Half-Skyrmions and the Equation of State for Compact-Star Matter
Dong, Huan; Lee, Hyun Kyu; Machleidt, R; Rho, Mannque
2012-01-01
The half-skyrmions that appear in dense baryonic matter when skyrmions are put on crystals modify drastically hadron properties in dense medium and affect strongly the nuclear tensor forces, thereby influencing the equation of state (EoS) of dense nuclear and asymmetric nuclear matter. The matter comprised of half skyrmions has vanishing quark condensate but non-vanishing pion decay constant and could be interpreted as a hadronic dual of strong-coupled quark matter. We infer from this observation a set of new scaling laws -- called "new-BR" -- for the parameters in nuclear effective field theory controlled by renormalization-group flow. They are subjected to the EoS of symmetric and asymmetric nuclear matter, and are then applied to nuclear symmetry energies and properties of compact stars. The changeover from the skyrmion matter to a half-skyrmion matter that takes place after the cross-over density $n_{1/2}$ makes the EoS stiffer and leads to a compact star as massive as $\\sim 2.4M_\\odot$. Cross-over densit...
Half-Skyrmion Theory for High-Temperature Superconductivity
Morinari, Takao
We review the half-Skyrmion theory for copper-oxide high-temperature superconductivity. In the theory, doped holes create a half-Skyrmion spin texture which is characterized by a topological charge. The formation of the half-Skyrmion is described in the single hole doped system, and then the half-Skyrmion excitation spectrum is compared with the angle-resolved photoemission spectroscopy results in the undoped system. Multi-half-Skyrmion configurations are studied by numerical simulations. We show that half-Skyrmions carry non-vanishing topological charge density below a critical hole doping concentration ~ 30% even in the absence of antiferromagnetic long-range order. The magnetic structure factor exhibits incommensurate peaks in stripe ordered configurations of half-Skyrmions and anti-half-Skyrmions. The interaction mediated by half-Skyrmions leads to dx2-y2-wave superconductivity. We also describe pseudogap behavior arising from the excitation spectrum of a composite particle of a half-Skyrmion and doped hole.
The quantum nature of skyrmions and half-skyrmions in Cu2OSeO3
Janson, Oleg; Rousochatzakis, Ioannis; Tsirlin, Alexander A; Belesi, Marilena; Leonov, Andrei A; Rößler, Ulrich K; van den Brink, Jeroen; Rosner, Helge
2014-01-01
.... Here, we achieve this for the first time in the skyrmionic Mott insulator Cu2OSeO3. We show that its magnetic building blocks are strongly fluctuating Cu4 tetrahedra, spawning a continuum theory that culminates in 51 nm large skyrmions, in striking agreement with experiment. One of the further predictions that ensues is the temperature-dependent decay of skyrmions into half-skyrmions.
Skyrmions, half-skyrmions and nucleon mass in dense baryonic matter
Ma, Yong-Liang; Harada, Masayasu; Lee, Hyun Kyu; Oh, Yongseok; Rho, Mannque
2014-04-01
We explore the hadron properties in dense baryonic matter in a unified way by using a Skyrme model constructed with an effective Lagrangian which includes the ρ and ω vector mesons as hidden gauge bosons and is valid up to O(p4) in chiral expansion including the homogeneous Wess-Zumino terms. With the two input values of pion decay constant and the lowest lying vector meson mass which can be fixed in free space, all the other low energy constants in the effective Lagrangian are determined by their master formulas derived from holographic QCD models, which allows us to study the baryonic matter properties with no additional free parameters and thus without ambiguities. We find that the ω field that figures in the homogeneous Wess-Zumino term plays a crucial role in the skyrmion structure and its matter properties. The most striking and intriguing observation is that the pion decay constant that smoothly drops with increasing density in the Skyrmion phase stops decreasing at n1/2 at which the skyrmions in medium fractionize into half-skyrmions and remains nearly constant in the half-skyrmion phase. In accordance with the large Nc consideration, the baryon mass also stays non-scaling in the half-skyrmion phase. This feature is supported by the nuclear effective field theory with the parameters of the Lagrangian scaling modified at the skyrmion-half-skyrmion phase transition. Our exploration also uncovers the crucial role of the ω meson in multi-baryon systems as well as in the structure of a single skyrmion.
Dense baryonic matter in the hidden local symmetry approach: Half-skyrmions and nucleon mass
Ma, Yong-Liang; Harada, Masayasu; Lee, Hyun Kyu; Oh, Yongseok; Park, Byung-Yoon; Rho, Mannque
2013-07-01
Hadron properties in dense medium are treated in a unified way in a skyrmion model constructed with an effective Lagrangian, in which the ρ and ω vector mesons are introduced as hidden gauge bosons, valid up to O(p4) terms in chiral expansion including the homogeneous Wess-Zumino terms. All the low energy constants of the Lagrangian—apart from the pion decay constant and the vector meson mass—are fixed by the master formula derived from the relation between the five-dimensional holographic QCD and the four-dimensional hidden local symmetry Lagrangian. This Lagrangian allows one to pin down the density n1/2 at which the skyrmions in medium fractionize into half-skyrmions, bringing in a drastic change in the equation of state of dense baryonic matter. We find that the U(1) field that figures in the Chern-Simons term in the five-dimensional holographic QCD action or equivalently the ω field in the homogeneous Wess-Zumino term in the dimensionally reduced hidden local symmetry action plays a crucial role in the half-skyrmion phase. The importance of the ω degree of freedom may be connected to what happens in the instanton structure of elementary baryon noticed in holographic QCD. The most striking and intriguing in what is found in the model is that the pion decay constant that smoothly drops with increasing density in the skyrmion phase stops decreasing at n1/2 and remains nearly constant in the half-skyrmion phase. In accordance with the large Nc consideration, the baryon mass also stays nonscaling in the half-skyrmion phase. This feature which is reminiscent of the parity-doublet baryon model with a chirally invariant mass m0 is supported by the nuclear effective field theory with the parameters of the Lagrangian scaling modified at the skyrmion-half-skyrmion phase transition. It also matches with one-loop renormalization group analysis based on hidden local symmetry. A link between a nonvanishing m0 and the origin of nucleon mass distinctive from
Topology Change and Tensor Forces for the EoS of Dense Baryonic Matter
Lee, Hyun Kyu
2013-01-01
When skyrmions representing nucleons are put on crystal lattice and compressed to simulate high density, there is a transition above the normal nuclear matter density $n_0$ from a matter consisting of skyrmions with integer baryon charge to a state of half-skyrmions with half-integer baryon charge. We exploit this observation in an effective field theory formalism to access dense baryonic system. We find that the topology change involved implies a changeover from a Fermi liquid structure to a non-Fermi liquid with the chiral condensate in the nucleon "melted off." The $\\sim 80%$ of the nucleon mass that remains, invariant under chiral transformation, points to the origin of the (bulk of) proton mass that is not encoded in the standard mechanism of spontaneously broken chiral symmetry. The topology change engenders a drastic modification of the nuclear tensor forces, thereby nontrivially affecting the EoS, in particular, the symmetry energy, for compact star matter. It brings in stiffening of the EoS needed to...
Dense Baryonic Matter in Hidden Local Symmetry Approach: Half-Skyrmions and Nucleon Mass
Ma, Yong-Liang; Lee, Hyun Kyu; Oh, Yongseok; Park, Byung-Yoon; Rho, Mannque
2013-01-01
Hadron properties in dense medium are treated in a unified way in a skyrmion model constructed with an effective Lagrangian, in which the rho and omega vector mesons are introduced as hidden gauge bosons, valid up to O(p^4) terms in chiral expansion including the homogeneous Wess-Zumino terms. All the low energy constants - apart from f_pi and m_rho - are fixed by the master formula derived from the relation between 5-D hQCD and 4-D HLS. This allows one to pin down the density n_1/2 at which the skyrmions in medium fractionize into half-skyrmions, bringing in a drastic change in the EoS of dense matter. We find that the U(1) field that figures in the CS term in the hQCD action or equivalently the omega field in the hWZ term in the dimensionally reduced HLS action plays a crucial role in the half-skyrmion phase. The importance of the omega degree of freedom may be connected to what happens in the instanton structure of elementary baryon noticed in hQCD. The most striking and intriguing in what is found in the ...
Radiation Forces and Torques without Stress (Tensors)
Bohren, Craig F.
2011-01-01
To understand radiation forces and torques or to calculate them does not require invoking photon or electromagnetic field momentum transfer or stress tensors. According to continuum electromagnetic theory, forces and torques exerted by radiation are a consequence of electric and magnetic fields acting on charges and currents that the fields induce…
Radiation Forces and Torques without Stress (Tensors)
Bohren, Craig F.
2011-01-01
To understand radiation forces and torques or to calculate them does not require invoking photon or electromagnetic field momentum transfer or stress tensors. According to continuum electromagnetic theory, forces and torques exerted by radiation are a consequence of electric and magnetic fields acting on charges and currents that the fields induce…
Renormalization persistency of tensor force in nuclei
Tsunoda, Naofumi; Tsukiyama, Koshiroh; Hjorth-Jensen, Morten
2011-01-01
In this work we analyze the tensor-force component of effective interactions appropriate for nuclear shell-model studies, with particular emphasis on the monopole term of the interactions. Standard nucleon-nucleon ($NN$) interactions such as AV8' and $\\chi$N$^3$LO are tailored to shell-model studies by employing $V_{low k}$ techniques to handle the short-range repulsion of the $NN$ interactions and by applying many-body perturbation theory to incorporate in-medium effects. We show, via numerical studies of effective interactions for the $sd$ and $pf$ shells, that the tensor-force contribution to the monopole term of the effective interaction is barely changed by these renormalization procedures, resulting in almost the same monopole term as the one of the bare $NN$ interactions. We propose to call this feature {\\it Renormalization Persistency} of the tensor force, as it is a remarkable property of the renormalization and should have many interesting consequences in nuclear systems. For higher multipole terms,...
Hypertriton: Λ⇆Σ conversion and tensor forces
Afnan, I. R.; Gibson, B. F.
1990-06-01
The separable potential equations that describe the hypertriton when ΛN-ΣN coupling and noncentral NN and YN forces are included, are formulated. Numerical solution of the equations for various potential models shows that Λ-Σ conversion in the YN interaction plays a significant role, even in the lightly bound 3ΛH. When the ΣN channel is formally eliminated, the dispersive energy dependence of the resulting ΛN effective interaction is repulsive, whereas the resulting ΛNN three-body force is attractive. The contribution of the ΛN tensor force is shown to depend upon the inclusion of the NN tensor force and the relative sign of the 3S1-3D1 NN and ΛN tensor coupling. Also, a model which supports a ΣN bound state in the continuum appears to severely overbind the 3ΛH system, indicating that such a phenomenon is not present in the K-d-->ΛNπ reaction.
Hypertriton:. Lambda. leftrightarrow. Sigma. conversion and tensor forces
Afnan, I.R. (School of Physical Sciences, The Flinders University of South Australia, Bedford Park, South Australia 5042, (Australia)); Gibson, B.F. (Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (USA))
1990-06-01
The separable potential equations that describe the hypertriton when {Lambda}{ital N}-{Sigma}{ital N} coupling and noncentral {ital NN} and {ital YN} forces are included, are formulated. Numerical solution of the equations for various potential models shows that {Lambda}-{Sigma} conversion in the {ital YN} interaction plays a significant role, even in the lightly bound {sub {Lambda}}{sup 3}H. When the {Sigma}{ital N} channel is formally eliminated, the dispersive energy dependence of the resulting {Lambda}{ital N} effective interaction is repulsive, whereas the resulting {Lambda}{ital NN} three-body force is attractive. The contribution of the {Lambda}{ital N} tensor force is shown to depend upon the inclusion of the {ital NN} tensor force and the relative sign of the {sup 3}{ital S}{sub 1}-{sup 3}{ital D}{sub 1} {ital NN} and {Lambda}{ital N} tensor coupling. Also, a model which supports a {Sigma}{ital N} bound state in the continuum appears to severely overbind the {sub {Lambda}}{sup 3}H system, indicating that such a phenomenon is not present in the {ital K}{sup {minus}}{ital d}{r arrow}{Lambda}{ital N}{pi} reaction.
The role of tensor force in nuclear matter saturation
Banerjee, M K; Banerjee, Manoj K.; Tjon, John A.
1998-01-01
Using a relativistic Dirac-Brueckner analysis the pion contribution to the ground state energy of nuclear matter is studied. Evidence is presented that the role of the tensor force in the saturation mechanism is substantially reduced compared to its dominant role in a usual non-relativistic treatment. The reduction of the pion contribution in nuclear matter is due to many-body effects present in a relativistic treatment. In particular, we show that the damping of OPEP is actually due to the decrease of $M^*/M$ with increasing density.
Interplay between tensor force and deformation in even–even nuclei
Bernard, Rémi N., E-mail: rbernard@ugr.es; Anguiano, Marta
2016-09-15
In this work we study the effect of the nuclear tensor force on properties related with deformation. We focus on isotopes in the Mg, Si, S, Ar, Sr and Zr chains within the Hartree–Fock–Bogoliubov theory using the D1ST2a Gogny interaction. Contributions to the tensor energy in terms of saturated and unsaturated subshells are analyzed. Like–particle and proton–neutron parts of the tensor term are independently examinated. We found that the tensor term may considerably modify the potential energy landscapes and change the ground state shape. We analyze too how the pairing characteristics of the ground state change when the tensor force is included.
Interplay between tensor force and deformation in even-even nuclei
Bernard, Rémi N.; Anguiano, Marta
2016-09-01
In this work we study the effect of the nuclear tensor force on properties related with deformation. We focus on isotopes in the Mg, Si, S, Ar, Sr and Zr chains within the Hartree-Fock-Bogoliubov theory using the D1ST2a Gogny interaction. Contributions to the tensor energy in terms of saturated and unsaturated subshells are analyzed. Like-particle and proton-neutron parts of the tensor term are independently examinated. We found that the tensor term may considerably modify the potential energy landscapes and change the ground state shape. We analyze too how the pairing characteristics of the ground state change when the tensor force is included.
Gravitational self-force in scalar-tensor gravity
Zimmerman, Peter
2015-01-01
Motivated by the discovery of floating orbits and the potential to provide extra constraints on alternative theories, in this paper we derive the self-force equation for a small compact object moving on an accelerated world line in a background spacetime which is a solution of the coupled gravitational and scalar field equations of scalar-tensor theory. In the Einstein frame, the coupled field equations governing the perturbations sourced by the particle share the same form as the field equations for perturbations of a scalarvac spacetime, with both falling under the general class of hyperbolic field equations studied by Zimmerman and Poisson. Here, we solve the field equations formally in terms of retarded Green functions, which have explicit representations as Hadamard forms in the neighbourhood of the world line. Using a quasi-local expansion of the Hadamard form, we derive the regular solutions in Fermi normal coordinates according to the Detweiler-Whiting prescription. To compute the equation of motion, ...
Nuclear response functions with finite range Gogny force: tensor terms and instabilities
De Pace, A
2016-01-01
A fully-antisymmetrized random phase approximation calculation employing the continued fraction technique is performed to study nuclear matter response functions with the finite range Gogny force. The most commonly used parameter sets of this force, as well as some recent generalizations that include the tensor terms are considered and the corresponding response functions are shown. The calculations are performed at the first and second order in the continued fraction expansion and the explicit expressions for the second order tensor contributions are given. Comparison between first and second order continued fraction expansion results are provided. The differences between the responses obtained at the two orders turn to be more pronounced for the forces including tensor terms than for the standard Gogny ones. In the vector channels the responses calculated with Gogny forces including tensor terms are characterized by a large heterogeneity, reflecting the different choices for the tensor part of the interacti...
Nuclear response functions with finite-range Gogny force: Tensor terms and instabilities
De Pace, A.; Martini, M.
2016-08-01
A fully antisymmetrized random phase approximation calculation employing the continued fraction technique is performed to study nuclear matter response functions with the finite-range Gogny force. The most commonly used parameter sets of this force, as well as some recent generalizations that include the tensor terms, are considered and the corresponding response functions are shown. The calculations are performed at first and second order in the continued fraction expansion and the explicit expressions for the second-order tensor contributions are given. Comparisons between first- and second-order continued fraction expansion results are provided. The differences between the responses obtained at the two orders turn out to be more pronounced for the forces including tensor terms than for the standard Gogny ones. In the vector channels the responses calculated with Gogny forces including tensor terms are characterized by a large heterogeneity, reflecting the different choices for the tensor part of the interaction. For the sake of comparison the response functions obtained considering a G -matrix-based nuclear interaction are also shown. As a first application of the present calculation, the possible existence of spurious finite-size instabilities of the Gogny forces with or without tensor terms has been investigated. The positive conclusion is that all the Gogny forces but the GT2 one are free of spurious finite-size instabilities. In perspective, the tool developed in the present paper can be inserted in the fitting procedure to construct new Gogny-type forces.
Systematic Study of Tensor Force Effect on Pseudospin Orbital Splittings in Sn Isotopes
WANG; Yan-zhao; YU; Guo-liang; LI; Zhen-yu; GU; Jian-zhong
2012-01-01
<正>The tensor force is a noncentral and nonlocal spin-spin coupling term of the nucleon-nucleon interaction whose effect on the nuclear structure has been discussed in the framework of the self-consistent mean field approaches and the shell model in the past a few years. Based on the Skyrme Hartree-Fock-Bogoliubov approach, we systematically investigated the role of the tensor force on the
The structure of the spherical tensor forces in the USD and GXPF1A shell model Hamiltonians
WANG Han-Kui; GAO Zao-Chun; CHEN Yong-Shou; GUO Jian-You; CHEN Yong-Jing; TU Ya
2011-01-01
The realistic shell model Hamiltonians, USD and GXPF1A, have been transformed from the particle-particle (normal) representation to the particle-hole representation (multipole-multipole)by using the known formulation in Ref. [1].The obtained multipole-multipole terms were compared with the known spherical tensor forces, including the coupled ones. It is the first time the contributions of the coupled tensor forces to the shell model Hamiltonian have been investigated. It has been shown that some coupled-tensor forces, such as [r2Y2σ]1,also give important contributions to the shell model Hamiltonian.
Pion tensor force and nuclear binding energy in the relativistic Hartree-Fock formalism
Marcos, S.; López-Quelle, M.; Niembro, R.; Savushkin, L. N.
2014-03-01
The binding energies of several isotopic families are studied within the relativistic Hartree-Fock approximation with the pseudovector coupling for the πN vertex, to find out a suitable strength for the effective pion tensor force (EPTF). An approximation for determining separately the contributions of the central and tensor forces generated by pion is considered. The results for heavy nuclei indicate that a realistic strength for the EPTF is smaller than a half of that appearing in the OPEP. This conclusion also applies to the results for the single-particle energies. Besides, it has been found that there is a genuine relativistic contribution of the EPTF in nuclear matter which is small but significant.
The force density and the kinetic energy-momentum tensor of electromagnetic fields in matter
Medina, Rodrigo
2014-01-01
We determine the invariant expression for the force density that the electromagnetic field exerts on dipolar matter. We construct the non-symmetric energy-momentum tensor of the electromagnetic field in matter which is consistent with that force and with Maxwell equations. We recover Minkowski's expression for the momentum density. We use our results to discuss momentum exchange of an electromagnetic wave-packet which falls into a dielectric block. In particular we show that the wave-packet pulls the block when it enters and drags it when it leaves.
Elking, Dennis M
2016-08-15
New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3(n) redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θpqr are given by introducing Cartesian tensor rotation matrix elements X(R). A polynomial expression and a recursion relation for X(R) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D(R). The expressions for X(R) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.
Delineating effects of tensor force on the density dependence of nuclear symmetry energy
Xu, Chang; Li, Bao-An
2012-01-01
In this talk, we report results of our recent studies to delineate effects of the tensor force on the density dependence of nuclear symmetry energy within phenomenological models. The tensor force active in the isosinglet neutron-proton interaction channel leads to appreciable depletion/population of nucleons below/above the Fermi surface in the single-nucleon momentum distribution in cold symmetric nuclear matter (SNM). We found that as a consequence of the high momentum tail in SNM the kinetic part of the symmetry energy $E^{kin}_{sym}(\\rho)$ is significantly below the well-known Fermi gas model prediction of approximately $12.5 (\\rho/\\rho_0)^{2/3}$. With about 15% nucleons in the high momentum tail as indicated by the recent experiments at J-Lab by the CLAS Collaboration, the $E^{kin}_{sym}(\\rho)$ is negligibly small. It even becomes negative when more nucleons are in the high momentum tail in SNM. These features have recently been confirmed by three independent studies based on the state-of-the-art micros...
Representation of light pressure resultant force and moment as a tensor series
Nerovny, Nikolay; Zimin, Vladimir; Fedorchuk, Sergey; Golubev, Evgeny
2017-08-01
In this article, we address the problem of the determination of light pressure upon space structures with a complex geometric shape. For each surface element, we enforce a condition that it can interact with light only from its front side, a condition represented in the form of series of Chebyshev polynomials of the first kind. This Chebyshev expansion enables the use of a series of tensors of increasing rank for determination of the force and moment acting on the sail. We obtain expressions for the determination of light pressure on space structures of complex geometry, taking into account self-shadowing and reflections within the structure. We also give the expressions for tensor parametrization using the specularity coefficient in case of specular -diffuse reflection. For these expressions, we calculated the principal moment and force upon two-sided flat solar sail, spherical and cylindrical bodies, and approximated light pressure upon the proposed space-based observatory Millimetron. The proposed expressions can be used in the ballistic analysis of solar sails and other space objects significantly affected by radiation pressure. Also, these results can be used to analyze the dynamics of large-scale space structures around their center of gravity under light pressure.
Minotti, F O
2013-01-01
The scalar-tensor theory of gravitation proposed by Mbelek and Lachi\\`{e}ze-Rey have been recently shown to lead to some remarkable effects beyond those initially explored by its authors. These new effects include a possible explanation of the forces measured in asymmetric resonant microwave cavities, and the variation of the amplitude of electromagnetic waves as they propagate in static electric or magnetic fields. These rather unique effects are excelent candidates for laboratory tests of this particular type of scalar-tensor theory. In the present work we introduce an additional remarkable effect of the theory: the generation of pulsed gravitational forces by transient, quasi-stationary electromagnetic fields. In particular, we explore the possible measurable effects of the simple experiment of turning on and off the current in a coil. We show that with the proposed values of the constant in the theory, this effect could resonantly excite a pendulum oscillation up to an easily measurable magnitude.
Central and tensor components of three-nucleon forces in low-energy proton-deuteron scattering
Ishikawa, S; Iseri, Y
2003-01-01
Contributions of three-nucleon forces (3NF) to proton-deuteron scattering observables at energies below the deuteron breakup threshold are studied by solving the Faddeev equation that includes the Coulomb interaction. At E_p=3.0 MeV, we find that the central part of a two-pion exchange 3NF removes the discrepancy between measured cross sections and the calculated ones by two-nucleon forces, and improves the agreement with T_{22} experimental data. However, the tensor part of the 3NF fails in reproducing data of the analyzing power T_{21} by giving worse agreement between the measured and the calculated. Detailed examinations of scattering amplitudes suggest that a P-wave contribution in spin quartet tensor amplitudes has unsuitable sign for reproducing the T_{21} data.
Neutron-proton scattering observables at 325 MeV, the epsilon/sub 1/ parameter, and the tensor force
Chulick, G.S.; Elster, C.; Machleidt, R.; Picklesimer, A.; Thaler, R.M.
1988-04-01
The sensitivity of neutron-proton elastic scattering observables to variations in the low angular momentum T = 0 phase shifts is studied at E/sub lab/ = 325 MeV. It is found that the J = 1 coupling parameter epsilon/sub 1/ is not well determined by existing data. This uncertainty in epsilon/sub 1/ permits models with quite different tensor forces to describe the extant data. Implications and possible experimental resolution of such ambiguities are discussed.
Neutron-proton scattering observables at 325 MeV, the ɛ1 parameter, and the tensor force
Chulick, G. S.; Elster, Ch.; Machleidt, R.; Picklesimer, A.; Thaler, R. M.
1988-04-01
The sensitivity of neutron-proton elastic scattering observables to variations in the low angular momentum T=0 phase shifts is studied at Elab=325 MeV. It is found that the J=1 coupling parameter ɛ1 is not well determined by existing data. This uncertainty in ɛ1 permits models with quite different tensor forces to describe the extant data. Implications and possible experimental resolution of such ambiguities are discussed.
Shape transitions in exotic Si and S isotopes and tensor-force-driven Jahn-Teller effect
Utsuno, Yutaka; Brown, B Alex; Honma, Michio; Mizusaki, Takahiro; Shimizu, Noritaka
2012-01-01
We show how shape transitions in the neutron-rich exotic Si and S isotopes occur in terms of shell-model calculations with a newly constructed Hamiltonian based on V_MU interaction. We first compare the calculated spectroscopic-strength distributions for the proton 0d_5/2,3/2 and 1s_1/2 orbitals with results extracted from a 48Ca(e,e'p) experiment to show the importance of the tensor-force component of the Hamiltonian. Detailed calculations for the excitation energies, B(E2) and two-neutron separation energies for the Si and S isotopes show excellent agreement with experimental data. The potential energy surface exhibits rapid shape transitions along the isotopic chains towards N=28 that are different for Si and S. We explain the results in terms of an intuitive picture involving a Jahn-Teller-type effect that is sensitive to the tensor-force-driven shell evolution. The closed sub-shell nucleus 42Si is a particularly good example of how the tensor-force-driven Jahn-Teller mechanism leads to a strong oblate ra...
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. Writte
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. Writt
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.
Effects of the tensor force on the ground and first $2^{+}$ states of the magic $^{54}$Ca nucleus
Yüksel, E; Khan, E; Bozkurt, K
2014-01-01
The magic nature of the $^{54}$Ca nucleus is investigated in the light of the recent experimental results. We employ both HFB and HF+BCS methods using Skyrme-type SLy5, SLy5+T and T44 interactions. The evolution of the single-particle spectra is studied for the N=34 isotones: $^{60}$Fe, $^{58}$Cr, $^{56}$Ti and $^{54}$Ca. An increase is obtained in the neutron spin-orbit splittings of $p$ and $f$ states due to the effect of the tensor force which also makes $^{54}$Ca a magic nucleus candidate. QRPA calculations on top of HF+BCS are performed to investigate the first $J^{\\pi}$=$2^{+}$ states of the calcium isotopic chain. A good agreement for excitation energies is obtained when we include the tensor force in the mean-field part of the calculations. The first $2^{+}$ states indicate a subshell closure for both $^{52}$Ca and $^{54}$Ca nuclei. We confirm that the tensor part of the interaction is quite essential in explaining the neutron subshell closure in $^{52}$Ca and $^{54}$Ca nuclei.
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. Written in Python, it aims at following the same standard adopted by the main projects of the Python scientific community and fully integrating with these. It allows for fast and straightforward tensor d...
S, Hajjawi; J, Buitrago [Instituto de AstrofIsica de Canarias, C/VIa Lactea, s/n 38205, La Laguna, Tenerife (Spain)
2007-05-15
We here show that the special, relativistic dynamical equation of the Lorentz-force-type, usually considered as a semi-empyrical result, arises from the geometry of the Minkowski-Space-Time. By re-formulating this result in spinorial lenguage, regarding spinors as being more fundamental objects than four-vectors, we obtain a set of dynamical Weyl spinors equations inducing this Lorentz-force-like-equation and geometrically obtain the spinorial form of the Electromagnetic Tensor. This representation alone actually reveals some properties of the free electromagnetic field that, within the context of the standard tensorial calculus, are known only by solving the second-order, partial-di.erential wave equation. Finally, we find that the spinorial equations of motion obtained, inducing the Lorentz-force-equation, do not only describe the evolution of the four-momentum but, surprisingly, also that of some additional degrees of freedom that may be associated to an intrinsic angular momentum.
Hajjawi, S.; Buitrago, J.
2007-05-01
We here show that the special, relativistic dynamical equation of the Lorentz-force-type, usually considered as a semi-empyrical result, arises from the geometry of the Minkowski-Space-Time. By re-formulating this result in spinorial lenguage, regarding spinors as being more fundamental objects than four-vectors, we obtain a set of dynamical Weyl spinors equations inducing this Lorentz-force-like-equation and geometrically obtain the spinorial form of the Electromagnetic Tensor. This representation alone actually reveals some properties of the free electromagnetic field that, within the context of the standard tensorial calculus, are known only by solving the second-order, partial-di.erential wave equation. Finally, we find that the spinorial equations of motion obtained, inducing the Lorentz-force-equation, do not only describe the evolution of the four-momentum but, surprisingly, also that of some additional degrees of freedom that may be associated to an intrinsic angular momentum.
Tensor-force-driven Jahn-Teller effect and shape transitions in exotic Si isotopes
Utsuno, Yutaka; Brown, B Alex; Honma, Michio; Mizusaki, Takahiro; Shimizu, Noritaka
2012-01-01
We show how the shape evolution of the neutron-rich exotic Si and S isotopes can be understood as a Jahn-Teller effect that comes in part from the tensor-driven evolution of single-particle energies. The detailed calculations we present are in excellent agreement with known experimental data, and we point out of new features that should be explored in new experiments. Potential energy surfaces are used to understand the shape evolutions. The sub-shell closed nucleus, $^{42}$Si, is shown to be a perfect example of a strongly oblate shape instead of a sphere through a robust Jahn-Teller mechanism. The distribution of spectroscopic factors measured by $^{48}$Ca(e,e'p) experiment is shown to be well described, providing a unique test on the tensor-driven shell evolution.
Milton, Kimball A; Parashar, Prachi; Kalauni, Pushpa; Murphy, Taylor
2016-01-01
Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporal-spatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth- and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a pote...
Milton, Kimball A.; Fulling, Stephen A.; Parashar, Prachi; Kalauni, Pushpa; Murphy, Taylor
2016-04-01
Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporal-spatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth- and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for z 0 , for z >0 . Previously, the stress tensor had been computed outside of the wall, whereas now we compute all components of the stress tensor in the interior of the wall. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, α =1 and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).
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.)
Budzanowski, A; Hawranek, P; Jahn, R; Jha, V; Kilian, K; Kirillov, Da; Kirillov, Di; Kliczewski, S; Kolev, D; Kravcikova, M; Lesiak, M; Lieb, J; Machner, H; Magiera, A; Maier, R; Martinská, G; Nedev, S; Niskanen, J A; Piskunov, N; Protic, D; Ritman, 6 J; Von Rossen, P; Roy, B J; Sitnik, I; Siudak, R; Stein, H J; Tsenov, R; Urbán, J; Vankova, 2 G; Wilkin, C
2009-01-01
Inclusive measurements of pion production in proton--proton collisions in the forward direction were undertaken at 400 and 600 MeV at COSY using the Big Karl spectrograph. The high resolution in the $\\pi^+$ momentum ensured that there was an unambiguous separation of the $pp\\to {\\pi}^+d/\\pi^+pn$ channels. Using these and earlier data, the ratio of the production cross sections could be followed through the $\\Delta$ region and compared with the predictions of final state interaction theory. Deviations are strongly influenced by long-range terms in the production operator and the tensor force in the final $pn$ system. These have been investigated in a realistic $pp\\to\\pi^+d/\\pi^+pn$ calculation that includes $S \\rightleftharpoons D$ channel coupling between the final nucleons. A semi-quantitative understanding of the observed effects is achieved.
Etingof, Pavel; Nikshych, Dmitri; Ostrik, Victor
2015-01-01
Is there a vector space whose dimension is the golden ratio? Of course not-the golden ratio is not an integer! But this can happen for generalizations of vector spaces-objects of a tensor category. The theory of tensor categories is a relatively new field of mathematics that generalizes the theory of group representations. It has deep connections with many other fields, including representation theory, Hopf algebras, operator algebras, low-dimensional topology (in particular, knot theory), homotopy theory, quantum mechanics and field theory, quantum computation, theory of motives, etc. This bo
Kukulin, V I; Cooper, S G; Dubovichenko, S B
1998-01-01
Improved potential solutions are presented for the inverse scattering problem for $d$+$^4$He data. The input for the inversions includes both the data of recent phase shift analyses and phase shifts from RGM coupled-channel calculations based on the NN Minnesota force. The combined calculations provide a more reliable estimate of the odd-even splitting of the potentials than previously found, suggesting a rather moderate role for this splitting in deuteron-nucleus scattering generally. The approximate parity-independence of the deuteron optical potentials is shown to arise from the nontrivial interference between antisymmetrization and channel coupling to the deuteron breakup channels. A further comparison of the empirical potentials established here and the double folding potential derived from the M3Y effective NN force (with the appropriate normalisation factor) reveals strong similarities. This result supports the application of the double folding model, combined with a small Majorana component, to the de...
Arai, K.; Aoyama, S.; Suzuki, Y.; Descouvemont, P.; Baye, D. [Division of General Education, Nagaoka National College of Technology, 888 Nishikatakai, Nagaoka, Niigata, 940-8532 (Japan); Center for Academic Information Service, Niigata University, Niigata 950-2181 (Japan); Department of Physics, Niigata University, Niigata 950-2181, Japan and RIKEN Nishina Center, Wako 351-0198 (Japan); Physique Nucleaire Theorique et Physique Mathematique, C.P.229, Universite Libre de Bruxelles, B 1050 Brussels (Belgium); Physique Quantique, CP165/82, Universite Libre de Bruxelles, B-1050 Brussels (Belgium)
2012-11-12
The {sup 2}H(d,p){sup 3}H, {sup 2}H(d,n){sup 3}He, and {sup 2}H(d,{gamma}){sup 4}He reactions at low energies are studied with realistic nucleon-nucleon interactions in an ab initio approach. The obtained astrophysical S-factors are all in very good agreement with experiment. The most important channels for both transfer and radiative capture are all found to dominate thanks to the tensor force.
Strictly nonnegative tensors and nonnegative tensor partition
HU ShengLong; HUANG ZhengHai; QI LiQun
2014-01-01
We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su？cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.
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....
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-...
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.
Gurau, Razvan
2016-09-01
This article is preface to the SIGMA special issue ''Tensor Models, Formalism and Applications'', http://www.emis.de/journals/SIGMA/Tensor_Models.html. The issue is a collection of eight excellent, up to date reviews on random tensor models. The reviews combine pedagogical introductions meant for a general audience with presentations of the most recent developments in the field. This preface aims to give a condensed panoramic overview of random tensors as the natural generalization of random matrices to higher dimensions.
Applications of tensor analysis
McConnell, A J
2011-01-01
Standard work applies tensorial methods to subjects within realm of advanced college mathematics. Text explains fundamental ideas and notation of tensor theory; covers geometrical treatment of tensor algebra; introduces theory of differentiation of tensors; and applies mathematics to dynamics, electricity, elasticity and hydrodynamics. 685 exercises, most with answers.
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.
Renormalizable Tensor Field Theories
Geloun, Joseph Ben
2016-01-01
Extending tensor models at the field theoretical level, tensor field theories are nonlocal quantum field theories with Feynman graphs identified with simplicial complexes. They become relevant for addressing quantum topology and geometry in any dimension and therefore form an interesting class of models for studying quantum gravity. We review the class of perturbatively renormalizable tensor field theories and some of their features.
Electromagnetic Stress Tensor in Ponderable Media
Mansuripur, Masud
2014-01-01
We derive an expression for the Maxwell stress tensor in a magnetic dielectric medium specified by its permittivity "epsilon" and permeability "mu." The derivation proceeds from the generalized form of the Lorentz law, which specifies the force exerted by the electromagnetic E and H fields on the polarization P and magnetization M of ponderable media. Our stress tensor differs from the well-known tensors of Abraham and Minkowski, which have been at the center of a century-old controversy surrounding the momentum of the electromagnetic field in transparent materials.
