4) mobile shopping, 5) entertainment, and 6) gaming at night. The membership vectors of the six patterns obtained a significantly better prediction performance than the raw data. For all patterns, the usage times of the SUD were much longer than those of the SUC. From our findings, we concluded that usage patterns and membership vectors were effective tools to assess and predict smartphone dependence and could provide an intervention guideline to predict and treat smartphone dependence based on usage data.
Tweeting Earthquakes using TensorFlow
Casarotti, E.; Comunello, F.; Magnoni, F.
2016-12-01
The use of social media is emerging as a powerful tool for disseminating trusted information about earthquakes. Since 2009, the Twitter account @INGVterremoti provides constant and timely details about M2+ seismic events detected by the Italian National Seismic Network, directly connected with the seismologists on duty at Istituto Nazionale di Geofisica e Vulcanologia (INGV). Currently, it updates more than 150,000 followers. Nevertheless, since it provides only the manual revision of seismic parameters, the timing (approximately between 10 and 20 minutes after an event) has started to be under evaluation. Undeniably, mobile internet, social network sites and Twitter in particular require a more rapid and "real-time" reaction. During the last 36 months, INGV tested the tweeting of the automatic detection of M3+ earthquakes, studying the reliability of the information both in term of seismological accuracy that from the point of view of communication and social research. A set of quality parameters (i.e. number of seismic stations, gap, relative error of the location) has been recognized to reduce false alarms and the uncertainty of the automatic detection. We present an experiment to further improve the reliability of this process using TensorFlow™ (an open source software library originally developed by researchers and engineers working on the Google Brain Team within Google's Machine Intelligence research organization).
An Introduction to Tensors for Students of Physics and Engineering
Kolecki, Joseph C.
2002-01-01
Tensor analysis is the type of subject that can make even the best of students shudder. My own post-graduate instructor in the subject took away much of the fear by speaking of an implicit rhythm in the peculiar notation traditionally used, and helped us to see how this rhythm plays its way throughout the various formalisms. Prior to taking that class, I had spent many years "playing" on my own with tensors. I found the going to be tremendously difficult but was able, over time, to back out some physical and geometrical considerations that helped to make the subject a little more transparent. Today, it is sometimes hard not to think in terms of tensors and their associated concepts. This article, prompted and greatly enhanced by Marlos Jacob, whom I've met only by e-mail, is an attempt to record those early notions concerning tensors. It is intended to serve as a bridge from the point where most undergraduate students "leave off" in their studies of mathematics to the place where most texts on tensor analysis begin. A basic knowledge of vectors, matrices, and physics is assumed. A semi-intuitive approach to those notions underlying tensor analysis is given via scalars, vectors, dyads, triads, and higher vector products. The reader must be prepared to do some mathematics and to think. For those students who wish to go beyond this humble start, I can only recommend my professor's wisdom: find the rhythm in the mathematics and you will fare pretty well.
Inference of segmented color and texture description by tensor voting.
Jia, Jiaya; Tang, Chi-Keung
2004-06-01
A robust synthesis method is proposed to automatically infer missing color and texture information from a damaged 2D image by (N)D tensor voting (N > 3). The same approach is generalized to range and 3D data in the presence of occlusion, missing data and noise. Our method translates texture information into an adaptive (N)D tensor, followed by a voting process that infers noniteratively the optimal color values in the (N)D texture space. A two-step method is proposed. First, we perform segmentation based on insufficient geometry, color, and texture information in the input, and extrapolate partitioning boundaries by either 2D or 3D tensor voting to generate a complete segmentation for the input. Missing colors are synthesized using (N)D tensor voting in each segment. Different feature scales in the input are automatically adapted by our tensor scale analysis. Results on a variety of difficult inputs demonstrate the effectiveness of our tensor voting approach.
Energy-momentum tensor in quantum field theory
Fujikawa, K.
1981-01-01
The definition of the energy-momentum tensor as a source current coupled to the background gravitational field receives an important modification in quantum theory. In the path-integral approach, the manifest covariance of the integral measure under general coordinate transformations dictates that field variables with weight 1/2 should be used as independent integration variables. An improved energy-momentum tensor is then generated by the variational derivative, and it gives rise to well-defined gravitational conformal (Weyl) anomalies. In the flat--space-time limit, all the Ward-Takahashi identities associated with space-time transformations including the global dilatation become free from anomalies in terms of this energy-momentum tensor, reflecting the general covariance of the integral measure; the trace of this tensor is thus finite at zero momentum transfer for renormalizable theories. The Jacobian for the local conformal transformation, however, becomes nontrivial, and it gives rise to an anomaly for the conformal identity. All the familiar anomalies are thus reduced to either chiral or conformal anomalies. The consistency of the dilatation and conformal identities at vanishing momentum transfer determines the trace anomaly of this energy-momentum tensor in terms of the renormalization-group b function and other parameters. In contrast, the trace of the conventional energy-momentum tensor generally diverges even at vanishing momentum transfer depending on the regularization scheme, and it is subtractively renormalized. We also explain how the apparently different renormalization properties of the chiral and trace anomalies arise
Atomic-batched tensor decomposed two-electron repulsion integrals
Schmitz, Gunnar; Madsen, Niels Kristian; Christiansen, Ove
2017-04-01
We present a new integral format for 4-index electron repulsion integrals, in which several strategies like the Resolution-of-the-Identity (RI) approximation and other more general tensor-decomposition techniques are combined with an atomic batching scheme. The 3-index RI integral tensor is divided into sub-tensors defined by atom pairs on which we perform an accelerated decomposition to the canonical product (CP) format. In a first step, the RI integrals are decomposed to a high-rank CP-like format by repeated singular value decompositions followed by a rank reduction, which uses a Tucker decomposition as an intermediate step to lower the prefactor of the algorithm. After decomposing the RI sub-tensors (within the Coulomb metric), they can be reassembled to the full decomposed tensor (RC approach) or the atomic batched format can be maintained (ABC approach). In the first case, the integrals are very similar to the well-known tensor hypercontraction integral format, which gained some attraction in recent years since it allows for quartic scaling implementations of MP2 and some coupled cluster methods. On the MP2 level, the RC and ABC approaches are compared concerning efficiency and storage requirements. Furthermore, the overall accuracy of this approach is assessed. Initial test calculations show a good accuracy and that it is not limited to small systems.
The normal conformal Cartan connection and the Bach tensor
Korzynski, Mikolaj; Lewandowski, Jerzy
2003-01-01
The goal of this paper is to express the Bach tensor of a four-dimensional conformal geometry of an arbitrary signature by the Cartan normal conformal (CNC) connection. We show that the Bach tensor can be identified with the Yang-Mills current of the connection. It follows from that result that a conformal geometry whose CNC connection is reducible in an appropriate way has a degenerate Bach tensor. As an example we study the case of a CNC connection which admits a twisting covariantly constant twistor field. This class of conformal geometries of this property is known as given by the Fefferman metric tensors. We use our result to calculate the Bach tensor of an arbitrary Fefferman metric and show that it is proportional to the tensorial square of the four-fold eigenvector of the Weyl tensor. Finally, we solve the Yang-Mills equations imposed on the CNC connection for all the homogeneous Fefferman metrics. The only solution is the Nurowski-Plebanski metric
Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination.
Zhao, Qibin; Zhang, Liqing; Cichocki, Andrzej
2015-09-01
CANDECOMP/PARAFAC (CP) tensor factorization of incomplete data is a powerful technique for tensor completion through explicitly capturing the multilinear latent factors. The existing CP algorithms require the tensor rank to be manually specified, however, the determination of tensor rank remains a challenging problem especially for CP rank . In addition, existing approaches do not take into account uncertainty information of latent factors, as well as missing entries. To address these issues, we formulate CP factorization using a hierarchical probabilistic model and employ a fully Bayesian treatment by incorporating a sparsity-inducing prior over multiple latent factors and the appropriate hyperpriors over all hyperparameters, resulting in automatic rank determination. To learn the model, we develop an efficient deterministic Bayesian inference algorithm, which scales linearly with data size. Our method is characterized as a tuning parameter-free approach, which can effectively infer underlying multilinear factors with a low-rank constraint, while also providing predictive distributions over missing entries. Extensive simulations on synthetic data illustrate the intrinsic capability of our method to recover the ground-truth of CP rank and prevent the overfitting problem, even when a large amount of entries are missing. Moreover, the results from real-world applications, including image inpainting and facial image synthesis, demonstrate that our method outperforms state-of-the-art approaches for both tensor factorization and tensor completion in terms of predictive performance.