Irreducible Killing Tensors from Third Rank Killing-Yano Tensors
Popa, Florian Catalin; Tintareanu-Mircea, Ovidiu
2006-01-01
We investigate higher rank Killing-Yano tensors showing that third rank Killing-Yano tensors are not always trivial objects being possible to construct irreducible Killing tensors from them. We give as an example the Kimura IIC metric were from two rank Killing-Yano tensors we obtain a reducible Killing tensor and from third rank Killing-Yano tensors we obtain three Killing tensors, one reducible and two irreducible.
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.
Metzler, S; Miettinen, P
2015-01-01
Tensor factorizations are computationally hard problems, and in particular, are often significantly harder than their matrix counterparts. In case of Boolean tensor factorizations -- where the input tensor and all the factors are required to be binary and we use Boolean algebra -- much of that hardness comes from the possibility of overlapping components. Yet, in many applications we are perfectly happy to partition at least one of the modes. In this paper we investigate what consequences doe...
Gurau, Razvan
2016-01-01
Preface to the SIGMA special issue "Tensor Models, Formalism and Applications." The SIGMA special issue "Tensor Models, Formalism and Applications" is a collection of eight excellent, up to date reviews \\cite{Ryan:2016sundry,Bonzom:2016dwy,Rivasseau:2016zco,Carrozza:2016vsq,Krajewski:2016svb,Rivasseau:2016rgt,Tanasa:2015uhr,Gielen:2016dss} on random tensor models. The reviews combine pedagogical introductions meant for a general audience with presentations of the most recent developments in the field. This preface aims to give a condensed panoramic overview of random tensors as the natural generalization of random matrices to higher dimensions.
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
On the tensor Permutation Matrices
Rakotonirina, Christian
2011-01-01
A property that tensor permutation matrices permutate tensor product of rectangle matrices is shown. Some examples, in the particular case of tensor commutation matrices, for studying some linear matricial equations are given.
Mantica, Carlo A
2012-01-01
The algebraic condition of Riemann compatibility for symmetric tensors generalizes the differential Codazzi condition, but preserves much of the geometric content. The compatibility condition can be extended to other curvature tensors. This paper is about Weyl compatible tensors and vectors. In particular it is shown that the existence of a Weyl compatible vector implies the Weyl tensor to be algebraically special, and it is a necessary and sufficient condition for the magnetic part to vanish. Some theorems (Derdzinski and Shen, Hall) are extended to the broader hypothesis of Weyl or Riemann compatibility. Weyl compatibility includes conditions that were investigated in the literature of general relativity (as McIntosh et al.). Hypersurfaces of pseudo Euclidean spaces provide a simple example of Weyl compatible Ricci tensor.
Skyrmions, half-skyrmions and nucleon mass in dense baryonic matter
Ma, Yong-Liang; Lee, Hyun Kyu; Oh, Yongseok; Rho, Mannque
2013-01-01
We explore the hadron properties in dense baryonic matter in a unified way by using a Skyrme model constructed with an effective Lagrangian which includes the $\\rho$ and $\\omega$ vector mesons as hidden gauge bosons and is valid up to $O(p^4)$ in chiral expansion including the homogeneous Wess-Zumino terms. With the two input values of pion decay constant and the lowest lying vector meson mass which can be fixed in free space, all the other low energy constants in the effective Lagrangian are determined by their master formulas derived from holographic QCD models, which allows us to study the baryonic matter properties with no additional free parameters and thus without ambiguities. We find that the $\\omega$ field that figures in the homogeneous Wess-Zumino term plays a crucial role in the skyrmion structure and its matter properties. The most striking and intriguing observation is that the pion decay constant that smoothly drops with increasing density in the Skyrmion phase stops decreasing at $n_{1/2}^{}$ a...
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.
Graybill, George
2007-01-01
Forces are at work all around us. Discover what a force is, and different kinds of forces that work on contact and at a distance. We use simple language and vocabulary to make this invisible world easy for students to ""see"" and understand. Examine how forces ""add up"" to create the total force on an object, and reinforce concepts and extend learning with sample problems.
Obtaining the Weyl tensor from the Bel-Robinson tensor
Ferrando, Joan J; 10.1007/s10714-009-0921-8
2010-01-01
The algebraic study of the Bel-Robinson tensor proposed and initiated in a previous work (Gen. Relativ. Gravit. {\\bf 41}, see ref [11]) is achieved. The canonical form of the different algebraic types is obtained in terms of Bel-Robinson eigen-tensors. An algorithmic determination of the Weyl tensor from the Bel-Robinson tensor is presented.
Tensor Network Skeletonization
Ying, Lexing
2016-01-01
We introduce a new coarse-graining algorithm, tensor network skeletonization, for the numerical computation of tensor networks. This approach utilizes a structure-preserving skeletonization procedure to remove short-range correlations effectively at every scale. This approach is first presented in the setting of 2D statistical Ising model and is then extended to higher dimensional tensor networks and disordered systems. When applied to the Euclidean path integral formulation, this approach also gives rise to new efficient representations of the ground states for 1D and 2D quantum Ising models.
Tensor Effect on Bubble Nuclei
WANG Yan-Zhao; GU Jian-Zhong; ZHANG Xi-Zhen; DONG Jian-Min
2011-01-01
In the framework of the Hartree-Fock-Bogoliubov (HFB) approach with Skyrme interactions SLy5+T, SLy5+Tw and several sets of TIJ parametrizations, I.e. The Skyrme interaction parametrizations including the tensor terms, the proton density distribution in 34Si and 46Ar nuclei is calculated with and without the tensor force. It is shown that the bubble effect in 34Si does not depend a great deal on the Skyrme parametrization and the proton density distribution in 34Si is hardly influenced by the tensor force. As to 46Ar, the SLy5+Tw parametrization favors the formation of the bubble structure due to the inversion between the 2s1/2 and 1d3/2 orbits (2s1/2-ld3/2 inversion). The inversion mechanism induced by the SLy5+Tw interaction is analyzed based on the proton single-particle spectra obtained from the SLy5 and SLy5+Tw interactions as well as the wave functions of the 2s1/2 and 1d3/2 states.%In the framework of the Hartree-Fock-Bogoliubov (HFB) approach with Skyrme interactions SLy5+ T,SLy5+ Tω and several sets of TIJ parametrizations,i.e.the Skyrme interaction pararmetrizations including the tensor terms,the proton density distribution in 34Si and 46 Ar nuclei is calculated with and without the tensor force.It is shown that the bubble effect in 34Si does not depend a great deal on the Skyrme parametrization and the proton density distribution in 34Si is hardly influenced by the tensor force.As to 46Ar,the SLy5+ Tω parametrization favors the formation of the bubble structure due to the inversion between the 2s1/2 and 1d3/2 orbits (2s1/2-1d3/2 inversion).The inversion mechanism induced by the SLy5+ Tω interaction is analyzed based on the proton single-particle spectra obtained from the SLy5 and SLy5+ Tω interactions as well as the wave functions of the 2s1/2 and 1d3/2 states.The study of exotic nuclear structures has been a hot topic in nuclear physics.[1-4] Exotic nuclei are unstabile,superheavy nuclei,halo nuclei and so forth,whose structures are quite different
Westerhof, E.
1996-01-01
The hot plasma dielectric tensor is discussed in its various approximations. Collisionless cyclotron resonant damping and ion/electron Bernstein waves are discussed to exemplify the significance of a kinetic description of plasma waves.
Chung, Daniel J H
2016-01-01
We reformulate gauge theories in analogy with the vierbein formalism of general relativity. More specifically, we reformulate gauge theories such that their gauge dynamical degrees of freedom are local fields that transform linearly under the dual representation of the charged matter field. These local fields, which naively have the interpretation of non-local operators similar to Wilson lines, satisfy constraint equations. A set of basis tensor fields are used to solve these constraint equations, and their field theory is constructed. A new local symmetry in terms of the basis tensor fields is used to make this field theory local and maintain a Hamiltonian that is bounded from below. The field theory of the basis tensor fields is what we call the basis tensor gauge theory.
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
Tensor Network Renormalization.
Evenbly, G; Vidal, G
2015-10-30
We introduce a coarse-graining transformation for tensor networks that can be applied to study both the partition function of a classical statistical system and the Euclidean path integral of a quantum many-body system. The scheme is based upon the insertion of optimized unitary and isometric tensors (disentanglers and isometries) into the tensor network and has, as its key feature, the ability to remove short-range entanglement or correlations at each coarse-graining step. Removal of short-range entanglement results in scale invariance being explicitly recovered at criticality. In this way we obtain a proper renormalization group flow (in the space of tensors), one that in particular (i) is computationally sustainable, even for critical systems, and (ii) has the correct structure of fixed points, both at criticality and away from it. We demonstrate the proposed approach in the context of the 2D classical Ising model.
Tensor-tensor theory of gravitation
Gogberashvili, Merab
1996-01-01
We consider the standard gauge theory of Poincar\\'{e} group, realizing as a subgroup of GL(5. R). The main problem of this theory was appearing of the fields connected with non-Lorentz symmetries, whose physical sense was unclear. In this paper we treat the gravitation as a Higgs-Goldstone field, and the translation gauge field as a new tensor field. The effective metric tensor in this case is hybrid of two tensor fields. In the linear approximation the massive translation gauge field can give the Yukava type correction to the Newtons potential. Also outer potentials of a sphere and ball of the same mass are different in this case. Corrections to the standard Einshtein post Newtonian formulas of the light deflection and radar echo delay is obtained. The string like solution of the nonlinear equations of the translation gauge fields is found. This objects can results a Aharonov-Bohm type effect even for the spinless particles. They can provide density fluctuations in the early universe, necessary for galaxy fo...
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 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...
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...
Symmetric tensor decomposition
Brachat, Jerome; Mourrain, Bernard; Tsigaridas, Elias
2009-01-01
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 total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on th...
Physical components of tensors
Altman, Wolf
2014-01-01
""This book provides a clear explanation of the mathematical properties of tensors, from a physical perspective. The book is rigorous and concise, yet easy to read and very accessible. The reader will enjoy the wide variety of examples and exercises with solution, which make the book very pedagogical. I believe this can be a very useful book for anyone interested in learning about the mathematics of tensors, no matter the field of study or research. I would definitely like to have this book on my shelf, and use it as a reference in my own lectures."" -Román Orús, Institut für Physik, Jo
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...
Heil, Konstantin; Moroianu, Andrei; Semmelmann, Uwe
2017-07-01
We show that Killing tensors on conformally flat n-dimensional tori whose conformal factor only depends on one variable, are polynomials in the metric and in the Killing vector fields. In other words, every first integral of the geodesic flow polynomial in the momenta on the sphere bundle of such a torus is linear in the momenta.
Palmkvist, Jakob, E-mail: palmkvist@ihes.fr [Institut des Hautes Etudes Scientifiques, 35 Route de Chartres, FR-91440 Bures-sur-Yvette (France)
2014-01-15
We introduce an infinite-dimensional Lie superalgebra which is an extension of the U-duality Lie algebra of maximal supergravity in D dimensions, for 3 ⩽ D ⩽ 7. The level decomposition with respect to the U-duality Lie algebra gives exactly the tensor hierarchy of representations that arises in gauge deformations of the theory described by an embedding tensor, for all positive levels p. We prove that these representations are always contained in those coming from the associated Borcherds-Kac-Moody superalgebra, and we explain why some of the latter representations are not included in the tensor hierarchy. The most remarkable feature of our Lie superalgebra is that it does not admit a triangular decomposition like a (Borcherds-)Kac-Moody (super)algebra. Instead the Hodge duality relations between level p and D − 2 − p extend to negative p, relating the representations at the first two negative levels to the supersymmetry and closure constraints of the embedding tensor.
The magnetic stress tensor in magnetized matter
Espinosa, Olivier R; Espinosa, Olivier; Reisenegger, Andreas
2003-01-01
We derive the form of the magnetic stress tensor in a completely general, stationary magnetic medium, with an arbitrary magnetization field $vec M(vec r)$ and free current density $vec j(vec r)$. We start with the magnetic force density $vec f$ acting on a matter element, modelled as a collection of microscopic magnetic dipoles in addition to the free currents. We show that there is a unique tensor ${bf T}$ quadratic in the magnetic flux density $vec B(vec r)$ and the magnetic field $vec H(vec r)=vec B-4pivec M$ whose divergence is $nablacdot{bf T}=vec f$. In the limit $vec M=0$, the well-known vacuum magnetic stress tensor is recovered. However, the general form of the tensor is asymmetric, leading to a divergent angular acceleration for matter elements of vanishing size. We argue that this is not inconsistent, because it occurs only if $vec M$ and $vec B$ are not parallel, in which case the macroscopic field does indeed exert a torque on each of the microscopic dipoles, so this state is only possible if the...
2010-11-30
ELECTROMAGNEETIC SURFACE IMPEDANCE PROPERTIES FA9550-09-C-0198 DR. ADOUR KABAKIAN HUGHES RESEARCH LABS AFOSR / RSE 875 North Randolph Street, Suit...325 Room 3112 Arlington, Virginia 22203-1768 AFOSR / RSE AFRL-OSR-VA-TR-2012-0770 Distribution A We have investigated and determined how the tensor...the case of a TM wave, which favors propagation along the shorter principal axis. Standard terms apply U U U UU Arje Nachman RSE (Program Manager
E6Tensors: A Mathematica Package for E6 Tensors
Deppisch, Thomas
2016-01-01
We present the Mathematica package E6Tensors, a tool for explicit tensor calculations in E6 gauge theories. In addition to matrix expressions for the group generators of E6, it provides structure constants, various higher rank tensors and expressions for the representations 27, 78, 351 and 351'. This paper comes along with a short manual including physically relevant examples. I further give a complete list of gauge invariant, renormalisable terms for superpotentials and Lagrangians.
Tensor classification of structure in smoothed particle hydrodynamics density fields
Forgan, Duncan; Bonnell, Ian; Lucas, William; Rice, Ken
2016-04-01
As hydrodynamic simulations increase in scale and resolution, identifying structures with non-trivial geometries or regions of general interest becomes increasingly challenging. There is a growing need for algorithms that identify a variety of different features in a simulation without requiring a `by eye' search. We present tensor classification as such a technique for smoothed particle hydrodynamics (SPH). These methods have already been used to great effect in N-Body cosmological simulations, which require smoothing defined as an input free parameter. We show that tensor classification successfully identifies a wide range of structures in SPH density fields using its native smoothing, removing a free parameter from the analysis and preventing the need for tessellation of the density field, as required by some classification algorithms. As examples, we show that tensor classification using the tidal tensor and the velocity shear tensor successfully identifies filaments, shells and sheet structures in giant molecular cloud simulations, as well as spiral arms in discs. The relationship between structures identified using different tensors illustrates how different forces compete and co-operate to produce the observed density field. We therefore advocate the use of multiple tensors to classify structure in SPH simulations, to shed light on the interplay of multiple physical processes.
Matsueda, Hiroaki; Hashizume, Yoichiro
2012-01-01
A tensor network formalism of thermofield dynamics is introduced. The formalism relates the original Hilbert space with its tilde space by a product of two copies of a tensor network. Then, their interface becomes an event horizon, and the logarithm of the tensor rank corresponds to the black hole entropy. Eventually, multiscale entanglement renormalization anzats (MERA) reproduces an AdS black hole at finite temperature. Our finding shows rich functionalities of MERA as efficient graphical representation of AdS/CFT correspondence.
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.
Symmetric Tensor Decomposition
Brachat, Jerome; Comon, Pierre; Mourrain, Bernard
2010-01-01
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 total degree d as a sum of powers of linear forms (Waring’s problem), incidence properties on secant varieties of the Veronese variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester’s approach from the dual point of view...
Hu, Yonggang; Wu, Yi; 10.3150/10-BEJ317
2012-01-01
The conventional definition of a depth function is vector-based. In this paper, a novel projection depth (PD) technique directly based on tensors, such as matrices, is instead proposed. Tensor projection depth (TPD) is still an ideal depth function and its computation can be achieved through the iteration of PD. Furthermore, we also discuss the cases for sparse samples and higher order tensors. Experimental results in data classification with the two projection depths show that TPD performs much better than PD for data with a natural tensor form, and even when the data have a natural vector form, TPD appears to perform no worse than PD.
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
Ridgely, Charles T.
2010-01-01
Many textbooks dealing with general relativity do not demonstrate the derivation of forces in enough detail. The analyses presented herein demonstrate straightforward methods for computing forces by way of general relativity. Covariant divergence of the stress-energy-momentum tensor is used to derive a general expression of the force experienced…
Ridgely, Charles T.
2010-01-01
Many textbooks dealing with general relativity do not demonstrate the derivation of forces in enough detail. The analyses presented herein demonstrate straightforward methods for computing forces by way of general relativity. Covariant divergence of the stress-energy-momentum tensor is used to derive a general expression of the force experienced…
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 independe...... of the small-scale structure of the Earth’s lithospheric field....
Tensor Network Contractions for #SAT
Biamonte, Jacob D.; Morton, Jason; Turner, Jacob
2015-09-01
The computational cost of counting the number of solutions satisfying a Boolean formula, which is a problem instance of #SAT, has proven subtle to quantify. Even when finding individual satisfying solutions is computationally easy (e.g. 2-SAT, which is in ), determining the number of solutions can be #-hard. Recently, computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics-inspired techniques. By these methods, we give an algorithm using an axiomatic tensor contraction language for n-variable #SAT instances with complexity where c is the number of COPY-tensors, g is the number of gates, and d is the maximal degree of any COPY-tensor. Thus, n-variable counting problems can be solved efficiently when their tensor network expression has at most COPY-tensors and polynomial fan-out. This framework also admits an intuitive proof of a variant of the Tovey conjecture (the r,1-SAT instance of the Dubois-Tovey theorem). This study increases the theory, expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science.
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.
Solving Tensor Structured Problems with Computational Tensor Algebra
Morozov, Oleksii
2010-01-01
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often originate from multidimensional data, might profit from even higher levels of abstraction. We developed a framework for solving tensor structured problems with tensor algebra that unifies concepts from tensor analysis, multilinear algebra and multidimensional signal processing. In contrast to the conventional matrix approach, it allows the formulation of multidimensional problems, in a multidimensional way, preserving structure and data coherence; and the implementation of automated optimizations of solving algorithms, based on the commutativity of all tensor operations. Its ability to handle large scientific tasks is showcased by a real-world, 4D medical imaging problem, with more than 30 million unknown parameters solved on a current, inexpensive hardware. This significantly...
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.
Invariant tensors for simple groups
De Azcarraga, J.A.; Macfarlane, A.J.; Mountain, A.J.; Perez Bueno, J.C. [Cambridge Univ. (United Kingdom). Dept. of Applied Mathematics and Theoretical Physics (DAMTP)
1998-01-26
The forms of the invariant primitive tensors for the simple Lie algebras A{sub l}, B{sub l}, C{sub l} and D{sub l} are investigated. A new family of symmetric invariant tensors is introduced using the non-trivial cocycles for the Lie algebra cohomology. For the A{sub l} algebra it is explicitly shown that the generic forms of these tensors become zero except for the l primitive ones and that they give rise to the l primitive Casimir operators. Some recurrence and duality relations are given for the Lie algebra cocycles. Tables for the 3- and 5-cocycles for su(3) and su(4) are also provided. Finally, new relations involving the d and f su(n) tensors are given. (orig.). 34 refs.
MACH: Fast Randomized Tensor Decompositions
Tsourakakis, Charalampos E
2009-01-01
Tensors naturally model many real world processes which generate multi-aspect data. Such processes appear in many different research disciplines, e.g, chemometrics, computer vision, psychometrics and neuroimaging analysis. Tensor decompositions such as the Tucker decomposition are used to analyze multi-aspect data and extract latent factors, which capture the multilinear data structure. Such decompositions are powerful mining tools, for extracting patterns from large data volumes. However, most frequently used algorithms for such decompositions involve the computationally expensive Singular Value Decomposition. In this paper we propose MACH, a new sampling algorithm to compute such decompositions. Our method is of significant practical value for tensor streams, such as environmental monitoring systems, IP traffic matrices over time, where large amounts of data are accumulated and the analysis is computationally intensive but also in "post-mortem" data analysis cases where the tensor does not fit in the availa...
Tensor computations in computer algebra systems
Korolkova, A V; Sevastyanov, L A
2014-01-01
This paper considers three types of tensor computations. On their basis, we attempt to formulate criteria that must be satisfied by a computer algebra system dealing with tensors. We briefly overview the current state of tensor computations in different computer algebra systems. The tensor computations are illustrated with appropriate examples implemented in specific systems: Cadabra and Maxima.
Tensor Product of Massey Products
Qi Bing ZHENG
2006-01-01
In this paper, we interpret Massey products in terms of realizations (twitsting cochains)of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative,and commutative up to homotopy.
Tensor 2-sums and entanglement
Klavzar, Sandi
2009-01-01
To define a minimal mathematical framework for isolating some of the characteristic properties of quantum entanglement, we introduce a generalization of the tensor product of graphs. Inspired by the notion of a density matrix, the generalization is a simple one: every graph can be obtained by addiction modulo two, possibly with many summands, of tensor products of adjacency matrices. In this picture, we are still able to prove a combinatorial analogue of the Peres-Horodecki criterion for testing separability.
Positivity and conservation of superenergy tensors
Pozo, J M
2002-01-01
Two essential properties of energy-momentum tensors T submu subnu are their positivity and conservation. This is mathematically formalized by, respectively, an energy condition, as the dominant energy condition, and the vanishing of their divergence nabla supmu T submu subnu = 0. The classical Bel and Bel-Robinson superenergy tensors, generated from the Riemann and Weyl tensors, respectively, are rank-4 tensors. But they share these two properties with energy-momentum tensors: the dominant property (DP) and the divergence-free property in the absence of sources (vacuum). Senovilla defined a universal algebraic construction which generates a basic superenergy tensor T left brace A right brace from any arbitrary tensor A. In this construction, the seed tensor A is structured as an r-fold multivector, which can always be done. The most important feature of the basic superenergy tensors is that they satisfy automatically the DP, independently of the generating tensor A. We presented a more compact definition of T...
Tensor-optimized shell model for the Li isotopes with a bare nucleon-nucleon interaction
Myo, Takayuki; Toki, Hiroshi; Ikeda, Kiyomi
2012-01-01
We study the Li isotopes systematically in terms of the tensor-optimized shell model (TOSM) by using a bare nucleon-nucleon interaction as the AV8' interaction. The short-range correlation is treated in the unitary correlation operator method (UCOM). Using the TOSM+UCOM approach, we investigate the role of the tensor force on each spectrum of the Li isotopes. It is found that the tensor force produces quite a characteristic effect on various states in each spectrum and those spectra are affected considerably by the tensor force. The energy difference between the spin-orbit partner, the p1/2 and p3/2 orbits of the last neutron, in 5Li is caused by opposite roles of the tensor correlation. In 6Li, the spin-triplet state in the LS coupling configuration is favored energetically by the tensor force in comparison with jj coupling shell model states. In 7,8,9Li, the low-lying states containing extra neutrons in the p3/2 orbit are favored energetically due to the large tensor contribution to allow the excitation fro...
Hypersurfaces with Isotropic Para-Blaschke Tensor
Jian Bo FANG; Kun ZHANG
2014-01-01
Let Mn be an n-dimensional submanifold without umbilical points in the (n+1)-dimen-sional unit sphere Sn+1. Four basic invariants of Mn under the Moebius transformation group of Sn+1 are a1-form Φ called moebius form, a symmetric (0, 2) tensor A called Blaschke tensor, a symmetric (0, 2) tensor B called Moebius second fundamental form and a positive definite (0, 2) tensor g called Moebius metric. A symmetric (0, 2) tensor D = A+μB called para-Blaschke tensor, where μ is constant, is also an Moebius invariant. We call the para-Blaschke tensor is isotropic if there exists a function λ such that D = λg. One of the basic questions in Moebius geometry is to classify the hypersurfaces with isotropic para-Blaschke tensor. When λ is not constant, all hypersurfaces with isotropic para-Blaschke tensor are explicitly expressed in this paper.
Compressive sensing of sparse tensors.
Friedland, Shmuel; Li, Qun; Schonfeld, Dan
2014-10-01
Compressive sensing (CS) has triggered an enormous research activity since its first appearance. CS exploits the signal's sparsity or compressibility in a particular domain and integrates data compression and acquisition, thus allowing exact reconstruction through relatively few nonadaptive linear measurements. While conventional CS theory relies on data representation in the form of vectors, many data types in various applications, such as color imaging, video sequences, and multisensor networks, are intrinsically represented by higher order tensors. Application of CS to higher order data representation is typically performed by conversion of the data to very long vectors that must be measured using very large sampling matrices, thus imposing a huge computational and memory burden. In this paper, we propose generalized tensor compressive sensing (GTCS)-a unified framework for CS of higher order tensors, which preserves the intrinsic structure of tensor data with reduced computational complexity at reconstruction. GTCS offers an efficient means for representation of multidimensional data by providing simultaneous acquisition and compression from all tensor modes. In addition, we propound two reconstruction procedures, a serial method and a parallelizable method. We then compare the performance of the proposed method with Kronecker compressive sensing (KCS) and multiway compressive sensing (MWCS). We demonstrate experimentally that GTCS outperforms KCS and MWCS in terms of both reconstruction accuracy (within a range of compression ratios) and processing speed. The major disadvantage of our methods (and of MWCS as well) is that the compression ratios may be worse than that offered by KCS.
Understanding the Magnetic Polarizability Tensor
Ledger, P D
2015-01-01
The aim of this paper is provide new insights into the properties of the rank 2 polarizability tensor $\\check{\\check{\\mathcal M}}$ proposed in (P.D. Ledger and W.R.B. Lionheart Characterising the shape and material properties of hidden targets from magnetic induction data, IMA Journal of Applied Mathematics, doi: 10.1093/imamat/hxv015) for describing the perturbation in the magnetic field caused by the presence of a conducting object in the eddy current regime. In particular, we explore its connection with the magnetic polarizability tensor and the P\\'olya-Szeg\\"o tensor and how, by introducing new splittings of $\\check{\\check{\\mathcal M}}$, they form a family of rank 2 tensors for describing the response from different categories of conducting (permeable) objects. We include new bounds on the invariants of the P\\'olya-Szeg\\"o tensor and expressions for the low frequency and high conductivity limiting coefficients of $\\check{\\check{\\mathcal M}}$. We show, for the high conductivity case (and for frequencies at...
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....
Carrozza, Sylvain; Tanasa, Adrian
2016-11-01
We define in this paper a class of three-index tensor models, endowed with {O(N)^{⊗ 3}} invariance ( N being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the U( N) invariant models. We first exhibit the existence of a large N expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large N expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.
Tensor interactions and {tau} decays
Godina Nava, J.J.; Lopez Castro, G. [Departamento de Fisica, Cinvestav del IPN, Apartado Postal 14-740, 07000 Mexico, D.F. (Mexico)
1995-09-01
We study the effects of charged tensor weak currents on the strangeness-changing decays of the {tau} lepton. First, we use the available information on the {ital K}{sub {ital e}3}{sup +} form factors to obtain {ital B}({tau}{sup {minus}}{r_arrow}{ital K}{sup {minus}}{pi}{sup 0}{nu}{sub {tau}}){similar_to}10{sup {minus}4} when the {ital K}{pi} system is produced in an antisymmetric tensor configuration. Then we propose a mechanism for the direct production of the {ital K}{sub 2}{sup *}(1430) in {tau} decays. Using the current upper limit on this decay we set a bound on the symmetric tensor interactions.
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.......e., the “cores”) comprising the functional TT decomposition. This result motivates an approximation scheme employing polynomial approximations of the cores. For functions with appropriate regularity, the resulting spectral tensor-train decomposition combines the favorable dimension-scaling of the TT...... decomposition with the spectral convergence rate of polynomial approximations, yielding efficient and accurate surrogates for high-dimensional functions. To construct these decompositions, we use the sampling algorithm \\tt TT-DMRG-cross to obtain the TT decomposition of tensors resulting from suitable...
Phase Transition in Tensor Models
Delepouve, Thibault
2015-01-01
Generalizing matrix models, tensor models generate dynamical triangulations in any dimension and support a $1/N$ expansion. Using the intermediate field representation we explicitly rewrite a quartic tensor model as a field theory for a fluctuation field around a vacuum state corresponding to the resummation of the entire leading order in $1/N$ (a resummation of the melonic family). We then prove that the critical regime in which the continuum limit in the sense of dynamical triangulations is reached is precisely a phase transition in the field theory sense for the fluctuation field.
Tensor networks for dynamic spacetimes
May, Alex
2016-01-01
Existing tensor network models of holography are limited to representing the geometry of constant time slices of static spacetimes. We study the possibility of describing the geometry of a dynamic spacetime using tensor networks. We find it is necessary to give a new definition of length in the network, and propose a definition based on the mutual information. We show that by associating a set of networks with a single quantum state and making use of the mutual information based definition of length, a network analogue of the maximin formula can be used to calculate the entropy of boundary regions.
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...
Catino, Giovanni; Mazzieri, Lorenzo
2012-01-01
We discuss a gap in Besse's book, recently pointed out by Merton, which concerns the classification of Riemannian manifolds admitting a Codazzi tensors with exactly two distinct eigenvalues. For such manifolds, we prove a structure theorem, without adding extra hypotheses and then we conclude with some application of this theory to the classification of three-dimensional gradient Ricci solitons.
Vector and tensor analysis with applications
Borisenko, A I; Silverman, Richard A
1979-01-01
Concise and readable, this text ranges from definition of vectors and discussion of algebraic operations on vectors to the concept of tensor and algebraic operations on tensors. It also includes a systematic study of the differential and integral calculus of vector and tensor functions of space and time. Worked-out problems and solutions. 1968 edition.
Objective Tinnitus and the Tensor Tympani Muscle.
Rock
1995-01-01
Objective tinnitus (OT) may be caused by contraction of the tensor tympani muscle (TTM). The more forcefully the TTM contracts, the greater the intensity of the OT heard. Forceful closure of both eyelids can reflexively cause OT by contracting the TTM. The Forceful Eyelid Closure Syndrome (FECS) was reported at the Proceedings of the Second International Tinnitus Seminar in 1983.(1) FECS consists of several factors: (1) Objective tinnitus (2) An associated waning of hearing primarily of the lower frequencies, as much as 45 dB at 125 Hz, 30 to 40 dB at 250 Hz ascending to the patient's norm at 2000 Hz and approximately a 5 to 10 dB at 4000 Hz and 5 to 20 dB at 8000 Hz (3) Retraction of the manubrium and posterior mid-third of the tympanic membrane (TM) at the malleus-umbo area as seen under the otomicroscope (OM) in 25% (108) of 432 ears examined (4) These same ears were 75% (324) positive for increased impedance at maximum compliance with FEC. Of the patients studied, 25% had no response under the otomicroscope or by impedance audiometry.
Bilayer linearized tensor renormalization group approach for thermal tensor networks
Dong, Yong-Liang; Chen, Lei; Liu, Yun-Jing; Li, Wei
2017-04-01
Thermal tensor networks constitute an efficient and versatile representation for quantum lattice models at finite temperatures. By Trotter-Suzuki decomposition, one obtains a D +1 dimensional TTN for the D -dimensional quantum system and then employs efficient renormalizaton group (RG) contractions to obtain the thermodynamic properties with high precision. The linearized tensor renormalization group (LTRG) method, which can be used to contract TTN efficiently and calculate the thermodynamics, is briefly reviewed and then generalized to a bilayer form. We dub this bilayer algorithm as LTRG++ and explore its performance in both finite- and infinite-size systems, finding the numerical accuracy significantly improved compared to single-layer algorithm. Moreover, we show that the LTRG++ algorithm in an infinite-size system is in essence equivalent to transfer-matrix renormalization group method, while reformulated in a tensor network language. As an application of LTRG++, we simulate an extended fermionic Hubbard model numerically, where the phase separation phenomenon, ground-state phase diagram, as well as quantum criticality-enhanced magnetocaloric effects, are investigated.
Killing(-Yano) Tensors in String Theory
Chervonyi, Yuri
2015-01-01
We construct the Killing(-Yano) tensors for a large class of charged black holes in higher dimensions and study general properties of such tensors, in particular, their behavior under string dualities. Killing(-Yano) tensors encode the symmetries beyond isometries, which lead to insights into dynamics of particles and fields on a given geometry by providing a set of conserved quantities. By analyzing the eigenvalues of the Killing tensor, we provide a prescription for constructing several conserved quantities starting from a single object, and we demonstrate that Killing tensors in higher dimensions are always associated with ellipsoidal coordinates. We also determine the transformations of the Killing(-Yano) tensors under string dualities, and find the unique modification of the Killing-Yano equation consistent with these symmetries. These results are used to construct the explicit form of the Killing(-Yano) tensors for the Myers-Perry black hole in arbitrary number of dimensions and for its charged version.
Causality and Primordial Tensor Modes
Baumann, Daniel
2009-01-01
We introduce the real space correlation function of $B$-mode polarization of the cosmic microwave background (CMB) as a probe of superhorizon tensor perturbations created by inflation. By causality, any non-inflationary mechanism for gravitational wave production after reheating, like global phase transitions or cosmic strings, must have vanishing correlations for angular separations greater than the angle subtended by the particle horizon at recombination, i.e. $\\theta \\gtrsim 2^\\circ$. Since ordinary $B$-modes are defined non-locally in terms of the Stokes parameters $Q$ and $U$ and therefore don't have to respect causality, special care is taken to define `causal $\\tilde B$-modes' for the analysis. We compute the real space $\\tilde B$-mode correlation function for inflation and discuss its detectability on superhorizon scales where it provides an unambiguous test of inflationary gravitational waves. The correct identification of inflationary tensor modes is crucial since it relates directly to the energy s...
Scalar-tensor linear inflation
Artymowski, Michal
2016-01-01
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 any form of the non-minimal coupling to gravity of the form of $f(\\varphi)R/2$; b) the particle physics approach, where we motivate the form of the Jordan frame potential by the loop corrections to the inflaton field. In both cases the Jordan frame potentials are modifications of the induced inflation, but instead of the Starobinsky attractor they lead to the linear inflation in the strong coupling limit.
Scalar-tensor linear inflation
Artymowski, Michał; Racioppi, Antonio
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(varphi)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.
Extended vector-tensor theories
Kimura, Rampei; Naruko, Atsushi; Yoshida, Daisuke
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.
Local virial and tensor theorems.
Cohen, Leon
2011-11-17
We show that for any wave function and potential the local virial theorem can always be satisfied 2K(r) = r·ΔV by choosing a particular expression for the local kinetic energy. In addition, we show that for each choice of local kinetic energy there are an infinite number of quasi-probability distributions which will generate the same expression. We also consider the local tensor virial theorem.
Causality and primordial tensor modes
Baumann, Daniel; Zaldarriaga, Matias, E-mail: dbaumann@physics.harvard.edu, E-mail: mzaldarriaga@cfa.harvard.edu [Department of Physics, Harvard University, 17 Oxford Street, Cambridge, MA 02138, U.S.A. and Center for Astrophysics, Harvard University, 60 Garden Street, Cambridge, MA 02138 (United States)
2009-06-01
We introduce the real space correlation function of B-mode polarization of the cosmic microwave background (CMB) as a probe of superhorizon tensor perturbations created by inflation. By causality, any non-inflationary mechanism for gravitational wave production after reheating, like global phase transitions or cosmic strings, must have vanishing correlations for angular separations greater than the angle subtended by the particle horizon at recombination, i.e. θ ∼> 2°. Since ordinary B-modes are defined non-locally in terms of the Stokes parameters Q and U and therefore don't have to respect causality, special care is taken to define 'causal B-tilde -modes' for the analysis. We compute the real space B-tilde -mode correlation function for inflation and discuss its detectability on superhorizon scales where it provides an unambiguous test of inflationary gravitational waves. The correct identification of inflationary tensor modes is crucial since it relates directly to the energy scale of inflation. Wrongly associating tensor modes from causal seeds with inflation would imply an incorrect inference of the energy scale of inflation. We find that the superhorizon B-tilde -mode signal is above cosmic variance for the angular range 2° < θ < 4° and is therefore in principle detectable. In practice, the signal will be challenging to measure since it requires accurately resolving the recombination peak of the B-mode power spectrum. However, a future CMB satellite (CMBPol), with noise level Δ{sub P} ≅ 1μK-arcmin and sufficient resolution to efficiently correct for lensing-induced B-modes, should be able to detect the signal at more than 3σ if the tensor-to-scalar ratio isn't smaller than r ≅ 0.01.
Tensor Interaction Effect in Dibaryon
CHEN Ling-Zhi; PANG Hou-Rong; PING Jia-Lun; WANG Fan
2005-01-01
The gluon and Goldstone boson induced tensor interaction effect on the dibaryon mass and the D-wave decay width has been studied in the quark delocalization, color screening model. The effective S-D wave transition interactions induced by gluon and Goldstone boson exchanges decrease quickly as the increasing of the channel strangeness. The K and η meson tensor contribution is negligible in this model. No six-quark state in the light flavor world can become a bound one by the help of these tensor interactions except the deuteron. The partial D-wave decay width of Ijp = 1/2 2+NΩ state to spin 0, 1 ∧([1]) final state is 20.7 keV and 63.1 keV respectively. It is a very narrow dibaryon resonance and might be detected in the relativistic heavy ion reaction by the existing RHIC detectors through the reconstruction of the ∧([1]) vertex mass and the future COMPAS detector at CERN and FAIR project in Germany.
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.
TIMER: tensor image morphing for elastic registration.
Yap, Pew-Thian; Wu, Guorong; Zhu, Hongtu; Lin, Weili; Shen, Dinggang
2009-08-15
We propose a novel diffusion tensor imaging (DTI) registration algorithm, called Tensor Image Morphing for Elastic Registration (TIMER), which leverages the hierarchical guidance of regional distributions and local boundaries, both extracted directly from the tensors. Currently available DTI registration methods generally extract tensor scalar features from each tensor to construct scalar maps. Subsequently, regional integration and other operations such as edge detection are performed to extract more features to guide the registration. However, there are two major limitations with these approaches. First, the computed regional features might not reflect the actual regional tensor distributions. Second, by the same token, gradient maps calculated from the tensor-derived scalar feature maps might not represent the actual tissue tensor boundaries. To overcome these limitations, we propose a new approach which extracts regional and edge information directly from a tensor neighborhood. Regional tensor distribution information, such as mean and variance, is computed in a multiscale fashion directly from the tensors by taking into account the voxel neighborhood of different sizes, and hence capturing tensor information at different scales, which in turn can be employed to hierarchically guide the registration. Such multiscale scheme can help alleviate the problem of local minimum and is also more robust to noise since one can better determine the statistical properties of each voxel by taking into account the properties of its surrounding. Also incorporated in our method is edge information extracted directly from the tensors, which is crucial to facilitate registration of tissue boundaries. Experiments involving real subjects, simulated subjects, fiber tracking, and atrophy detection indicate that TIMER performs better than the other methods (Yang et al., 2008; Zhang et al., 2006).
Monitoring the refinement of crystal structures with (15)N solid-state NMR shift tensor data.
Kalakewich, Keyton; Iuliucci, Robbie; Mueller, Karl T; Eloranta, Harriet; Harper, James K
2015-11-21
The (15)N chemical shift tensor is shown to be extremely sensitive to lattice structure and a powerful metric for monitoring density functional theory refinements of crystal structures. These refinements include lattice effects and are applied here to five crystal structures. All structures improve based on a better agreement between experimental and calculated (15)N tensors, with an average improvement of 47.0 ppm. Structural improvement is further indicated by a decrease in forces on the atoms by 2-3 orders of magnitude and a greater similarity in atom positions to neutron diffraction structures. These refinements change bond lengths by more than the diffraction errors including adjustments to X-Y and X-H bonds (X, Y = C, N, and O) of 0.028 ± 0.002 Å and 0.144 ± 0.036 Å, respectively. The acquisition of (15)N tensors at natural abundance is challenging and this limitation is overcome by improved (1)H decoupling in the FIREMAT method. This decoupling dramatically narrows linewidths, improves signal-to-noise by up to 317%, and significantly improves the accuracy of measured tensors. A total of 39 tensors are measured with shifts distributed over a range of more than 400 ppm. Overall, experimental (15)N tensors are at least 5 times more sensitive to crystal structure than (13)C tensors due to nitrogen's greater polarizability and larger range of chemical shifts.
Reconstruction of convex bodies from surface tensors
Kousholt, Astrid; Kiderlen, Markus
The set of all surface tensors of a convex body K (Minkowski tensors derived from the surface area measure of K) determine K up to translation, and hereby, the surface tensors of K contain all information on the shape of K. Here, shape means the equivalence class of all convex bodies...... 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....... 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...
Killing tensors in pp-wave spacetimes
Keane, Aidan J [87 Carlton Place, Glasgow G5 9TD, Scotland (United Kingdom); Tupper, Brian O J, E-mail: aidan@countingthoughts.co, E-mail: bt32@rogers.co [Department of Mathematics and Statistics, University of New Brunswick, Fredericton, New Brunswick, E3B 5A3 (Canada)
2010-12-21
The formal solution of the second-order Killing tensor equations for the general pp-wave spacetime is given. The Killing tensor equations are integrated fully for some specific pp-wave spacetimes. In particular, the complete solution is given for the conformally flat plane wave spacetimes and we find that irreducible Killing tensors arise for specific classes. The maximum number of independent irreducible Killing tensors admitted by a conformally flat plane wave spacetime is shown to be six. It is shown that every pp-wave spacetime that admits an homothety will admit a Killing tensor of Koutras type and, with the exception of the singular scale-invariant plane wave spacetimes, this Killing tensor is irreducible.
Derivatives on the isotropic tensor functions
DUI; Guansuo; WANG; Zhengdao; JIN; Ming
2006-01-01
The derivative of the isotropic tensor function plays an important part in continuum mechanics and computational mechanics, and also it is still an opening problem. By means of a scalar response function and solving a tensor equation, this problem is well studied. A compact explicit expression for the derivative of the isotropic tensor function is presented, which is valid for both distinct and repeated eigenvalue cases. Throughout the analysis, the formulation holds for general isotropic tensor functions without need to solve eigenvector problems or determine coefficients. On the theoretical side, a very simple solution of a tensor equation is obtained. As an application to continuum mechanics, a base-free expression for the Hill's strain rate is given, which is more compact than the existent results. Finally, with an example we compute the derivative of an exponent tensor function. And the efficiency of the present formulations is demonstrated.
Tensor eigenvalues and entanglement of symmetric states
Bohnet-Waldraff, F.; Braun, D.; Giraud, O.
2016-10-01
Tensor eigenvalues and eigenvectors have been introduced in the recent mathematical literature as a generalization of the usual matrix eigenvalues and eigenvectors. We apply this formalism to a tensor that describes a multipartite symmetric state or a spin state, and we investigate to what extent the corresponding tensor eigenvalues contain information about the multipartite entanglement (or, equivalently, the quantumness) of the state. This extends previous results connecting entanglement to spectral properties related to the state. We show that if the smallest tensor eigenvalue is negative, the state is detected as entangled. While for spin-1 states the positivity of the smallest tensor eigenvalue is equivalent to separability, we show that for higher values of the angular momentum there is a correlation between entanglement and the value of the smallest tensor eigenvalue.
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...
Hard Exclusive Production of Tensor Mesons
Braun, V M
2001-01-01
We point out that hard exclusive production of tensor mesons $f_2(1270)$ with helicity $\\lambda=\\pm 2$ is dominated by the gluon component in the meson wave function and can be used to determine gluon admixture in tensor mesons in a theoretically clean manner. We present a detailed analysis of the tensor meson distribution amplitudes and calculate the transition form factor $\\gamma+\\gamma^*\\to f_2(1270)$ for one real and one virtual photon.
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.
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
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.)
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.
Tensor power spectrum and disformal transformations
Fumagalli, Jacopo; Postma, Marieke
2016-01-01
In a general effective theory description of inflation a disformal transformation can be used to set the tensor sound speed to one. After the transformation, the tensor power spectrum then automatically only depends on the Hubble parameter. We show that this disformal transformation, however, is nothing else than a change of units. It is a very useful tool for simplifying and interpreting computations, but it cannot change any physics. While the apparent parametrical dependence of the tensor power spectrum does change under a disformal transformation, the physics described is frame invariant. We further illustrate the frame invariance of the tensor power spectrum by writing it exclusively in terms of separately invariant quantities.
Loop Optimization for Tensor Network Renormalization
Yang, Shuo; Gu, Zheng-Cheng; Wen, Xiao-Gang
2017-03-01
We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the 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.
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...
Loop Optimization for Tensor Network Renormalization.
Yang, Shuo; Gu, Zheng-Cheng; Wen, Xiao-Gang
2017-03-17
We introduce a tensor renormalization group scheme for coarse graining a two-dimensional tensor network that can be successfully applied to both classical and quantum systems on and off criticality. The key innovation in our scheme is to deform a 2D tensor network into small loops and then optimize the 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.
A uniform parameterization of moment tensors
Tape, C.; Tape, W.
2015-12-01
A moment tensor is a 3 x 3 symmetric matrix that expresses an earthquake source. We construct a parameterization of the five-dimensional space of all moment tensors of unit norm. The coordinates associated with the parameterization are closely related to moment tensor orientations and source types. The parameterization is uniform, in the sense that equal volumes in the coordinate domain of the parameterization correspond to equal volumes of moment tensors. Uniformly distributed points in the coordinate domain therefore give uniformly distributed moment tensors. A cartesian grid in the coordinate domain can be used to search efficiently over moment tensors. We find that uniformly distributed moment tensors have uniformly distributed orientations (eigenframes), but that their source types (eigenvalue triples) are distributed so as to favor double couples. An appropriate choice of a priori moment tensor probability is a prerequisite for parameter estimation. As a seemingly sensible choice, we consider the homogeneous probability, in which equal volumes of moment tensors are equally likely. We believe that it will lead to improved characterization of source processes.
Groupoid normalizers of tensor products
Fang, Junsheng; White, Stuart A; Wiggins, Alan D
2008-01-01
We consider an inclusion $B\\subseteq M$ of finite von Neumann algebras satisfying $B'\\cap M\\subseteq B$. A partial isometry $v\\in M$ is called a groupoid normalizer if $vBv^*, v^*Bv\\subseteq B$. Given two such inclusions $B_i\\subseteq M_i$, $i=1,2$, we find approximations to the groupoid normalizers of $B_1 \\vnotimes B_2$ in $M_1\\vnotimes M_2$, from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis $B_i'\\cap M_i\\subseteq B_i$, $i=1,2$. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries $v\\in M$ satisfying $vBv^*\\subseteq B$ and $v^*v, vv^*\\in B$.
3D reconstruction of tensors and vectors
Defrise, Michel; Gullberg, Grant T.
2005-02-17
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.
Bayesian regularization of diffusion tensor images
Frandsen, Jesper; Hobolth, Asger; Østergaard, Leif;
2007-01-01
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...
Tensor Products of Random Unitary Matrices
Tkocz, Tomasz; Kus, Marek; Zeitouni, Ofer; Zyczkowski, Karol
2012-01-01
Tensor products of M random unitary matrices of size N from the circular unitary ensemble are investigated. We show that the spectral statistics of the tensor product of random matrices becomes Poissonian if M=2, N become large or M become large and N=2.
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.
Kuang-dai Leng
2012-01-01
Full Text Available Fabric tensor has proved to be an effective tool statistically characterizing directional data in a smooth and frame-indifferent form. Directional data arising from microscopic physics and mechanics can be summed up as tensor-valued orientation distribution functions (ODFs. Two characterizations of the tensor-valued ODFs are proposed, using the asymmetric and symmetric fabric tensors respectively. The later proves to be nonconvergent and less accurate but still an available solution for where fabric tensors are required in full symmetry. Analytic solutions of the two types of fabric tensors characterizing centrosymmetric and anticentrosymmetric tensor-valued ODFs are presented in terms of orthogonal irreducible decompositions in both two- and three-dimensional (2D and 3D spaces. Accuracy analysis is performed on normally distributed random ODFs to evaluate the approximation quality of the two characterizations, where fabric tensors of higher orders are employed. It is shown that the fitness is dominated by the dispersion degree of the original ODFs rather than the orders of fabric tensors. One application of tensor-valued ODF and fabric tensor in continuum damage mechanics is presented.
On Lovelock analogs of the Riemann tensor
Camanho, Xian O. [Albert-Einstein-Institut, Max-Planck-Institut fuer Gravitationsphysik, Golm (Germany); Dadhich, Naresh [Jamia Millia Islamia, Centre for Theoretical Physics, New Delhi (India); Inter-University Centre for Astronomy and Astrophysics, Pune (India)
2016-03-15
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. (orig.)
Algebraically contractible topological tensor network states
Denny, S J; Jaksch, D; Clark, S R
2011-01-01
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with $Z_2$ topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-half lattice to a class of exactly contractible states on spin-S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how perturbations l...
An Algorithm to Simplify Tensor Expressions
Portugal, R
1998-01-01
The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations and the renaming of dummy indices. The tensor indices are split into classes and a natural place for them is defined. The canonical form is the closest configuration to the natural configuration. In the second part, the Groebner basis method is used to simplify tensor expressions which obey the linear identities that come from cyclic symmetries (or more general tensor identities, including non-linear identities). The algorithm is suitable for implementation in general purpose computer algebra systems. Some timings of an experimental implementation over the Riemann package are shown.
Fluid Registration of Diffusion Tensor Images Using Information Theory
Chiang, Ming-Chang; Leow, Alex D.; Klunder, Andrea D.; Dutton, Rebecca A.; Barysheva, Marina; Rose, Stephen E.; McMahon, Katie L.; de Zubicaray, Greig I.; Toga, Arthur W.; Thompson, Paul M.
2008-01-01
We apply an information-theoretic cost metric, the symmetrized Kullback-Leibler (sKL) divergence, or J-divergence, to fluid registration of diffusion tensor images. The difference between diffusion tensors is quantified based on the sKL-divergence of their associated probability density functions (PDFs). Three-dimensional DTI data from 34 subjects were fluidly registered to an optimized target image. To allow large image deformations but preserve image topology, we regularized the flow with a large-deformation diffeomorphic mapping based on the kinematics of a Navier-Stokes fluid. A driving force was developed to minimize the J-divergence between the deforming source and target diffusion functions, while reorienting the flowing tensors to preserve fiber topography. In initial experiments, we showed that the sKL-divergence based on full diffusion PDFs is adaptable to higher-order diffusion models, such as high angular resolution diffusion imaging (HARDI). The sKL-divergence was sensitive to subtle differences between two diffusivity profiles, showing promise for nonlinear registration applications and multisubject statistical analysis of HARDI data. PMID:18390342
Interpretation of the Weyl Tensor
Hofmann, Stefan; Schneider, Robert
2013-01-01
According to folklore in general relativity, the Weyl tensor can be decomposed into parts corresponding to Newton-like, incoming- and outgoing wave-like field components. It is shown here that this interpretation cannot be applied to space-time geometries with cylindrical isometries. This is done by investigating some well-known exact solutions of Einstein's field equations with whole-cylindrical symmetry, for which the physical interpretation is very clear, but for which the standard Weyl interpretation would give contradictory results. For planar or spherical geometries, however, the standard interpretation works for both, static and dynamical space-times. It is argued that one reason for the failure in the cylindrical case is that for waves spreading in two spatial dimensions there is no local criterion to distinguish incoming and outgoing waves already at the linear level. It turns out that Thorne's local energy notion, subject to certain qualifications, provides an efficient diagnostic tool to extract th...
$\\rho$-Nucleon Tensor Coupling and Charge-Exchange Resonances
De Conti, C; Krmpotic, F
2000-01-01
The Gamow-Teller resonances are discussed in the context of a self-consistentRPA, based on the relativistic mean field theory. We inquire on the possibilityof substituting the phenomenological Landau-Migdal force by a microscopicnucleon-nucleon interaction, generated from the rho-nucleon tensor coupling.The effect of this coupling turns out to be very small when the short rangecorrelations are not taken into account, but too large when these correlationsare simulated by the simple extraction of the contact terms from the resultingnucleon-nucleon interaction.
Semi-empirical refinements of crystal structures using (17)O quadrupolar-coupling tensors.
Holmes, Sean T; Iuliucci, Robbie J; Mueller, Karl T; Dybowski, Cecil
2017-02-14
We demonstrate a modification of Grimme's two-parameter empirical dispersion force field (referred to as the PW91-D2* method), in which the damping function has been optimized to yield geometries that result in predictions of the principal values of (17)O quadrupolar-coupling tensors that are systematically in close agreement with experiment. The predictions of (17)O quadrupolar-coupling tensors using PW91-D2*-refined structures yield a root-mean-square deviation (RMSD) (0.28 MHz) for twenty-two crystalline systems that is smaller than the RMSD for predictions based on X-ray diffraction structures (0.58 MHz) or on structures refined with PW91 (0.53 MHz). In addition, (13)C, (15)N, and (17)O chemical-shift tensors and (35)Cl quadrupolar-coupling tensors determined with PW91-D2*-refined structures are compared to the experiment. Errors in the prediction of chemical-shift tensors and quadrupolar-coupling tensors are, in these cases, substantially lowered, as compared to predictions based on PW91-refined structures. With this PW91-D2*-based method, analysis of 42 (17)O chemical-shift-tensor principal components gives a RMSD of only 18.3 ppm, whereas calculations on unrefined X-ray structures give a RMSD of 39.6 ppm and calculations of PW91-refined structures give an RMSD of 24.3 ppm. A similar analysis of (35)Cl quadrupolar-coupling tensor principal components gives a RMSD of 1.45 MHz for the unrefined X-ray structures, 1.62 MHz for PW91-refined structures, and 0.59 MHz for the PW91-D2*-refined structures.
About Advances in Tensor Data Denoising Methods
Salah Bourennane
2008-10-01
Full Text Available Tensor methods are of great interest since the development of multicomponent sensors. The acquired multicomponent data are represented by tensors, that is, multiway arrays. This paper presents advances on filtering methods to improve tensor data denoising. Channel-by-channel and multiway methods are presented. The first multiway method is based on the lower-rank (K1,Ã¢Â€Â¦,KN truncation of the HOSVD. The second one consists of an extension of Wiener filtering to data tensors. When multiway tensor filtering is performed, the processed tensor is flattened along each mode successively, and singular value decomposition of the flattened matrix is performed. Data projection on the singular vectors associated with dominant singular values results in noise reduction. We propose a synthesis of crucial issues which were recently solved, that is, the estimation of the number of dominant singular vectors, the optimal choice of flattening directions, and the reduction of the computational load of multiway tensor filtering methods. The presented methods are compared through an application to a color image and a seismic signal, multiway Wiener filtering providing the best denoising results. We apply multiway Wiener filtering and its fast version to a hyperspectral image. The fast multiway filtering method is 29 times faster and yields very close denoising results.
Goto, Shin-itiro; Walton, Timothy J
2010-01-01
This is paper I of a series of two papers, offering a self-contained analysis of the role of electromagnetic stress-energy-momentum tensors in the classical description of continuous polarizable perfectly insulating media. While acknowledging the primary role played by the total stress-energy-momentum tensor on spacetime, we argue that it is meaningful and useful in the context of covariant constitutive theory to assign preferred status to particular parts of this total tensor, when defined with respect to a particular splitting. The notion of a force density that arises from the divergence of these tensors is strictly defined relative to some inertial property of the medium. Consistency with the laws of Newtonian continuum mechanics demands that the total force density on any element of a medium be proportional to the local linear acceleration field of that element in an inertial frame and must also arise as part of the divergence of the total stress-energy-momentum tensor. In this paper we explore how elect...
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....
Incremental Discriminant Analysis in Tensor Space
Chang, Liu; Weidong, Zhao; Tao, Yan; Qiang, Pu; Xiaodan, Du
2015-01-01
To study incremental machine learning in tensor space, this paper proposes incremental tensor discriminant analysis. The algorithm employs tensor representation to carry on discriminant analysis and combine incremental learning to alleviate the computational cost. This paper proves that the algorithm can be unified into the graph framework theoretically and analyzes the time and space complexity in detail. The experiments on facial image detection have shown that the algorithm not only achieves sound performance compared with other algorithms, but also reduces the computational issues apparently. PMID:26339229
Global nuclear structure aspects of tensor interaction
Satula, W; Dobaczewski, J; Olbratowski, P; Rafalski, M; Werner, T R; Wyss, R A
2008-01-01
A direct fit of the isoscalar spin-orbit and both isoscalar and isovector tensor coupling constants to the f5/2-f7/2 SO splittings in 40Ca, 56Ni, and 48Ca requires: (i) a significant reduction of the standard isoscalar spin-orbit strength and (ii) strong attractive tensor coupling constants. The aim of this paper is to address the consequences of these strong attractive tensor and weak spin-orbit fields on total binding energies, two-neutron separation energies and nuclear deformability.
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 methods for large, sparse unconstrained optimization
Bouaricha, A.
1996-11-01
Tensor methods for unconstrained optimization were first introduced by Schnabel and Chow [SIAM J. Optimization, 1 (1991), pp. 293-315], who describe these methods for small to moderate size problems. This paper extends these methods to large, sparse unconstrained optimization problems. This requires an entirely new way of solving the tensor model that makes the methods suitable for solving large, sparse optimization problems efficiently. We present test results for sets of problems where the Hessian at the minimizer is nonsingular and where it is singular. These results show that tensor methods are significantly more efficient and more reliable than standard methods based on Newton`s method.
Tensor network and a black hole
Matsueda, Hiroaki; Ishihara, Masafumi; Hashizume, Yoichiro
2013-03-01
A tensor-network variational formalism of thermofield dynamics is introduced. The formalism relates the original Hilbert space with its tilde space by a product of two copies of a tensor network. Then, their interface becomes an event horizon, and the logarithm of the tensor rank corresponds to the black hole entropy. Eventually, a multiscale entanglement renormalization ansatz reproduces an anti-de Sitter black hole at finite temperature. Our finding shows rich functionalities of multiscale entanglement renormalization ansatz as efficient graphical representation of AdS/CFT correspondence.
[Face rejuvenation with tensor threads].
Cornette de Saint Cyr, B; Benouaiche, L
2017-08-25
The last decades has seen new priorities in treatment of a flabby, ageing face towards minimally invasive aesthetic surgery, to be accompanied and followed by the requirements to perform such interventions with the maximally reduced health hazards, with inconsiderable injury, without cuts and, respectively, to be followed by no resulting scars, as well as a short postoperative period. We propose a new reviewing presentation of the tensor threads. After having explained the technology of the threads, we will discuss the good patient indication, the criteria which determine the choice of the threads and methods for each type of patient. There are many techniques, which we will present. Then, we will discuss the results, unsatisfactory outcomes obtained and complications encountered, as well as how to improve the cosmetic outcomes to be obtained. To conclude, we will propose a strategy for the long-term treatment of the neck and the face, preventing surgical management of the aging process. Copyright © 2017. Published by Elsevier Masson SAS.
Diffusion Tensor Imaging of Pedophilia.
Cantor, James M; Lafaille, Sophie; Soh, Debra W; Moayedi, Massieh; Mikulis, David J; Girard, Todd A
2015-11-01
Pedophilia is a principal motivator of child molestation, incurring great emotional and financial burdens on victims and society. Even among pedophiles who never commit any offense,the condition requires lifelong suppression and control. Previous comparison using voxel-based morphometry (VBM)of MR images from a large sample of pedophiles and controls revealed group differences in white matter. The present study therefore sought to verify and characterize white matter involvement using diffusion tensor imaging (DTI), which better captures the microstructure of white matter than does VBM. Pedophilics ex offenders (n=24) were compared with healthy, age-matched controls with no criminal record and no indication of pedophilia (n=32). White matter microstructure was analyzed with Tract-Based Spatial Statistics, and the trajectories of implicated fiber bundles were identified by probabilistic tractography. Groups showed significant, highly focused differences in DTI parameters which related to participants’ genital responses to sexual depictions of children, but not to measures of psychopathy or to childhood histories of physical abuse, sexual abuse, or neglect. Some previously reported gray matter differences were suggested under highly liberal statistical conditions (p(uncorrected)<.005), but did not survive ordinary statistical correction (whole brain per voxel false discovery rate of 5%). These results confirm that pedophilia is characterized by neuroanatomical differences in white matter microstructure, over and above any neural characteristics attributable to psychopathy and childhood adversity, which show neuroanatomic footprints of their own. Although some gray matter structures were implicated previously, only few have emerged reliably.
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
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...... deformations in an attempt to ensure that the local deformations in the warped image remains true to the orientation of the underlying fibers; forward mapping, however, can also create "seams" or gaps and consequently artifacts in the warped image by failing to define accurately the voxels in the template...... space where the magnitude of the deformation is large (e.g., |Jacobian| > 1). Backward mapping, in contrast, defines voxels in the template space by mapping them back to locations in the original imaging space. Backward mapping allows every voxel in the template space to be defined without the creation...
Shifted power method for computing tensor eigenpairs.
Mayo, Jackson R.; Kolda, Tamara Gibson
2010-10-01
Recent work on eigenvalues and eigenvectors for tensors of order m {>=} 3 has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form Ax{sup m-1} = {lambda}x subject to {parallel}x{parallel} = 1, which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a novel shifted symmetric higher-order power method (SS-HOPM), which we showis guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to fnding complex eigenpairs.
Quantum theory with bold operator tensors.
Hardy, Lucien
2015-08-06
In this paper, we present a formulation of quantum theory in terms of bold operator tensors. A circuit is built up of operations where an operation corresponds to a use of an apparatus. We associate collections of operator tensors (which together comprise a bold operator) with these apparatus uses. We give rules for combining bold operator tensors such that, for a circuit, they give a probability distribution over the possible outcomes. If we impose certain physicality constraints on the bold operator tensors, then we get exactly the quantum formalism. We provide both symbolic and diagrammatic ways to represent these calculations. This approach is manifestly covariant in that it does not require us to foliate the circuit into time steps and then evolve a state. Thus, the approach forms a natural starting point for an operational approach to quantum field theory.
The Weyl tensor correlator in cosmological spacetimes
Fröb, Markus B
2014-01-01
We give a general expression for the Weyl tensor two-point function in a general Friedmann-Lema\\^itre-Robertson-Walker spacetime. We work in reduced phase space for the perturbations, i.e., quantize only the dynamical degrees of freedom without adding any gauge-fixing term. The general formula is illustrated by a calculation in slow-roll single-field inflation to first order in the slow-roll parameters $\\epsilon$ and $\\delta$, and the result is shown to have the correct de Sitter limit as $\\epsilon, \\delta \\to 0$. Furthermore, it is seen that the Weyl tensor correlation function does not suffer from infrared divergences, unlike the two-point functions of the metric and scalar field perturbations. Lastly, we show how to recover the usual tensor power spectrum from the Weyl tensor correlation function.
Symbolic Tensor Calculus -- Functional and Dynamic Approach
Woszczyna, A; Czaja, W; Golda, Z A
2016-01-01
In this paper, we briefly discuss the dynamic and functional approach to computer symbolic tensor analysis. The ccgrg package for Wolfram Language/Mathematica is used to illustrate this approach. Some examples of applications are attached.
Multipartite Entanglement in Stabilizer Tensor Networks
Nezami, Sepehr
2016-01-01
Tensor network models reproduce important structural features of holography, including the Ryu-Takayanagi formula for the entanglement entropy and quantum error correction in the entanglement wedge. In contrast, only little is known about their multipartite entanglement structure, which has been of considerable recent interest. In this work, we study random stabilizer tensor networks and show that here the tripartite entanglement question has a sharp answer: The average number of GHZ triples that can be extracted from a stabilizer tensor network is small, implying that the entanglement is predominantly bipartite. As a consequence, we obtain a new operational interpretation of the monogamy of the Ryu-Takayanagi mutual information and an entropic diagnostic for higher-partite entanglement. Our technical contributions include a spin model for evaluating the average GHZ content of stabilizer tensor networks and a novel formula for the third moment of random stabilizer states.
Primordial tensor modes of the early Universe
Martínez, Florencia Benítez
2016-01-01
We study cosmological tensor perturbations on a quantized background within the hybrid quantization approach. In particular, we consider a flat, homogeneous and isotropic spacetime and small tensor inhomogeneities on it. We truncate the action to second order in the perturbations. The dynamics is ruled by a homogeneous scalar constraint. We carry out a canonical transformation in the system where the Hamiltonian for the tensor perturbations takes a canonical form. The new tensor modes now admit a standard Fock quantization with a unitary dynamics. We then combine this representation with a generic quantum scheme for the homogeneous sector. We adopt a Born-Oppenheimer ansatz for the solutions to the constraint operator, previously employed to study the dynamics of scalar inhomogeneities. We analyze the approximations that allow us to recover, on the one hand, a Schr\\"odinger equation similar to the one emerging in the dressed metric approach, and, on the other hand, the ones necessary for the effective evoluti...
Entangled scalar and tensor fluctuations during inflation
Collins, Hael; Vardanyan, Tereza [Department of Physics, Carnegie Mellon University,5000 Forbes Avenue, Pittsburgh, Pennsylvania (United States)
2016-11-29
We show how the choice of an inflationary state that entangles scalar and tensor fluctuations affects the angular two-point correlation functions of the T, E, and B modes of the cosmic microwave background. The propagators for a state starting with some general quadratic entanglement are solved exactly, leading to predictions for the primordial scalar-scalar, tensor-tensor, and scalar-tensor power spectra. These power spectra are expressed in terms of general functions that describe the entangling structure of the initial state relative to the standard Bunch-Davies vacuum. We illustrate how such a state would modify the angular correlations in the CMB with a simple example where the initial state is a small perturbation away from the Bunch-Davies state. Because the state breaks some of the rotational symmetries, the angular power spectra no longer need be strictly diagonal.
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.
A nonlinear theory of tensor distributions
Vickers, J A
1998-01-01
The coordinate invariant theory of generalised functions of Colombeau and Meril is reviewed and extended to enable the construction of multi-index generalised tensor functions whose transformation laws coincide with their counterparts in classical distribution theory.
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.
Nonlocal elasticity tensors in dislocation and disclination cores
Taupin, V.; Gbemou, K.; Fressengeas, C.; Capolungo, L.
2017-03-01
Nonlocal elastic constitutive laws are introduced for crystals containing defects such as dislocations and disclinations. In addition to pointwise elastic moduli tensors adequately reflecting the elastic response of defect-free regions by relating stresses to strains and couple-stresses to curvatures, elastic cross-moduli tensors relating strains to couple-stresses and curvatures to stresses within convolution integrals are derived from a nonlocal analysis of strains and curvatures in the defects cores. Sufficient conditions are derived for positive-definiteness of the resulting free energy, and stability of elastic solutions is ensured. The elastic stress/couple stress fields associated with prescribed dislocation/disclination density distributions and solving the momentum and moment of momentum balance equations in periodic media are determined by using a Fast Fourier Transform spectral method. The convoluted cross-moduli bring the following results: (i) Nonlocal stresses and couple stresses oppose their local counterparts in the defects core regions, playing the role of restoring forces and possibly ensuring spatio-temporal stability of the simulated defects, (ii) The couple stress fields are strongly affected by nonlocality. Such effects favor the stability of the simulated grain boundaries and allow investigating their elastic interactions with extrinsic defects, (iii) Driving forces inducing grain growth or refinement derive from the self-stress and couple stress fields of grain boundaries in nanocrystalline configurations.
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.
The Energy-Momentum Tensor(s) in Classical Gauge Theories
Blaschke, Daniel N; Reboud, Meril; Schweda, Manfred
2016-01-01
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.
Analysis of the tensor-tensor type scalar tetraquark states with QCD sum rules
Wang, Zhi-Gang
2016-01-01
In this article, we study the ground states and the first radial excited states of the tensor-tensor type scalar hidden-charm tetraquark states with the QCD sum rules. We separate the ground state contributions from the first radial excited state contributions unambiguously, and obtain the QCD sum rules for the ground states and the first radial excited states, respectively. Then we search for the Borel parameters and continuum threshold parameters according to four criteria and obtain the masses of the tensor-tensor type scalar hidden-charm tetraquark states, which can be confronted to the experimental data in the future.
Renormalization procedure for random tensor networks and the canonical tensor model
Sasakura, Naoki
2015-01-01
We discuss a renormalization procedure for random tensor networks, and show that the corresponding renormalization-group flow is given by the Hamiltonian vector flow of the canonical tensor model, which is a discretized model of quantum gravity. The result is the generalization of the previous one concerning the relation between the Ising model on random networks and the canonical tensor model with N=2. We also prove a general theorem which relates discontinuity of the renormalization-group flow and the phase transitions of random tensor networks.
Novel Physics with Tensor Polarized Deuteron Targets
Slifer, Karl J. [UNH; Long, Elena A. [UNH
2013-09-01
Development of solid spin-1 polarized targets will open the study of tensor structure functions to precise measurement, and holds the promise to enable a new generation of polarized scattering experiments. In this talk we will discuss a measurement of the leading twist tensor structure function b1, along with prospects for future experiments with a solid tensor polarized target. The recently approved JLab experiment E12-13-011 will measure the lead- ing twist tensor structure function b1, which provides a unique tool to study partonic effects, while also being sensitive to coherent nuclear properties in the simplest nuclear system. At low x, shadowing effects are expected to dominate b1, while at larger values, b1 provides a clean probe of exotic QCD effects, such as hidden color due to 6-quark configuration. Since the deuteron wave function is relatively well known, any non-standard effects are expected to be readily observable. All available models predict a small or vanishing value of b1 at moderate x. However, the first pioneer measurement of b1 at HERMES revealed a crossover to an anomalously large negative value in the region 0.2 < x < 0.5, albeit with relatively large experimental uncertainty. E12-13-011 will perform an inclusive measurement of the deuteron tensor asymmetry in the region 0.16 < x < 0.49, for 0.8 < Q2 < 5.0 GeV2. The UVa solid polarized ND3 target will be used, along with the Hall C spectrometers, and an unpolarized 115 nA beam. This measurement will provide access to the tensor quark polarization, and allow a test of the Close-Kumano sum rule, which vanishes in the absence of tensor polarization in the quark sea. Until now, tensor structure has been largely unexplored, so the study of these quantities holds the potential of initiating a new field of spin physics at Jefferson Lab.
Phylogenetic estimation with partial likelihood tensors
Sumner, J G
2008-01-01
We present an alternative method for calculating likelihoods in molecular phylogenetics. Our method is based on partial likelihood tensors, which are generalizations of partial likelihood vectors, as used in Felsenstein's approach. Exploiting a lexicographic sorting and partial likelihood tensors, it is possible to obtain significant computational savings. We show this on a range of simulated data by enumerating all numerical calculations that are required by our method and the standard approach.
Effects of tensor interactions in {tau} decays
Lopez Castro, G.; Godina Nava, J.J. [Departamento de Fisica, Cinvestav del IPN, Mexico DF. (MEXICO)
1996-02-01
Recent claims for the observation of antisymmetric weak tensor currents in {pi} and {ital K} decays are considered for the case of {tau}{r_arrow}{ital K}{pi}{nu} transitions. Assuming the existence of symmetric tensor currents, a mechanism for the direct production of the {ital K}{sub 2}{sup {asterisk}}(1430) spin-2 meson in {tau} decays is proposed. {copyright} {ital 1996 American Institute of Physics.}
A Comparison of Geodetic Strain Rates With Earthquake Moment Tensors
Zhu, W.; Holt, W. E.