Dibb, Russell; Liu, Chunlei
2017-06-01
To develop a susceptibility-based MRI technique for probing microstructure and fiber architecture of magnetically anisotropic tissues-such as central nervous system white matter, renal tubules, and myocardial fibers-in three dimensions using susceptibility tensor imaging (STI) tools. STI can probe tissue microstructure, but is limited by reconstruction artifacts because of absent phase information outside the tissue and noise. STI accuracy may be improved by estimating a joint eigenvector from mutually anisotropic susceptibility and relaxation tensors. Gradient-recalled echo image data were simulated using a numerical phantom and acquired from the ex vivo mouse brain, kidney, and heart. Susceptibility tensor data were reconstructed using STI, regularized STI, and the proposed algorithm of mutually anisotropic and joint eigenvector STI (MAJESTI). Fiber map and tractography results from each technique were compared with diffusion tensor data. MAJESTI reduced the estimated susceptibility tensor orientation error by 30% in the phantom, 36% in brain white matter, 40% in the inner medulla of the kidney, and 45% in myocardium. This improved the continuity and consistency of susceptibility-based fiber tractography in each tissue. MAJESTI estimation of the susceptibility tensors yields lower orientation errors for susceptibility-based fiber mapping and tractography in the intact brain, kidney, and heart. Magn Reson Med 77:2331-2346, 2017. © 2016 International Society for Magnetic Resonance in Medicine. © 2016 International Society for Magnetic Resonance in Medicine.
Retinal Vessel Segmentation via Structure Tensor Coloring and Anisotropy Enhancement
Mehmet Nergiz
2017-11-01
Full Text Available Retinal vessel segmentation is one of the preliminary tasks for developing diagnosis software systems related to various retinal diseases. In this study, a fully automated vessel segmentation system is proposed. Firstly, the vessels are enhanced using a Frangi Filter. Afterwards, Structure Tensor is applied to the response of the Frangi Filter and a 4-D tensor field is obtained. After decomposing the Eigenvalues of the tensor field, the anisotropy between the principal Eigenvalues are enhanced exponentially. Furthermore, this 4-D tensor field is converted to the 3-D space which is composed of energy, anisotropy and orientation and then a Contrast Limited Adaptive Histogram Equalization algorithm is applied to the energy space. Later, the obtained energy space is multiplied by the enhanced mean surface curvature of itself and the modified 3-D space is converted back to the 4-D tensor field. Lastly, the vessel segmentation is performed by using Otsu algorithm and tensor coloring method which is inspired by the ellipsoid tensor visualization technique. Finally, some post-processing techniques are applied to the segmentation result. In this study, the proposed method achieved mean sensitivity of 0.8123, 0.8126, 0.7246 and mean specificity of 0.9342, 0.9442, 0.9453 as well as mean accuracy of 0.9183, 0.9442, 0.9236 for DRIVE, STARE and CHASE_DB1 datasets, respectively. The mean execution time of this study is 6.104, 6.4525 and 18.8370 s for the aforementioned three datasets respectively.
Complete algebraic reduction of one-loop tensor Feynman integrals
Fleischer, J.; Riemann, T.
2011-01-01
We set up a new, flexible approach for the tensor reduction of one-loop Feynman integrals. The 5-point tensor integrals up to rank R=5 are expressed by 4-point tensor integrals of rank R-1, such that the appearance of the inverse 5-point Gram determinant is avoided. The 4-point tensor coefficients are represented in terms of 4-point integrals, defined in d dimensions, 4-2ε≤d≤4-2ε+2(R-1), with higher powers of the propagators. They can be further reduced to expressions which stay free of the inverse 4-point Gram determinants but contain higher-dimensional 4-point integrals with only the first power of scalar propagators, plus 3-point tensor coefficients. A direct evaluation of the higher-dimensional 4-point functions would avoid the appearance of inverse powers of the Gram determinants completely. The simplest approach, however, is to apply here dimensional recurrence relations in order to reduce them to the familiar 2- to 4-point functions in generic dimension d=4-2ε, introducing thereby coefficients with inverse 4-point Gram determinants up to power R for tensors of rank R. For small or vanishing Gram determinants--where this reduction is not applicable--we use analytic expansions in positive powers of the Gram determinants. Improving the convergence of the expansions substantially with Pade approximants we close up to the evaluation of the 4-point tensor coefficients for larger Gram determinants. Finally, some relations are discussed which may be useful for analytic simplifications of Feynman diagrams.
Tucker Tensor analysis of Matern functions in spatial statistics
Litvinenko, Alexander
2018-03-09
In this work, we describe advanced numerical tools for working with multivariate functions and for the analysis of large data sets. These tools will drastically reduce the required computing time and the storage cost, and, therefore, will allow us to consider much larger data sets or finer meshes. Covariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to compute and store, especially in 3D. Therefore, we approximate covariance functions by cheap surrogates in a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of Matern- and Slater-type functions with varying parameters and demonstrate numerically that their approximations exhibit exponentially fast convergence. We prove the exponential convergence of the Tucker and canonical approximations in tensor rank parameters. Several statistical operations are performed in this low-rank tensor format, including evaluating the conditional covariance matrix, spatially averaged estimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse, and Cholesky decomposition of a large covariance matrix. Low-rank tensor approximations reduce the computing and storage costs essentially. For example, the storage cost is reduced from an exponential O(n^d) to a linear scaling O(drn), where d is the spatial dimension, n is the number of mesh points in one direction, and r is the tensor rank. Prerequisites for applicability of the proposed techniques are the assumptions that the data, locations, and measurements lie on a tensor (axes-parallel) grid and that the covariance function depends on a distance, ||x-y||.
Walder, Brennan J.; Davis, Michael C.; Grandinetti, Philip J. [Department of Chemistry, Ohio State University, 100 West 18th Avenue, Columbus, Ohio 43210 (United States); Dey, Krishna K. [Department of Physics, Dr. H. S. Gour University, Sagar, Madhya Pradesh 470003 (India); Baltisberger, Jay H. [Division of Natural Science, Mathematics, and Nursing, Berea College, Berea, Kentucky 40403 (United States)
2015-01-07
A new two-dimensional Nuclear Magnetic Resonance (NMR) experiment to separate and correlate the first-order quadrupolar and chemical/paramagnetic shift interactions is described. This experiment, which we call the shifting-d echo experiment, allows a more precise determination of tensor principal components values and their relative orientation. It is designed using the recently introduced symmetry pathway concept. A comparison of the shifting-d experiment with earlier proposed methods is presented and experimentally illustrated in the case of {sup 2}H (I = 1) paramagnetic shift and quadrupolar tensors of CuCl{sub 2}⋅2D{sub 2}O. The benefits of the shifting-d echo experiment over other methods are a factor of two improvement in sensitivity and the suppression of major artifacts. From the 2D lineshape analysis of the shifting-d spectrum, the {sup 2}H quadrupolar coupling parameters are 〈C{sub q}〉 = 118.1 kHz and 〈η{sub q}〉 = 0.88, and the {sup 2}H paramagnetic shift tensor anisotropy parameters are 〈ζ{sub P}〉 = − 152.5 ppm and 〈η{sub P}〉 = 0.91. The orientation of the quadrupolar coupling principal axis system (PAS) relative to the paramagnetic shift anisotropy principal axis system is given by (α,β,γ)=((π)/2 ,(π)/2 ,0). Using a simple ligand hopping model, the tensor parameters in the absence of exchange are estimated. On the basis of this analysis, the instantaneous principal components and orientation of the quadrupolar coupling are found to be in excellent agreement with previous measurements. A new point dipole model for predicting the paramagnetic shift tensor is proposed yielding significantly better agreement than previously used models. In the new model, the dipoles are displaced from nuclei at positions associated with high electron density in the singly occupied molecular orbital predicted from ligand field theory.
Ultrasound elastic tensor imaging: comparison with MR diffusion tensor imaging in the myocardium
Lee, Wei-Ning; Larrat, Benoît; Pernot, Mathieu; Tanter, Mickaël
2012-08-01
We have previously proven the feasibility of ultrasound-based shear wave imaging (SWI) to non-invasively characterize myocardial fiber orientation in both in vitro porcine and in vivo ovine hearts. The SWI-estimated results were in good correlation with histology. In this study, we proposed a new and robust fiber angle estimation method through a tensor-based approach for SWI, coined together as elastic tensor imaging (ETI), and compared it with magnetic resonance diffusion tensor imaging (DTI), a current gold standard and extensively reported non-invasive imaging technique for mapping fiber architecture. Fresh porcine (n = 5) and ovine (n = 5) myocardial samples (20 × 20 × 30 mm3) were studied. ETI was firstly performed to generate shear waves and to acquire the wave events at ultrafast frame rate (8000 fps). A 2.8 MHz phased array probe (pitch = 0.28 mm), connected to a prototype ultrasound scanner, was mounted on a customized MRI-compatible rotation device, which allowed both the rotation of the probe from -90° to 90° at 5° increments and co-registration between two imaging modalities. Transmural shear wave speed at all propagation directions realized was firstly estimated. The fiber angles were determined from the shear wave speed map using the least-squares method and eigen decomposition. The test myocardial sample together with the rotation device was then placed inside a 7T MRI scanner. Diffusion was encoded in six directions. A total of 270 diffusion-weighted images (b = 1000 s mm-2, FOV = 30 mm, matrix size = 60 × 64, TR = 6 s, TE = 19 ms, 24 averages) and 45 B0 images were acquired in 14 h 30 min. The fiber structure was analyzed by the fiber-tracking module in software, MedINRIA. The fiber orientation in the overlapped myocardial region which both ETI and DTI accessed was therefore compared, thanks to the co-registered imaging system. Results from all ten samples showed good correlation (r2 = 0.81, p 0.05, unpaired, one-tailed t-test, N = 10). In
Tensor Completion for Estimating Missing Values in Visual Data
Liu, Ji
2012-01-25
In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependant relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between Fa
Tensor Completion for Estimating Missing Values in Visual Data
Liu, Ji; Musialski, Przemyslaw; Wonka, Peter; Ye, Jieping
2012-01-01
In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependant relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between Fa
Tensor completion for estimating missing values in visual data.