2004-12-01
In this paper we compare the global model from interpolation of GPS data with the global model inferred from earthquake moment tensors. We use the Harvard CMT catalog to calculate moment rates based on 3 assumptions: a. we assume earthquakes are self-similar; b. we assume a uniform Beta value of the Gutenberg-Richter distribution; c. we assume that all of the long-term strain is accommodated seismically. If these assumptions are correct then the seismicity rate is proportional to the tectonic moment rate. We then inferred a long-term moment rate tensor field estimate for all plate boundary zones from which we inferred a long-term seismic strain rate estimate. Using this estimate we solved for a self-consistent kinematic global solution (motions of rigid spherical caps and motions within plate boundary zones) using bi-cubic spline interpolation of the inferred strain rates. We tested the above assumptions by comparing the global kinematic model obtained from earthquake data with a global model inferred from interpolation of space geodetic data [Kreemer et al., 2003]. A comparison between the two models shows good agreement for motion directions of the North American, and Eurasian plates and for the plate boundary zones within these regions (e.g., Tibet). Problems arise, and our assumptions break down, for plates adjacent to fast spreading ridges where divergence of plates appears to be accommodated aseismically. We next investigated the correlation of strain rate tensor inferred from the interpolation of GPS observations within deforming Asia with the earthquake moment tensors, using both elastic and viscous rheologies. Our solutions satisfy the force balance equations for a given rheology. Our goal for this exercise is to investigate whether the interseismic signal, inferred from GPS, correlates better with moment tensor style for an elastic rheology as opposed to a viscous rheology. Results to date suggest that the viscous models only provide a better agreement
C%2B%2B tensor toolbox user manual.
Plantenga, Todd D.; Kolda, Tamara Gibson
2012-04-01
The C++ Tensor Toolbox is a software package for computing tensor decompositions. It is based on the Matlab Tensor Toolbox, and is particularly optimized for sparse data sets. This user manual briefly overviews tensor decomposition mathematics, software capabilities, and installation of the package. Tensors (also known as multidimensional arrays or N-way arrays) are used in a variety of applications ranging from chemometrics to network analysis. The Tensor Toolbox provides classes for manipulating dense, sparse, and structured tensors in C++. The Toolbox compiles into libraries and is intended for use with custom applications written by users.
Algebraically contractible topological tensor network states
Denny, S J; Jaksch, D; Clark, S R [Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU (United Kingdom); Biamonte, J D, E-mail: s.denny1@physics.ox.ac.uk [Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543 (Singapore)
2012-01-13
We adapt the bialgebra and Hopf relations to expose internal structure in the ground state of a Hamiltonian with Z{sub 2} topological order. Its tensor network description allows for exact contraction through simple diagrammatic rewrite rules. The contraction property does not depend on specifics such as geometry, but rather originates from the non-trivial algebraic properties of the constituent tensors. We then generalise the resulting tensor network from a spin-1/2 lattice to a class of exactly contractible states on spin-S degrees of freedom, yielding the most efficient tensor network description of finite Abelian lattice gauge theories. We gain a new perspective on these states as examples of two-dimensional quantum states with algebraically contractible tensor network representations. The introduction of local perturbations to the network is shown to reduce the von Neumann entropy of string-like regions, creating an unentangled sub-system within the bulk in a certain limit. We also show how local perturbations induce finite-range correlations in this system. This class of tensor networks is readily translated onto any lattice, and we differentiate between the physical consequences of bipartite and non-bipartite lattices on the properties of the corresponding quantum states. We explicitly show this on the hexagonal, square, kagome and triangular lattices. (paper)
TWIN SUPPORT TENSOR MACHINES FOR MCS DETECTION
Zhang Xinsheng; Gao Xinbo; Wang Ying
2009-01-01
Tensor representation is useful to reduce the overfitting problem in vector-based learning algorithm in pattern recognition.This is mainly because the structure information of objects in pattern analysis is a reasonable constraint to reduce the number of unknown parameters used to model a classifier.In this paper,we generalize the vector-based learning algorithm TWin Support Vector Machine (TWSVM)to the tensor-based method TWin Support Tensor Machines(TWSTM),which accepts general tensors as input.To examine the effectiveness of TWSTM,we implement the TWSTM method for Microcalcification Clusters (MCs) detection.In the tensor subspace domain,the MCs detection procedure is formulated as a supervised learning and classification problem.and TWSTM is used as a classifier to make decision for the presence of MCs or not.A large number of experiments were carried out to evaluate and compare the performance of the proposed MCs detection algorithm.By comparison with TWSVM,the tensor version reduces the overfitting problem.
Moment tensors of a dislocation in a porous medium
Wang, Zhi; Hu, Hengshan
2016-06-01
A dislocation can be represented by a moment tensor for calculating seismic waves. However, the moment tensor expression was derived in an elastic medium and cannot completely describe a dislocation in a porous medium. In this paper, effective moment tensors of a dislocation in a porous medium are derived. It is found that the dislocation is equivalent to two independent moment tensors, i.e., the bulk moment tensor acting on the bulk of the porous medium and the isotropic fluid moment tensor acting on the pore fluid. Both of them are caused by the solid dislocation as well as the fluid-solid relative motion corresponding to fluid injection towards the surrounding rocks (or fluid outflow) through the fault plane. For a shear dislocation, the fluid moment tensor is zero, and the dislocation is equivalent to a double couple acting on the bulk; for an opening dislocation or fluid injection, the two moment tensors are needed to describe the source. The fluid moment tensor only affects the radiated compressional waves. By calculating the ratio of the radiation fields generated by unit fluid moment tensor and bulk moment tensor, it is found that the fast compressional wave radiated by the bulk moment tensor is much stronger than that radiated by the fluid moment tensor, while the slow compressional wave radiated by the fluid moment tensor is several times stronger than that radiated by the bulk moment tensor.
Scalar perturbations in a Friedmann-like metric with non-null Weyl tensor
Santos, Grasiele B; Salim, José M
2013-01-01
In a previous work some of the authors have solved the Einstein equations of General Relativity for a class of metrics with constant spatial curvature, where it was found a non vanishing Weyl tensor in the presence of an energy-momentum tensor with an anisotropic pressure component. Here, we perform the perturbative analysis of this model in order to study the gravitational stability under linear scalar perturbations. For this purpose, we take the Quasi-Maxwellian formalism of General Relativity as our framework, which offers a naturally covariant and gauge-invariant approach to deal with perturbations that are directly linked to observational quantities. We also consider a generalization of the causal thermodynamics to include the effect of the non-null Weyl tensor, which introduces a "viscosity" due solely to the gravitational tidal forces.
Shape evolution of Ne isotopes and Ne hypernuclei: The interplay of pairing and tensor interactions
Li A.
2014-03-01
Full Text Available We study tensor and pairing effects on the quadruple deformation of neon isotopes based on a deformed Skyrme-Hartree-Fock model with BCS approximation for the pairing channel. We extend the Skyrme-Hartree-Fock formalism for the description of hypernuclei adopting the recently-proposed ESC08b hyperon-nucleon interaction. It is found that the interplay of pairing and tensor interactions is crucial to derive the deformations in several neon isotopes. Especially, the shapes of 26,30Ne are studied in details in comparisons with experimentally observed shapes. Furthermore the deformations of the hypernuclei are compared with the corresponding neon isotopic cores in the presence of tensor force. We find the same shapes with somewhat smaller deformations for single Λ-hypernuclei compared with their core deformations.
Vacuum stress tensor of a scalar field in a rectangular waveguide
Rodrigues, R.B.; Svaiter, N.F. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)]. E-mail: robson@cbpf.br; svaiter@lns.mit.edu; Paola, R.D.M. de [Escola Federal de Engenharia de Itajuba, MG (Brazil). Inst. de Ciencias]. E-mail: rpaola@efei.br
2001-11-01
Using the heat Kernel method and the analytical continuation of the zeta function, we calculate the canonical and improved vacuum stress tensors, {l_brace}T{sub {mu}}{sub {nu}}(vector x){r_brace} and {l_brace}{theta}{sub {mu}}{sub {nu}} (vector x){r_brace}, associated with a massless scalar field confined in the interior of an infinity long rectangular waveguide. The local depence of the renormalized energy for two special configurations when the total energy is positive and negative are presented using {l_brace}T{sub 00}(vector x){r_brace} and {l_brace}{theta}{sub 00}(vector x){r_brace}. From the stress tensors we obtain the local casimir forces in all walls by introducing a particular external configuration. It is hown that this external configuration cannot give account of the edge divergences of the local forces. The local form of the forces is obtained for three special configurations. (author)
Grekova, E. F.
2012-09-01
We consider a linear reduced Cosserat medium: a linear elastic continuum, whose point bodies possess kinematically independent translational and rotational degrees of freedom, but the strain energy does not depend on the gradient of rotation of particles. In such a medium the force stress tensor is asymmetric, but the couple stress tensor is zero. This model can be applied for description of soils and granular media. Since for the time being the experimental technique for measurement of rotational deformations is not well developed, we investigate how the presence of rotational degrees of freedom affects the dynamics of translational displacements. We consider the case of the spherical tensor of inertia and isotropy with respect to the rotational degrees of freedom. Integration of the equation of balance of torques lets us in several cases to put in correspondence a linear reduced Cosserat continuum with the spherical tensor of inertia with a classical (non-polar elastic linear) medium with memory with the same equation for the balance of forces, written in terms of translational displacements. This is possible for the isotropic case and also if the anisotropy is present only in the tensor of elastic constants corresponding to the classical strain tensor. If the material is isotropic with respect to rotational deformations but the (anisotropic) coupling between rotational and classical translational strains is present, then the corresponding classical medium does not exist. If we ignore the rotational degrees of freedom when this coupling is present, this will lead us to the conclusion that the principle of material objectivity is violated.
Bound-Preserving Reconstruction of Tensor Quantities for Remap in ALE Fluid Dynamics
Klima, Matej [Czech Technical Univ. in Prague, Praha (Czech Republic); Kucharik, MIlan [Czech Technical Univ. in Prague, Praha (Czech Republic); Shashkov, Mikhail Jurievich [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Velechovsky, Jan [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2017-01-06
We analyze several new and existing approaches for limiting tensor quantities in the context of deviatoric stress remapping in an ALE numerical simulation of elastic flow. Remapping and limiting of the tensor component-by-component is shown to violate radial symmetry of derived variables such as elastic energy or force. Therefore, we have extended the symmetry-preserving Vector Image Polygon algorithm, originally designed for limiting vector variables. This limiter constrains the vector (in our case a vector of independent tensor components) within the convex hull formed by the vectors from surrounding cells – an equivalent of the discrete maximum principle in scalar variables. We compare this method with a limiter designed specifically for deviatoric stress limiting which aims to constrain the J_{2} invariant that is proportional to the specific elastic energy and scale the tensor accordingly. We also propose a method which involves remapping and limiting the J_{2} invariant independently using known scalar techniques. The deviatoric stress tensor is then scaled to match this remapped invariant, which guarantees conservation in terms of elastic energy.
[Tensor veli palatini and tensor tympani muscles: anatomical, functional and symptomatic links].
Ramirez Aristeguieta, Luis Miguel; Ballesteros Acuña, Luis Ernesto; Sandoval Ortiz, Germán Pablo
2010-01-01
Temporomandibular disorders are associated with symptoms such as tinnitus, vertigo, sensation of hearing loss, ear fullness and otalgia. The connection and dysfunction of the tensor tympani and tensor veli palatini muscles seems to be associated with the aforementioned symptoms. We seek to demonstrate and explain this connection through the morphometry of these structures. We studied 22 paired blocks and 1 left side of human temporal bone. Digital measurements were made of the tensor tympani muscles and stapes. The average length of the stapedial muscle was 5.8 mm SD 0.61, and that of the tensor tympani was 19.69 mm SD 1.07. Anatomical connections were found in all the samples between the tensor veli palatini muscles through a common tendon. There is a need for an interdisciplinary management between physician and specialized dentist in cases of craniofacial pain. 2009 Elsevier España, S.L. All rights reserved.
Tensor network algorithm by coarse-graining tensor renormalization on finite periodic lattices
Zhao, Hui-Hai; Xie, Zhi-Yuan; Xiang, Tao; Imada, Masatoshi
2016-03-01
We develop coarse-graining tensor renormalization group algorithms to compute physical properties of two-dimensional lattice models on finite periodic lattices. Two different coarse-graining strategies, one based on the tensor renormalization group and the other based on the higher-order tensor renormalization group, are introduced. In order to optimize the tensor network model globally, a sweeping scheme is proposed to account for the renormalization effect from the environment tensors under the framework of second renormalization group. We demonstrate the algorithms by the classical Ising model on the square lattice and the Kitaev model on the honeycomb lattice, and show that the finite-size algorithms achieve substantially more accurate results than the corresponding infinite-size ones.
Electronic stress tensor analysis of molecules in gas phase of CVD process for GeSbTe alloy
Nozaki, Hiroo; Ichikawa, Kazuhide; Tachibana, Akitomo
2015-01-01
We analyze the electronic structure of molecules which may exist in gas phase of chemical vapor deposition process for GeSbTe alloy using the electronic stress tensor, with special focus on the chemical bonds between Ge, Sb and Te atoms. We find that, from the viewpoint of the electronic stress tensor, they have intermediate properties between alkali metals and hydrocarbon molecules. We also study the correlation between the bond order which is defined based on the electronic stress tensor, and energy-related quantities. We find that the correlation with the bond dissociation energy is not so strong while one with the force constant is very strong. We interpret these results in terms of the energy density on the "Lagrange surface", which is considered to define the boundary surface of atoms in a molecule in the framework of the electronic stress tensor analysis.
Electronic stress tensor analysis of molecules in gas phase of CVD process for GeSbTe alloy.
Nozaki, Hiroo; Ikeda, Yuji; Ichikawa, Kazuhide; Tachibana, Akitomo
2015-06-15
We analyze the electronic structure of molecules which may exist in gas phase of chemical vapor deposition process for GeSbTe alloy using the electronic stress tensor, with special focus on the chemical bonds between Ge, Sb, and Te atoms. We find that, from the viewpoint of the electronic stress tensor, they have intermediate properties between alkali metals and hydrocarbon molecules. We also study the correlation between the bond order which is defined based on the electronic stress tensor, and energy-related quantities. We find that the correlation with the bond dissociation energy is not so strong while one with the force constant is very strong. We interpret these results in terms of the energy density on the "Lagrange surface," which is considered to define the boundary surface of atoms in a molecule in the framework of the electronic stress tensor analysis.
Testing gravity theories using tensor perturbations
Lin, Weikang; Ishak-Boushaki, Mustapha B.
2017-01-01
Primordial gravitational waves constitute a promising probe of the very early universe physics and the laws of gravity. We study the changes to tensor-mode perturbations that can arise in various modified gravity theories. These include a modified friction and a nonstandard dispersion relation. We introduce a physically motivated parametrization of these effects and use current data to obtain excluded parameter spaces. Taking into account the foreground subtraction, we then perform a forecast analysis focusing on the tensor-mode modified-gravity parameters as constrained by future experiments COrE, Stage-IV and PIXIE. For the tensor-to-scalar ratio r=0.01, we find the minimum detectible modified-gravity effects. In particular, the minimum detectable graviton mass is about 7.8˜9.7×10-33 eV, which is of the same order of magnitude as the graviton mass that allows massive gravity to produce late-time cosmic acceleration. Finally, we study the tensor-mode perturbations in modified gravity during inflation. We find that, the tensor spectral index would be additionally related to the friction parameter ν0 by nT=-3ν0-r/8. In some cases, the future experiments will be able to distinguish this relation from the standard one. In sum, primordial gravitational waves provide a complementary avenue to test gravity theories.
Estimates of the Nucleon Tensor Charge
Gamberg, L P; Gamberg, Leonard; Goldstein, Gary R.
2001-01-01
Like the axial vector charges, defined from the forward nucleon matrix element of the axial vector current on the light cone, the nucleon tensor charge, defined from the corresponding matrix element of the tensor current, is essential for characterizing the momentum and spin structure of the nucleon. Because there must be a helicity flip of the struck quark in order to probe the transverse spin polarization of the nucleon, the transversity distribution (and thus the tensor charge) decouples at leading twist in deep inelastic scattering, although no such suppression appears in Drell-Yan processes. This makes the tensor charge difficult to measure and its non-conservation makes its prediction model dependent. We present a different approach. Exploiting an approximate SU(6)xO(3) symmetric mass degeneracy of the light axial vector mesons (a1(1260), b1(1235) and h1(1170)) and using pole dominance, we calculate the tensor charge. The result is simple in form and depends on the decay constants of the axial vector me...
Global nuclear structure effects of tensor interaction
Zalewski, M; Rafalski, M; Satula, W; Werner, T R; Wyss, R A
2009-01-01
A direct fit of the isoscalar spin-orbit (SO) and both isoscalar and isovector tensor coupling constants to the f5/2-f7/2 SO splittings in 40Ca, 56Ni, and 48Ca nuclei requires a drastic reduction of the isoscalar SO strength and strong attractive tensor coupling constants. The aim of this work is to address further consequences of these strong attractive tensor and weak SO fields on binding energies, nuclear deformability, and high-spin states. In particular, we show that contribution to the nuclear binding energy due to the tensor field shows generic magic structure with tensorial magic numbers at N(Z)=14, 32, 56, or 90 corresponding to the maximum spin-asymmetries in 1d5/2, 1f7/2-2p3/2, 1g9/2-2d5/2 and 1h11/2-2f7/2 single-particle configurations and that these numbers are smeared out by pairing correlations and deformation effects. We also examine the consequences of strong attractive tensor fields and weak SO interaction on nuclear stability at the drip lines, in particular close to the tensorial doubly ma...
Tensor scale-based image registration
Saha, Punam K.; Zhang, Hui; Udupa, Jayaram K.; Gee, James C.
2003-05-01
Tangible solutions to image registration are paramount in longitudinal as well as multi-modal medical imaging studies. In this paper, we introduce tensor scale - a recently developed local morphometric parameter - in rigid image registration. A tensor scale-based registration method incorporates local structure size, orientation and anisotropy into the matching criterion, and therefore, allows efficient multi-modal image registration and holds potential to overcome the effects of intensity inhomogeneity in MRI. Two classes of two-dimensional image registration methods are proposed - (1) that computes angular shift between two images by correlating their tensor scale orientation histogram, and (2) that registers two images by maximizing the similarity of tensor scale features. Results of applications of the proposed methods on proton density and T2-weighted MR brain images of (1) the same slice of the same subject, and (2) different slices of the same subject are presented. The basic superiority of tensor scale-based registration over intensity-based registration is that it may allow the use of local Gestalts formed by the intensity patterns over the image instead of simply considering intensities as isolated events at the pixel level. This would be helpful in dealing with the effects of intensity inhomogeneity and noise in MRI.
Particle creation from the quantum stress tensor
Firouzjaee, Javad T
2015-01-01
Among the different methods to derive particle creation, finding the quantum stress tensor expectation value gives a covariant quantity which can be used for examining the back-reaction issue. However this tensor also includes vacuum polarization in a way that depends on the vacuum chosen. Here we review different aspects of particle creation by looking at energy conservation and at the quantum stress tensor. It will be shown that in the case of general spherically symmetric black holes that have a \\emph{dynamical horizon}, as occurs in a cosmological context, one cannot have pair creation on the horizon because this violates energy conservation. This confirms the results obtained in other ways in a previous paper [25]. Looking at the expectation value of the quantum stress tensor with three different definitions of the vacuum state, we study the nature of particle creation and vacuum polarization in black hole and cosmological models, and the associated stress energy tensors. We show that the thermal tempera...
Entanglement, tensor networks and black hole horizons
Molina-Vilaplana, J.; Prior, J.
2014-11-01
We elaborate on a previous proposal by Hartman and Maldacena on a tensor network which accounts for the scaling of the entanglement entropy in a system at a finite temperature. In this construction, the ordinary entanglement renormalization flow given by the class of tensor networks known as the Multi Scale Entanglement Renormalization Ansatz (MERA), is supplemented by an additional entanglement structure at the length scale fixed by the temperature. The network comprises two copies of a MERA circuit with a fixed number of layers and a pure matrix product state which joins both copies by entangling the infrared degrees of freedom of both MERA networks. The entanglement distribution within this bridge state defines reduced density operators on both sides which cause analogous effects to the presence of a black hole horizon when computing the entanglement entropy at finite temperature in the AdS/CFT correspondence. The entanglement and correlations during the thermalization process of a system after a quantum quench are also analyzed. To this end, a full tensor network representation of the action of local unitary operations on the bridge state is proposed. This amounts to a tensor network which grows in size by adding succesive layers of bridge states. Finally, we discuss on the holographic interpretation of the tensor network through a notion of distance within the network which emerges from its entanglement distribution.
The pressure tensor in tangential equilibria
F. Mottez
2004-09-01
Full Text Available The tangential equilibria are characterized by a bulk plasma velocity and a magnetic field that are perpendicular to the gradient direction. Such equilibria can be spatially periodic (like waves, or they can separate two regions with asymptotic uniform conditions (like MHD tangential discontinuities. It is possible to compute the velocity moments of the particle distribution function. Even in very simple cases, the pressure tensor is not isotropic and not gyrotropic. The differences between a scalar pressure and the pressure tensor derived in the frame of the Maxwell-Vlasov theory are significant when the gradient scales are of the order of the Larmor radius; they concern mainly the ion pressure tensor.
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...
Spectral analysis of the full gravity tensor
Rummel, R.; van Gelderen, M.
1992-10-01
It is shown that, when the five independent components of the gravity tensor are grouped into (Gamma-zz), (Gamma-xz, Gamma-yz), and (Gamma-xx - Gamma-yy, 2Gamma-xy) sets and expanded into an infinite series of pure-spin spherical harmonic tensors, it is possible to derive simple eigenvalue connections between these three sets and the spherical harmonic expansion of the gravity potential. The three eigenvalues are (n + 1)(n + 2), -(n + 2) sq rt of n(n + 1), and sq rt of (n - 1)n(n + 1)(n + 2). The joint ESA and NASA Aristoteles mission is designed to measure with high precision the tensor components Gamma-zz, Gamma-yz, and Gamma-yy, which will make it possible to determine the global gravity field in six months time with a high precision.
Carrozza, Sylvain
2015-01-01
We define in this paper a class of three indices tensor models, endowed with $O(N)^{\\otimes 3}$ invariance ($N$ being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the $U(N)$ invariant models. We first exhibit the existence of a large $N$ expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large $N$ expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.
Carrozza, Sylvain; Tanasa, Adrian
2016-08-01
We define in this paper a class of three-index tensor models, endowed with {O(N)^{⊗ 3}} invariance (N being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the U(N) invariant models. We first exhibit the existence of a large N expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large N expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.
Inflatonic baryogenesis with large tensor mode
Naoyuki Takeda
2015-06-01
Full Text Available We consider a complex inflaton field with a CP asymmetric term for its potential. This CP asymmetric term produces the global charge of the inflaton after inflation. With the assignment of the baryon number to the inflaton, the baryon asymmetry of the universe is produced by inflaton's decay. In addition to this, the U(1 breaking term modulates the curvature of the inflaton radial direction depending on its phase, which affects the tensor-to-scalar ratio. In this paper, we have studied the relation between the baryon asymmetry and the tensor-to-scalar ratio, then verified that the future CMB observation could test this baryogenesis scenario with large tensor modes.
Quantum Fluctuations Of The Stress Tensor
Wu, C
2002-01-01
Quantum fluctuations of the stress tensor are important in many branches of physics, including the study of the validity of semiclassical gravity and the backreaction problem in stochastic semiclassical gravity. The geometry fluctuations induced by stress tensor fluctuations are important to understand quantum gravity and the problem of lightcone fluctuations. Stress tensor fluctuations also hold the key to understand fundamental physical effects like quantum fluctuations of radiation pressure, and that is crucial to the sensitivity of interferometers and the limitations on the detection of gravitational waves. Even the wave-particle duality of light can be better understood by the study of quantum fluctuations of thermal radiation. It is well known in quantum field theory that the expectation value of the energy density, which contains quadratic field operators (e.g. E2 and B2 in the electromagnatic field case), is divergent and can be renormalized simply by normal ordering, which is subtracting out the vac...
List Decoding Tensor Products and Interleaved Codes
Gopalan, Parikshit; Raghavendra, Prasad
2008-01-01
We design the first efficient algorithms and prove new combinatorial bounds for list decoding tensor products of codes and interleaved codes. We show that for {\\em every} code, the ratio of its list decoding radius to its minimum distance stays unchanged under the tensor product operation (rather than squaring, as one might expect). This gives the first efficient list decoders and new combinatorial bounds for some natural codes including multivariate polynomials where the degree in each variable is bounded. We show that for {\\em every} code, its list decoding radius remains unchanged under $m$-wise interleaving for an integer $m$. This generalizes a recent result of Dinur et al \\cite{DGKS}, who proved such a result for interleaved Hadamard codes (equivalently, linear transformations). Using the notion of generalized Hamming weights, we give better list size bounds for {\\em both} tensoring and interleaving of binary linear codes. By analyzing the weight distribution of these codes, we reduce the task of boundi...
Codazzi Tensors with Two Eigenvalue Functions
Merton, Gabe
2011-01-01
This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is $n-1$, then the metric is a warped product where the base is an open interval- a conclusion we will show to be true under a milder trace condition. Furthermore, we construct examples of Codazzi tensors having two eigenvalue functions, one of which has eigenspace dimension $n-1$, where the metric is not a warped product with interval base, refuting a remark in \\cite{Besse} that the warped product conclusion holds without any restriction on the trace.
Permittivity and permeability tensors for cloaking applications
Choudhury, Balamati; Jha, Rakesh Mohan
2016-01-01
This book is focused on derivations of analytical expressions for stealth and cloaking applications. An optimal version of electromagnetic (EM) stealth is the design of invisibility cloak of arbitrary shapes in which the EM waves can be controlled within the cloaking shell by introducing a prescribed spatial variation in the constitutive parameters. The promising challenge in design of invisibility cloaks lies in the determination of permittivity and permeability tensors for all the layers. This book provides the detailed derivation of analytical expressions of the permittivity and permeability tensors for various quadric surfaces within the eleven Eisenhart co-ordinate systems. These include the cylinders and the surfaces of revolutions. The analytical modeling and spatial metric for each of these surfaces are provided along with their tensors. This mathematical formulation will help the EM designers to analyze and design of various quadratics and their hybrids, which can eventually lead to design of cloakin...
Friction tensor concept for textured surfaces
K R Y Simha; Anirudhan Pottirayil; Pradeep L Menezes; Satish V Kailas
2008-06-01
Directionality of grinding marks inﬂuences the coefﬁcient of friction during sliding. Depending on the sliding direction the coefﬁcient of friction varies between maximum and minimum for textured surfaces. For random surfaces without any texture the friction coefﬁcient becomes independent of the sliding direction. This paper proposes the concept of a friction tensor analogous to the heat conduction tensor in anisotropic media. This implies that there exists two principal friction coefﬁcients $\\mu_{1,2}$ analogous to the principal conductivities $k_{1,2}$. For symmetrically textured surfaces the principal directions are orthogonal with atleast one plane of symmetry. However, in the case of polished single crystalline solids in relative sliding motion, crystallographic texture controls the friction tensor.
Holographic duality from random tensor networks
Hayden, Patrick; Qi, Xiao-Liang; Thomas, Nathaniel; Walter, Michael; Yang, Zhao
2016-01-01
Tensor networks provide a natural framework for exploring holographic duality because they obey entanglement area laws. They have been used to construct explicit simple 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 obey the Ryu-Takayanagi entropy formula for all boundary regions, whether connected or not, a fact closely related to known properties of the multipartite entanglement of assistance. Moreover, we find that all boundary regions faithfully encode the physics of their entire bulk entanglement wedges, not just their smaller causal wedges. 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 bu...
Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images.
Ennis, Daniel B; Kindlmann, Gordon
2006-01-01
This paper outlines the mathematical development and application of two analytically orthogonal tensor invariants sets. Diffusion tensors can be mathematically decomposed into shape and orientation information, determined by the eigenvalues and eigenvectors, respectively. The developments herein orthogonally decompose the tensor shape using a set of three orthogonal invariants that characterize the magnitude of isotropy, the magnitude of anisotropy, and the mode of anisotropy. The mode of anisotropy is useful for resolving whether a region of anisotropy is linear anisotropic, orthotropic, or planar anisotropic. Both tensor trace and fractional anisotropy are members of an orthogonal invariant set, but they do not belong to the same set. It is proven that tensor trace and fractional anisotropy are not mutually orthogonal measures of the diffusive process. The results are applied to the analysis and visualization of diffusion tensor magnetic resonance images of the brain in a healthy volunteer. The theoretical developments provide a method for generating scalar maps of the diffusion tensor data, including novel fractional anisotropy maps that are color encoded for the mode of anisotropy and directionally encoded colormaps of only linearly anisotropic structures, rather than of high fractional anisotropy structures.
Role of the tensor interaction in He isotopes with a tensor-optimized shell model
Myo, Takayuki; Toki, Hiroshi; Ikeda, Kiyomi
2011-01-01
We study the role of the tensor interaction in He isotopes systematically on the basis of the tensor-optimized shell model (TOSM). We use a bare nucleon-nucleon interaction AV8 obtained from nucleon-nucleon scattering data. The short-range correlation is treated in the unitary correlation operator method (UCOM). Using the TOSM+UCOM approach, we investigate the role of tensor interaction on each spectrum in He isotopes. It is found that the tensor interaction enhances the LS splitting energy observed in 5He, in which the p1/2 and p3/2 orbits play different roles on the tensor correlation. In {6,7,8}He, the low-lying states containing extra neutrons in the p3/2 orbit gain the tensor contribution. On the other hand, the excited states containing extra neutrons in the p1/2 orbit lose the tensor contribution due to the Pauli-blocking effect with the 2p2h states in the 4He core configuration.
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...
Vacuum stress-tensor in SSB theories
Asorey, Manuel; Ribeiro, Baltazar J; Shapiro, Ilya L
2012-01-01
The renormalized energy-momentum tensor of vacuum has been deeply explored many years ago. The main result of these studies was that such a tensor should satisfy the conservation laws which reflects the covariance of the theory in the presence of loop corrections. In view of this general result we address two important questions, namely how to implement the momentum cut-off in a covariant way and whether this general result holds in the theory with Spontaneous Symmetry Breaking. In the last case some new interesting details arise and although the calculations are more involved we show that the final result satisfies the conservation laws.
Raman Tensor Formalism for Optically Anisotropic Crystals.
Kranert, Christian; Sturm, Chris; Schmidt-Grund, Rüdiger; Grundmann, Marius
2016-03-25
We present a formalism for calculating the Raman scattering intensity dependent on the polarization configuration for optically anisotropic crystals. It can be applied to crystals of arbitrary orientation and crystal symmetry measured in normal incidence backscattering geometry. The classical Raman tensor formalism cannot be used for optically anisotropic materials due to birefringence causing the polarization within the crystal to be depth dependent. We show that in the limit of averaging over a sufficiently large scattering depth, the observed Raman intensities converge and can be described by an effective Raman tensor given here. Full agreement with experimental results for uniaxial and biaxial crystals is demonstrated.
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...
TIAN Dongyan; JIN Ming; DUI Guansuo
2006-01-01
A new approach for the derivation of the principal invariants of the stretch tensor with respect to the right Cauchy Green tensor is presented in this paper. According to the definition of the derivation of tensor function, the three first-order derivatives for the principal invariants of the stretch tensor are obtained through derivation directly to the right Cauchy-Green tensor by incremental method. Then the three second-order derivatives are yielded by the derivation to the right Cauchy-Green strain tensor directly. Furthermore, an explicit expression of the tangent modulus of the general Varga material is given as an example.
Tensor Fields in Relativistic Quantum Mechanics
Dvoeglazov, Valeriy V
2015-01-01
We re-examine the theory of antisymmetric tensor fields and 4-vector potentials. We discuss corresponding massless limits. We analize the quantum field theory taking into account the mass dimensions of the notoph and the photon. Next, we deduced the gravitational field equations from relativistic quantum mechanics.
Holographic coherent states from random tensor networks
Qi, Xiao-Liang; Yang, Zhao; You, Yi-Zhuang
2017-08-01
Random tensor networks provide useful models that incorporate various important features of holographic duality. A tensor network is usually defined for a fixed graph geometry specified by the connection of tensors. In this paper, we generalize the random tensor network approach to allow quantum superposition of different spatial geometries. We setup a framework in which all possible bulk spatial geometries, characterized by weighted adjacient matrices of all possible graphs, are mapped to the boundary Hilbert space and form an overcomplete basis of the boundary. We name such an overcomplete basis as holographic coherent states. A generic boundary state can be expanded in this basis, which describes the state as a superposition of different spatial geometries in the bulk. We discuss how to define distinct classical geometries and small fluctuations around them. We show that small fluctuations around classical geometries define "code subspaces" which are mapped to the boundary Hilbert space isometrically with quantum error correction properties. In addition, we also show that the overlap between different geometries is suppressed exponentially as a function of the geometrical difference between the two geometries. The geometrical difference is measured in an area law fashion, which is a manifestation of the holographic nature of the states considered.
Quantum tensor product structures are observable induced.
Zanardi, Paolo; Lidar, Daniel A; Lloyd, Seth
2004-02-13
It is argued that the partition of a quantum system into subsystems is dictated by the set of operationally accessible interactions and measurements. The emergence of a multipartite tensor product structure of the state space and the associated notion of quantum entanglement are then relative and observable induced. We develop a general algebraic framework aimed to formalize this concept.
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.
Pomeron as a Reggeized Tensor Glueball
MA Wei-Xing; A.W.Thomas; SHEN Peng-Nian; ZHOU Li-Juan
2001-01-01
We study gluonic content of the pomeron and propose that the pomeron could be a reggeized tensor glueball ζ(2230) with quantum numbers IG JPc = 0+2++.This conjecture is examined in high energy proton-proton elastic scattering,and the calculations lend a favorable support to our physical idea.``
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.)
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.
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...
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.
Weinberg's Approach and Antisymmetric Tensor Fields
Dvoeglazov, V V
2002-01-01
We extend the previous series of articles \\cite{HPA} devoted to finding mappings between the Weinberg-Tucker-Hammer formalism and antisymmetric tensor fields. Now we take into account solutions of different parities of the Weinberg-like equations. Thus, the Proca, Duffin-Kemmer and Bargmann-Wigner formalisms are generalized.
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.
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 Squeezed Limits and the Higuchi Bound
Bordin, Lorenzo; Mirbabayi, Mehrdad; Noreña, Jorge
2016-01-01
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 con- sistency 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...
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 correlations in nuclei and exlusive electron scattering
Ryckebusch, J; Van Nespen, W; Debruyne, D
2000-01-01
The effect of tensor nucleon-nucleon correlations upon exclusive and semi-exclusive electronuclear reactions is studied. Differential cross sections for the semi-exclusive ^{16}O(e,e'p) and exclusive ^{16}O(e,e'pn) processes are computed by explicitly evaluating the dynamical electromagnetic coupling to a tensor correlated nucleon pair. In both reaction channels the tensor correlations contribute in a very substantial way. Tensor correlations are found to generate more electronuclear strength than central Jastrow correlations do.
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.
Radiation force and balance of electromagnetic momentum
Campos, I.; Jiménez, J. L.; Roa-Neri, J. A. E.
2016-07-01
Some force densities can be expressed as a divergence of a stress tensor, as is the case with the electromagnetic force density. We have shown elsewhere that from the Maxwell equations several balance equations of electromagnetic momentum can be derived, depending on the form these equations are expressed in terms of fields E, D, B, H, and polarisations P and M. These balance equations imply different force densities and different stress tensors, providing a great flexibility to solve particular problems. Among these force densities we have found some proposed in the past with plausibility arguments, like the Einstein-Laub force density, while other proposed force densities appear as particular or limit cases of these general force densities, like the Helmholtz force density. We calculate the radiation force of an electromagnetic wave incident on a semi-infinite negligibly absorbing material using these balance equations, corroborating in this way that the surface integration of the stress tensor gives the same result that the calculation made through a volume integration of the force density, as done by Bohren. As is usual in applications of Gauss’s theorem, the surface on which the surface integral is to be performed must be chosen judiciously, and due care of discontinuities on the boundary conditions must be taken. Advanced undergraduates and graduate students will find a different approach to new aspects of the interaction of radiation with matter.
Conductivity tensor mapping of the human brain using diffusion tensor MRI
Tuch, David S.; Wedeen, Van J.; Dale, Anders M.; George, John S.; Belliveau, John W.
2001-09-01
Knowledge of the electrical conductivity properties of excitable tissues is essential for relating the electromagnetic fields generated by the tissue to the underlying electrophysiological currents. Efforts to characterize these endogenous currents from measurements of the associated electromagnetic fields would significantly benefit from the ability to measure the electrical conductivity properties of the tissue noninvasively. Here, using an effective medium approach, we show how the electrical conductivity tensor of tissue can be quantitatively inferred from the water self-diffusion tensor as measured by diffusion tensor magnetic resonance imaging. The effective medium model indicates a strong linear relationship between the conductivity and diffusion tensor eigenvalues (respectively, σ and d) in agreement with theoretical bounds and experimental measurements presented here (σ/d ≈ 0.844 ± 0.0545 S·s/mm3, r2 = 0.945). The extension to other biological transport phenomena is also discussed.
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.
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.
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...
The operator tensor formulation of quantum theory.
Hardy, Lucien
2012-07-28
In this paper, we provide what might be regarded as a manifestly covariant presentation of discrete quantum theory. A typical quantum experiment has a bunch of apparatuses placed so that quantum systems can pass between them. We regard each use of an apparatus, along with some given outcome on the apparatus (a certain detector click or a certain meter reading for example), as an operation. An operation (e.g. B(b(2)a(3))(a(1))) can have zero or more quantum systems inputted into it and zero or more quantum systems outputted from it. The operation B(b(2)a(3))(a(1)) has one system of type a inputted, and one system of type b and one system of type a outputted. We can wire together operations to form circuits, for example, A(a(1))B(b(2)a(3))(a(1))C(b(2)a(3)). Each repeated integer label here denotes a wire connecting an output to an input of the same type. As each operation in a circuit has an outcome associated with it, a circuit represents a set of outcomes that can happen in a run of the experiment. In the operator tensor formulation of quantum theory, each operation corresponds to an operator tensor. For example, the operation B(b(2)a(3))(a(1)) corresponds to the operator tensor B(b(2)a(3))(a(1)). Further, the probability for a general circuit is given by replacing operations with corresponding operator tensors as in Prob(A(a(1))B(b(2)a(3))(a(1))C(b(2)a(3))) = Â(a(1))B(b(2)a(3))(a(1))C(b(2)a(3)). Repeated integer labels indicate that we multiply in the associated subspace and then take the partial trace over that subspace. Operator tensors must be physical (namely, they must have positive input transpose and satisfy a certain normalization condition).
Multidimensional seismic data reconstruction using tensor analysis
Kreimer, Nadia
Exploration seismology utilizes the seismic wavefield for prospecting oil and gas. The seismic reflection experiment consists on deploying sources and receivers in the surface of an area of interest. When the sources are activated, the receivers measure the wavefield that is reflected from different subsurface interfaces and store the information as time-series called traces or seismograms. The seismic data depend on two source coordinates, two receiver coordinates and time (a 5D volume). Obstacles in the field, logistical and economical factors constrain seismic data acquisition. Therefore, the wavefield sampling is incomplete in the four spatial dimensions. Seismic data undergoes different processes. In particular, the reconstruction process is responsible for correcting sampling irregularities of the seismic wavefield. This thesis focuses on the development of new methodologies for the reconstruction of multidimensional seismic data. This thesis examines techniques based on tensor algebra and proposes three methods that exploit the tensor nature of the seismic data. The fully sampled volume is low-rank in the frequency-space domain. The rank increases when we have missing traces and/or noise. The methods proposed perform rank reduction on frequency slices of the 4D spatial volume. The first method employs the Higher-Order Singular Value Decomposition (HOSVD) immersed in an iterative algorithm that reinserts weighted observations. The second method uses a sequential truncated SVD on the unfoldings of the tensor slices (SEQ-SVD). The third method formulates the rank reduction problem as a convex optimization problem. The measure of the rank is replaced by the nuclear norm of the tensor and the alternating direction method of multipliers (ADMM) minimizes the cost function. All three methods have the interesting property that they are robust to curvature of the reflections, unlike many reconstruction methods. Finally, we present a comparison between the methods
Regularity Criteria for a Coupled Navier-Stokes and Q-Tensor System
Jishan Fan
2013-01-01
Full Text Available We study a system describing the evolution of a nematic liquid crystal flow. The system couples a forced Navier-Stokes system describing the flow with a parabolic-type system describing the evolution of the nematic crystal director fields (Q-tensors. We prove some regularity criteria for the local strong solutions. However, we do not provide estimates on the rates of increase of high norms.
A preliminary report on the development of MATLAB tensor classes for fast algorithm prototyping.
Bader, Brett William; Kolda, Tamara Gibson (Sandia National Laboratories, Livermore, CA)
2004-07-01
We describe three MATLAB classes for manipulating tensors in order to allow fast algorithm prototyping. A tensor is a multidimensional or N-way array. We present a tensor class for manipulating tensors which allows for tensor multiplication and 'matricization.' We have further added two classes for representing tensors in decomposed format: cp{_}tensor and tucker{_}tensor. We demonstrate the use of these classes by implementing several algorithms that have appeared in the literature.
On some properties of nonnegative weakly irreducible tensors
Yang, Yuning
2011-01-01
In this paper, we mainly focus on how to generalize some conclusions from \\emph{nonnegative irreducible tensors} to \\emph{nonnegative weakly irreducible tensors}. To do so, a basic lemma as Lemma 3.1 of \\cite{s11} is proven using new tools. First, we define the stochastic tensor. Then we show that every nonnegative weakly irreducible tensor with spectral radius be 1 is diagonally similar to a unique weakly irreducible stochastic tensor. Based on it, we prove some important lemmas, which help us to generalize the results.
Testing gravity theories using tensor perturbations
Lin, Weikang; Ishak, Mustapha
2016-12-01
Primordial gravitational waves constitute a promising probe of the very early Universe and the laws of gravity. We study in this work changes to tensor-mode perturbations that can arise in various proposed modified gravity theories. These include additional friction effects, nonstandard dispersion relations involving a massive graviton, a modified speed, and a small-scale modification. We introduce a physically motivated parametrization of these effects and use current available data to obtain exclusion regions in the parameter spaces. Taking into account the foreground subtraction, we then perform a forecast analysis focusing on the tensor-mode modified-gravity parameters as constrained by the future experiments COrE, Stage-IV and PIXIE. For a fiducial value of the tensor-to-scalar ratio r =0.01 , we find that an additional friction of 3.5-4.5% compared to GR will be detected at 3 -σ by these experiments, while a decrease in friction will be more difficult to detect. The speed of gravitational waves needs to be by 5-15% different from the speed of light for detection. We find that the minimum detectable graviton mass is about 7.8 - 9.7 ×10-33 eV , which is of the same order of magnitude as the graviton mass that allows massive gravity theories to produce late-time cosmic acceleration. Finally, we study the tensor-mode perturbations in modified gravity during inflation using our parametrization. We find that, in addition to being related to r , the tensor spectral index would be related to the friction parameter ν0 by nT=-3 ν0-r /8 . Assuming that the friction parameter is unchanged throughout the history of the Universe, and that ν0 is much larger than r , the future experiments considered here will be able to distinguish this modified-gravity consistency relation from the standard inflation consistency relation, and thus can be used as a further test of modified gravity. In summary, tensor-mode perturbations and cosmic-microwave-background B
Exploring the Tensor Networks/AdS Correspondence
Bhattacharyya, Arpan; Hung, Ling-Yan; Liu, Si-Nong
2016-01-01
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. Study- ing 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, the leading contribution is given by structures related to geodesics connecting the operators inserted at the boundary physical dofs. Such considerations admits 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 ...
Tensors the mathematics of relativity and continuum mechanics
Das, A J
2007-01-01
Tensors: The Mathematics of Relativity Theory and Continuum Mechanics, by Anadijiban Das, emerged from courses taught over the years at the University College of Dublin, Carnegie-Mellon University and Simon Fraser University. This book will serve readers well as a modern introduction to the theories of tensor algebra and tensor analysis. Throughout Tensors, examples and worked-out problems are furnished from the theory of relativity and continuum mechanics. Topics covered in this book include, but are not limited to: -tensor algebra -differential manifold -tensor analysis -differential forms -connection forms -curvature tensors -Riemannian and pseudo-Riemannian manifolds The extensive presentation of the mathematical tools, examples and problems make the book a unique text for the pursuit of both the mathematical relativity theory and continuum mechanics.
A gravito-electromagnetic analogy based on tidal tensors
Costa, L F; Herdeiro, Carlos A. R.
2006-01-01
We propose a new approach to a physical analogy between General Relativity and Electromagnetism, based on comparing tidal tensors of both theories. Using this approach we write a covariant form for the gravitational analogues of the Maxwell equations, from which the regime of validity of the analogy becomes manifest. Two explicit realisations of the analogy are given. The first one matches linearised gravitational tidal tensors to exact electromagnetic tidal tensors in Minkwoski spacetime. The second one matches exact magnetic gravitational tidal tensors for ultra-stationary metrics to exact magnetic tidal tensors of electromagnetism in curved spaces. We then establish a new proof for a class of tensor identities that define invariants of the type $\\vec{E}^2-\\vec{B}^2$ and $\\vec{E}\\cdot\\vec{B}$, and we exhibit the invariants built from tidal tensors in both gravity and electromagnetism. We contrast our approach with the two gravito-electromagnetic analogies commonly found in the literature, which are reviewed...
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...
Extended Scalar-Tensor Theories of Gravity
Crisostomi, Marco; Tasinato, Gianmassimo
2016-01-01
We determine new consistent scalar-tensor theories of gravity, with potentially interesting cosmological applications. We develop a general method to find the conditions for the existence of a primary constraint, which is necessary to prevent the propagation of an additional dangerous mode associated with higher order equations of motion. We then classify the most general, consistent scalar-tensor theories that are at most quadratic in the second derivatives of the scalar field. In addition, we investigate the possible connection between these theories and (beyond) Horndeski through conformal and disformal transformations. Finally, we point out that these theories can be associated with new operators in the effective field theory of dark energy, which might open up new possibilities to test dark energy models in future surveys.
Oscillating Chiral Tensor Spectrum from Axionic Inflation
Obata, Ippei
2016-01-01
We study the axionic inflation with a modulated potential and examine if the primordial tensor power spectrum exhibits oscillatory feature, which is testable with future space-based gravitational wave experiments such as DECIGO and BBO. In the case of the single-field axion monodromy inflation, it turns out that it is difficult to detect the oscillation in the spectrum due to suppression of the sub-Planckian decay constant of axion. On the other hand, in the case of aligned chromo-natural inflation where the axion is coupled to a SU(2) gauge field, it turns out that the sizable oscillation in the tensor spectrum can occur due to the enhancement of chiral gravitational waves sourced by the gauge field. We expect that this feature will be a new probe to axion phenomenologies in early universe through the chiral gravitational waves.
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.
Tensor modes on the string theory landscape
Westphal, Alexander
2012-01-01
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.
Scalable Tensor Factorizations with Missing Data
Acar, Evrim; Dunlavy, Daniel M.; Kolda, Tamara G.
2010-01-01
The problem of missing data is ubiquitous in domains such as biomedical signal processing, network trac analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networks|all domains in which data collection is subject to occasional errors. Moreover......, these data sets can be quite large and have more than two axes of variation, e.g., sender, receiver, time. Many applications in those domains aim to capture the underlying latent structure of the data; in other words, they need to factorize data sets with missing entries. If we cannot address the problem...... 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...
Damage Tensor Analysis on Regional Seismic Status
Zhong Jimao; Cheng Wanzheng
2006-01-01
In this paper, we researched the regional seismic status by using theories of the Damage Mechanics. The macroscopic damage status of the earth crust block, which is caused by earthquake fracture, is described with several concepts-the damage degree, the damage rate and the strain rate. In the earthquake process, the average strain rate of the studied block is equal to the sum of all seismic moment tensors of the earthquakes taking place in unit time and physical volume. To describe the anisotropy of microdamage of the crust block, we use the damage tensor that is expressed in the fissure density. By means of the transformation from the focal coordinate system to the observation system, we obtained the external normal vector of the focal fault plane expressed in its observation system and obtained the macrodamage degree of the researched block, which is calculated in dyadic. This provides a new analysis method for recognizing the underground damage status and the stress status.
Fermionic orbital optimisation in tensor network states
Krumnow, C; Eisert, J
2015-01-01
Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such non-local fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this work, we propose a way to overcome this challenge. We suggest a method intertwining the optimisation over matrix product states with suitable fermionic Gaussian mode transformations, hence bringing the advantages of both approaches together. The described algorithm generalises basis changes in the spirit of the Hartree-Fock methods to matrix-product states, and provides a black box tool for basis optimisations in tensor network methods.
Review of Scalars, Vectors, Tensors, and Dyads
Schnack, Dalton D.
In MHD, we will deal with relationships between quantities such as the magnetic field and the velocity that have both magnitude and direction. These quantities are examples of vectors (or, as we shall soon see, pseudovectors). The basic concepts of scalar and vector quantities are introduced early in any scientific education. However, to formulate the laws of MHD precisely, it will be necessary to generalize these ideas and to introduce the less familiar concepts of matrices, tensors, and dyads. The ability to understand and manipulate these abstract mathematical concepts is essential to learning MHD. Therefore, for the sake of both reference and completeness, this lecture is about the mathematical properties of scalars, vectors, matrices, tensors, and dyads. If you are already an expert, or think you are, please skip class and go on to Lecture 3. You can always refer back here if needed!
Scalable tensor factorizations for incomplete data
Acar, Evrim; Dunlavy, Daniel M.; KOlda, Tamara G.
2011-01-01
The problem of incomplete data—i.e., data with missing or unknown values—in multi-way arrays is ubiquitous in biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, communication networks, etc. We consider the problem of how...... 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......-WOPT on two real-world applications: a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes and the problem of modeling computer network traffic where data may be absent due to the expense of the data collection process....
Scalable Tensor Factorizations with Missing Data
Acar, Evrim; Dunlavy, Daniel M.; Kolda, Tamara G.
2010-01-01
The problem of missing data is ubiquitous in domains such as biomedical signal processing, network trac analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networks|all domains in which data collection is subject to occasional errors. Moreover......, these data sets can be quite large and have more than two axes of variation, e.g., sender, receiver, time. Many applications in those domains aim to capture the underlying latent structure of the data; in other words, they need to factorize data sets with missing entries. If we cannot address the problem...... 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...
Derived Metric Tensors for Flow Surface Visualization.
Obermaier, H; Joy, K I
2012-12-01
Integral flow surfaces constitute a widely used flow visualization tool due to their capability to convey important flow information such as fluid transport, mixing, and domain segmentation. Current flow surface rendering techniques limit their expressiveness, however, by focusing virtually exclusively on displacement visualization, visually neglecting the more complex notion of deformation such as shearing and stretching that is central to the field of continuum mechanics. To incorporate this information into the flow surface visualization and analysis process, we derive a metric tensor field that encodes local surface deformations as induced by the velocity gradient of the underlying flow field. We demonstrate how properties of the resulting metric tensor field are capable of enhancing present surface visualization and generation methods and develop novel surface querying, sampling, and visualization techniques. The provided results show how this step towards unifying classic flow visualization and more advanced concepts from continuum mechanics enables more detailed and improved flow analysis.
Entanglement, Tensor Networks and Black Hole Horizons
Molina-Vilaplana, Javier
2014-01-01
We elaborate on a previous proposal by Hartman and Maldacena on a tensor network which accounts for the scaling of the entanglement entropy in a system at a finite temperature. In this construction, the ordinary entanglement renormalization flow given by the class of tensor networks known as the Multi Scale Entanglement Renormalization Ansatz (MERA), is supplemented by an additional entanglement structure at the length scale fixed by the temperature. The network comprises two copies of a MERA circuit with a fixed number of layers and a pure matrix product state which joins both copies by entangling the infrared degrees of freedom of both MERA networks. The entanglement distribution within this bridge state defines reduced density operators on both sides which cause analogous effects to the presence of a black hole horizon when computing the entanglement entropy at finite temperature in the AdS/CFT correspondence. The entanglement and correlations during the thermalization process of a system after a quantum q...
Translationally invariant treatment of pair correlations in nuclei; 2, Tensor correlations
Bishop, R F; Moliner, I; Navarro, J; Portesi, M; Puente, A; Walet, N R
1998-01-01
We study the extension of our translationally invariant treatment of few-body nuclear systems to include tensor forces and correlations. It is shown that a direct application of our method is not as successful for realistic V6 interactions as our previous results for V4 potentials suggested. We investigate the cause in detail for the case of $^4$He, and show that a combination of our method with that of Jastrow-correlated wave functions seems to be a lot more powerful, thereby suggesting that for mildly to strongly repulsive forces such a hybrid procedure may be an appropriate description.
Pressure Tensor of Nanoscopic Liquid Drops
José G. Segovia-López
2015-04-01
Full Text Available This study describes the structure of an inhomogeneous fluid of one or several components that forms a spherical interface. Using the stress tensor of Percus–Romero, which depends on the density of one particle and the intermolecular potential, it provides an analytical development leading to the microscopic expressions of the pressure differences and the interfacial properties of both systems. The results are compared with a previous study and agree with the description of the mean field.
Inflation in anisotropic scalar-tensor theories
Pimentel, L.O.; Stein-Schabes, J.
1989-01-05
The existence of an inflationary phase in anisotropic scalar-tensor theories is investigated by means of a conformal transformation that allows us to rewrite these theories as gravity minimally coupled to a scalar field with a non-trivial potential. We then use the explicit form of the potential and the no hair theorem to conclude that there is an inflationary phase in all open or flat anisotropic spacetimes in these theories. Several examples are constructed where the effect becomes manifest.
Inflation in anisotropic scalar-tensor theories
Pimentel, Luis O.; Stein-Schabes, Jaime
1988-01-01
The existence of an inflationary phase in anisotropic Scalar-Tensor Theories is investigated by means of a conformal transformation that allows us to rewrite these theories as gravity minimally coupled to a scalar field with a nontrivial potential. The explicit form of the potential is then used and the No Hair Theorem concludes that there is an inflationary phase in all open or flat anisotropic spacetimes in these theories. Several examples are constructed where the effect becomes manifest.
Tensor integrand reduction via Laurent expansion
Hirschi, Valentin
2016-01-01
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_aMC@NLO framework. We performed a detailed performance study, comparing against other public reduction tools, namely CutTools, Samurai, IREGI, PJFry++ and Golem. 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 Golem which is however more limited and slower than Ninja. We considered many benchmark multi-scale processes of increasing ...
Tensors, BICEP2, prior dependence, and dust
Cortês, Marina; Parkinson, David
2014-01-01
We investigate the prior dependence on the inferred spectrum of primordial tensor perturbations, in light of recent results from BICEP2 and taking into account a possible dust contribution to polarized anisotropies. We highlight an optimized parameterization of the tensor power spectrum, and adoption of a logarithmic prior on its amplitude $A_T$, leading to results that transform more evenly under change of pivot scale. In the absence of foregrounds the tension between the results of BICEP2 and Planck drives the tensor spectral index $n_T$ to be blue-tilted in a joint analysis, which would be in contradiction to the standard inflation prediction ($n_T<0$). When foregrounds are accounted for, the BICEP2 results no longer require non-standard inflationary parameter regions. We present limits on primordial $A_T$ and $n_T$, adopting foreground scenarios put forward by Mortonson & Seljak and motivated by Planck 353 GHz observations, and assess what dust contribution leaves a detectable cosmological signal. ...
Myelin water weighted diffusion tensor imaging.
Avram, Alexandru V; Guidon, Arnaud; Song, Allen W
2010-10-15
In this study we describe our development and implementation of a magnetization transfer (MT) prepared stimulated-echo diffusion tensor imaging (DTI) technique that can be made sensitive to the microanatomy of myelin tissue. The short echo time (TE) enabled by the stimulated-echo acquisition preserves significant signal from the short T(2) component (myelin water), and the MT preparation further provides differentiating sensitization to this signal. It was found that this combined strategy could provide sufficient sensitivity in our first attempt to image myelin microstructure. Compared to the diffusion tensor derived from the conventional DTI technique, the myelin water weighted (MWW) tensor has the same principal diffusion direction but exhibits a significant increase in fractional anisotropy (FA), which is mainly due to a decrease in radial diffusivity. These findings are consistent with the microstructural organization of the myelin sheaths that wrap around the axons in the white matter and therefore hinder radial diffusion. Given that many white matter diseases (e.g. multiple sclerosis) begin with a degradation of myelin microanatomy but not a loss of myelin content (e.g. loosening of the myelin sheaths), our newly implemented MWW DTI has the potential to lead to improved assessment of myelin pathology and early detection of demyelination.
Testing gravity theories using tensor perturbations
Lin, Weikang
2016-01-01
Primordial gravitational waves constitute a promising probe of the very-early universe and the laws of gravity. We study changes to tensor mode perturbations that can arise in various proposed modified gravity (MG) theories. These include additional friction effects, non-standard dispersion relations involving a massive graviton, a modified speed, and a small-scale modification. We introduce a physically-motivated parameterization of these effects and use current available data to obtain exclusion regions in the parameter spaces. Taking into account the foreground subtraction, we then perform a forecast analysis focusing on the tensor mode MG parameters as constrained by the future experiments COrE, Stage-IV and PIXIE. For a fiducial value of the tensor-to-scalar ratio r=0.01, we find that an additional friction of 3.5-4.5% compared to GR will be detected at $3\\sigma$ by these experiments while a decrease in friction will be more difficult to detect. The speed of gravitational waves needs to be 5-15% differen...
Irreducible Virasoro modules from tensor products
Tan, Haijun; Zhao, Kaiming
2016-04-01
In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules Ω(λ,b) with irreducible highest weight modules V(θ ,h) or with irreducible Virasoro modules Ind_{θ}(N) defined in Mazorchuk and Zhao (Selecta Math. (N.S.) 20:839-854, 2014). We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of Ω(λ,b) with a classical Whittaker module is isomorphic to the module Ind_{θ, λ}({C}_{{m}}) defined in Mazorchuk and Weisner (Proc. Amer. Math. Soc. 142:3695-3703, 2014). As a by-product we obtain the necessary and sufficient conditions for the module Ind_{θ, λ}({C}_{{m}}) to be irreducible. We also generalize the module Ind_{θ, λ}({C}_{{m}}) to Ind_{θ,λ}({B}^{(n)}_{{s}}) for any non-negative integer n and use the above results to completely determine when the modules Ind_{θ,λ}({B}^{(n)}_{{s}}) are irreducible. The submodules of Ind_{θ,λ}({B}^{(n)}_{{s}}) are studied and an open problem in Guo et al. (J. Algebra 387:68-86, 2013) is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper.
Effective elasticity tensor of a periodic composite
Nunan, Kevin C.; Keller, Joseph B.
THE EFFECTIVE elasticity tensor of a composite is defined to be the four-tensor C which relates the average stress to the average strain. We determine it for an array of rigid spheres centered on the points of a periodic lattice in a homogeneous isotropic elastic medium. We first express C in terms of the traction exerted on a single sphere by the medium, and then derive an integral equation for this traction. We solve this equation numerically for simple, body-centered and face-centered cubic lattices with inclusion concentrations up to 90% of the close-packing concentration. For lattices with cubic symmetry the effective elasticity tensor involves just three parameters, which we compute from the solution for the traction. We obtain approximate asymptotic formulas for low concentrations which agree well with the numerical results. We also derive asymptotic results for C at high inclusion concentrations for arbitrary lattice geometries. We find them to be in good agreement with the numerical results for cubic lattices. For low and moderate concentrations the approximate results of NEMAT- NASSERet al., also agree well with the numerical results for cubic lattices.
On Tensors, Sparsity, and Nonnegative Factorizations
Chi, Eric C
2011-01-01
Tensors have found application in a variety of fields, ranging from chemometrics to signal processing and beyond. In this paper, we consider the problem of multilinear modeling of sparse count data. Our goal is to develop a descriptive tensor factorization model of such data, along with appropriate algorithms and theory. To do so, we propose that the random variation is best described via a Poisson distribution, which better describes the zeros observed in the data as compared to the typical assumption of a Gaussian distribution. Under a Poisson assumption, we fit a model to observed data using the negative log-likelihood score. We present a new algorithm for Poisson tensor factorization called CANDECOMP--PARAFAC Alternating Poisson Regression (CP-APR) that is based on a majorization-minimization approach. It can be shown that CP-APR is a generalization of the Lee-Seung multiplicative updates. We show how to prevent the algorithm from converging to non-KKT points and prove convergence of CP-APR under mild con...
Scalable tensor factorizations with incomplete data.
Morup, Morten (Technical University of Denmark); Dunlavy, Daniel M. (Sandia National Laboratories, Albuquerque, NM); Acar, Evrim (Information Technologies Institute, Turkey); Kolda, Tamara Gibson
2010-07-01
The problem of incomplete data - i.e., data with missing or unknown values - in multi-way arrays is ubiquitous in biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, communication networks, etc. We consider the problem of how 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). In the presence of missing data, CP can be formulated as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) that uses a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical 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 x 1000 x 1000 with five million known entries (0.5% dense). We further demonstrate the usefulness of CP-WOPT on two real-world applications: a novel EEG (electroencephalogram) application where missing data is frequently encountered due to disconnections of electrodes and the problem of modeling computer network traffic where data may be absent due to the expense of the data collection process.
An $OSp$ extension of Canonical Tensor Model
Narain, Gaurav
2015-01-01
Tensor models are generalizations of matrix models, and are studied as discrete models of quantum gravity for arbitrary dimensions. Among them, the canonical tensor model (CTM for short) is a rank-three tensor model formulated as a totally constrained system with a number of first-class constraints, which have a similar algebraic structure as the constraints of the ADM formalism of general relativity. In this paper, we formulate a super-extension of CTM as an attempt to incorporate fermionic degrees of freedom. The kinematical symmetry group is extended from $O(N)$ to $OSp(N,\\tilde N)$, and the constraints are constructed so that they form a first-class constraint super-Poisson algebra. This is a straightforward super-extension, and the constraints and their algebraic structure are formally unchanged from the purely bosonic case, except for the additional signs associated to the order of the fermionic indices and dynamical variables. However, this extension of CTM leads to the existence of negative norm state...
Murg, V; Verstraete, F; Schneider, R; Nagy, P R; Legeza, Ö
2015-03-10
We study the tree-tensor-network-state (TTNS) method with variable tensor orders for quantum chemistry. TTNS is a variational method to efficiently approximate complete active space (CAS) configuration interaction (CI) wave functions in a tensor product form. TTNS can be considered as a higher order generalization of the matrix product state (MPS) method. The MPS wave function is formulated as products of matrices in a multiparticle basis spanning a truncated Hilbert space of the original CAS-CI problem. These matrices belong to active orbitals organized in a one-dimensional array, while tensors in TTNS are defined upon a tree-like arrangement of the same orbitals. The tree-structure is advantageous since the distance between two arbitrary orbitals in the tree scales only logarithmically with the number of orbitals N, whereas the scaling is linear in the MPS array. It is found to be beneficial from the computational costs point of view to keep strongly correlated orbitals in close vicinity in both arrangements; therefore, the TTNS ansatz is better suited for multireference problems with numerous highly correlated orbitals. To exploit the advantages of TTNS a novel algorithm is designed to optimize the tree tensor network topology based on quantum information theory and entanglement. The superior performance of the TTNS method is illustrated on the ionic-neutral avoided crossing of LiF. It is also shown that the avoided crossing of LiF can be localized using only ground state properties, namely one-orbital entanglement.
Using Perturbation theory to reduce noise in diffusion tensor fields.
Bansal, Ravi; Staib, Lawrence H; Xu, Dongrong; Laine, Andrew F; Liu, Jun; Peterson, Bradley S
2009-08-01
We propose the use of Perturbation theory to reduce noise in Diffusion Tensor (DT) fields. Diffusion Tensor Imaging (DTI) encodes the diffusion of water molecules along different spatial directions in a positive definite, 3 x 3 symmetric tensor. Eigenvectors and eigenvalues of DTs allow the in vivo visualization and quantitative analysis of white matter fiber bundles across the brain. The validity and reliability of these analyses are limited, however, by the low spatial resolution and low Signal-to-Noise Ratio (SNR) in DTI datasets. Our procedures can be applied to improve the validity and reliability of these quantitative analyses by reducing noise in the tensor fields. We model a tensor field as a three-dimensional Markov Random Field and then compute the likelihood and the prior terms of this model using Perturbation theory. The prior term constrains the tensor field to be smooth, whereas the likelihood term constrains the smoothed tensor field to be similar to the original field. Thus, the proposed method generates a smoothed field that is close in structure to the original tensor field. We evaluate the performance of our method both visually and quantitatively using synthetic and real-world datasets. We quantitatively assess the performance of our method by computing the SNR for eigenvalues and the coherence measures for eigenvectors of DTs across tensor fields. In addition, we quantitatively compare the performance of our procedures with the performance of one method that uses a Riemannian distance to compute the similarity between two tensors, and with another method that reduces noise in tensor fields by anisotropically filtering the diffusion weighted images that are used to estimate diffusion tensors. These experiments demonstrate that our method significantly increases the coherence of the eigenvectors and the SNR of the eigenvalues, while simultaneously preserving the fine structure and boundaries between homogeneous regions, in the smoothed tensor
Physical states in the canonical tensor model from the perspective of random tensor networks
Narain, Gaurav; Sato, Yuki
2014-01-01
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 physical wave-functions, which do not seem straightforward to solve 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 con...
Component reduction in N=2 supergravity: the vector, tensor, and vector-tensor multiplets
Butter, Daniel
2012-01-01
Recent advances in curved N=2 superspace methods have rendered the component reduction of superspace actions more feasible than in the past. In this paper, we consider models involving both vector and tensor multiplets coupled to supergravity and demonstrate explicitly how component actions may be efficiently obtained. In addition, tensor multiplets coupled to conformal supergravity are considered directly within projective superspace, where their formulation is most natural. We then demonstrate how the inverse procedure -- the lifting of component results to superspace -- can simplify the analysis of complicated multiplets. We address the off-shell N=2 vector-tensor multiplet coupled to conformal supergravity with a central charge and demonstrate explicitly how its constraints and Lagrangian can be written in a simpler way using superfields.
Li, Junning; Shi, Yonggang; Tran, Giang; Dinov, Ivo; Wang, Danny J J; Toga, Arthur
2014-05-01
Diffusion tensor imaging is widely used in brain connectivity research. As more and more studies recruit large numbers of subjects, it is important to design registration methods which are not only theoretically rigorous, but also computationally efficient. However, the requirement of reorienting diffusion tensors complicates and considerably slows down registration procedures, due to the correlated impacts of registration forces at adjacent voxel locations. Based on the diffeomorphic Demons algorithm (Vercauteren , 2009), we propose a fast local trust region algorithm for handling inseparable registration forces for quadratic energy functions. The method guarantees that, at any time and at any voxel location, the velocity is always within its local trust region. This local regularization allows efficient calculation of the transformation update with numeric integration instead of completely solving a large linear system at every iteration. It is able to incorporate exact reorientation and regularization into the velocity optimization, and preserve the linear complexity of the diffeomorphic Demons algorithm. In an experiment with 84 diffusion tensor images involving both pair-wise and group-wise registrations, the proposed algorithm achieves better registration in comparison with other methods solving large linear systems (Yeo , 2009). At the same time, this algorithm reduces the computation time and memory demand tenfold.
Global strong solution to the three-dimensional liquid crystal flows of Q-tensor model
Xiao, Yao
2017-02-01
A complex hydrodynamic system that models the fluid of nematic liquid crystals in a bounded domain in R3 is studied. The system is a forced incompressible Navier-Stokes equation coupled with a parabolic type equation of Q-tensors. We invoke the maximal regularity of the Stokes operators and parabolic operators in Besov spaces to obtain the local strong solution if the initial Q-tensor is not too "wild". In addition, it is showed that such solution can be extended to a global one if the initial data is a sufficiently small perturbation around the trivial equilibrium state. Finally, it is proved that the global strong solution obtained here is identical to those weak solutions obtained in Paicu and Zarnescu [26].
Thermodynamic driving force for rafting in superalloys
Nabarro, FRN
1996-08-01
Full Text Available Eshelby’s energy-momentum tensor is used to provide an analytical expression for the driving force for rafting in the elastic regime in a super alloy with a high volume fraction of gamma'. The structure is modeled as a simple cubic array of gamma...
InfTucker: t-Process based Infinite Tensor Decomposition
Xu, Zenglin; Yuan,; Qi,
2011-01-01
Tensor decomposition is a powerful tool for multiway data analysis. Many popular tensor decomposition approaches---such as the Tucker decomposition and CANDECOMP/PARAFAC (CP)---conduct multi-linear factorization. They are insufficient to model (i) complex interactions between data entities, (ii) various data types (e.g. missing data and binary data), and (iii) noisy observations and outliers. To address these issues, we propose a tensor-variate latent $t$ process model, InfTucker, for robust multiway data analysis: it conducts robust Tucker decomposition in an infinite feature space. Unlike classical tensor decomposition models, it handles both continuous and binary data in a probabilistic framework. Unlike previous nonparametric Bayesian models on matrices and tensors, our latent $t$-process model focuses on multiway analysis and uses nonlinear covariance functions. To efficiently learn InfTucker from data, we develop a novel variational inference technique on tensors. Compared with classical implementation,...
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...
Reduction of noise in diffusion tensor images using anisotropic smoothing.
Ding, Zhaohua; Gore, John C; Anderson, Adam W
2005-02-01
To improve the accuracy of tissue structural and architectural characterization with diffusion tensor imaging, a novel smoothing technique is developed for reducing noise in diffusion tensor images. The technique extends the traditional anisotropic diffusion filtering method by allowing isotropic smoothing within homogeneous regions and anisotropic smoothing along structure boundaries. This is particularly useful for smoothing diffusion tensor images in which direction information contained in the tensor needs to be restored following noise corruption and preserved around tissue boundaries. The effectiveness of this technique is quantitatively studied with experiments on simulated and human in vivo diffusion tensor data. Illustrative results demonstrate that the anisotropic smoothing technique developed can significantly reduce the impact of noise on the direction as well as anisotropy measures of the diffusion tensor images.
On the algebraic types of the Bel-Robinson tensor
Ferrando, Joan J
2008-01-01
The Bel-Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov-Bel classification of the Weyl tensor. The new classes correspond to degenerate type I space-times which have already been introduced in literature from another point of view. The Petrov-Bel types and the additional ones are intrinsically characterized in terms of the sole Bel-Robinson tensor, and an algorithm is proposed that enables the different classes to be distinguished. Results are presented that solve the problem of obtaining the Weyl tensor from the Bel-Robinson tensor in regular cases.
Emergent classical geometries on boundaries of randomly connected tensor networks
Chen, Hua; Sasakura, Naoki; Sato, Yuki
2016-03-01
It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. With variation of the tensors the dimensions of the spaces can be freely chosen, and the geometries—which are curved in general—can be varied. We give the explicit solvable examples of emergent flat tori in arbitrary dimensions, and the correspondence from the tensors to the geometries for general curved cases. The perturbative dynamics in the emergent space is shown to be described by an effective action which is invariant under the spatial diffeomorphism due to the underlying orthogonal group symmetry of the randomly connected tensor network. It is also shown that there are various phase transitions among spaces, including extended and point-like ones, under continuous change of the tensors.