Liu, Ji; Musialski, Przemyslaw; Wonka, Peter; Ye, Jieping
2013-01-01
In this paper, we propose an algorithm to estimate missing values in tensors of visual data. The values can be missing due to problems in the acquisition process or because the user manually identified unwanted outliers. Our algorithm works even with a small amount of samples and it can propagate structure to fill larger missing regions. Our methodology is built on recent studies about matrix completion using the matrix trace norm. The contribution of our paper is to extend the matrix case to the tensor case by proposing the first definition of the trace norm for tensors and then by building a working algorithm. First, we propose a definition for the tensor trace norm that generalizes the established definition of the matrix trace norm. Second, similarly to matrix completion, the tensor completion is formulated as a convex optimization problem. Unfortunately, the straightforward problem extension is significantly harder to solve than the matrix case because of the dependency among multiple constraints. To tackle this problem, we developed three algorithms: simple low rank tensor completion (SiLRTC), fast low rank tensor completion (FaLRTC), and high accuracy low rank tensor completion (HaLRTC). The SiLRTC algorithm is simple to implement and employs a relaxation technique to separate the dependent relationships and uses the block coordinate descent (BCD) method to achieve a globally optimal solution; the FaLRTC algorithm utilizes a smoothing scheme to transform the original nonsmooth problem into a smooth one and can be used to solve a general tensor trace norm minimization problem; the HaLRTC algorithm applies the alternating direction method of multipliers (ADMMs) to our problem. Our experiments show potential applications of our algorithms and the quantitative evaluation indicates that our methods are more accurate and robust than heuristic approaches. The efficiency comparison indicates that FaLTRC and HaLRTC are more efficient than SiLRTC and between FaLRTC an
Diffusion tensor MR microscopy of tissues with low diffusional anisotropy.
Bajd, Franci; Mattea, Carlos; Stapf, Siegfried; Sersa, Igor
2016-06-01
Diffusion tensor imaging exploits preferential diffusional motion of water molecules residing within tissue compartments for assessment of tissue structural anisotropy. However, instrumentation and post-processing errors play an important role in determination of diffusion tensor elements. In the study, several experimental factors affecting accuracy of diffusion tensor determination were analyzed. Effects of signal-to-noise ratio and configuration of the applied diffusion-sensitizing gradients on fractional anisotropy bias were analyzed by means of numerical simulations. In addition, diffusion tensor magnetic resonance microscopy experiments were performed on a tap water phantom and bovine articular cartilage-on-bone samples to verify the simulation results. In both, the simulations and the experiments, the multivariate linear regression of the diffusion-tensor analysis yielded overestimated fractional anisotropy with low SNRs and with low numbers of applied diffusion-sensitizing gradients. An increase of the apparent fractional anisotropy due to unfavorable experimental conditions can be overcome by applying a larger number of diffusion sensitizing gradients with small values of the condition number of the transformation matrix. This is in particular relevant in magnetic resonance microscopy, where imaging gradients are high and the signal-to-noise ratio is low.
Tensor perturbations during inflation in a spatially closed Universe
Bonga, Béatrice; Gupt, Brajesh; Yokomizo, Nelson, E-mail: bpb165@psu.edu, E-mail: bgupt@gravity.psu.edu, E-mail: yokomizo@gravity.psu.edu [Institute for Gravitation and the Cosmos and Physics Department, The Pennsylvania State University, 104 Lavey Lab, University Park, PA 16802 (United States)
2017-05-01
In a recent paper [1], we studied the evolution of the background geometry and scalar perturbations in an inflationary, spatially closed Friedmann-Lemaȋtre-Robertson-Walker (FLRW) model having constant positive spatial curvature and spatial topology S{sup 3}. Due to the spatial curvature, the early phase of slow-roll inflation is modified, leading to suppression of power in the scalar power spectrum at large angular scales. In this paper, we extend the analysis to include tensor perturbations. We find that, similarly to the scalar perturbations, the tensor power spectrum also shows suppression for long wavelength modes. The correction to the tensor spectrum is limited to the very long wavelength modes, therefore the resulting observable CMB B-mode polarization spectrum remains practically the same as in the standard scenario with flat spatial sections. However, since both the tensor and scalar power spectra are modified, there are scale dependent corrections to the tensor-to-scalar ratio that leads to violation of the standard slow-roll consistency relation.
Nonperturbative loop quantization of scalar-tensor theories of gravity
Zhang Xiangdong; Ma Yongge
2011-01-01
The Hamiltonian formulation of scalar-tensor theories of gravity is derived from their Lagrangian formulation by Hamiltonian analysis. The Hamiltonian formalism marks off two sectors of the theories by the coupling parameter ω(φ). In the sector of ω(φ)=-(3/2), the feasible theories are restricted and a new primary constraint generating conformal transformations of spacetime is obtained, while in the other sector of ω(φ)≠-(3/2), the canonical structure and constraint algebra of the theories are similar to those of general relativity coupled with a scalar field. By canonical transformations, we further obtain the connection-dynamical formalism of the scalar-tensor theories with real su(2) connections as configuration variables in both sectors. This formalism enables us to extend the scheme of nonperturbative loop quantum gravity to the scalar-tensor theories. The quantum kinematical framework for the scalar-tensor theories is rigorously constructed. Both the Hamiltonian constraint operator and master constraint operator are well defined and proposed to represent quantum dynamics. Thus the loop quantum gravity method is also valid for general scalar-tensor theories.
Simultaneous tensor decomposition and completion using factor priors.
Chen, Yi-Lei; Hsu, Chiou-Ting; Liao, Hong-Yuan Mark
2014-03-01
The success of research on matrix completion is evident in a variety of real-world applications. Tensor completion, which is a high-order extension of matrix completion, has also generated a great deal of research interest in recent years. Given a tensor with incomplete entries, existing methods use either factorization or completion schemes to recover the missing parts. However, as the number of missing entries increases, factorization schemes may overfit the model because of incorrectly predefined ranks, while completion schemes may fail to interpret the model factors. In this paper, we introduce a novel concept: complete the missing entries and simultaneously capture the underlying model structure. To this end, we propose a method called simultaneous tensor decomposition and completion (STDC) that combines a rank minimization technique with Tucker model decomposition. Moreover, as the model structure is implicitly included in the Tucker model, we use factor priors, which are usually known a priori in real-world tensor objects, to characterize the underlying joint-manifold drawn from the model factors. By exploiting this auxiliary information, our method leverages two classic schemes and accurately estimates the model factors and missing entries. We conducted experiments to empirically verify the convergence of our algorithm on synthetic data and evaluate its effectiveness on various kinds of real-world data. The results demonstrate the efficacy of the proposed method and its potential usage in tensor-based applications. It also outperforms state-of-the-art methods on multilinear model analysis and visual data completion tasks.
Energy-momentum tensor in quantum field theory
Fujikawa, Kazuo.
1980-12-01
The definition of the energy-momentum tensor as a source current coupled to the background gravitational field receives an important modification in quantum theory. In the path integral approach, the manifest covariance of the integral measure under general coordinate transformations dictates that field variables with weight 1/2 should be used as independent integration variables. An improved energy-momentum tensor is then generated by the variational derivative, and it gives rise to well-defined gravitational conformal (Weyl) anomalies. In the flat space-time limit, all the Ward-Takahashi identities associate with space-time transformations including the global dilatation become free from anomalies, reflecting the general covariance of the integral measure; the trace of this energy-momentum tensor is thus finite at the zero momentum transfer. The Jacobian for the local conformal transformation however becomes non-trivial, and it gives rise to an anomaly for the conformal identity. All the familiar anomalies are thus reduced to either chiral or conformal anomalies. The consistency of the dilatation and conformal identities at the vanishing momentum transfer determines the trace anomaly of this energy-momentum tensor in terms of the renormalization group β-function and other parameters. In contrast, the trace of the conventional energy-momentum tensor generally diverges even at the vanishing momentum transfer depending on the regularization scheme, and it is subtractively renormalized. We also explain how the apparently different renormalization properties of the chiral and trace anomalies arise. (author)
Saur, R. [Sektion fuer Experimentelle Kernspinresonanz des ZNS, Abt. Neuroradiologie, Universitaetsklinikum Tuebingen (Germany); Augenklinik des Universitaetsklinikums Tuebingen (Germany); Klinik fuer Psychiatrie und Psychotherapie des Universitaetsklinikums Tuebingen (Germany); Gharabaghi, A. [Klinik fuer Neurochirurgie des Universitaetsklinikums Tuebingen (Germany); Erb, M. [Sektion fuer Experimentelle Kernspinresonanz des ZNS, Abt. Neuroradiologie, Universitaetsklinikum Tuebingen (Germany)
2007-07-01
Knowledge about integrity and location of fibre tracts arising from eloquent cortical areas is important to plan neurosurgical interventions and to allow maximization of resection of pathological tissue while preserving vital white matter tracts. Diffusion Tensor Imaging (DTI) is so far the only method to get preoperatively an impression of the individual complexity of nerve bundles. Thereby nerve fibres are not mapped directly. They are derived indirectly by analysis of the directional distribution of diffusion of water molecules which is influenced mainly by large fibre tracts. From acquisition to reconstruction and visualisation of the fibre tracts many representational stages and working steps have to be passed. Exact knowledge about problems of Diffusion Imaging is important for interpretation of the results. Particularly, brain tumor edema, intraoperative brain shift, MR-artefacts and limitations of the mathematical models and algorithms challenge DTI-developers and applicants. (orig.)