Emergent classical geometries on boundaries of randomly connected tensor networks
Chen, Hua; Sato, Yuki
2016-01-01
It is shown that classical spaces with geometries emerge on boundaries of randomly connected tensor networks with appropriately chosen tensors in the thermodynamic limit. With variation of the tensors, the dimensions of the spaces can be freely chosen, and the geometries, which are curved in general, can be varied. We give the explicit solvable examples of emergent flat tori in arbitrary dimensions, and the correspondence from the tensors to the geometries for general curved cases. The perturbative dynamics in the emergent space is shown to be described by an effective action which is invariant under the spatial diffeomorphism due to the underlying orthogonal group symmetry of the randomly connected tensor network. It is also shown that there are various phase transitions among spaces, including extended and point-like ones, under continuous change of the tensors.
Grasso, Marcella
2014-01-01
We investigate the magicity of the isotopes $^{52}$Ca and $^{54}$Ca, that was recently confirmed by two experimental measurements, and relate it to like--particle and neutron--proton tensor effects within a mean--field description. By analyzing Ca isotopes, we show that the like--particle tensor contribution induces shell effects that render these nuclei more magic than they would be predicted by neglecting it. In particular, such induced shell effects are stronger in the nucleus $^{52}$Ca and the single--particle gaps are increased in both isotopes due to the tensor force. By studying $N=32$ and $N=34$ isotones, neutron--proton tensor effects may be isolated and their role analyzed. It is shown that neutron--proton tensor effects lead to increasing $N=32$ and $N=34$ gaps, when going along isotonic chains, from $^{58}$Fe to $^{52}$Ca, and from $^{60}$Fe to $^{54}$Ca, respectively. The mean--field calculations are perfomed by employing one Skyrme parameter set, that was introduced in a previous work by fitting...
Irreducible Cartesian tensors of highest weight, for arbitrary order
Mane, S. R.
2016-03-01
A closed form expression is presented for the irreducible Cartesian tensor of highest weight, for arbitrary order. Two proofs are offered, one employing bookkeeping of indices and, after establishing the connection with the so-called natural tensors and their projection operators, the other one employing purely coordinate-free tensor manipulations. Some theorems and formulas in the published literature are generalized from SO(3) to SO(n), for dimensions n ≥ 3.
Construction of energy-momentum tensor of gravitation
Bamba, Kazuharu; Shimizu, Katsutaro
2016-10-01
We construct the gravitational energy-momentum tensor in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy-momentum conservation is satisfied because one of the two indices of the energy-momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.
Deformaciones miniversales de parejas de tensores de segundo orden
Clotet Juan, Josep; Magret Planas, María Dolors; Peña Carrera, Marta
2007-01-01
Consideramos en el espacio de parejas de tensores tension y deformacion la relacion de equivalencia que se corresponde con cambios de base ortonormales. Identificandolas con parejas de matrices cuadradas, podemos utilizar la tecnica de las deformaciones miniversales para averiguar, dada una pareja de tensores cualquiera, cuales son las parejas de tensores que se pueden obtener al perturbar ligeramente la dada, puesto que las clases de equivalencia se pueden identificar con las ...
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.
Scalable tensor factorizations with missing data.
Morup, Morten (Technical University of Denmark); Dunlavy, Daniel M.; Acar, Evrim (Turkish National Research Institute of Electronics and Cryptology); Kolda, Tamara Gibson
2010-04-01
The problem of missing data is ubiquitous in domains such as biomedical signal processing, network traffic analysis, bibliometrics, social network analysis, chemometrics, computer vision, and communication networks|all domains in which data collection is subject to occasional errors. Moreover, these data sets can be quite large and have more than two axes of variation, e.g., sender, receiver, time. Many applications in those domains aim to capture the underlying latent structure of the data; in other words, they need to factorize data sets with missing entries. If we cannot address the problem 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), and formulate the CP model as a weighted least squares problem that models only the known entries. We develop an algorithm called CP-WOPT (CP Weighted OPTimization) using a first-order optimization approach to solve the weighted least squares problem. Based on extensive numerical experiments, our algorithm 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) application where missing data is frequently encountered due to disconnections of electrodes.
Tensor Decompositions for Learning Latent Variable Models
2012-12-08
Isotropic PCA and affine-invariant clustering. In FOCS, 2008. [Car91] J.-F. Cardoso . Super-symmetric decomposition of the fourth-order cumulant tensor...3109–3112. IEEE, 1991. [Car94] J.-F. Cardoso . Perturbation of joint diagonalizers. ref# 94d027. Technical report, Télécom Paris, 1994. [Cat44] R. B...CC96] J.-F. Cardoso and Pierre Comon. Independent component analysis, a survey of some algebraic methods. In IEEE International Symposium on Circuits
Beam-plasma dielectric tensor with Mathematica
Bret, A.
2007-03-01
We present a Mathematica notebook allowing for the symbolic calculation of the 3×3 dielectric tensor of an electron-beam plasma system in the fluid approximation. Calculation is detailed for a cold relativistic electron beam entering a cold magnetized plasma, and for arbitrarily oriented wave vectors. We show how one can elaborate on this example to account for temperatures, arbitrarily oriented magnetic field or a different kind of plasma. Program summaryTitle of program: Tensor Catalog identifier: ADYT_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADYT_v1_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Computer for which the program is designed and others on which it has been tested: Computers: Any computer running Mathematica 4.1. Tested on DELL Dimension 5100 and IBM ThinkPad T42. Installations: ETSI Industriales, Universidad Castilla la Mancha, Ciudad Real, Spain Operating system under which the program has been tested: Windows XP Pro Programming language used: Mathematica 4.1 Memory required to execute with typical data: 7.17 Mbytes No. of bytes in distributed program, including test data, etc.: 33 439 No. of lines in distributed program, including test data, etc.: 3169 Distribution format: tar.gz Nature of the physical problem: The dielectric tensor of a relativistic beam plasma system may be quite involved to calculate symbolically when considering a magnetized plasma, kinetic pressure, collisions between species, and so on. The present Mathematica notebook performs the symbolic computation in terms of some usual dimensionless variables. Method of solution: The linearized relativistic fluid equations are directly entered and solved by Mathematica to express the first-order expression of the current. This expression is then introduced into a combination of Faraday and Ampère-Maxwell's equations to give the dielectric tensor. Some additional manipulations are needed to express the result in terms of the
Hyperbolicity of Scalar Tensor Theories of Gravity
Salgado, Marcelo; Alcubierre, Miguel; Núñez, Dario
2008-01-01
Two first order strongly hyperbolic formulations of scalar tensor theories of gravity (STT) allowing non-minimal couplings (Jordan frame) are presented along the lines of the 3+1 decomposition of spacetime. One is based on the Bona-Masso formulation while the other one employs a conformal decomposition similar to that of Baumgarte-Shapiro-Shibata-Nakamura. A modified Bona-Masso slicing condition adapted to the STT is proposed for the analysis. This study confirms that STT posses a well posed Cauchy problem even when formulated in the Jordan frame.
A Rastall Scalar-Tensor theory
Caramês, T; Oliveira, A M; Piattella, O F; Strokov, V
2015-01-01
We formulate a theory combining the principles of a scalar-tensor gravity and the Rastall proposal of a violation of the usual conservation laws. In the resulting Brans-Dicke-Rastall (BDR) theory the only exact, static, spherically symmetric solution is a Robinson-Bertotti type solution besides the trivial Schwarzschild one. The PPN constraints can be completely satisfied for some values of the free parameters.The cosmological solutions display, among others, a decelerate-accelerate transition in the matter dominated phase.
Tensor networks for gauge field theories
Buyens, Boye; Verstraete, Frank; Van Acoleyen, Karel
2015-01-01
Over the last decade tensor network states (TNS) have emerged as a powerful tool for the study of quantum many body systems. The matrix product states (MPS) are one particular class of TNS and are used for the simulation of (1+1)-dimensional systems. In this proceeding we use MPS to determine the elementary excitations of the Schwinger model in the presence of an electric background field. We obtain an estimate for the value of the background field where the one-particle excitation with the largest energy becomes unstable and decays into two other elementary particles with smaller energy.
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
Link prediction via generalized coupled tensor factorisation
Ermiş, Beyza; Evrim, Acar Ataman; Taylan Cemgil, A.
2012-01-01
This study deals with the missing link prediction problem: the problem of predicting the existence of missing connections between entities of interest. We address link prediction using coupled analysis of relational datasets represented as heterogeneous data, i.e., datasets in the form of matrices...... 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....
Full Elasticity Tensor from Thermal Diffuse Scattering
Wehinger, Björn; Mirone, Alessandro; Krisch, Michael; Bosak, Alexeï
2017-01-01
We present a method for the precise determination of the full elasticity tensor from a single crystal diffraction experiment using monochromatic x rays. For the two benchmark systems calcite and magnesium oxide, we show that the measurement of thermal diffuse scattering in the proximity of Bragg reflections provides accurate values of the complete set of elastic constants. This approach allows for a reliable and model-free determination of the elastic properties and can be performed together with crystal structure investigation in the same experiment.
Modelamento mecanico-quantico de tensores polares
Harley Paiva Martins Filho
1994-01-01
Resumo: Os tensores polares de onze moléculas de halometanos e haletos de carbonila e tiocarbonila ( CH2Cl2, CF2Cl2, CF3Cl, CFCl3, CH3F, CH3Cl, CH3I, Cl2CO, F2CO, Cl2CS e F2CS ) foram determinados com o objetivo de verificar a validade de modelos de eletronegatividade para previsão de invariantes tensoriais (momento dipolar médio e carga efetiva) e somas de intensidades previamente estabelecidos. O método de determinação foi desenvolvido recentemente, baseando-se essencialmente na comparação ...
Scalar-Tensor Bianchi VI Models
J. A. Belinchón
2013-01-01
Full Text Available We study how may vary the gravitational and the cosmological “constants,” ( and in several scalar-tensor theories with Bianchi III, , and symmetries. By working under the hypothesis of self-similarity we find exact solutions for two different theoretical models, which are the Jordan-Brans-Dicke (JBD with and the usual JBD model with potential (that mimics the behaviour of . We compare both theoretical models, and some physical and geometrical properties of the solutions are also discussed putting special emphasis on the study of the isotropization of the solutions.
On the dynamic viscous permeability tensor symmetry.
Perrot, Camille; Chevillotte, Fabien; Panneton, Raymond; Allard, Jean-François; Lafarge, Denis
2008-10-01
Based on a direct generalization of a proof given by Torquato for symmetry property in static regime, this express letter clarifies the reasons why the dynamic permeability tensor is symmetric for spatially periodic structures having symmetrical axes which do not coincide with orthogonal pairs being perpendicular to the axis of three-, four-, and sixfold symmetry. This somewhat nonintuitive property is illustrated by providing detailed numerical examples for a hexagonal lattice of solid cylinders in the asymptotic and frequency dependent regimes. It may be practically useful for numerical implementation validation and/or convergence assessment.
Tensor Modeling Based for Airborne LiDAR Data Classification
Li, N.; Liu, C.; Pfeifer, N.; Yin, J. F.; Liao, Z. Y.; Zhou, Y.
2016-06-01
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.
Practical definition of averages of tensors in general relativity
Boero, Ezequiel F
2016-01-01
We present a definition of tensor fields which are average of tensors over a manifold, with a straightforward and natural definition of derivative for the averaged fields; which in turn makes a suitable and practical construction for the study of averages of tensor fields that satisfy differential equations. Although we have in mind applications to general relativity, our presentation is applicable to a general n-dimensional manifold. The definition is based on the integration of scalars constructed from a physically motivated basis, making use of the least amount of geometrical structure. We also present definitions of covariant derivative of the averaged tensors and Lie derivative.
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.
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.
Physical properties of crystals their representation by tensors and matrices
Nye, J F
1985-01-01
First published in 1957, this classic study has been reissued in a paperback version that includes an additional chapter bringing the material up to date. The author formulates the physical properties of crystals systematically in tensor notation, presenting tensor properties in terms of their common mathematical basis and the thermodynamic relations between them. The mathematical groundwork is laid in a discussion of tensors of the first and second ranks. Tensors of higher ranks and matrix methods are then introduced as natural developments of the theory. A similar pattern is followed in discussing thermodynamic and optical aspects.
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...
Vertical variations of wave-induced radiation stress tensor
Zheng Jinhai; Yan Yixin
2001-01-01
The distributions of the wave-induced radiation stress tensor over depth are studied by using the linear wave theory, which are divided into three regions, i.e., above the mean water level, below the wave trough level, and between these two levels. The computational expressions of the wave-induced radiation stress tensor at the arbitrary wave angle are established by means of the Eulerian coordinate transformation, and the asymptotic forms for deep and shallow water are also presented. The vertical variations of a 30° incident wave-induced radiation stress tensor in deep water, intermediate water and shallow water are calculated respectively. The following conclusions are obtained from computations.The wave-induced radiation stress tensor below the wave trough level is induced by the water wave particle velocities only, whereas both the water wave particle velocities and the wave pressure contribute to the tensor above the wave trough level. The vertical variations of the wave-induced radiation stress tensor are influenced substantially by the velocity component in the direction of wave propagation. The distributions of the wave-induced radiation stress tensor over depth are nonuniform and the proportion of the tensor below the wave trough level becomes considerable in the shallow water. From the water surface to the seabed, the reversed variations occur for the predominant tensor components.
Physical states in the canonical tensor model from the perspective of random tensor networks
Narain, Gaurav; Sasakura, Naoki; Sato, Yuki
2015-01-01
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.
A general theory of linear cosmological perturbations: scalar-tensor and vector-tensor theories
Lagos, Macarena; Ferreira, Pedro G; Noller, Johannes
2016-01-01
We present a method for parametrizing linear cosmological perturbations of theories of gravity, around homogeneous and isotropic backgrounds. The method is sufficiently general and systematic that it can be applied to theories with any degrees of freedom (DoFs) and arbitrary gauge symmetries. In this paper, we focus on scalar-tensor and vector-tensor theories, invariant under linear coordinate transformations. In the case of scalar-tensor theories, we use our framework to recover the simple parametrizations of linearized Horndeski and "Beyond Horndeski" theories, and also find higher-derivative corrections. In the case of vector-tensor theories, we first construct the most general quadratic action for perturbations that leads to second-order equations of motion, which propagates two scalar DoFs. Then we specialize to the case in which the vector field is time-like (\\`a la Einstein-Aether gravity), where the theory only propagates one scalar DoF. As a result, we identify the complete forms of the quadratic act...
Microscopic derivation of electromagnetic force density in magnetic dielectric media
Shevchenko, A.; Hoenders, B. J.
2010-01-01
Macroscopic force density imposed on a linear isotropic magnetic dielectric medium by an arbitrary electromagnetic field is derived by spatially averaging the microscopic Lorentz force density. The obtained expression differs from the commonly used expressions, but the energy-momentum tensor derived
Saliency Mapping Enhanced by Structure Tensor
Zhiyong He
2015-01-01
Full Text Available We propose a novel efficient algorithm for computing visual saliency, which is based on the computation architecture of Itti model. As one of well-known bottom-up visual saliency models, Itti method evaluates three low-level features, color, intensity, and orientation, and then generates multiscale activation maps. Finally, a saliency map is aggregated with multiscale fusion. In our method, the orientation feature is replaced by edge and corner features extracted by a linear structure tensor. Following it, these features are used to generate contour activation map, and then all activation maps are directly combined into a saliency map. Compared to Itti method, our method is more computationally efficient because structure tensor is more computationally efficient than Gabor filter that is used to compute the orientation feature and our aggregation is a direct method instead of the multiscale operator. Experiments on Bruce’s dataset show that our method is a strong contender for the state of the art.
Stability of Horndeski vector-tensor interactions
Jiménez, Jose Beltrán [Centre for Cosmology, Particle Physics and Phenomenology, Institute of Mathematics and Physics, Louvain University, 2 Chemin du Cyclotron, Louvain-la-Neuve, 1348 (Belgium); Durrer, Ruth; Heisenberg, Lavinia [Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ansermet, Genève 4, CH-1211 (Switzerland); Thorsrud, Mikjel, E-mail: jose.beltran@uclouvain.be, E-mail: ruth.durrer@unige.ch, E-mail: lavinia.heisenberg@unige.ch, E-mail: mikjel.thorsrud@astro.uio.no [Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, Oslo, N-0315 (Norway)
2013-10-01
We study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field. This theory is remarkably simple and consists of only 2 terms for the vector field, namely: the standard Maxwell kinetic term and a coupling to the dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge symmetry and the theory contains only one free parameter, M{sup 2}, that controls the strength of the non-minimal coupling. We explore the theory in a de Sitter spacetime and study the presence of instabilities and show that it corresponds to an attractor solution in the presence of the vector field. We also investigate the cosmological evolution and stability of perturbations in a general FLRW spacetime. We find that a sufficient condition for the absence of ghosts is M{sup 2} > 0. Moreover, we study further constraints coming from imposing the absence of Laplacian instabilities. Finally, we study the stability of the theory in static and spherically symmetric backgrounds (in particular, Schwarzschild and Reissner-Nordström-de Sitter). We find that the theory, quite generally, do have ghosts or Laplacian instabilities in regions of spacetime where the non-minimal interaction dominates over the Maxwell term. We also calculate the propagation speed in these spacetimes and show that superluminality is a quite generic phenomenon in this theory.
Stability of Horndeski vector-tensor interactions
Jiménez, Jose Beltrán; Heisenberg, Lavinia; Thorsrud, Mikjel
2013-01-01
We study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field. This theory is remarkably simple and consists of only 2 terms for the vector field, namely: the standard Maxwell kinetic term and a coupling to the dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge symmetry and the theory contains only one free parameter, M^2, that controls the strength of the non-minimal coupling. We explore the theory in a de Sitter spacetime and study the presence of instabilities and show that it corresponds to an attractor solution in the presence of the vector field. We also investigate the cosmological evolution and stability of perturbations in a general FLRW spacetime. We find that a sufficient condition for the absence of ghosts is M^2>0. Moreover, we study further constraints coming from imposing the absence of Laplacian instabilities. Finally, we study the stability of the theory in static and spherica...
Stability of Horndeski vector-tensor interactions
Beltrán Jiménez, Jose; Durrer, Ruth; Heisenberg, Lavinia; Thorsrud, Mikjel
2013-10-01
We study the Horndeski vector-tensor theory that leads to second order equations of motion and contains a non-minimally coupled abelian gauge vector field. This theory is remarkably simple and consists of only 2 terms for the vector field, namely: the standard Maxwell kinetic term and a coupling to the dual Riemann tensor. Furthermore, the vector sector respects the U(1) gauge symmetry and the theory contains only one free parameter, M2, that controls the strength of the non-minimal coupling. We explore the theory in a de Sitter spacetime and study the presence of instabilities and show that it corresponds to an attractor solution in the presence of the vector field. We also investigate the cosmological evolution and stability of perturbations in a general FLRW spacetime. We find that a sufficient condition for the absence of ghosts is M2 > 0. Moreover, we study further constraints coming from imposing the absence of Laplacian instabilities. Finally, we study the stability of the theory in static and spherically symmetric backgrounds (in particular, Schwarzschild and Reissner-Nordström-de Sitter). We find that the theory, quite generally, do have ghosts or Laplacian instabilities in regions of spacetime where the non-minimal interaction dominates over the Maxwell term. We also calculate the propagation speed in these spacetimes and show that superluminality is a quite generic phenomenon in this theory.
A Renormalizable 4-Dimensional Tensor Field Theory
Geloun, Joseph Ben
2011-01-01
We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on $U(1)^4$ is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the $\\phi^6$ rather than of the $\\phi^4$ type, since two different $\\phi^6$-type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new localit...
Phases of antisymmetric tensor field theories
Quevedo, Fernando; Quevedo, Fernando; Trugenberger, Carlo
1997-01-01
We study the different phases of field theories of compact antisymmetric tensors of rank h-1 in arbitrary space-time dimensions D=d+1. Starting in a `Coulomb' phase, topological defects of dimension d-h-1 ((d-h-1)-branes) may condense leading to a generalized `confinement' phase. If the dual theory is also compact the model may also have a third, generalized `Higgs' phase, driven by the condensation of the dual (h-2)-branes. Developing on the work of Julia and Toulouse for ordered solid-state media, we obtain the low energy effective action for these phases. Each phase has two dual descriptions in terms of antisymmetric tensors of different ranks, which are massless for the Coulomb phase but massive for the Higgs and confinement phases. We illustrate our prescription in detail for compact QED in 4D. Compact QED and O(2) models in 3D, as well as a periodic scalar field in 2D (strings on a circle), are also discussed. In this last case we show how T-duality is maintained if one considers both worldsheet instant...
Black holes in vector-tensor theories
Heisenberg, Lavinia; Kase, Ryotaro; Minamitsuji, Masato; Tsujikawa, Shinji
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.
Quantum chaos and holographic tensor models
Krishnan, Chethan; Sanyal, Sambuddha; Subramanian, P. N. Bala
2017-03-01
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.
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].
Nucleon tensor charges and electric dipole moments
Pitschmann, Mario; Seng, Chien-Yeah; Roberts, Craig D.; Schmidt, Sebastian M.
2015-04-01
A symmetry-preserving Dyson-Schwinger equation treatment of a vector-vector contact interaction is used to compute dressed-quark-core contributions to the nucleon σ -term and tensor charges. The latter enable one to directly determine the effect of dressed-quark electric dipole moments (EDMs) on neutron and proton EDMs. The presence of strong scalar and axial-vector diquark correlations within ground-state baryons is a prediction of this approach. These correlations are active participants in all scattering events and thereby modify the contribution of the singly represented valence quark relative to that of the doubly represented quark. Regarding the proton σ -term and that part of the proton mass which owes to explicit chiral symmetry breaking, with a realistic d -u mass splitting, the singly represented d quark contributes 37% more than the doubly represented u quark; and in connection with the proton's tensor charges, δTu , δTd , the ratio δTd /δTu is 18% larger than anticipated from simple quark models. Of particular note, the size of δTu is a sensitive measure of the strength of dynamical chiral symmetry breaking; and δTd measures the amount of axial-vector diquark correlation within the proton, vanishing if such correlations are absent.
Microscopic origin of Casimir-Polder forces
2006-01-01
We establish a general relation between dispersion forces. First, based on QED in causal media, leading-order perturbation theory is used to express both the single-atom Casimir-Polder and the two-atom van der Waals potentials in terms of the atomic polarizabilities and the Green tensor for the body-assisted electromagnetic field. Endowed with this geometry-independent framework, we then employ the Born expansion of the Green tensor together with the Clausius-Mosotti relation to prove that th...
The Lorentz Force and Energy-Momentum for Off-Shell Electromagnetism
Land, M C
2016-01-01
The kinematics of pre-Maxwell electrodynamics is examined and interpretations of these fields is found through an examination of the associated Lorentz force and the structure of the energy-momentum tensor.
Embedding DBI inflation in scalar-tensor theory
Mota, David F; Weller, Joel M
2010-01-01
The Dirac-Born-Infeld (DBI) action has been widely studied as an interesting example of a model of k-inflation in which the sound speed of the cosmological perturbations differs from unity. In this article we consider a scalar-tensor theory in which the matter component is a field with a DBI action. Transforming to the Einstein frame, we explore the effect of the resulting coupling on the background dynamics of the fields and the first-order perturbations. We find that the coupling forces the scalar field into the minimum of its effective potential. While the additional scalar field contributes significantly to the energy density during inflation, the dynamics are determined by the DBI field, which has the interesting effect of increasing the number of efolds of inflation and decreasing the boost factor of the DBI field. Focusing on this case, we show, with the benefit of numerical examples, that the power spectrum of the primordial perturbations is determined by the behaviour of the perturbations of the modi...
The Kummer tensor density in electrodynamics and in gravity
Baekler, Peter [University of Appl. Sciences, 40474 Düsseldorf (Germany); Favaro, Alberto [Inst. Physics, Carl-von-Ossietzky-Univ., 26111 Oldenburg (Germany); Itin, Yakov [Inst. Mathematics, Hebrew University of Jerusalem (Israel); Jerusalem College of Technology (Israel); Hehl, Friedrich W., E-mail: hehl@thp.uni-koeln.de [Inst. Theor. Physics, University of Cologne, 50923 Köln (Germany); Department of Physics and Astron., University of Missouri, Columbia, MO 65211 (United States)
2014-10-15
Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, K{sup ijkl}. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four T{sup ijkl}, which is antisymmetric in its first two and its last two indices: T{sup ijkl}=−T{sup jikl}=−T{sup ijlk}. Thus, K∼T{sup 3}, see Eq. (46). (i) If T is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized Fresnel wave surfaces for propagating light. In the reversible case, the wave surfaces turn out to be Kummer surfaces as defined in algebraic geometry (Bateman 1910). (ii) If T is identified with the curvature tensor R{sup ijkl} of a Riemann–Cartan spacetime, then K∼R{sup 3} and, in the special case of general relativity, K reduces to the Kummer tensor of Zund (1969). This K is related to the principal null directions of the curvature. We discuss the properties of the general Kummer tensor density. In particular, we decompose K irreducibly under the 4-dimensional linear group GL(4,R) and, subsequently, under the Lorentz group SO(1,3)
One-loop tensor Feynman integral reduction with signed minors
Fleischer, Jochem; Riemann, Tord; Yundin, Valery
2012-01-01
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 ter...
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.
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.
Tensor DoA estimation with directional elements
Raimondi, Francesca; DELEVOYE, Elisabeth; Comon, Pierre
2016-01-01
This paper introduces directivity gain pattern as a physical diversity for tensor array processing, in addition to time and space shift diversities. We show that tensor formulation allows to estimate Directions of Arrival (DoAs) under the assumption of unknown gain pattern, improving the performance of the omnidirectional case. The proposed approach is then applied to biologically inspired acoustic elements.
Electromagnetic Energy Momentum Tensor in a Spatially Dispersive Medium
Fietz, Chris
2016-01-01
We derive a generalized Minkowski Energy Momentum Tensor for a monochromatic wave in a lossless medium exhibiting temporal and spatial dispersion. The Energy Momentum Tensor is then related to familiar expressions for energy density and energy flux, as well as new expressions for momentum density and momentum flux.
Dynamic rotation and stretch tensors from a dynamic polar decomposition
Haller, George
2016-01-01
The local rigid-body component of continuum deformation is typically characterized by the rotation tensor, obtained from the polar decomposition of the deformation gradient. Beyond its well-known merits, the polar rotation tensor also has a lesser known dynamical inconsistency: it does not satisfy the fundamental superposition principle of rigid-body rotations over adjacent time intervals. As a consequence, the polar rotation diverts from the observed mean material rotation of fibers in fluids, and introduces a purely kinematic memory effect into computed material rotation. Here we derive a generalized polar decomposition for linear processes that yields a unique, dynamically consistent rotation component, the dynamic rotation tensor, for the deformation gradient. The left dynamic stretch tensor is objective, and shares the principal strain values and axes with its classic polar counterpart. Unlike its classic polar counterpart, however, the dynamic stretch tensor evolves in time without spin. The dynamic rotation tensor further decomposes into a spatially constant mean rotation tensor and a dynamically consistent relative rotation tensor that is objective for planar deformations. We also obtain simple expressions for dynamic analogues of Cauchy's mean rotation angle that characterize a deforming body objectively.
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...
A characterization of the Einstein tensor in terms of spinors
Anderson, I.M.; Lovelock, D.
1976-06-01
All tensors of contravariant rank two which are divergence-free on one index, concomitants of a spinor field sigma/sub iAX'/ together with its first two partial derivatives, and scalars under spin transformations are constructed. The Einstein and metric tensors are the only candidates. (AIP)
U-dual branes and mixed symmetry tensor fields
Chatzistavrakidis, A. [Institut fuer Theoretische Physik, Appelstrasse 2, 30167 Hannover (Germany); Gautason, F.F. [Institut fuer Theoretische Physik, Appelstrasse 2, 30167 Hannover (Germany); Center for Quantum Engineering and Spacetime Research, Appelstrasse 2, 30167 Hannover (Germany)
2014-09-11
We review and explain the relation between U-dual branes in string theory and mixed symmetry tensors of various degrees. In certain cases these mixed symmetry tensors can be related to diverse types of fluxes that play an important role in compactifications of string theory. (Copyright copyright 2014 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)
Chow groups and intersection products for tensor triangulated categories
Klein, S.A.
2014-01-01
This thesis deals with a generalization of an important family of invariants from algebraic geometry (the Chow groups of an algebraic variety) to the setting of tensor triangulated categories. It is shown that these tensor triangular Chow groups recover the usual notion of Chow groups of an
Special properties of Eshelby tensor for a regular polygonal inclusion
Baixiang Xu; Minzhong Wang
2005-01-01
When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore,in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain,and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.
Transversely isotropic higher-order averaged structure tensors
Hashlamoun, Kotaybah; Federico, Salvatore
2017-08-01
For composites or biological tissues reinforced by statistically oriented fibres, a probability distribution function is often used to describe the orientation of the fibres. The overall effect of the fibres on the material response is accounted for by evaluating averaging integrals over all possible directions in space. The directional average of the structure tensor (tensor product of the unit vector describing the fibre direction by itself) is of high significance. Higher-order averaged structure tensors feature in several models and carry similarly important information. However, their evaluation has a quite high computational cost. This work proposes to introduce mathematical techniques to minimise the computational cost associated with the evaluation of higher-order averaged structure tensors, for the case of a transversely isotropic probability distribution of orientation. A component expression is first introduced, using which a general tensor expression is obtained, in terms of an orthonormal basis in which one of the vectors coincides with the axis of symmetry of transverse isotropy. Then, a higher-order transversely isotropic averaged structure tensor is written in an appropriate basis, constructed starting from the basis of the space of second-order transversely isotropic tensors, which is constituted by the structure tensor and its complement to the identity.
Chern-Simons Couplings and Inequivalent Vector-Tensor Multiplets
Claus, P.; Wit, B. de; Faux, M.; Termonia, P.
1996-01-01
The off-shell vector-tensor multiplet is considered in an arbitrary background of N=2 vector supermultiplets. We establish the existence of two inequivalent versions, characterized by different Chern-Simons couplings. In one version the vector field of the vector-tensor multiplet is contained
Exploring the tensor networks/AdS correspondence
Bhattacharyya, Arpan; Gao, Zhe-Shen; Hung, Ling-Yan; Liu, Si-Nong
2016-08-01
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.
Coordinate independent expression for transverse trace-free tensors
Conboye, Rory
2015-01-01
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 3-space possess only two degrees of freedom, they cannot ordinarily be given solely by two differential functions. However, coordinate expressions depending on two scalar potentials alone have been derived for all TT tensors in flat space (Conboye and \\'O Murchadha 2014 Class. Quantum Grav. 31, 085019), with either a linear or axial symmetry. Since TT tensors are conformally covariant, these also give TT tensors in conformally-flat space. Here, this work has been extended to give a coordinate-independent expression for these TT tensors, also generalizing the symmetry conditions to invariance along any hypersurface orthogonal Killing vector.
On the energy-momentum tensor in Moyal space
Balasin, Herbert; Schweda, Manfred [Vienna University of Technology, Institute for Theoretical Physics, Vienna (Austria); Blaschke, Daniel N. [Los Alamos National Laboratory, Theory Division, Los Alamos, NM (United States); Gieres, Francois [Universite de Lyon, Universite Claude Bernard Lyon 1 et CNRS/IN2P3, Institut de Physique Nucleaire de Lyon, Villeurbanne (France)
2015-06-15
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.)
Tensor classification of structure in smoothed particle hydrodynamics density fields
Forgan, Duncan; Lucas, William; Rice, Ken
2016-01-01
As hydrodynamic simulations increase in scale and resolution, identifying structures with non-trivial geometries or regions of general interest becomes increasingly challenging. There is a growing need for algorithms that identify a variety of different features in a simulation without requiring a "by-eye" search. We present tensor classification as such a technique for smoothed particle hydrodynamics (SPH). These methods have already been used to great effect in N-Body cosmological simulations, which require smoothing defined as an input free parameter. We show that tensor classification successfully identifies a wide range of structures in SPH density fields using its native smoothing, removing a free parameter from the analysis and preventing the need for tesselation of the density field, as required by some classification algorithms. As examples, we show that tensor classification using the tidal tensor and the velocity shear tensor successfully identifies filaments, shells and sheet structures in giant m...
Lagrange Multipliers and Third Order Scalar-Tensor Field Theories
Horndeski, Gregory W
2016-01-01
In a space of 4-dimensions, I will examine constrained variational problems in which the Lagrangian, and constraint scalar density, are concomitants of a (pseudo-Riemannian) metric tensor and its first two derivatives. The Lagrange multiplier for these constrained extremal problems will be a scalar field. For suitable choices of the Lagrangian, and constraint, we can obtain Euler-Lagrange equations which are second order in the scalar field and third order in the metric tensor. The effect of disformal transformations on the constraint Lagrangians, and their generalizations, is examined. This will yield other second order scalar-tensor Lagrangians which yield field equations which are at most of third order. No attempt is made to construct all possible third order scalar-tensor Euler-Lagrange equations in a 4-space, although nine classes of such field equations are presented. Two of these classes admit subclasses which yield conformally invariant field equations. A few remarks on scalar-tensor-connection theor...
Inflationary tensor fossils in large-scale structure
Dimastrogiovanni, Emanuela; Jeong, Donghui; Kamionkowski, Marc
2014-01-01
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-attrac...
Tensor-optimized antisymmetrized molecular dynamics in nuclear physics
Myo, Takayuki; Ikeda, Kiyomi; Horiuchi, Hisashi; Suhara, Tadahiro
2015-01-01
We develop a new formalism to treat nuclear many-body systems using bare nucleon-nucleon interaction. It has become evident that the tensor interaction plays important role in nuclear many-body systems due to the role of the pion in strongly interacting system. We take the antisymmetrized molecular dynamics (AMD) as a basic framework and add a tensor correlation operator acting on the AMD wave function using the concept of the tensor-optimized shell model (TOSM). We demonstrate a systematical and straightforward formulation utilizing the Gaussian integration and differentiation method and the antisymmetrization technique to calculate all the matrix elements of the many-body Hamiltonian. We can include the three-body interaction naturally and calculate the matrix elements systematically in the progressive order of the tensor correlation operator. We call the new formalism "tensor-optimized antisymmetrized molecular dynamics".
A recursive approach to the reduction of tensor Feynman integrals
Diakonidis, Theodoros; Riemann, Tord; Tausk, Bas
2010-01-01
We describe a new, convenient, recursive tensor integral reduction scheme for one-loop $n$-point Feynman integrals. The reduction is based on the algebraic Davydychev-Tarasov formalism where the tensors are represented by scalars with shifted dimensions and indices, and then expressed by conventional scalars with generalized recurrence relations. The scheme is worked out explicitly for up to $n=6$ external legs and for tensor ranks $R\\leq n$. The tensors are represented by scalar one- to four-point functions in $d$ dimensions. For the evaluation of them, the Fortran code for the tensor reductions has to be linked with a package like QCDloop or LoopTools/FF. Typical numerical results are presented.