Micromechanics based framework with second-order damage tensors
Desmorat, R.; Desmorat, B.; Olive, M.; Kolev, B.
2018-05-01
The harmonic product of tensors---leading to the concept of harmonic factorization---has been defined in a previous work (Olive et al, 2017). In the practical case of 3D crack density measurements on thin or thick walled structures, this mathematical tool allows us to factorize the harmonic (irreducible) part of the fourth-order damage tensor as an harmonic square: an exact harmonic square in 2D, an harmonic square over the set of so-called mechanically accessible directions for measurements in the 3D case. The corresponding micro-mechanics framework based on second---instead of fourth---order damage tensors is derived. An illustrating example is provided showing how the proposed framework allows for the modeling of the so-called hydrostatic sensitivity up to high damage levels.
Diffusion tensor and diffusion weighted imaging. Pictorial mathematics
Nakada, Tsutomu [California Univ., Davis, CA (United States)
1995-06-01
A new imaging algorithm for the treatment of a second order apparent diffusion tensor, D{sub app}{sup {xi}} is described. The method calls for only mathematics of images (pictorial mathematics) without necessity of eigenvalues/eigenvectors estimation. Nevertheless, it is capable of extracting properties of D{sub app}{sup {xi}} invariant to observation axes. While trace image is an example of images weighted by invariance of the tensor matrix, three dimensional anisotropy (3DAC) contrast represents the imaging method making use to anisotropic direction of tensor ellipsoid producing color coded contrast of exceptionally high anatomic resolution. Contrary to intuition, the processes require only a simple algorithm directly applicable to clinical magnetic resonance imaging (MRI). As a contrast method which precisely represents physical characteristics of a target tissue, invariant D{sub app}{sup {xi}} images produced by pictorial mathematics possess significant potential for a number of biological and clinical applications. (author).
Entanglement and tensor product decomposition for two fermions
Caban, P; Podlaski, K; Rembielinski, J; Smolinski, K A; Walczak, Z
2005-01-01
The problem of the choice of tensor product decomposition in a system of two fermions with the help of Bogoliubov transformations of creation and annihilation operators is discussed. The set of physical states of the composite system is restricted by the superselection rule forbidding the superposition of fermions and bosons. It is shown that the Wootters concurrence is not the proper entanglement measure in this case. The explicit formula for the entanglement of formation is found. This formula shows that the entanglement of a given state depends on the tensor product decomposition of a Hilbert space. It is shown that the set of separable states is narrower than in the two-qubit case. Moreover, there exist states which are separable with respect to all tensor product decompositions of the Hilbert space. (letter to the editor)
TENSOR DECOMPOSITIONS AND SPARSE LOG-LINEAR MODELS
Johndrow, James E.; Bhattacharya, Anirban; Dunson, David B.
2017-01-01
Contingency table analysis routinely relies on log-linear models, with latent structure analysis providing a common alternative. Latent structure models lead to a reduced rank tensor factorization of the probability mass function for multivariate categorical data, while log-linear models achieve dimensionality reduction through sparsity. Little is known about the relationship between these notions of dimensionality reduction in the two paradigms. We derive several results relating the support of a log-linear model to nonnegative ranks of the associated probability tensor. Motivated by these findings, we propose a new collapsed Tucker class of tensor decompositions, which bridge existing PARAFAC and Tucker decompositions, providing a more flexible framework for parsimoniously characterizing multivariate categorical data. Taking a Bayesian approach to inference, we illustrate empirical advantages of the new decompositions. PMID:29332971
Ambiguities and symmetry relations associated with fermionic tensor densities
Dallabona, G.; Battistel, O. A.
2004-01-01
We consider the consistent evaluation of perturbative (divergent) Green functions associated with fermionic tensor densities and the derivation of symmetry relations for them. We show that, in spite of current algebra methods being not applicable, it is possible to derive symmetry properties analogous to the Ward identities of vector and axial-vector densities. The proposed method, which is applicable to any previously chosen order of perturbative calculation, gives the same results as those of current algebra when such a tool is applicable. By using a very general calculational strategy, concerning the manipulations and calculations involving divergent Feynman integrals, we evaluate the purely fermionic two-point functions containing tensor vertices and derive their symmetry properties. The present investigation is the first step in the study and characterization of possible anomalies involving fermionic tensor densities, particularly in purely fermionic three-point functions
On the axial anomalies in external tensor fields
Khudaverdyan, O.M.; Mkrtchyan, R.L.; Zurabyan, L.A.
1985-01-01
Computation of the axial anomaly for Dirac fermions in external tensor fields is studied. The sequence of the supersymmetric one-dimensional models is presented. Their supercharges are equal, after quantization, to Dirac operators in external tensor fields, and the density of Witten's partition function gives the anomaly. It is shown that action in the corresponding path integral differs from the classical one. Gaussian approximation gives the anomaly only in the case of third-rank tensor with zero exterior derivative and in that case anomaly is calculated in all dimensions. The interpretation of that field as the torsion of gravitational field and also connection with the results of Witten and Alvarez-Gaume and Atiyah-Singer index theorem are discussed
Tensor analysis and elementary differential geometry for physicists and engineers
Nguyen-Schäfer, Hung
2014-01-01
Tensors and methods of differential geometry are very useful mathematical tools in many fields of modern physics and computational engineering including relativity physics, electrodynamics, computational fluid dynamics (CFD), continuum mechanics, aero and vibroacoustics, and cybernetics. This book comprehensively presents topics, such as bra-ket notation, tensor analysis, and elementary differential geometry of a moving surface. Moreover, authors intentionally abstain from giving mathematically rigorous definitions and derivations that are however dealt with as precisely as possible. The reader is provided with hands-on calculations and worked-out examples at which he will learn how to handle the bra-ket notation, tensors and differential geometry and to use them in the physical and engineering world. The target audience primarily comprises graduate students in physics and engineering, research scientists, and practicing engineers.
Endomorphism Algebras of Tensor Powers of Modules for Quantum Groups
Andersen, Therese Søby
We determine the ring structure of the endomorphism algebra of certain tensor powers of modules for the quantum group of sl2 in the case where the quantum parameter is allowed to be a root of unity. In this case there exists -- under a suitable localization of our ground ring -- a surjection from...... the group algebra of the braid group to the endomorphism algebra of any tensor power of the Weyl module with highest weight 2. We take a first step towards determining the kernel of this map by reformulating well-known results on the semisimplicity of the Birman-Murakami-Wenzl algebra in terms of the order...... of the quantum parameter. Before we arrive at these main results, we investigate the structure of the endomorphism algebra of the tensor square of any Weyl module....
The 1/ N Expansion of Tensor Models Beyond Perturbation Theory
Gurau, Razvan
2014-09-01
We analyze in full mathematical rigor the most general quartically perturbed invariant probability measure for a random tensor. Using a version of the Loop Vertex Expansion (which we call the mixed expansion) we show that the cumulants write as explicit series in 1/ N plus bounded rest terms. The mixed expansion recasts the problem of determining the subleading corrections in 1/ N into a simple combinatorial problem of counting trees decorated by a finite number of loop edges. As an aside, we use the mixed expansion to show that the (divergent) perturbative expansion of the tensor models is Borel summable and to prove that the cumulants respect an uniform scaling bound. In particular the quartically perturbed measures fall, in the N→ ∞ limit, in the universality class of Gaussian tensor models.
Gradients estimation from random points with volumetric tensor in turbulence
Watanabe, Tomoaki; Nagata, Koji
2017-12-01
We present an estimation method of fully-resolved/coarse-grained gradients from randomly distributed points in turbulence. The method is based on a linear approximation of spatial gradients expressed with the volumetric tensor, which is a 3 × 3 matrix determined by a geometric distribution of the points. The coarse grained gradient can be considered as a low pass filtered gradient, whose cutoff is estimated with the eigenvalues of the volumetric tensor. The present method, the volumetric tensor approximation, is tested for velocity and passive scalar gradients in incompressible planar jet and mixing layer. Comparison with a finite difference approximation on a Cartesian grid shows that the volumetric tensor approximation computes the coarse grained gradients fairly well at a moderate computational cost under various conditions of spatial distributions of points. We also show that imposing the solenoidal condition improves the accuracy of the present method for solenoidal vectors, such as a velocity vector in incompressible flows, especially when the number of the points is not large. The volumetric tensor approximation with 4 points poorly estimates the gradient because of anisotropic distribution of the points. Increasing the number of points from 4 significantly improves the accuracy. Although the coarse grained gradient changes with the cutoff length, the volumetric tensor approximation yields the coarse grained gradient whose magnitude is close to the one obtained by the finite difference. We also show that the velocity gradient estimated with the present method well captures the turbulence characteristics such as local flow topology, amplification of enstrophy and strain, and energy transfer across scales.
Tensor fields on orbits of quantum states and applications
Volkert, Georg Friedrich
2010-07-19
On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes of tensor field constructions. First, as pull-back tensor fields of order two from modified Hermitian tensor fields, constructed on Hilbert spaces by means of the property of having the vertical distributions of the C{sub 0}-principal bundle H{sub 0} {yields} P(H) over the projective Hilbert space P(H) in the kernel. And second, directly constructed on the Lie group, as left-invariant representation-dependent operator-valued tensor fields (LIROVTs) of arbitrary order being evaluated on a quantum state. Within the NP-hard problem of deciding whether a given state in a n-level bi-partite quantum system is entangled or separable (Gurvits, 2003), we show that both tensor field constructions admit a geometric approach to this problem, which evades the traditional ambiguity on defining metrical structures on the convex set of mixed states. In particular by considering manifolds associated to orbits passing through a selected state when acted upon by the local unitary group U(n) x U(n) of Schmidt coefficient decomposition inducing transformations, we find the following results: In the case of pure states we show that Schmidt-equivalence classes which are Lagrangian submanifolds define maximal entangled states. This implies a stronger statement as the one proposed by Bengtsson (2007). Moreover, Riemannian pull-back tensor fields split on orbits of separable states and provide a quantitative characterization of entanglement which recover the entanglement measure proposed by Schlienz and Mahler (1995). In the case of mixed states we highlight a relation between LIROVTs of order two and a class of computable separability criteria based on the Bloch-representation (de Vicente, 2007). (orig.)
Tensor fields on orbits of quantum states and applications
Volkert, Georg Friedrich
2010-01-01
On classical Lie groups, which act by means of a unitary representation on finite dimensional Hilbert spaces H, we identify two classes of tensor field constructions. First, as pull-back tensor fields of order two from modified Hermitian tensor fields, constructed on Hilbert spaces by means of the property of having the vertical distributions of the C 0 -principal bundle H 0 → P(H) over the projective Hilbert space P(H) in the kernel. And second, directly constructed on the Lie group, as left-invariant representation-dependent operator-valued tensor fields (LIROVTs) of arbitrary order being evaluated on a quantum state. Within the NP-hard problem of deciding whether a given state in a n-level bi-partite quantum system is entangled or separable (Gurvits, 2003), we show that both tensor field constructions admit a geometric approach to this problem, which evades the traditional ambiguity on defining metrical structures on the convex set of mixed states. In particular by considering manifolds associated to orbits passing through a selected state when acted upon by the local unitary group U(n) x U(n) of Schmidt coefficient decomposition inducing transformations, we find the following results: In the case of pure states we show that Schmidt-equivalence classes which are Lagrangian submanifolds define maximal entangled states. This implies a stronger statement as the one proposed by Bengtsson (2007). Moreover, Riemannian pull-back tensor fields split on orbits of separable states and provide a quantitative characterization of entanglement which recover the entanglement measure proposed by Schlienz and Mahler (1995). In the case of mixed states we highlight a relation between LIROVTs of order two and a class of computable separability criteria based on the Bloch-representation (de Vicente, 2007). (orig.)
Inflationary tensor fossils in large-scale structure
Dimastrogiovanni, Emanuela [School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 (United States); Fasiello, Matteo [Department of Physics, Case Western Reserve University, Cleveland, OH 44106 (United States); Jeong, Donghui [Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802 (United States); Kamionkowski, Marc, E-mail: ema@physics.umn.edu, E-mail: mrf65@case.edu, E-mail: duj13@psu.edu, E-mail: kamion@jhu.edu [Department of Physics and Astronomy, 3400 N. Charles St., Johns Hopkins University, Baltimore, MD 21218 (United States)
2014-12-01
Inflation models make specific predictions for a tensor-scalar-scalar three-point correlation, or bispectrum, between one gravitational-wave (tensor) mode and two density-perturbation (scalar) modes. This tensor-scalar-scalar correlation leads to a local power quadrupole, an apparent departure from statistical isotropy in our Universe, as well as characteristic four-point correlations in the current mass distribution in the Universe. So far, the predictions for these observables have been worked out only for single-clock models in which certain consistency conditions between the tensor-scalar-scalar correlation and tensor and scalar power spectra are satisfied. Here we review the requirements on inflation models for these consistency conditions to be satisfied. We then consider several examples of inflation models, such as non-attractor and solid-inflation models, in which these conditions are put to the test. In solid inflation the simplest consistency conditions are already violated whilst in the non-attractor model we find that, contrary to the standard scenario, the tensor-scalar-scalar correlator probes directly relevant model-dependent information. We work out the predictions for observables in these models. For non-attractor inflation we find an apparent local quadrupolar departure from statistical isotropy in large-scale structure but that this power quadrupole decreases very rapidly at smaller scales. The consistency of the CMB quadrupole with statistical isotropy then constrains the distance scale that corresponds to the transition from the non-attractor to attractor phase of inflation to be larger than the currently observable horizon. Solid inflation predicts clustering fossils signatures in the current galaxy distribution that may be large enough to be detectable with forthcoming, and possibly even current, galaxy surveys.
Mohammadi, Siawoosh; Hutton, Chloe; Nagy, Zoltan; Josephs, Oliver; Weiskopf, Nikolaus
2013-01-01
Diffusion tensor imaging is widely used in research and clinical applications, but this modality is highly sensitive to artefacts. We developed an easy-to-implement extension of the original diffusion tensor model to account for physiological noise in diffusion tensor imaging using measures of peripheral physiology (pulse and respiration), the so-called extended tensor model. Within the framework of the extended tensor model two types of regressors, which respectively modeled small (linear) and strong (nonlinear) variations in the diffusion signal, were derived from peripheral measures. We tested the performance of four extended tensor models with different physiological noise regressors on nongated and gated diffusion tensor imaging data, and compared it to an established data-driven robust fitting method. In the brainstem and cerebellum the extended tensor models reduced the noise in the tensor-fit by up to 23% in accordance with previous studies on physiological noise. The extended tensor model addresses both large-amplitude outliers and small-amplitude signal-changes. The framework of the extended tensor model also facilitates further investigation into physiological noise in diffusion tensor imaging. The proposed extended tensor model can be readily combined with other artefact correction methods such as robust fitting and eddy current correction. PMID:22936599
Analytical effective tensor for flow-through composites
Sviercoski, Rosangela De Fatima [Los Alamos, NM
2012-06-19
A machine, method and computer-usable medium for modeling an average flow of a substance through a composite material. Such a modeling includes an analytical calculation of an effective tensor K.sup.a suitable for use with a variety of media. The analytical calculation corresponds to an approximation to the tensor K, and follows by first computing the diagonal values, and then identifying symmetries of the heterogeneity distribution. Additional calculations include determining the center of mass of the heterogeneous cell and its angle according to a defined Cartesian system, and utilizing this angle into a rotation formula to compute the off-diagonal values and determining its sign.
Properties of the stress tensor in more than two dimensions
Cappelli, A.
1988-03-01
Some aspects of conformal invariance in more than two dimensions are analysed. In this case conformal (Weyl) transformations of the metric are not realized in general by coordinate transformations. The operator product expansion of the stress tensor is investigated by means of examples in the free bosonic and fermionic theories. The effective action for the general form of the trace anomaly is built in four dimensions and the Wess-Zumino consistency conditions are discussed. This gives the anomalous transformation law of the stress tensor and the relation to the Casimir effect in the geometry R x S 3 . The explicit computation of the bosonic partition function provides a check
Tensor calculus, relativity, and cosmology a first course
Dalarsson, M
2005-01-01
This book combines relativity, astrophysics, and cosmology in a single volume, providing an introduction to each subject that enables students to understand more detailed treatises as well as the current literature. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The next section on relativistic astrophysics discusses
Tensor and vector analysis with applications to differential geometry
Springer, C E
2012-01-01
Concise and user-friendly, this college-level text assumes only a knowledge of basic calculus in its elementary and gradual development of tensor theory. The introductory approach bridges the gap between mere manipulation and a genuine understanding of an important aspect of both pure and applied mathematics.Beginning with a consideration of coordinate transformations and mappings, the treatment examines loci in three-space, transformation of coordinates in space and differentiation, tensor algebra and analysis, and vector analysis and algebra. Additional topics include differentiation of vect
Tensor calculus for supergravity on a manifold with boundary
Belyaev, Dmitry V.; Nieuwenhuizen, Peter van
2008-01-01
Using the simple setting of 3D N = 1 supergravity, we show how the tensor calculus of supergravity can be extended to manifolds with boundary. We present an extension of the standard F-density formula which yields supersymmetric bulk-plus-boundary actions. To construct additional separately supersymmetric boundary actions, we decompose bulk supergravity and bulk matter multiplets into co-dimension one submultiplets. As an illustration we obtain the supersymmetric extension of the York-Gibbons-Hawking extrinsic curvature boundary term. We emphasize that our construction does not require any boundary conditions on off-shell fields. This gives a significant improvement over the existing orbifold supergravity tensor calculus
Numerical evaluation of the tensor bispectrum in two field inflation
Raveendran, Rathul Nath [The Institute of Mathematical Sciences, HBNI, CIT Campus, Chennai, 600113 India (India); Sriramkumar, L., E-mail: rathulnr@imsc.res.in, E-mail: sriram@physics.iitm.ac.in [Department of Physics, Indian Institute of Technology Madras, Chennai, 600036 India (India)
2017-07-01
We evaluate the dimensionless non-Gaussianity parameter h {sub NL}, that characterizes the amplitude of the tensor bispectrum, numerically for a class of two field inflationary models such as double inflation, hybrid inflation and aligned natural inflation. We compare the numerical results with the slow roll results which can be obtained analytically. In the context of double inflation, we also investigate the effects on h {sub NL} due to curved trajectories in the field space. We explicitly examine the validity of the consistency relation governing the tensor bispectrum in the squeezed limit. Lastly, we discuss the contribution to h {sub NL} due to the epoch of preheating in two field models.