The atomistic representation of first strain-gradient elastic tensors
Admal, Nikhil Chandra; Marian, Jaime; Po, Giacomo
2017-02-01
We derive the atomistic representations of the elastic tensors appearing in the linearized theory of first strain-gradient elasticity for an arbitrary multi-lattice. In addition to the classical second-Piola) stress and elastic moduli tensors, these include the rank-three double-stress tensor, the rank-five tensor of mixed elastic moduli, and the rank-six tensor of strain-gradient elastic moduli. The atomistic representations are closed-form analytical expressions in terms of the first and second derivatives of the interatomic potential with respect to interatomic distances, and dyadic products of relative atomic positions. Moreover, all expressions are local, in the sense that they depend only on the atomic neighborhood of a lattice site. Our results emanate from the condition of energetic equivalence between continuum and atomistic representations of a crystal, when the kinematics of the latter is governed by the Cauchy-Born rule. Using the derived expressions, we prove that the odd-order tensors vanish if the lattice basis admits central-symmetry. The analytical expressions are implemented as a KIM compliant algorithm to compute the strain gradient elastic tensors for various materials. Numerical results are presented to compare representative interatomic potentials used in the literature for cubic crystals, including simple lattices (fcc Al and Cu and bcc Fe and W) and multi-lattices (diamond-cubic Si). We observe that central potentials exhibit generalized Cauchy relations for the rank-six tensor of strain-gradient elastic moduli. In addition, this tensor is found to be indefinite for many potentials. We discuss the relationship between indefiniteness and material stability. Finally, the atomistic representations are specialized to central potentials in simple lattices. These expressions are used with analytical potentials to study the sensitivity of the elastic tensors to the choice of the cutoff radius.
The formalism of invariants in scalar-tensor and multiscalar-tensor theories of gravitation
Jarv, Laur; Saal, Margus; Vilson, Ott
2016-01-01
We give a brief summary of the formalism of invariants in general scalar-tensor and multiscalar-tensor gravities without derivative couplings. By rescaling of the metric and reparametrization of the scalar fields, the theory can be presented in different conformal frames and parametrizations. Due to this freedom in transformations, the scalar fields themselves do not carry independent physical meaning (in a generic parametrization). However, there are functions of the scalar fields and their derivatives which remain invariant under the transformations, providing a set of physical variables for the theory. We indicate how to construct such invariants and show how the observables like parametrized post-Newtonian parameters and characteristics of Friedmann-Lemaitre-Robertson-Walker cosmology can be neatly expressed in terms of the invariants.
Tensor Algebra and Tensor Analysis for Engineers With Applications to Continuum Mechanics
Itskov, Mikhail
2013-01-01
There is a large gap between the engineering course in tensor algebra on the one hand and the treatment of linear transformations within classical linear algebra on the other hand. The aim of this modern textbook is to bridge this gap by means of the consequent and fundamental exposition. The book primarily addresses engineering students with some initial knowledge of matrix algebra. Thereby the mathematical formalism is applied as far as it is absolutely necessary. Numerous exercises are provided in the book and are accompanied by solutions, enabling self-study. The last chapters of the book deal with modern developments in the theory of isotropic and anisotropic tensor functions and their applications to continuum mechanics and are therefore of high interest for PhD-students and scientists working in this area. This third edition is completed by a number of additional figures, examples and exercises. The text and formulae have been revised and improved where necessary.
Gravity Gradient Tensor Eigendecomposition for Spacecraft Positioning
Chen, Pei; Han, Chao
2016-01-01
In this Note, a new approach to spacecraft positioning based on GGT inversion is presented. The gravity gradient tensor is initially measured in the gradiometer reference frame (GRF) and then transformed to the Earth-Centered Earth-Fixed (ECEF) frame via attitude information as well as Earth rotation parameters. Matrix Eigen-Decomposition is introduced to directly translate GGT into position based on the fact that the eigenvalues and eigenvectors of GGT are simplespecific functions of spherical coordinates of the observation position. without the need of an initial position. Unlike the strategy of inertial navigation aiding, no prediction or first guess of the spacecraft position is needed. The method makes use of the J2 gravity model, and is suitable for space navigation where higher frequency terrain contributions to the GGT signals can be neglected.
Chiral perturbation theory with tensor sources
Cata, Oscar; Cata, Oscar; Mateu, Vicent
2007-05-21
We construct the most general chirally-invariant Lagrangian for mesons in the presence of external sources coupled to the tensor current \\bar psi sigma_mu nu psi. In order to have only even terms in the chiral expansion, we consider the new source of O(p2). With this choice, we build the even-parity effective Lagrangian up to the p6-order (NLO). While there are only 4 new terms at the p4-order, at p6-order we find 78 terms for n_f=2 and 113 terms for n_f=3. We provide a detailed discussion on the different mechanisms that ensure that our final set of operators is complete and non-redundant. We also examine the odd-parity sector, to conclude that the first operators appear at the p8-order (NNLO).
Electroproduction of tensor mesons in QCD
Braun, V M; Strohmaier, M; Vladimirov, A A
2016-01-01
Due to multiple possible polarizations hard exclusive production of tensor mesons by virtual photons or in heavy meson decays offers interesting possibilities to study the helicity structure of the underlying short-distance process. Motivated by the first measurement of the transition form factor $\\gamma^*\\gamma \\to f_2(1270)$ at large momentum transfers by the BELLE collaboration we present an improved QCD analysis of this reaction in the framework of collinear factorization including contributions of twist-three quark-antiquark-gluon operators and an estimate of soft end-point corrections using light-cone sum rules. The results appear to be in a very good agreement with the data, in particular the predicted scaling behavior is reproduced in all cases.
f(R)-gravity from Killing tensors
Paliathanasis, Andronikos
2016-04-01
We consider f(R)-gravity in a Friedmann-Lemaître-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of f(R) and for the field equations to be invariant under Lie-Bäcklund transformations, which are linear in momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether’s theorem. We find three new integrable f(R)-models, for which, with the application of the conservation laws, we reduce the field equations to a system of two first-order ordinary differential equations. For each model we study the evolution of the cosmological fluid. We find that for each integrable model the cosmological fluid has an equation of state parameter, in which there is linear behavior in terms of the scale factor which describes the Chevallier, Polarski and Linder parametric dark energy model.
Tensor products of commutative Banach algebras
U. B. Tewari
1982-01-01
Full Text Available Let A1, A2 be commutative semisimple Banach algebras and A1⊗∂A2 be their projective tensor product. We prove that, if A1⊗∂A2 is a group algebra (measure algebra of a locally compact abelian group, then so are A1 and A2. As a consequence, we prove that, if G is a locally compact abelian group and A is a comutative semi-simple Banach algebra, then the Banach algebra L1(G,A of A-valued Bochner integrable functions on G is a group algebra if and only if A is a group algebra. Furthermore, if A has the Radon-Nikodym property, then the Banach algebra M(G,A of A-valued regular Borel measures of bounded variation on G is a measure algebra only if A is a measure algebra.
Diffusion tensor imaging of peripheral nerves.
Jambawalikar, Sachin; Baum, Jeremy; Button, Terry; Li, Haifang; Geronimo, Veronica; Gould, Elaine S
2010-11-01
Magnetic resonance diffusion tensor imaging (DTI) allows the directional dependence of water diffusion to be studied. Analysis of the resulting image data allows for the determination of fractional anisotropy (FA), apparent diffusion coefficient (ADC), as well as allowing three-dimensional visualization of the fiber tract (tractography). We visualized the ulnar nerve of ten healthy volunteers with DTI. We found FA to be 0.752 ± 0.067 and the ADC to be 0.96 ± 0.13 × 10(-3) mm(2)/s. A nuts-and-bolts description of the physical aspects of DTI is provided as an educational process for readers.
Estimation of tensors and tensor-derived measures in diffusional kurtosis imaging.
Tabesh, Ali; Jensen, Jens H; Ardekani, Babak A; Helpern, Joseph A
2011-03-01
This article presents two related advancements to the diffusional kurtosis imaging estimation framework to increase its robustness to noise, motion, and imaging artifacts. The first advancement substantially improves the estimation of diffusion and kurtosis tensors parameterizing the diffusional kurtosis imaging model. Rather than utilizing conventional unconstrained least squares methods, the tensor estimation problem is formulated as linearly constrained linear least squares, where the constraints ensure physically and/or biologically plausible tensor estimates. The exact solution to the constrained problem is found via convex quadratic programming methods or, alternatively, an approximate solution is determined through a fast heuristic algorithm. The computationally more demanding quadratic programming-based method is more flexible, allowing for an arbitrary number of diffusion weightings and different gradient sets for each diffusion weighting. The heuristic algorithm is suitable for real-time settings such as on clinical scanners, where run time is crucial. The advantage offered by the proposed constrained algorithms is demonstrated using in vivo human brain images. The proposed constrained methods allow for shorter scan times and/or higher spatial resolution for a given fidelity of the diffusional kurtosis imaging parametric maps. The second advancement increases the efficiency and accuracy of the estimation of mean and radial kurtoses by applying exact closed-form formulae.
Estimation of Tensors and Tensor-Derived Measures in Diffusional Kurtosis Imaging1
Tabesh, Ali; Jensen, Jens H.; Ardekani, Babak A.; Helpern, Joseph A.
2010-01-01
This paper presents two related advancements to the diffusional kurtosis imaging (DKI) estimation framework to increase its robustness to noise, motion, and imaging artifacts. The first advancement substantially improves the estimation of diffusion and kurtosis tensors parameterizing the DKI model. Rather than utilizing conventional unconstrained least squares (LS) methods, the tensor estimation problem is formulated as linearly constrained linear LS, where the constraints ensure physically and/or biologically plausible tensor estimates. The exact solution to the constrained problem is found via convex quadratic programming methods or, alternatively, an approximate solution is determined through a fast heuristic algorithm. The computationally more demanding quadratic programming-based method is more flexible, allowing for an arbitrary number of diffusion weightings and different gradient sets for each diffusion weighting. The heuristic algorithm is suitable for real-time settings such as on clinical scanners, where run time is crucial. The advantage offered by the proposed constrained algorithms is demonstrated using in vivo human brain images. The proposed constrained methods allow for shorter scan times and/or higher spatial resolution for a given fidelity of the DKI parametric maps. The second advancement increases the efficiency and accuracy of the estimation of mean and radial kurtoses by applying exact closed-form formulae. PMID:21337412
On the Decomposition of the Spacetime Metric Tensor and of Tensor Fields in Strained Spacetime
Millette P. A.
2012-10-01
Full Text Available We propose a natural decomposition of the spacetime metric tensor of General Relativ- ity into a background and a dynamical part based on an analysis from first principles of the effect of a test mass on the background metric. We find that the presence of mass results in strains in the spacetime continuum. Those strains correspond to the dy- namical part of the spacetime metric tensor. We then apply the stress-strain relation of Continuum Mechanics to the spacetime continuum to show that rest-mass energy den- sity arises from the volume dilatation of the spacetime continuum. Finally we propose a natural decomposition of tensor fields in strained spacetime, in terms of dilatations and distortions. We show that dilatations correspond to rest-mass energy density, while distortions correspond to massless shear transverse waves. We note that this decom- position in a massive dilatation and a massless transverse wave distortion, where both are present in spacetime continuum deformations, is somewhat reminiscent of wave- particle duality. We note that these results are considered to be local effects in the particular reference frame of the observer. In addition, the applicability of the proposed metric to the Einstein field equations remains open.
Measurement of the vector and tensor analyzing powers for dp- elastic scattering at 880 MeV
Kurilkin, P K; Uesaka, T; Suda, K; Gurchin, Yu V; Isupov, A Yu; Itoh, K; Janek, M; Karachuk, J -T; Kawabata, T; Khrenov, A N; Kiselev, A S; Kizka, V A; Krasnov, V A; Ladygina, N B; Livanov, A N; Maeda, Y; Malakhov, A I; Piyadin, S M; Reznikov, S G; Sakaguchi, S; Sakai, H; Sasamoto, Y; Sekiguchi, K; Shikhalev, M A; Vasiliev, T A; Witala, H
2012-01-01
The vector Ay and tensor analyzing powers Ayy and Axx for dp- elastic scattering were measured at Td = 880 MeV over the c.m. angular range from 60 to 140 degrees at the JINR Nuclotron. The data are compared with predictions of different theoretical models based on the use of nucleon-nucleon forces only. The observed discrepancies of the measured analyzing powers from the calculations require the consideration of additional mechanisms.
Seismic moment tensors and estimated uncertainties in southern Alaska
Silwal, Vipul; Tape, Carl
2016-04-01
We present a moment tensor catalog of 106 earthquakes in southern Alaska, and we perform a conceptually based uncertainty analysis for 21 of them. For each earthquake, we use both body waves and surface waves to do a grid search over double couple moment tensors and source depths in order to find the minimum of the misfit function. Our uncertainty parameter or, rather, our confidence parameter is the average value of the curve 𝒫 (V), where 𝒫 (V) is the posterior probability as a function of the fractional volume V of moment tensor space surrounding the minimum misfit moment tensor. As a supplemental means for characterizing and visualizing uncertainties, we generate moment tensor samples of the posterior probability. We perform a series of inversion tests to quantify the impact of certain decisions made within moment tensor inversions and to make comparisons with existing catalogs. For example, using an L1 norm in the misfit function provides more reliable solutions than an L2 norm, especially in cases when all available waveforms are used. Using body waves in addition to surface waves, as well as using more stations, leads to the most accurate moment tensor solutions.
Generalized Einstein Tensor for a Weyl Manifold and Its Applications
Abdülkadir （O）ZDE（G）ER
2013-01-01
It is well known that the Einstein tensor G for a Piemannian manifold defined by Gβα =Rβα-1/2Rδα,Rβα =gβγRγα where Rγα and R are respectively the Ricci tensor and the scalar curvature of the manifold,plays an important part in Einstein's theory of gravitation as well as in proving some theorems in Riemannian geometry.In this work,we first obtain the generalized Einstein tensor for a Weyl manifold.Then,after studying some properties of generalized Einstein tensor,we prove that the conformed invariance of the generalized Einstein tensor implies the conformed invariance of the curvature tensor of the Weyl manifold and conversely.Moreover,we show that such Weyl manifolds admit a one-parameter family of hypersurfaces the orthogoned trajectories of which are geodesics.Finally,a necessary and sufficient condition in order that the generalized circles of a Weyl manifold be preserved by a conformal mapping is stated in terms of generalized Einstein tensors at corresponding points.
On the energy-momentum tensor in Moyal space
Balasin, Herbert; Gieres, Francois; Schweda, Manfred
2015-01-01
After reviewing the known results for the definition and properties of the energy-momentum tensor(s) in Minkowski space, 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 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 gauge invariant Wilson line. We show that the latter 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 prod...
The tensor bi-spectrum in a matter bounce
Sreenath, V.; Chowdhury, Debika; Sriramkumar, L.
2016-03-01
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 hNL 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.
Tensor-based dynamic reconstruction method for electrical capacitance tomography
Lei, J.; Mu, H. P.; Liu, Q. B.; Li, Z. H.; Liu, S.; Wang, X. Y.
2017-03-01
Electrical capacitance tomography (ECT) is an attractive visualization measurement method, in which the acquisition of high-quality images is beneficial for the understanding of the underlying physical or chemical mechanisms of the dynamic behaviors of the measurement objects. In real-world measurement environments, imaging objects are often in a dynamic process, and the exploitation of the spatial-temporal correlations related to the dynamic nature will contribute to improving the imaging quality. Different from existing imaging methods that are often used in ECT measurements, in this paper a dynamic image sequence is stacked into a third-order tensor that consists of a low rank tensor and a sparse tensor within the framework of the multiple measurement vectors model and the multi-way data analysis method. The low rank tensor models the similar spatial distribution information among frames, which is slowly changing over time, and the sparse tensor captures the perturbations or differences introduced in each frame, which is rapidly changing over time. With the assistance of the Tikhonov regularization theory and the tensor-based multi-way data analysis method, a new cost function, with the considerations of the multi-frames measurement data, the dynamic evolution information of a time-varying imaging object and the characteristics of the low rank tensor and the sparse tensor, is proposed to convert the imaging task in the ECT measurement into a reconstruction problem of a third-order image tensor. An effective algorithm is developed to search for the optimal solution of the proposed cost function, and the images are reconstructed via a batching pattern. The feasibility and effectiveness of the developed reconstruction method are numerically validated.
Lorentz force and ponderomotive force in the presence of a minimal length
Khosropour, Behrooz
2017-09-01
In this work, according to the electromagnetic field tensor in the framework of generalized uncertainty principle (GUP), we obtain the Lorentz force and Faraday’s law of induction in the presence of a minimal length. Also, the ponderomotive force and ponderomotive pressure in the presence of a measurable minimal length are found. It is shown that in the limit β → 0, the generalized Lorentz force and ponderomotive force become the usual forms. The upper bound on the isotropic minimal length is estimated.
The general dielectric tensor for bi-kappa magnetized plasmas
Gaelzer, Rudi; Meneses, Anelise Ramires
2016-01-01
In this paper we derive the dielectric tensor for a plasma containing particles described by an anisotropic superthermal (bi-kappa) velocity distribution function. The tensor components are written in terms of the two-variables kappa plasma special functions, recently defined by Gaelzer and Ziebell [Phys. Plasmas 23, 022110 (2016)]. We also obtain various new mathematical properties for these functions, which are useful for the analytical treatment, numerical implementation and evaluation of the functions and, consequently, of the dielectric tensor. The formalism developed here and in the previous paper provides a mathematical framework for the study of electromagnetic waves propagating at arbitrary angles and polarizations in a superthermal plasma.
Standard cosmological model with non vanishing Weyl tensor
Bittencourt, E
2013-01-01
We have solved Einstein's equations of general relativity for a homogeneous and isotropic metric with constant spatial curvature and found a non vanishing Weyl tensor in the presence of an anisotropic pressure component of the energy-momentum tensor. The time evolution of the space-time is guided by the usual Friedman equations and the properties of the spatial components comprise a separated system of equations that can be independently solved. The physical features of this solution are elucidated by using the Quasi-Maxwellian equations of general relativity which directly connect the anisotropic pressure to the electric part of the Weyl tensor for the cosmological fluid.
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 confi...... 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....
A Proposal of Proper Gravitational Energy Momentum Tensor
Shimizu, Katsutaro
2016-01-01
We propose a gravitational energy momentum tensor of the general relativity by using Noether theorem. It changes as an tensor under the general coordinate transformations. One of the two indices of the gravitational energy momentum tensor is a local Lorentz frame to satisfy an energy momentum conservation law. The energies of a gravitational wave, Schwarzschild black hole and Friedman-Lemertre-Robertoson-Walker universe are calculated as examples. The gravitational energy of Schwarzschild black hole exists only out of a horizon. Its amount is -M.
On Semi-tensor Product of Matrices and Its Applications
Dai-zhan Cheng; Li-jun Zhang
2003-01-01
The left semi-tensor product of matrices was proposed in [2]. In this paper the right semi-tensor product is introduced first. Some basic properties are presented and compared with those of the left semi-tensor product.Then two new applications are investigated. Firstly, its applications to connection, an important concept in differential geometry, is considered. The structure matrix and the Christoffel matrix are introduced. The transfer formulas under coordinate transformation are expressed in matrix form. Certain new results are obtained.Secondly, the structure of finite dimensional Lie algebra, etc. are investigated under the matrix expression.These applications demonstrate the usefulness of the new matrix products.
Tensor-decomposed vibrational coupled-cluster theory
Madsen, Niels Kristian; Godtliebsen, Ian Heide; Christiansen, Ove
can be reduced by decomposing the VCC amplitudes and error vectors to the CANDECOMP/PARAFAC (CP) tensor format. Using the CP format allows us to automatically adapt the size of the parameter space as well as the computational effort to the strength of the physical interactions in the molecule while...... maintaining the same accuracy as the standard VCC method. We have implemented our VCC algorithms and equation solvers such that the VCC equations can be solved without constructing any tensors in full dimension. The tensors are automatically recompressed during the summation of the many terms in the VCC...
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.
Finding of electromagnetic field by energy-momentum tensor
Mitrofanova, T G
2002-01-01
One of the reverse problems on the electrodynamics consists in reducing the electromagnetic field by the known energy-momentum tensor of this field. The energy-momentum tensor aspect is of essential importance by developing new methods for analytical integration of field equations. Thereby there appears the question, whether the energy-momentum tensor corresponds to any physical system and if so - to which one namely. The formulated reverse problem in this paper is solved as applied to the electromagnetic field in the absence of charges and currents
Tensor analyzing power in πd elastic scattering
Smith, G. R.; Altman, A.; Delheij, P.; Gill, D. R.; Healey, D.; Johnson, R. R.; Jones, G.; Ottewell, D.; Rozon, F. M.; Sevior, M. E.; Tervisidis, F.; Trelle, R. P.; Wait, G. D.; Walden, P.; Mathie, E. L.; Lolos, G. J.; Naqvi, S. I.; Boschitz, E. T.; Ottermann, C. R.; Kyle, G. S.; Amaudruz, P. A.
1986-08-01
A tensor-polarized deuteron target has been employed for the first measurements of the tensor analyzing power, T20, in πd elastic scattering. Data at six angles were measured at pion bombarding energies of 133.8 and 150.9 MeV. The results settle a long-standing controversy over conflicting measurements of the tensor polarization t20, and dispute evidence for dibaryon resonances predicated on one of these t20 measurements. The data are shown to be in reasonable agreement with recent Faddeev calculations which have reduced contributions from pion absorption.
Construction of energy-momentum tensor of gravitation
Bamba, Kazuharu
2015-01-01
We argue the possibility that the gravitational energy-momentum tensor is constructed in general relativity through the Noether theorem. In particular, we explicitly demonstrate that the constructed quantity can vary as a tensor under the general coordinate transformation. Furthermore, we verify that the energy-momentum conservation is satisfied because one of the two indices of the energy-momentum tensor should be in the local Lorentz frame. It is also shown that the gravitational energy and the matter one cancel out in certain space-times.
Grasso, M
2015-01-01
Neutron $2p$ and $1f$ spin--orbit splittings were recently measured in the isotones $^{37}$S and $^{35}$Si by $(d,p)$ transfer reactions. Values were reported by using the major fragments of the states. An important reduction of the $p$ splitting was observed, from $^{37}$S to $^{35}$Si, associated to a strong modification of the spin--orbit potential in the central region of the nucleus $^{35}$Si. We analyze $2p$ and $1f$ neutron spin--orbit splittings in the $N=20$ isotones $^{40}$Ca, $^{36}$S, and $^{34}$Si. We employ several Skyrme and Gogny interactions, to reliably isolate pure spin--orbit and tensor--induced contributions, within the mean--field approximation. We use interactions (i) without the tensor force; (ii) with the tensor force and with tensor parameters adjusted on top of existing parametrizations; (iii) with the tensor force and with tensor and spin--orbit parameters adjusted simultaneously on top of existing parametrizations. We predict in cases (ii) and (iii) a non negligible reduction of b...
The theory of intermolecular forces
Stone, Anthony J
2013-01-01
The theory of intermolecular forces has advanced very greatly in recent years. It has become possible to carry out accurate calculations of intermolecular forces for molecules of useful size, and to apply the results to important practical applications such as understanding protein structure and function, and predicting the structures of molecular crystals. The Theory of Intermolecular Forces sets out the mathematical techniques that are needed to describe and calculate intermolecular interactions and to handle the more elaborate mathematical models. It describes the methods that are used to calculate them, including recent developments in the use of density functional theory and symmetry-adapted perturbation theory. The use of higher-rank multipole moments to describe electrostatic interactions is explained in both Cartesian and spherical tensor formalism, and methods that avoid the multipole expansion are also discussed. Modern ab initio perturbation theory methods for the calculation of intermolecular inte...
Mixed symmetry tensors in the worldline formalism
Corradini, Olindo; Edwards, James P.
2016-05-01
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 world-line 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.
Scalar-tensor gravity and conformal continuations
Bronnikov, K A
2002-01-01
Global properties of vacuum static, spherically symmetric configurations are studied in a general class of scalar-tensor theories (STT) of gravity in various dimensions. The conformal mapping between the Jordan and Einstein frames is used as a tool. Necessary and sufficient conditions are found for the existence of solutions admitting a conformal continuation (CC). The latter means that a singularity in the Einstein-frame manifold maps to a regular surface S_(trans) in the Jordan frame, and the solution is then continued beyond this surface. S_(trans) can be an ordinary regular sphere or a horizon. In the second case, S_(trans) proves to connect two epochs of a Kantowski-Sachs type cosmology. It is shown that, in an arbitrary STT, with arbitrary potential functions $U(\\phi)$, the list of possible types of causal structures of vacuum space-times is the same as in general relativity with a cosmological constant. This is true even for conformally continued solutions. It is found that when S_(trans) is an ordinar...
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
f(R)-gravity from Killing Tensors
Paliathanasis, Andronikos
2015-01-01
We consider $f\\left( R\\right) $-gravity in a Friedmann-Lema\\^{\\i}tre-Robertson-Walker spacetime with zero spatial curvature. We apply the Killing tensors of the minisuperspace in order to specify the functional form of $f\\left( R\\right) $ and the field equations to be invariant under Lie-B\\"{a}cklund transformations which are linear in the momentum (contact symmetries). Consequently, the field equations to admit quadratic conservation laws given by Noether's Theorem. We find three new integrable $f\\left( R\\right) $ models, for which with the application of the conservation laws we reduce the field equations to a system of two first-order ordinary differential equations. For each model we study the evolution of the cosmological fluid. Where we find that for the one integrable model the cosmological fluid has an equation of state parameter, in which in the latter there is a linear behavior in terms of the scale factor which describes the CPL parametric dark energy model.
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).
Photo-production of tensor mesons
Xie Ju-Jun
2016-01-01
Full Text Available Assuming that the f2(1270, f′2(1525, a2(1320, and K*2(1430 resonances are dynamically generated states from the vector meson-vector meson interactions in L = 0 and spin 2, we study the γp → f2(1270[f′2(1525]p, γp → a02 (1320p, and γp → K*2(1430Λ(Σ reactions. For the γp → f2(1270p reaction, we find that the theoretical results for the differential cross sections are in good agreement with the experimental measurements and provide support for the molecular picture of the f2(1270 in the first baryonic reaction where it has been tested. Furthermore, we predict also the total and differential cross sections for other reactions. The results can be tested in future experiments and therefore offer new clues on the nature of these tensor states.
Mixed symmetry tensors in the worldline formalism
Corradini, Olindo
2016-01-01
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....
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.
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...
Quantum canonical tensor model and an exact wave function
Sasakura, Naoki
2013-01-01
Tensor models in various forms are being studied as models of quantum gravity. Among them the canonical tensor model has a canonical pair of rank-three tensors as dynamical variables, and is a pure constraint system with first-class constraints. The Poisson algebra of the first-class constraints has structure functions, and provides an algebraically consistent way of discretizing the Dirac first-class constraint algebra for general relativity. This paper successfully formulates the Wheeler-DeWitt scheme of quantization of the canonical tensor model; the ordering of operators in the constraints is determined without ambiguity by imposing Hermiticity and covariance on the constraints, and the commutation algebra of constraints takes essentially the same from as the classical Poisson algebra, i.e. is first-class. Thus one could consistently obtain, at least locally in the configuration space, wave functions of "universe" by solving the partial differential equations representing the constraints, i.e. the Wheeler...
The tensor hierarchy of 8-dimensional field theories
Andino, Óscar Lasso; Ortín, Tomás
2016-10-01
We construct the tensor hierarchy of generic, bosonic, 8-dimensional field theories. We first study the form of the most general 8-dimensional bosonic theory with Abelian gauge symmetries only and no massive deformations. This study determines the tensors that occur in the Chern-Simons terms of the (electric and magnetic) field strengths and the action for the electric fields, which we determine. Having constructed the most general Abelian theory we study the most general gaugings of its global symmetries and the possible massive deformations using the embedding tensor formalism, constructing the complete tensor hierarchy using the Bianchi identities. We find the explicit form of all the field strengths of the gauged theory up to the 6-forms. Finally, we find the equations of motion comparing the Noether identities with the identities satisfied by the Bianchi identities themselves. We find that some equations of motion are not simply the Bianchi identities of the dual fields, but combinations of them.
The tensor hierarchy of 8-dimensional field theories
Andino, Oscar Lasso
2016-01-01
We construct the tensor hierarchy of generic, bosonic, 8-dimensional field theories. We first study the form of the most general 8-dimensional bosonic theory with Abelian gauge symmetries only and no massive deformations. This study determines the tensors that occur in the Chern-Simons terms of the (electric and magnetic) field strengths and the action for the electric fields, which we determine. Having constructed the most general Abelian theory we study the most general gaugings of its global symmetries and the possible massive deformations using the embedding tensor formalism, constructing the complete tensor hierarchy using the Bianchi identities. We find the explicit form of all the field strengths of the gauged theory up to the 6-forms. Finally, we find the equations of motion comparing the Noether identities with the identities satisfied by the Bianchi identities themselves. We find that some equations of motion are not simply the Bianchi identities of the dual fields, but combinations of them.
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.
Tensor perturbations in a general class of Palatini theories
Jiménez, Jose Beltrán; Olmo, Gonzalo J
2015-01-01
We study a general class of gravitational theories formulated in the Palatini approach and derive the equations governing the evolution of tensor perturbations. In the absence of torsion, the connection can be solved as the Christoffel symbols of an auxiliary metric which is non-trivially related to the space-time metric. We then consider background solutions corresponding to a perfect fluid and show that the tensor perturbations equations (including anisotropic stresses) for the auxiliary metric around such a background take an Einstein-like form. This facilitates the study in a homogeneous and isotropic cosmological scenario where we explicitly establish the relation between the auxiliary metric and the space-time metric tensor perturbations. As a general result, we show that both tensor perturbations coincide in the absence of anisotropic stresses.
Tensor perturbations in a general class of Palatini theories
Beltrán Jiménez, Jose; Heisenberg, Lavinia; Olmo, Gonzalo J.
2015-06-01
We study a general class of gravitational theories formulated in the Palatini approach and derive the equations governing the evolution of tensor perturbations. In the absence of torsion, the connection can be solved as the Christoffel symbols of an auxiliary metric which is non-trivially related to the space-time metric. We then consider background solutions corresponding to a perfect fluid and show that the tensor perturbations equations (including anisotropic stresses) for the auxiliary metric around such a background take an Einstein-like form. This facilitates the study in a homogeneous and isotropic cosmological scenario where we explicitly establish the relation between the auxiliary metric and the space-time metric tensor perturbations. As a general result, we show that both tensor perturbations coincide in the absence of anisotropic stresses.
Tensor Models: extending the matrix models structures and methods
Dartois, Stephane
2016-01-01
In this text we review a few structural properties of matrix models that should at least partly generalize to random tensor models. We review some aspects of the loop equations for matrix models and their algebraic counterpart for tensor models. Despite the generic title of this review, we, in particular, invoke the Topological Recursion. We explain its appearance in matrix models. Then we state that a family of tensor models provides a natural example which satisfies a version of the most general form of the topological recursion, named the blobbed topological recursion. We discuss the difficulties of extending the technical solutions existing for matrix models to tensor models. Some proofs are not published yet but will be given in a coming paper, the rest of the results are well known in the literature.
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.
Gravitational wave stress tensor from the linearised field equations
Balbus, Steven A
2016-01-01
A conserved stress energy tensor for weak field gravitational waves in standard general relativity is derived directly from the linearised wave equation alone, for an arbitrary gauge. The form of the tensor leads directly to the classical expression for the outgoing wave energy in any harmonic gauge. The method described here, however, is a much simpler, shorter, and more physically motivated approach than is the customary procedure, which involves a lengthy and cumbersome second-order (in wave-amplitude) calculation starting with the Einstein tensor. Our method has the added advantage of exhibiting the direct coupling between the outgoing energy flux in gravitational waves and the work done by the gravitational field on the sources. For nonharmonic gauges, the derived wave stress tensor has an index asymmetry. This coordinate artefact may be removed by techniques similar to those used in classical electrodynamics (where this issue also arises), but only by appeal to a more lengthy calculation. For any harmon...
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 tensor approach to the estimation of hydraulic conductivities in ...
2006-07-03
Jul 3, 2006 ... The HC values computed from the data measured on the weathered or ... Keywords: hydraulic conductivity tensor, roughness, combined stress, hydraulic aperture, Table Mountain ... the anisotropic nature of studied media.
Cosmic no-hair conjecture in scalar–tensor theories
T Singh; R Chaubey
2006-12-01
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 is no difference between scalar–tensor theories, Lyra's manifold and general relativity (GR).
Lagrange Multipliers and Third Order Scalar-Tensor Field Theories
Horndeski, Gregory W.
2016-01-01
In a space of 4-dimensions, I will examine constrained variational problems in which the Lagrangian, and constraint scalar density, are concomitants of a (pseudo-Riemannian) metric tensor and its first two derivatives. The Lagrange multiplier for these constrained extremal problems will be a scalar field. For suitable choices of the Lagrangian, and constraint, we can obtain Euler-Lagrange equations which are second order in the scalar field and third order in the metric tensor. The effect of ...
Irreducible decomposition for tensor prodect representations of Jordanian quantum algebras
Aizawa, N
1997-01-01
Tensor products of irreducible representations of the Jordanian quantum algebras U_h(sl(2)) and U_h(su(1,1)) are considered. For both the highest weight finite dimensional representations of U_h(sl(2)) and lowest weight infinite dimensional ones of U_h(su(1,1)), it is shown that tensor product representations are reducible and that the decomposition rules to irreducible representations are exactly the same as those of corresponding Lie algebras.
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.)
Expression of strain tensor in orthogonal curvilinear coordinates
Xuyan Liu
2010-01-01
Full Text Available Based on an analysis of connotation and extension of the concept of the orthogonal curvilinear coordinates, we have deduced a platform of strain tensor expression of Cartesian coordinates, which turns out to be a function of Lame coefficient and unit vector. By using transform matrix between Cartesian coordinates and orthogonal curvilinear coordinates, we have deduced a mathematical expression for correcting displacement vector differential in orthogonal curvilinear coordinates, and given a general expression of strain tensor in orthogonal curvilinear coordinates.
One-loop tensor Feynman integral reduction with signed minors
Fleischer, Jochem; Riemann, Tord; Yundin, Valery
2012-01-01
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...... in a systematic way. The final expressions for the numerical evaluation are then compact combinations of the contributing basic scalar functions....
Exact and Approximate Quadratures for Curvature Tensor Estimation
Langer, Torsten; Belyaev, Alexander; Seidel, Hans-Peter; Greiner, Günther; Hornegger, Joachim; Niemann, Heinrich; Stamminger, Marc
2005-01-01
Accurate estimations of geometric properties of a surface from its discrete approximation are important for many computer graphics and geometric modeling applications. In this paper, we derive exact quadrature formulae for mean curvature, Gaussian curvature, and the Taubin integral representation of the curvature tensor. The exact quadratures are then used to obtain reliable estimates of the curvature tensor of a smooth surface approximated by a dense triangle me...
Symmetric and conserved energy-momentum tensors in moving media
Ravndal, Finn
2011-01-01
A symmetric and conserved energy-momentum tensor for a scalar field in a moving medium is derived using the Gordon metric. When applied to an electromagnetic field, the method gives a similar result. This approach thus points a way out of the old Abraham-Minkowski controversy around the question about the correct energy-momentum tensor for the electromagnetic field in a material medium.
Coupling the antisymmetric tensor to the supergravity-matter system
Binetruy, P.; Girardi, G.; Grimm, R.; Mueller, M.
1987-09-10
The description of the antisymmetric tensor gauge field with Chern-Simons in Kaehler superspace is used to derive a particular coupling of the antisymmetric tensor to the general supergravity-matter system in terms of superfields as well as component fields. The construction is performed directly in terms of the linear multiplet. The proper duality transformations are presented at the full superfield level. General couplings are shortly discussed.
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, M; 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 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.