The metric theory of tensor products Grothendieck's resume revisited
Diestel, Joe; Swart, Johan; Swarte, Johannes Laurentius; Diestel, Joseph
2008-01-01
Grothendieck's Resumé is a landmark in functional analysis. Despite having appeared more than a half century ago, its techniques and results are still not widely known nor appreciated. This is due, no doubt, to the fact that Grothendieck included practically no proofs, and the presentation is based on the theory of the very abstract notion of tensor products. This book aims at providing the details of Grothendieck's constructions and laying bare how the important classes of operators are a consequence of the abstract operations on tensor norms. Particular attention is paid to how the classical
Tensor rank of the tripartite state |W>xn
Yu Nengkun; Guo Cheng; Duan Runyao; Chitambar, Eric
2010-01-01
Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its nonadditivity as an entanglement measure has recently been observed. In this Brief Report, we estimate the tensor rank of multiple copies of the tripartite state |W>=(1/√(3))(|100>+|010>+|001>). Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the rank of |W> x 2 is 7, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between |W> xn and multiple copies of the state |GHZ>=(1/√(2))(|000>+|111>).
STRUCTURAL CONNECTIVITY VIA THE TENSOR-BASED MORPHOMETRY.
Kim, Seung-Goo; Chung, Moo K; Hanson, Jamie L; Avants, Brian B; Gee, James C; Davidson, Richard J; Pollak, Seth D
2011-01-01
The tensor-based morphometry (TBM) has been widely used in characterizing tissue volume difference between populations at voxel level. We present a novel computational framework for investigating the white matter connectivity using TBM. Unlike other diffusion tensor imaging (DTI) based white matter connectivity studies, we do not use DTI but only T1-weighted magnetic resonance imaging (MRI). To construct brain network graphs, we have developed a new data-driven approach called the ε -neighbor method that does not need any predetermined parcellation. The proposed pipeline is applied in detecting the topological alteration of the white matter connectivity in maltreated children.
General scalar-tensor theories for induced gravity inflation
Boutaleb J, H.; Marrakchi, A.L.
1992-07-01
Some cosmological implications of a general scalar-tensor theory for induced gravity are discussed. The model exhibits a slow-rolling phase provided that the coupling function ε(φ) varies slowly enough such that φ dlnε(φ)/dφ much less than 2 during almost the inflationary epoch. It is then shown that, as in the ordinary induced gravity inflation, the chaotic scenario is more natural than the new scenario which proves to be even not self-consistent. The results are applied, for illustration, to a scalar-tensor theory of the Barker type. (author). 25 refs
Mesh Denoising based on Normal Voting Tensor and Binary Optimization.
Yadav, Sunil Kumar; Reitebuch, Ulrich; Polthier, Konrad
2017-08-17
This paper presents a two-stage mesh denoising algorithm. Unlike other traditional averaging approaches, our approach uses an element-based normal voting tensor to compute smooth surfaces. By introducing a binary optimization on the proposed tensor together with a local binary neighborhood concept, our algorithm better retains sharp features and produces smoother umbilical regions than previous approaches. On top of that, we provide a stochastic analysis on the different kinds of noise based on the average edge length. The quantitative results demonstrate that the performance of our method is better compared to state-of-the-art smoothing approaches.
Torsion tensor and covector in a unified field theory
Chernikov, N.A.
1976-01-01
The Einstein unified field theory is used to solve a tensor equation to provide the unambiguous definition of affine connectedness. In the process of solving the Einstein equation limitations imposed by symmetry on the tensor and the torsion covector as well as on affine connectedness are elucidated. It is demonstrated that in a symmetric case the connectedness is unambiguously determined by the Einstein equation. By means of the Riemann geometry a formula for the torsion covector is derived. The equivalence of Einstein equations to those of the nonlinear Born-Infeld electrodynamics is proved
3D inversion of full tensor magnetic gradiometry (FTMG) data
Zhdanov, Michael; Cai, Hongzhu; Wilson, Glenn
2011-01-01
Following recent advances in SQUID technology, full tensor magnetic gradiometry (FTMG) is emerging as a practical exploration method. We introduce 3D regularized focusing inversion for FTMG data. Our model studies show that inversion of magnetic tensor data can significantly improve resolution...... compared to inversion of magnetic vector data for the same model. We present a case study for the 3D inversion of GETMAG® FTMG data acquired over a magnetite skarn at Tallawang, Australia. The results obtained from our 3D inversion agree very well with the known geology of the area....
Feynman path integral in area tensor Regge calculus and positivity
Khatsymovsky, V.M.
2004-01-01
The versions of quantum measure in the area tensor Regge calculus constructed in the previous paper are studied on the simplest configurations of the system. These are found to be positively defined in the Euclidean case on physical surface corresponding to the ordinary Regge calculus (but not outside this surface), that is, adopt probabilistic interpretation. (Since Euclidean measure is defined via analytical continuation, positivity is not evident property.) An argument for positivity on physical surface on general configurations of area tensor Regge calculus is given
Validation of buoyancy driven spectral tensor model using HATS data
Chougule, A.; Mann, Jakob; Kelly, Mark C.
2016-01-01
We present a homogeneous spectral tensor model for wind velocity and temperature fluctuations, driven by mean vertical shear and mean temperature gradient. Results from the model, including one-dimensional velocity and temperature spectra and the associated co-spectra, are shown in this paper....... The model also reproduces two-point statistics, such as coherence and phases, via cross-spectra between two points separated in space. Model results are compared with observations from the Horizontal Array Turbulence Study (HATS) field program (Horst et al. 2004). The spectral velocity tensor in the model...
Tables of Products of Tensor Operators and Stevens Operators
Lindgård, Per-Anker
1975-01-01
Numerical tables of products of tensor (Racah) operators, Rl,m(J), and Stevens operators Olm(J), working within a J-multiplet are given as a function of X=J(J+1). Examples of the use of the tables, such as the calculation of commutation relations and thermal averages are given.......Numerical tables of products of tensor (Racah) operators, Rl,m(J), and Stevens operators Olm(J), working within a J-multiplet are given as a function of X=J(J+1). Examples of the use of the tables, such as the calculation of commutation relations and thermal averages are given....
The tensor product in Wadler's analysis of lists
Nielson, Flemming; Nielson, Hanne Riis
1992-01-01
We consider abstract interpretation (in particular strictness analysis) for pairs and lists. We begin by reviewing the well-known fact that the best known description of a pair of elements is obtained using the tensor product rather than the cartesian product. We next present a generalisation...... of Wadler's strictness analysis for lists using the notion of open set. Finally, we illustrate the intimate connection between the case analysis implicit in Wadler's strictness analysis and the precision that the tensor product allows for modelling the inverse cons operation....
The tensor product in Wadler's analysis of lists
Nielson, Flemming; Nielson, Hanne Riis
1994-01-01
We consider abstract interpretation (in particular strictness analysis) for pairs and lists. We begin by reviewing the well-known fact that the best known description of a pair of elements is obtained using the tensor product rather than the cartesian product. We next present a generalisation...... of Wadler's strictness analysis for lists (1987) using the notion of open set. Finally, we illustrate the intimate connection between the case analysis implicit in Wadler's strictness analysis and the precision that the tensor product allows for modelling the inverse cons operation...
Perron-Frobenius Theorem for Rectangular Tensors and Directed Hypergraphs
Lu, Linyuan; Yang, Arthur L. B.; Zhao, James J. Y.
2018-01-01
For any positive integers $r$, $s$, $m$, $n$, an $(r,s)$-order $(n,m)$-dimensional rectangular tensor ${\\cal A}=(a_{i_1\\cdots i_r}^{j_1\\cdots j_s}) \\in ({\\mathbb R}^n)^r\\times ({\\mathbb R}^m)^s$ is called partially symmetric if it is invariant under any permutation on the lower $r$ indexes and any permutation on the upper $s$ indexes. Such partially symmetric rectangular tensor arises naturally in studying directed hypergraphs. Ling and Qi [Front. Math. China, 2013] first studied the $(p,q)$-...
Numerical evaluation of the tensor bispectrum in two field inflation
Raveendran, Rathul Nath; Sriramkumar, L.
2017-01-01
We evaluate the dimensionless non-Gaussianity parameter h NL , that characterizes the amplitude of the tensor bispectrum, numerically for a class of two field inflationary models such as double inflation, hybrid inflation and aligned natural inflation. We compare the numerical results with the slow roll results which can be obtained analytically. In the context of double inflation, we also investigate the effects on h NL due to curved trajectories in the field space. We explicitly examine the validity of the consistency relation governing the tensor bispectrum in the squeezed limit. Lastly, we discuss the contribution to h NL due to the epoch of preheating in two field models.
Quark-gluon mixing in pseudoscalar and tensor mesons
Eremyan, Sh.S.; Nazaryan, A.E.
1986-01-01
A mixing model of quark-antiquark ang gluonium states in η, η', i(1440) pseudoscalar and f, f', Θ(1690) tensor mesons is considered. Description of and predictions for 68 two-particle decays with these particles taking part in them are obtained. It is shown that i(1440) by 85% consists of gluonium and Θ(1690) is a pure gluonic state. The quark-gluon and gluon-gluon couplings in the pseudoscalar sector are obtained to be stronger as compared to the corresponding ones in the tensor case
The deuteron bound state wave function with tensor forces
Takemasa, Tadashi
1991-01-01
A FORTRAN program named DEUTERON is developed to calculate the binding energy and wave function of a deuteron, when the interaction between two nucleons is described in terms of central, tensor, spin-orbit, and quadratic LS potentials with or without a hard core. An important use of the program is to provide the deuteron wave function required in nuclear reaction calculations involving a deuteron. Also, this program may be employed in nuclear Hartree-Fock calculations using an effective nucleon-nucleon interaction with a tensor component. (author)
Relativistic New Yukawa-Like Potential and Tensor Coupling
Ikhdair, S.M.; Hamzavi, M.
2012-01-01
We approximately solve the Dirac equation for a new suggested generalized inversely quadratic Yukawa potential including a Coulomb-like tensor interaction with arbitrary spin-orbit coupling quantum number κ. In the framework of the spin and pseudo spin (p-spin) symmetry, we obtain the energy eigenvalue equation and the corresponding eigenfunctions, in closed form, by using the parametric Nikiforov-Uvarov method. The numerical results show that the Coulomb-like tensor interaction, -T/r, removes degeneracies between spin and p-spin state doublets. The Dirac solutions in the presence of exact spin symmetry are reduced to Schroedinger solutions for Yukawa and inversely quadratic Yukawa potentials. (author)
Nenadić, Igor; Hoof, Anna; Dietzek, Maren; Langbein, Kerstin; Reichenbach, Jürgen R; Sauer, Heinrich; Güllmar, Daniel
2017-08-30
Both schizophrenia and bipolar disorder show abnormalities of white matter, as seen in diffusion tensor imaging (DTI) analyses of major brain fibre bundles. While studies in each of the two conditions have indicated possible overlap in anatomical location, there are few direct comparisons between the disorders. Also, it is unclear whether phenotypically similar subgroups (e.g. patients with bipolar disorder and psychotic features) might share white matter pathologies or be rather similar. Using region-of-interest (ROI) analysis of white matter with diffusion tensor imaging (DTI) at 3 T, we analysed fractional anisotropy (FA), radial diffusivity (RD), and apparent diffusion coefficient (ADC) of the corpus callosum and cingulum bundle in 33 schizophrenia patients, 17 euthymic (previously psychotic) bipolar disorder patients, and 36 healthy controls. ANOVA analysis showed significant main effects of group for RD and ADC (both elevated in schizophrenia). Across the corpus callosum ROIs, there was not group effect on FA, but for RD (elevated in schizophrenia, lower in bipolar disorder) and ADC (higher in schizophrenia, intermediate in bipolar disorder). Our findings show similarities and difference (some gradual) across regions of the two major fibre tracts implicated in these disorders, which would be consistent with a neurobiological overlap of similar clinical phenotypes. Copyright © 2017 Elsevier Ireland Ltd. All rights reserved.
Q-creation and annihilation tensors for the two parameters deformation of U(SU(2))
Wehrhahn, R.F.; Vraceanu, D.
1993-03-01
The Jordan-Schwinger construction for the Hopf algebra U qp (su(2)) is realized. The creation and annihilation tensor operators together with their tensor products including the Casimir operators are calculated. (orig.)
a tensor theory of gravitation in a curved metric on a flat background
Drummond, J.E.
1979-01-01
A theory of gravity is proposed using a tensor potential for the field on a flat metric. This potential cannot be isolated by local observations, but some details can be deduced from measurements at a distance. The requirement that the field equations for the tensor potential shall be deducible from an action integral, that the action and field equations are gauge invariant, and, conversely, that the Lagrangian in the action integral can be integrated from the field equations leads to Einstein's field equations. The requirement that the field energy-momentum tensor exists leads to a constraint on the tensor potential. If the constraint is a differential gauge condition, then it can only be the Hilbert condition giving a unique background tensor, metric tensor and tensor potential. For a continuous field inside a solid sphere the metric must be homogeneous in the spatial coordinates, and the associated field energy-momentum tensor has properties consistent with Newtonian dynamics. (author)
Kim, Youn Hwan; Kim, Sang Wha; Kim, Jeong Tae; Kim, Chang Yeon
2013-06-01
Tensor fascia lata (TFL) musculocutaneous flaps often require a donor site graft when harvesting a large flap. However, a major drawback is that it also sacrifices the muscle. To overcome this disadvantage, we designed a TFL perforator-based island flap that was harvested from a site near the defect and involved transposition within 90 degrees without full isolation of the pedicles. We performed procedures on 17 musculocutaneous flaps and 23 perforator-based island flaps, and compared the outcomes of these surgeries. The overall complication rate was 27.5% (11 regions). There were 7 complications related to the musculocutaneous flaps and 4 complications related to the perforator flaps. Although there were no statistical differences between those groups, lower complication rates were associated with procedures involving perforator flaps. The TFL perforator procedure is a simple and fast operation that avoids sacrificing muscle. This decreases complication rates compared to true perforator flap techniques that require dissection around the perforator or pedicle.
Top-down approach in protein RDC data analysis: de novo estimation of the alignment tensor
Chen Kang; Tjandra, Nico
2007-01-01
In solution NMR spectroscopy the residual dipolar coupling (RDC) is invaluable in improving both the precision and accuracy of NMR structures during their structural refinement. The RDC also provides a potential to determine protein structure de novo. These procedures are only effective when an accurate estimate of the alignment tensor has already been made. Here we present a top-down approach, starting from the secondary structure elements and finishing at the residue level, for RDC data analysis in order to obtain a better estimate of the alignment tensor. Using only the RDCs from N-H bonds of residues in α-helices and CA-CO bonds in β-strands, we are able to determine the offset and the approximate amplitude of the RDC modulation-curve for each secondary structure element, which are subsequently used as targets for global minimization. The alignment order parameters and the orientation of the major principal axis of individual helix or strand, with respect to the alignment frame, can be determined in each of the eight quadrants of a sphere. The following minimization against RDC of all residues within the helix or strand segment can be carried out with fixed alignment order parameters to improve the accuracy of the orientation. For a helical protein Bax, the three components A xx , A yy and A zz , of the alignment order can be determined with this method in average to within 2.3% deviation from the values calculated with the available atomic coordinates. Similarly for β-sheet protein Ubiquitin they agree in average to within 8.5%. The larger discrepancy in β-strand parameters comes from both the diversity of the β-sheet structure and the lower precision of CA-CO RDCs. This top-down approach is a robust method for alignment tensor estimation and also holds a promise for providing a protein topological fold using limited sets of RDCs
A Closed-Form Solution to Tensor Voting: Theory and Applications
Wu, Tai-Pang; Yeung, Sai-Kit; Jia, Jiaya; Tang, Chi-Keung; Medioni, Gerard
2016-01-01
We prove a closed-form solution to tensor voting (CFTV): given a point set in any dimensions, our closed-form solution provides an exact, continuous and efficient algorithm for computing a structure-aware tensor that simultaneously achieves salient structure detection and outlier attenuation. Using CFTV, we prove the convergence of tensor voting on a Markov random field (MRF), thus termed as MRFTV, where the structure-aware tensor at each input site reaches a stationary state upon convergence...
MathGR: a tensor and GR computation package to keep it simple
Wang, Yi
2013-01-01
We introduce the MathGR package, written in Mathematica. The package can manipulate tensor and GR calculations with either abstract or explicit indices, simplify tensors with permutational symmetries, decompose tensors from abstract indices to partially or completely explicit indices and convert partial derivatives into total derivatives. Frequently used GR tensors and a model of FRW universe with ADM type perturbations are predefined. The package is built around the philosophy to "keep it si...
Theoretical study of the relativistic molecular rotational g-tensor
Aucar, I. Agustín, E-mail: agustin.aucar@conicet.gov.ar; Gomez, Sergio S., E-mail: ssgomez@exa.unne.edu.ar [Institute for Modeling and Technological Innovation, IMIT (CONICET-UNNE) and Faculty of Exact and Natural Sciences, Northeastern University of Argentina, Avenida Libertad 5400, W3404AAS Corrientes (Argentina); Giribet, Claudia G.; Ruiz de Azúa, Martín C. [Physics Department, Faculty of Exact and Natural Sciences, University of Buenos Aires and IFIBA CONICET, Ciudad Universitaria, Pab. I, 1428 Buenos Aires (Argentina)
2014-11-21
An original formulation of the relativistic molecular rotational g-tensor valid for heavy atom containing compounds is presented. In such formulation, the relevant terms of a molecular Hamiltonian for non-relativistic nuclei and relativistic electrons in the laboratory system are considered. Terms linear and bilinear in the nuclear rotation angular momentum and an external uniform magnetic field are considered within first and second order (relativistic) perturbation theory to obtain the rotational g-tensor. Relativistic effects are further analyzed by carrying out the linear response within the elimination of the small component expansion. Quantitative results for model systems HX (X=F, Cl, Br, I), XF (X=Cl, Br, I), and YH{sup +} (Y=Ne, Ar, Kr, Xe, Rn) are obtained both at the RPA and density functional theory levels of approximation. Relativistic effects are shown to be small for this molecular property. The relation between the rotational g-tensor and susceptibility tensor which is valid in the non-relativistic theory does not hold within the relativistic framework, and differences between both molecular parameters are analyzed for the model systems under study. It is found that the non-relativistic relation remains valid within 2% even for the heavy HI, IF, and XeH{sup +} systems. Only for the sixth-row Rn atom a significant deviation of this relation is found.
Spinors, tensors and the covariant form of Dirac's equation
Chen, W.Q.; Cook, A.H.
1986-01-01
The relations between tensors and spinors are used to establish the form of the covariant derivative of a spinor, making use of the fact that certain bilinear combinations of spinors are vectors. The covariant forms of Dirac's equation are thus obtained and examples in specific coordinate systems are displayed. (author)
Tensor Susceptibilities of the Vacuum from Constituent Quarks
Broniowski, W.; Polyakov, M.; Goeke, K.; Hyun-Chul Kim
1998-01-01
We show that the constituent quark model leads to simple expression for the isoscalar and isovector tensor susceptibilities of the vacuum. The found values are negative and of magnitude compatible with QCD-sum-rule parametrizations of spectral densities in appropriate L=1-meson channels. (author)
Theoretical study of the relativistic molecular rotational g-tensor
Aucar, I. Agustín; Gomez, Sergio S.; Giribet, Claudia G.; Ruiz de Azúa, Martín C.
2014-01-01
An original formulation of the relativistic molecular rotational g-tensor valid for heavy atom containing compounds is presented. In such formulation, the relevant terms of a molecular Hamiltonian for non-relativistic nuclei and relativistic electrons in the laboratory system are considered. Terms linear and bilinear in the nuclear rotation angular momentum and an external uniform magnetic field are considered within first and second order (relativistic) perturbation theory to obtain the rotational g-tensor. Relativistic effects are further analyzed by carrying out the linear response within the elimination of the small component expansion. Quantitative results for model systems HX (X=F, Cl, Br, I), XF (X=Cl, Br, I), and YH + (Y=Ne, Ar, Kr, Xe, Rn) are obtained both at the RPA and density functional theory levels of approximation. Relativistic effects are shown to be small for this molecular property. The relation between the rotational g-tensor and susceptibility tensor which is valid in the non-relativistic theory does not hold within the relativistic framework, and differences between both molecular parameters are analyzed for the model systems under study. It is found that the non-relativistic relation remains valid within 2% even for the heavy HI, IF, and XeH + systems. Only for the sixth-row Rn atom a significant deviation of this relation is found
Relations among the crack growth modes resulting from tensor splitting
Kafka, Vratislav
2015-01-01
Roč. 60, č. 4 (2015), s. 319-335 ISSN 0001-7043 Institutional support: RVO:68378297 Keywords : fracture mechanics * combination of crack-growth modes * non-local effect * tensor splitting Subject RIV: JL - Materials Fatigue, Friction Mechanics http://journal.it.cas.cz/60(15)4-Contents/60(15)4a.pdf
Quasilocal energy and the Bel-Robinson tensor
Krishnasamy, Ilangkovan
1985-01-01
The general-relativistic field equations are examined from the point of view of a local inertial observer and a quasilocal definitions of energy-momentum is thereby obtained. This definition relates to the Bel-Robinson tensor and the approach is shown to be consistent with the result obtained from the definition of energy given by Hawking. (author)
Anisotropic cosmological models and generalized scalar tensor theory
Abstract. In this paper generalized scalar tensor theory has been considered in the background of anisotropic cosmological models, namely, axially symmetric Bianchi-I, Bianchi-III and Kortowski–. Sachs space-time. For bulk viscous fluid, both exponential and power-law solutions have been stud- ied and some assumptions ...
Anisotropic cosmological models and generalized scalar tensor theory
In this paper generalized scalar tensor theory has been considered in the background of anisotropic cosmological models, namely, axially symmetric Bianchi-I, Bianchi-III and Kortowski–Sachs space-time. For bulk viscous ﬂuid, both exponential and power-law solutions have been studied and some assumptions among the ...
Collineations of the curvature tensor in general relativity
Curvature collineations for the curvature tensor, constructed from a fundamental Bianchi Type-V metric, are studied. We are concerned with a symmetry property of space-time which is called curvature collineation, and we briefly discuss the physical and kinematical properties of the models.
Cosmic no-hair conjecture in scalar–tensor theories
We have shown that, within the context of scalar–tensor theories, the anisotropic Bianchi-type cosmological models evolve towards de Sitter Universe. A similar result holds in the case of cosmology in Lyra manifold. Thus the analogue of cosmic no-hair theorem of Wald [1] hold in both the cases. In fact, during inflation there ...
Tensor Susceptibilities of the Vacuum from Constituent Quarks
Broniowski, W [The H. Niewodniczanski Institute of Nuclear Physics, Cracow (Poland); Polyakov, M; Goeke, K [Institute for Theoretical Physics II, Ruhr-Universitaet-Bochum, Bochum (Germany); Kim, Hyun-Chul [Pusan National University, Department of Physics, Pusan (Korea, Republic of)
1998-05-15
We show that the constituent quark model leads to simple expression for the isoscalar and isovector tensor susceptibilities of the vacuum. The found values are negative and of magnitude compatible with QCD-sum-rule parametrizations of spectral densities in appropriate L=1-meson channels. (author)
Dilaton and second-rank tensor fields as supersymmetric compensators
Nishino, Hitoshi; Rajpoot, Subhash
2007-01-01
We formulate a supersymmetric theory in which both a dilaton and a second-rank tensor play roles of compensators. The basic off-shell multiplets are a linear multiplet (B μν ,χ,φ) and a vector multiplet (A μ ,λ;C μνρ ), where φ and B μν are, respectively, a dilaton and a second-rank tensor. The third-rank tensor C μνρ in the vector multiplet is ''dual'' to the conventional D field with 0 on-shell or 1 off-shell degree of freedom. The dilaton φ is absorbed into one longitudinal component of A μ , making it massive. Initially, B μν has 1 on-shell or 3 off-shell degrees of freedom, but it is absorbed into the longitudinal components of C μνρ . Eventually, C μνρ with 0 on-shell or 1 off-shell degree of freedom acquires in total 1 on-shell or 4 off-shell degrees of freedom, turning into a propagating massive field. These basic multiplets are also coupled to chiral multiplets and a supersymmetric Dirac-Born-Infeld action. Some of these results are also reformulated in superspace. The proposed mechanism may well provide a solution to the long-standing puzzle of massless dilatons and second-rank tensors in supersymmetric models inspired by string theory
A scalar-tensor bimetric brane world cosmology
Youm, Donam
2001-08-01
We study a scalar-tensor bimetric cosmology in the Randall-Sundrum model with one positive tension brane, where the biscalar field is assumed to be confined on the brane. The effective Friedmann equations on the brane are obtained and analyzed. We comment on resolution of cosmological problems in this bimetric model. (author)
Visualization of Plant Fibres via Difusion Tensor Imaging
Gescheidtová, E.; Marcon, P.; Bartušek, Karel
2011-01-01
Roč. 7, č. 6 (2011), s. 543-546 ISSN 1931-7360 R&D Projects: GA ČR GAP102/11/0318 Institutional research plan: CEZ:AV0Z20650511 Keywords : plant fibres * difusion weighted imaging * difusion tensor imaging * MR imaging Subject RIV: JA - Electronics ; Optoelectronics, Electrical Engineering
New results for algebraic tensor reduction of Feynman integrals
Fleischer, Jochem [Bielefeld Univ. (Germany). Fakultaet fuer Physik; Riemann, Tord [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Yundin, Valery [Copenhagen Univ. (Denmark). Niels Bohr International Academy and Discovery Center
2012-02-15
We report on some recent developments in algebraic tensor reduction of one-loop Feynman integrals. For 5-point functions, an efficient tensor reduction was worked out recently and is now available as numerical C++ package, PJFry, covering tensor ranks until five. It is free of inverse 5- point Gram determinants and inverse small 4-point Gram determinants are treated by expansions in higher-dimensional 3-point functions. By exploiting sums over signed minors, weighted with scalar products of chords (or, equivalently, external momenta), extremely efficient expressions for tensor integrals contracted with external momenta were derived. The evaluation of 7-point functions is discussed. In the present approach one needs for the reductions a (d +2)-dimensional scalar 5-point function in addition to the usual scalar basis of 1- to 4-point functions in the generic dimension d=4-2{epsilon}. When exploiting the four-dimensionality of the kinematics, this basis is sufficient. We indicate how the (d+2)-dimensional 5-point function can be evaluated. (orig.)