The Kummer tensor density in electrodynamics and in gravity
Baekler, Peter; Itin, Yakov; Hehl, Friedrich W
2014-01-01
Guided by results in the premetric electrodynamics of local and linear media, we introduce on 4-dimensional spacetime the new abstract notion of a Kummer tensor density of rank four, ${\\cal K}^{ijkl}$. This tensor density is, by definition, a cubic algebraic functional of a tensor density of rank four ${\\cal T}^{ijkl}$, which is antisymmetric in its first two and its last two indices: ${\\cal T}^{ijkl} = - {\\cal T}^{jikl} = - {\\cal T}^{ijlk}$. Thus, ${\\cal K}\\sim {\\cal T}^3$, see Eq.(46). (i) If $\\cal T$ is identified with the electromagnetic response tensor of local and linear media, the Kummer tensor density encompasses the generalized {\\it Fresnel wave surfaces} for propagating light. In the reversible case, the wave surfaces turn out to be {\\it Kummer surfaces} as defined in algebraic geometry (Bateman 1910). (ii) If $\\cal T$ is identified with the {\\it curvature} tensor $R^{ijkl}$ of a Riemann-Cartan spacetime, then ${\\cal K}\\sim R^3$ and, in the special case of general relativity, ${\\cal K}$ reduces to t...
A ten year Moment Tensor database for Western Greece
Serpetsidaki, Anna; Sokos, Efthimios; Tselentis, G.-Akis
2016-10-01
Moment Tensors (MTs) provide important information for seismotectonic, stress distribution and source studies. It is also important as a real time or near real time information in shakemaps, tsunami warning, and stress transfer. Therefore a reliable and rapid MT computation is a routine task for modern seismic networks with broadband sensors and real-time digital telemetry. In this paper we present the database of Moment Tensor solutions computed during the last ten years in Western Greece by the University of Patras, Seismological Laboratory (UPSL). The data from UPSL broad band network were used together with the ISOLA Moment Tensor inversion package for routine MT calculation. The procedures followed and the comparison of UPSL derived solutions with the ones provided by other agencies for Western Greece region are presented as well. The Moment Tensor database includes solutions for events in the magnitude range 2.8-6.8 and provides a unique insight into the faulting characteristics of Western Greece. Moreover it paves the way for detailed studies of stress tensor and stress transfer. The weak events' Moment Tensor included in UPSL's database are important for the comprehension of local seismotectonics and reveal the role of minor faults, which may be critical in seismic hazard estimation.
Renormalization Group Flows of Hamiltonians Using Tensor Networks
Bal, M.; Mariën, M.; Haegeman, J.; Verstraete, F.
2017-06-01
A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015), 10.1103/PhysRevLett.115.180405; S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017), 10.1103/PhysRevLett.118.110504], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.
Charmless Hadronic B Decays into a Tensor Meson
Cheng, Hai-Yang
2010-01-01
Two-body charmless hadronic B decays involving a tensor meson in the final state are studied within the framework of QCD factorization (QCDF). Due to the G-parity of the tensor meson, both the chiral-even and chiral-odd two-parton light-cone distribution amplitudes of the tensor meson are antisymmetric under the interchange of momentum fractions of the quark and anti-quark in the SU(3) limit. Our main results are: (i) In the naive factorization approach, the decays such as $B^-\\to \\bar K_2^{*0}\\pi^-$ and $\\bar B^0\\to K_2^{*-}\\pi^+$ with a tensor meson emitted are prohibited owing to the fact that a tensor meson cannot be created from the local V-A or tensor current. Nevertheless, they receive nonfactorizable contributions in QCDF from vertex, penguin and hard spectator corrections. The experimental observation of $B^-\\to \\bar K_2^{*0}\\pi^-$ indicates the importance of nonfactorizable effects. (ii) For penguin-dominated $B\\to TP$ and $TV$ decays, the predicted rates in naive factorization are usually too small...
Tensor Decomposition for Signal Processing and Machine Learning
Sidiropoulos, Nicholas D.; De Lathauwer, Lieven; Fu, Xiao; Huang, Kejun; Papalexakis, Evangelos E.; Faloutsos, Christos
2017-07-01
Tensors or {\\em multi-way arrays} are functions of three or more indices $(i,j,k,\\cdots)$ -- similar to matrices (two-way arrays), which are functions of two indices $(r,c)$ for (row,column). Tensors have a rich history, stretching over almost a century, and touching upon numerous disciplines; but they have only recently become ubiquitous in signal and data analytics at the confluence of signal processing, statistics, data mining and machine learning. This overview article aims to provide a good starting point for researchers and practitioners interested in learning about and working with tensors. As such, it focuses on fundamentals and motivation (using various application examples), aiming to strike an appropriate balance of breadth {\\em and depth} that will enable someone having taken first graduate courses in matrix algebra and probability to get started doing research and/or developing tensor algorithms and software. Some background in applied optimization is useful but not strictly required. The material covered includes tensor rank and rank decomposition; basic tensor factorization models and their relationships and properties (including fairly good coverage of identifiability); broad coverage of algorithms ranging from alternating optimization to stochastic gradient; statistical performance analysis; and applications ranging from source separation to collaborative filtering, mixture and topic modeling, classification, and multilinear subspace learning.
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.
Renormalization Group Flows of Hamiltonians Using Tensor Networks.
Bal, M; Mariën, M; Haegeman, J; Verstraete, F
2017-06-23
A renormalization group flow of Hamiltonians for two-dimensional classical partition functions is constructed using tensor networks. Similar to tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.180405; S. Yang, Z.-C. Gu, and X.-G. Wen, Phys. Rev. Lett. 118, 110504 (2017)PRLTAO0031-900710.1103/PhysRevLett.118.110504], we obtain approximate fixed point tensor networks at criticality. Our formalism, however, preserves positivity of the tensors at every step and hence yields an interpretation in terms of Hamiltonian flows. We emphasize that the key difference between tensor network approaches and Kadanoff's spin blocking method can be understood in terms of a change of the local basis at every decimation step, a property which is crucial to overcome the area law of mutual information. We derive algebraic relations for fixed point tensors, calculate critical exponents, and benchmark our method on the Ising model and the six-vertex model.
Computer Tensor Codes to Design the War Drive
Maccone, C.
To address problems in Breakthrough Propulsion Physics (BPP) and design the Warp Drive one needs sheer computing capabilities. This is because General Relativity (GR) and Quantum Field Theory (QFT) are so mathematically sophisticated that the amount of analytical calculations is prohibitive and one can hardly do all of them by hand. In this paper we make a comparative review of the main tensor calculus capabilities of the three most advanced and commercially available “symbolic manipulator” codes. We also point out that currently one faces such a variety of different conventions in tensor calculus that it is difficult or impossible to compare results obtained by different scholars in GR and QFT. Mathematical physicists, experimental physicists and engineers have each their own way of customizing tensors, especially by using different metric signatures, different metric determinant signs, different definitions of the basic Riemann and Ricci tensors, and by adopting different systems of physical units. This chaos greatly hampers progress toward the design of the Warp Drive. It is thus suggested that NASA would be a suitable organization to establish standards in symbolic tensor calculus and anyone working in BPP should adopt these standards. Alternatively other institutions, like CERN in Europe, might consider the challenge of starting the preliminary implementation of a Universal Tensor Code to design the Warp Drive.
The tensor bi-spectrum in a matter bounce
Chowdhury, Debika; Sriramkumar, L
2015-01-01
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 $h_{_{\\rm NL}}$ 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 scenari...
The Hamiltonian structure of Dirac's equation in tensor form and its Fermi quantization
Reifler, Frank; Morris, Randall
1992-01-01
Currently, there is some interest in studying the tensor forms of the Dirac equation to elucidate the possibility of the constrained tensor fields admitting Fermi quantization. We demonstrate that the bispinor and tensor Hamiltonian systems have equivalent Fermi quantizations. Although the tensor Hamiltonian system is noncanonical, representing the tensor Poisson brackets as commutators for the Heisenberg operators directly leads to Fermi quantization without the use of bispinors.
Electrostatic forces in the Poisson-Boltzmann systems.
Xiao, Li; Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2013-09-07
Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.
Topological conformal defects with tensor networks
Hauru, Markus; Evenbly, Glen; Ho, Wen Wei; Gaiotto, Davide; Vidal, Guifre
2016-09-01
The critical two-dimensional classical Ising model on the square lattice has two topological conformal defects: the Z2 symmetry defect Dɛ and the Kramers-Wannier duality defect Dσ. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function ZD of the critical Ising model in the presence of a topological conformal defect D is expressed in terms of the scaling dimensions Δα and conformal spins sα of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising conformal field theory. This characteristic conformal data {Δα,sα}D can be extracted from the eigenvalue spectrum of a transfer matrix MD for the partition function ZD. In this paper, we investigate the use of tensor network techniques to both represent and coarse grain the partition functions ZDɛand ZD σ of the critical Ising model with either a symmetry defect Dɛ or a duality defect Dσ. We also explain how to coarse grain the corresponding transfer matrices MDɛand MD σ, from which we can extract accurate numerical estimates of {Δα,sα}Dɛ and {Δα,sα}Dσ. Two key ingredients of our approach are (i) coarse graining of the defect D , which applies to any (i.e., not just topological) conformal defect and yields a set of associated scaling dimensions Δα, and (ii) construction and coarse graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins sα.
Centroid moment tensor catalogue for Indonesia
Nakano, M.; Yamashina, T.; Kumagai, H.; Inoue, H.; Sunarjo
2010-12-01
We developed a centroid moment tensor (CMT) catalogue of earthquakes in and around Indonesia (InaCMT) using data from the nationwide broadband seismograph network in Indonesia. We obtained CMT solutions for about 500 earthquakes that occurred in and around Indonesia between 2006 and 2009. The moment magnitudes ranged from 4.5 to 8.3. We examined the accuracy of the CMT solutions of the InaCMT catalogue by comparing them with those obtained by the Global CMT (GCMT) Project. The seismic moments and focal mechanisms of these catalogues were highly consistent with each other, but we found systematic differences between the catalogues in the source centroid locations of earthquakes off Sumatra. The InaCMT source centroid locations were closer to the hypocenter locations in the Preliminary Determination of Earthquakes (PDE) of the U.S. Geological Survey compared to those of GCMT. The systematic deviations in the GCMT source centroid locations may mainly reflect insufficient azimuthal coverage of the stations used for the inversions as well as uncertainties in the Earth model. Using the InaCMT catalogue, we investigated seismic activity related to the off Bengkulu seismic sequence on 12 September 2007 ( Mw = 8.3, 7.9, and 6.8), southwest of Sumatra, and the earthquakes northwest of the island of New Guinea on 3 January 2009 ( Mw = 7.7 and 7.4). In the aftershock activity of the 2007 off Bengkulu seismic sequence, we found that shallow earthquakes were aligned along the eastern coast of Siberut Island, located between the Sunda trench and Sumatra. These earthquakes may have occurred along the Mentawai fault or another unknown fault. The focal mechanisms of the earthquakes were dominantly reverse slip, although the Mentawai fault has been considered to be a strike-slip fault. Shallow large earthquakes along this fault may cause damage above the source region and generate large tsunamis. We found that the 2009 earthquakes northwest of New Guinea occurred along the Manokwari
Diffusion tensor imaging of hippocampal network plasticity.
Sierra, Alejandra; Laitinen, Teemu; Gröhn, Olli; Pitkänen, Asla
2015-03-01
Diffusion tensor imaging (DTI) has become a valuable tool to investigate white matter integrity in the brain. DTI also gives contrast in gray matter, which has been relatively little explored in studies assessing post-injury structural abnormalities. The present study was designed to compare white and gray matter reorganization in the rat hippocampus after two epileptogenic brain injuries, status epilepticus (SE) and traumatic brain injury (TBI), using ex vivo high-resolution DTI. Imaging was performed at 6-12 months post-injury and findings were compared to histological analyses of Nissl, myelin, and Timm-stained preparations from the same animals. In agreement with the severity of histological damage, fractional anisotropy (FA), axial (D ||) and radial (D ⊥) diffusivities, and mean diffusivity (MD) measurements were altered in the order SE > TBI ipsilaterally > TBI contralaterally. After SE, the most severe abnormalities were found in the dentate gyrus and CA3b-c subfields, in which the mean FA was increased to 125 % (p < 0.001) and 143 % (p < 0.001) of that in controls, respectively. In both subfields, the change in FA was associated with an increase in D || (p < 0.01). In the stratum radiatum of the CA1, FA was decreased to 81 % of that in controls (p < 0.05) which was associated with an increase in D ⊥ (p < 0.01). After TBI, DTI did not reveal any major abnormalities in the dentate gyrus. In the ipsilateral CA3b-c, however, FA was increased to 126 % of that in controls (p < 0.01) and associated with a mild decrease in D ⊥ (p < 0.05). In the stratum radiatum of the ipsilateral CA1, FA was decreased to 88 % of that in controls (p < 0.05). Our data demonstrate that DTI reveals subfield-specific abnormalities in the hippocampus with remarkable qualitative and quantitative differences between the two epileptogenic etiologies, suggesting that DTI could be a valuable tool for follow-up of focal circuitry reorganization during the post
Tensor completion for estimating missing values in visual data
Liu, Ji
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 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
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.)
Structure of the velocity gradient tensor in turbulent shear flows
Pumir, Alain
2017-07-01
The expected universality of small-scale properties of turbulent flows implies isotropic properties of the velocity gradient tensor in the very large Reynolds number limit. Using direct numerical simulations, we determine the tensors formed by n =2 and 3 velocity gradients at a single point in turbulent homogeneous shear flows and in the log-layer of a turbulent channel flow, and we characterize the departure of these tensors from the corresponding isotropic prediction. Specifically, we separate the even components of the tensors, invariant under reflexion with respect to all axes, from the odd ones, which identically vanish in the absence of shear. Our results indicate that the largest deviation from isotropy comes from the odd component of the third velocity gradient correlation function, especially from the third moment of the derivative along the normal direction of the streamwise velocity component. At the Reynolds numbers considered (Reλ≈140 ), we observe that these second- and third-order correlation functions are significantly larger in turbulent channel flows than in homogeneous shear flow. Overall, our work demonstrates that a mean shear leads to relatively simple structure of the velocity gradient tensor. How isotropy is restored in the very large Reynolds limit remains to be understood.
Role of tensor operators in RK and RK*
Bardhan, Debjyoti; Byakti, Pritibhajan; Ghosh, Diptimoy
2017-10-01
The recent LHCb measurement of RK* in two q2 bins, when combined with the earlier measurement of RK, strongly suggests lepton flavour non-universal new physics in semi-leptonic B meson decays. Motivated by these intriguing hints of new physics, several authors have considered vector, axial vector, scalar and pseudo scalar operators as possible explanations of these measurements. However, tensor operators have widely been neglected in this context. In this paper, we consider the effect of tensor operators in RK and RK*. We find that, unlike other local operators, tensor operators can comfortably produce both of RK*low and RK*central close to their experimental central values. However, a simultaneous explanation of RK is not possible with only Tensor operators, and other vector or axial vector operators are needed. In fact, we find that combination of vector and tensor operators can provide simultaneous explanations of all the anomalies comfortably at the 1σ level, a scenario which is hard to achieve with only vector or axial vector operators. We also comment on the compatibility of the various new physics solutions with the measurements of the inclusive decay Bd →Xsℓ+ℓ-.
White matter degeneration in schizophrenia: a comparative diffusion tensor analysis
Ingalhalikar, Madhura A.; Andreasen, Nancy C.; Kim, Jinsuh; Alexander, Andrew L.; Magnotta, Vincent A.
2010-03-01
Schizophrenia is a serious and disabling mental disorder. Diffusion tensor imaging (DTI) studies performed on schizophrenia have demonstrated white matter degeneration either due to loss of myelination or deterioration of fiber tracts although the areas where the changes occur are variable across studies. Most of the population based studies analyze the changes in schizophrenia using scalar indices computed from the diffusion tensor such as fractional anisotropy (FA) and relative anisotropy (RA). The scalar measures may not capture the complete information from the diffusion tensor. In this paper we have applied the RADTI method on a group of 9 controls and 9 patients with schizophrenia. The RADTI method converts the tensors to log-Euclidean space where a linear regression model is applied and hypothesis testing is performed between the control and patient groups. Results show that there is a significant difference in the anisotropy between patients and controls especially in the parts of forceps minor, superior corona radiata, anterior limb of internal capsule and genu of corpus callosum. To check if the tensor analysis gives a better idea of the changes in anisotropy, we compared the results with voxelwise FA analysis as well as voxelwise geodesic anisotropy (GA) analysis.
The dissipation tensor $\\varepsilon_{ij}$ in wall turbulence
Gerolymos, G A
2016-01-01
The paper investigates the dissipation tensor $\\varepsilon_{ij}$ in wall turbulence. Available \\tsn{DNS} data are examined to illustrate the differences in the anisotropy of the dissipation tensor $\\varepsilon_{ij}$ with respect to the anisotropy of the Reynolds-stresses $r_{ij}$. The budgets of the transport equations of the dissipation tensor $\\varepsilon_{ij}$ are studied using novel \\tsn{DNS} data of low-Reynolds-number turbulent plane channel flow with spatial resolution sufficiently fine to accurately determine the correlations of products of 2-derivatives of fluctuating velocities $u_i'$ and pressure $p'$ which appear in various terms. Examination of the anisotropy of the destruction-of-dissipation tensor $\\varepsilon_{\\varepsilon_{ij}}$ reveals a very different behaviour, never approaching the 2-component (2-C) state at the solid-wall. The wall-asymptotics of different terms in the transport equations are studied in detail. The dissipation tensor $\\varepsilon_{ij}$ is also studied in terms of 2-point ...
Thought Experiments on Gravitational Forces
Lynden-Bell, Donald
2013-01-01
Large contributions to the near closure of the Universe and to the acceleration of its expansion are due to the gravitation of components of the stress-energy tensor other than its mass density. To familiarise astronomers with the gravitation of these components we conduct thought experiments on gravity, analogous to the real experiments that our forebears conducted on electricity. By analogy to the forces due to electric currents we investigate the gravitational forces due to the flows of momentum, angular momentum, and energy along a cylinder. Under tension the gravity of the cylinder decreases but the 'closure' of the 3-space around it increases. When the cylinder carries a torque the flow of angular momentum along it leads to a novel helical interpretation of Levi-Civita's external metric and a novel relativistic effect. Energy currents give gravomagnetic effects in which parallel currents repel and antiparallel currents attract, though such effects must be added to those of static gravity. The gravity of...
Gravitational Forces on the Branes
Arnowitt, R L
2005-01-01
We examine the gravitational forces in a brane-world scenario felt by point particles on two 3-branes bounding a 5-dimensional AdS space with $S^{1}/Z_2$ symmetry. The particles are treated as perturbations on the vacuum metric and coordinate conditions are chosen so that no brane bending effects occur. We make an ADM type decomposition of the metric tensor and solve Einstein's equations to linear order in the static limit. While no stabilization mechanism is assumed, all the 5D Einstein equations are solved and are seen to have a consistent solution. We find that Newton's law is reproduced on the Planck brane at the origin while particles on the TeV brane a distance $y_2$ from the origin experience an attractive force that has a growing exponential dependence on the brane position.
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.
Tensor Network Renormalization Yields the Multiscale Entanglement Renormalization Ansatz
Evenbly, G.; Vidal, G.
2015-11-01
We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] to the Euclidean time evolution operator e-β H for infinite β . This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β , produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.
High spatial resolution diffusion tensor imaging and its applications
Wang, J J
2002-01-01
Introduction Magnetic Resonance Imaging is at present the only imaging technique available to measure diffusion of water and metabolites in humans. It provides vital insights to brain connectivity and has proved to be an important tool in diagnosis and therapy planning in many neurological diseases such as brain tumour, ischaemia and multiple sclerosis. This project focuses on the development of a high resolution diffusion tensor imaging technique. In this thesis, the basic theory of diffusion tensor MR Imaging is presented. The technical challenges encountered during development of these techniques will be discussed, with proposed solutions. New sequences with high spatial resolution have been developed and the results are compared with the standard technique more commonly used. Overview The project aims at the development of diffusion tensor imaging techniques with a high spatial resolution. Chapter 2 will describe the basic physics of MRI, the phenomenon of diffusion and the measurement of diffusion by MRI...
Group-theoretical method for physical property tensors of quasicrystals
Gong Ping; Hu Cheng-Zheng; Zhou Xiang; Wang Ai-Jun; Miao Ling
2006-01-01
In addition to the phonon variable there is the phason variable in hydrodynamics for quasicrystals. These two kinds of hydrodynamic variables have different transformation properties. The phonon variable transforms under the vector representation, whereas the phason variable transforms under another related representation. Thus, a basis (or a set of basis functions) in the representation space should include such two kinds of variables. This makes it more difficult to determine the physical property tensors of quasicrystals. In this paper the group-theoretical method is given to determine the physical property tensors of quasicrystals. As an illustration of this method we calculate the third-order elasticity tensors of quasicrystals with five-fold symmetry by means of basis functions. It follows that the linear phonon elasticity is isotropic, but the nonlinear phonon elasticity is anisotropic for pentagonal quasicrystals. Meanwhile, the basis functions are constructed for all noncrystallographic point groups of quasicrystals.
GUT-scale inflation with sizeable tensor modes
Brummer, Felix; Sanz, Veronica
2014-01-01
A sizeable tensor-to-scalar ratio, such as recently claimed by BICEP2, would imply a scale of inflation at the typical scale of supersymmetric grand unification. This could be an accident, or strong support for supersymmetric theories. Models of F-term hybrid inflation naturally connect the GUT scale with the inflationary scale, but they also predict the tensor-to-scalar ratio to be unmeasurably small. In this work we analyze a general UV embedding of F-term hybrid inflation into a supergravity theory with a general Kahler potential. The CMB observables are generated during the early phase of inflation, at large inflaton values, where the potential is dominated by Planck-suppressed operators. Tuning the leading higher-order terms can give an inflaton potential with sizeable tensor fluctuations and a field excursion which is still sub-Planckian but close to the Planck scale, as expected from the Lyth bound.
Calculation of nonlinear magnetic susceptibility tensors for a uniaxial antiferromagnet
Lim, Siew-Choo; Osman, Junaidah; Tilley, D. R.
2000-11-01
In this paper, we present a derivation of the nonlinear susceptibility tensors for a two-sublattice uniaxial antiferromagnet up to the third-order effects within the standard definition by which the rf magnetization m is defined as a power series expansion in the rf fields h with the susceptibility tensors χ(q) as the coefficients. The starting point is the standard set of torque equations of motion for this problem. A complete set of tensor elements is derived for the case of a single-frequency input wave. Within a circular polarization frame (pnz) expressions are given for the first-order susceptibility, second-harmonic generation, optical rectification, third-harmonic generation and intensity-dependent susceptibility. Some of the coefficients with representative resonance features in the far infrared are illustrated graphically and we conclude with a brief discussion of the implications of the resonance features arising from the calculations and their potential applications.
Incremental dimension reduction of tensors with random index
Sandin, Fredrik; Sahlgren, Magnus
2011-01-01
We present an incremental, scalable and efficient dimension reduction technique for tensors that is based on sparse random linear coding. Data is stored in a compactified representation with fixed size, which makes memory requirements low and predictable. Component encoding and decoding are performed on-line without computationally expensive re-analysis of the data set. The range of tensor indices can be extended dynamically without modifying the component representation. This idea originates from a mathematical model of semantic memory and a method known as random indexing in natural language processing. We generalize the random-indexing algorithm to tensors and present signal-to-noise-ratio simulations for representations of vectors and matrices. We present also a mathematical analysis of the approximate orthogonality of high-dimensional ternary vectors, which is a property that underpins this and other similar random-coding approaches to dimension reduction. To further demonstrate the properties of random ...
Partial wave decomposition of finite-range effective tensor interaction
Davesne, D; Pastore, A; Navarro, J
2016-01-01
We perform a detailed analysis of the properties of the finite-range tensor term associated with the Gogny and M3Y effective interactions. In particular, by using a partial wave decomposition of the equation of state of symmetric nuclear matter, we show how we can extract their tensor parameters directly from microscopic results based on bare nucleon-nucleon interactions. Furthermore, we show that the zero-range limit of both finite-range interactions has the form of the N3LO Skyrme pseudo-potential, which thus constitutes a reliable approximation in the density range relevant for finite nuclei. Finally, we use Brueckner-Hartree-Fock results to fix the tensor parameters for the three effective interactions.
Vector field instability and the primordial tensor spectrum
Eccles, Stefan; Lorshbough, Dustin; Stephens, Benjamin A
2015-01-01
It has recently been shown that the presence of a spectator pseudoscalar field, coupled to photons through a Chern-Simons term, can amplify the primordial tensor spectrum without observationally disrupting the primordial scalar spectrum. The amplification occurs due to an instability that develops for the vector fields. We extend previous studies to account for the contribution arising from an inhomogeneous vector background, which emerges as the dominant correction to the primordial tensor spectrum. These semiclassical contributions dominate over the quantum loop contributions and possibly enhance the primordial tensor spectrum such as to have observational effects even though the loop corrections might be undetectable. A similar effect would occur by replacing the visible electromagnetic U(1) by an unbroken dark U(1).
Three—Dimensional Vector Field Visualization Based on Tensor Decomposition
梁训东; 李斌; 等
1996-01-01
This paper presents a visualization method called the deformed cube for visualizing 3D velocity vector field.Based on the decomposition of the tensor which describes the changes of the velocity,it provides a technique for visualizing local flow.A deformed cube,a cube transformed by a tensor in a local coordinate frame,shows the local stretch,shear and rigid body rotation of the local flow corresponding to the decomposed component of the tensor.Users can interactively view the local deformation or any component of the changes.The animation of the deformed cube moving along a streamline achieves a more global impression of the flow field.This method is intended as a complement to global visualization methods.
Block Tensor Decomposition for Source Apportionment of Air Pollution
Hopke, Philip K; Li, Na; Navasca, Carmeliza
2011-01-01
The ambient particulate chemical composition data with three particle diameter sizes (2.5mm
Agyrotropic pressure tensor induced by the plasma velocity shear
Pegoraro, Francesco; Del Sarto, Danele; Califano, Francesco
2016-10-01
We show that the spatial inhomogeneity of a shear flow in a fluid plasma is transferred to a pressure anisotropy that has both a gyrotropic and a non gyrotropic component. We investigate this process both analytically and numerically by including the full pressure tensor dynamics. We determine the time evolution of the pressure agyrotropy and in general of the pressure tensor anisotropization which arise from the action of both the magnetic eld and the flow strain tensor. This mechanism can affect the onset and development of shear-induced fluid instabilities in plasmas and is relevant to the understanding of the origin of some of the non-Maxwellian distribution functions evidenced both in Vlasov simulations and in space plasma measurements that exhibit pressure agyrotropy.
A DISCUSSION ABOUT SCALE INVARIANTS FOR TENSOR FUNCTIONS
Huang Yongnian; Luo Xiongping; Emily S.C.Ching
2000-01-01
It is found that in some cases the complete and irreducible scale invariants given by Ref.[1]are not independent.There are some implicit functional relations among them.The scale invariants for two different cases are calculated.The first case is an arbitrary second order tensor.The second case includes a symmetric tensor,an antisymmetric tensor and a vector.By using the eigentensor notation it is proved that in the first case there are only six independent scale invariants rather than seven as reported in Ref.[1]and in the second case there are only nine independent scale invariants which are leas than that obtained in Ref.[1].
Robust Estimation of Trifocal Tensor Using Messy Genetic Algorithm
HUMingxing; YUANBaozong; TANGXiaofang
2003-01-01
Given three partially overlapping views of a scene from which a set of point or line correspondences have been extracted, 3D structure and camera motion pa-rameters can be represented by the trifocal tensor, which is the key to many problems of computer vision among three views. This paper addresses the problem of robust esti-mating the trifocal tensor employing a new method based on messy genetic algorithm, which uses each gene to stand for a triplet of correspondences, and takes every chromo-some as a minimum subset for trifocal tensor estimation.The method would eventually converge to a near optimal solution and is relatively unaffected by the outliers. Exper-iments with both synthetic data and real images show that our method is more robust and precise than other typical methods because it can efficiently detect and delete the bad corresponding points, which include both bad loca-tions and false matches.
Photon polarization tensor in a magnetized plasma system
Chao, Jingyi
2016-01-01
We investigate the photon polarization tensor at finite temperature in the presence of a static and homogeneous external magnetic field. In our scheme, the Matsubara frequency summation is performed after Poisson summation, which will be taken easily and convergent quickly in the frame of proper time representation. Moreover, the dependence of Landau levels is expressed explicitly. It demonstrates the convergence of summing Landau levels as it has to be. Consequently, there is no necessary to truncate the Landau level in a numerical estimation. At zero temperature, the Lowest Landau Level (LLL) approximation is analytically satisfied for the imaginary parts of the vacuum photon polarization tensor. Our results examine that, the LLL approximation is not enough for the thermal photon polarization tensor, it gains the contribution not only from the lowest Landau level but also up to the finite-$n$ levels. Such large imaginary ones only show up at finite temperatures, which is the so called Landau damping. It ori...
A geometrical approach to degenerate scalar-tensor theories
Chagoya, Javier
2016-01-01
Degenerate scalar-tensor theories are recently proposed covariant theories of gravity coupled with a scalar field. Despite being characterised by higher order equations of motion, they do not propagate more than three degrees of freedom, thanks to the existence of constraints. We discuss a geometrical approach to degenerate scalar-tensor systems, and analyse its consequences. We show that some of these theories emerge as a certain limit of DBI Galileons. In absence of dynamical gravity, these systems correspond to scalar theories enjoying a symmetry which is different from Galileon invariance. The scalar theories have however problems concerning the propagation of fluctuations around a time dependent background. These issues can be tamed by breaking the symmetry by hand, or by minimally coupling the scalar with dynamical gravity in a way that leads to degenerate scalar-tensor systems. We show that distinct theories can be connected by a relation which generalizes Galileon duality, in certain cases also when g...
The Invar tensor package: Differential invariants of Riemann
Martín-García, J. M.; Yllanes, D.; Portugal, R.
2008-10-01
The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6ṡ10 objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6ṡ10 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica) and Canon (for Maple). Program summaryProgram title:Invar Tensor Package v2.0 Catalogue identifier:ADZK_v2_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADZK_v2_0.html Program obtainable from:CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions:Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.:3 243 249 No. of bytes in distributed program, including test data, etc.:939 Distribution format:tar.gz Programming language:Mathematica and Maple Computer:Any computer running Mathematica versions 5.0 to 6.0 or Maple versions 9 and 11 Operating system:Linux, Unix, Windows XP, MacOS RAM:100 Mb Word size:64 or 32 bits Supplementary material:The new database of relations is much larger than that for the previous version and therefore has not been included in
Multi-Mode Tensor Representation of Motion Data
Meinard Müller
2008-08-01
Full Text Available In this paper, we investigate how a multilinear model can be used to represent human motion data. Based on technical modes (referring to degrees of freedom and number of frames and natural modes that typically appear in the context of a motion capture session (referring to actor, style, and repetition, the motion data is encoded in form of a high-order tensor. This tensor is then reduced by using N-mode singular value decomposition. Our experiments show that the reduced model approximates the original motion better then previously introduced PCA-based approaches. Furthermore, we discuss how the tensor representation may be used as a valuable tool for the synthesis of new motions.
Tensor Network Renormalization Yields the Multiscale Entanglement Renormalization Ansatz.
Evenbly, G; Vidal, G
2015-11-13
We show how to build a multiscale entanglement renormalization ansatz (MERA) representation of the ground state of a many-body Hamiltonian H by applying the recently proposed tensor network renormalization [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] to the Euclidean time evolution operator e(-βH) for infinite β. This approach bypasses the costly energy minimization of previous MERA algorithms and, when applied to finite inverse temperature β, produces a MERA representation of a thermal Gibbs state. Our construction endows tensor network renormalization with a renormalization group flow in the space of wave functions and Hamiltonians (and not merely in the more abstract space of tensors) and extends the MERA formalism to classical statistical systems.
Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States
Chen, Lin; Duan, Runyao; Ji, Zhengfeng; Winter, Andreas
2010-01-01
The tensor rank (aka generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state $\\ket{W_3}=\\tfrac{1}{\\sqrt{3}}(\\ket{100}+\\ket{010}+\\ket{001})$ and its $N$-partite generalization $\\ket{W_N}$. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of $\\ket{W_3}$ has rank either 15 or 16, (ii) two copies of $\\ket{W_N}$ has rank $3N-2$, and (iii) $n$ copies of $\\ket{W_N}$ has rank O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple copy bunches or when assisted by some catalyzing state. This novel effect is impossible for bipartite pure states.
Three-dimensional diffusion tensor microscopy of fixed mouse hearts.
Jiang, Yi; Pandya, Kumar; Smithies, Oliver; Hsu, Edward W
2004-09-01
The relative utility of 3D, microscopic resolution assessments of fixed mouse myocardial structure via diffusion tensor imaging is demonstrated in this study. Isotropic 100-microm resolution fiber orientation mapping within 5.5 degrees accuracy was achieved in 9.1 hr scan time. Preliminary characterization of the diffusion tensor primary eigenvector reveals a smooth and largely linear angular rotation across the left ventricular wall. Moreover, a higher level of structural hierarchy is evident from the organized secondary and tertiary eigenvector fields. These findings are consistent with the known myocardial fiber and laminar structures reported in the literature and suggest an essential role of diffusion tensor microscopy in developing quantitative atlases for studying the structure-function relationships of mouse hearts.
Reduced Stress Tensor and Dissipation and the Transport of Lamb Vector
Wu, Jie-Zhi; Zhou, Ye; Wu, Jian-Ming
1996-01-01
We develop a methodology to ensure that the stress tensor, regardless of its number of independent components, can be reduced to an exactly equivalent one which has the same number of independent components as the surface force. It is applicable to the momentum balance if the shear viscosity is constant. A direct application of this method to the energy balance also leads to a reduction of the dissipation rate of kinetic energy. Following this procedure, significant saving in analysis and computation may be achieved. For turbulent flows, this strategy immediately implies that a given Reynolds stress model can always be replaced by a reduced one before putting it into computation. Furthermore, we show how the modeling of Reynolds stress tensor can be reduced to that of the mean turbulent Lamb vector alone, which is much simpler. As a first step of this alternative modeling development, we derive the governing equations for the Lamb vector and its square. These equations form a basis of new second-order closure schemes and, we believe, should be favorably compared to that of traditional Reynolds stress transport equation.
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.
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.)
Occupational Outlook Quarterly, 2012
2012-01-01
The labor force is the number of people ages 16 or older who are either working or looking for work. It does not include active-duty military personnel or the institutionalized population, such as prison inmates. Determining the size of the labor force is a way of determining how big the economy can get. The size of the labor force depends on two…
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
Extended Tensor Products and Generalization of the Notion of Entanglement
Khrennikov, Andrei
2012-01-01
Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., $p$-adic) and complex Hilbert spaces.
Primordial power spectrum of tensor perturbations in Finsler spacetime
Li, Xin [Chongqing University, Department of Physics, Chongqing (China); Chinese Academy of Sciences, State Key Laboratory Theoretical Physics, Institute of Theoretical Physics, Beijing (China); Wang, Sai [Chinese Academy of Sciences, State Key Laboratory Theoretical Physics, Institute of Theoretical Physics, Beijing (China)
2016-02-15
We first investigate the gravitational wave in the flat Finsler spacetime. In the Finslerian universe, we derive the perturbed gravitational field equation with tensor perturbations. The Finslerian background spacetime breaks rotational symmetry and induces parity violation. Then we obtain the modified primordial power spectrum of the tensor perturbations. The parity violation feature requires that the anisotropic effect contributes to the TT, TE, EE, BB angular correlation coefficients with l{sup '} = l + 1 and TB, EB with l{sup '} = l. The numerical results show that the anisotropic contributions to the angular correlation coefficients depend on m, and TE and ET angular correlation coefficients are different. (orig.)
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
Analytical stress tensor and pressure calculations with the CRYSTAL code
Doll, K.
2010-02-01
The calculation of the stress tensor and related properties and its implementation in the CRYSTAL code are described. The stress tensor is obtained from the earlier implemented analytical gradients with respect to the cell parameters. Subsequently, the pressure and enthalpy are computed, and a test concerning the pressure-driven phase transition in KI is used as an illustration. Finally, the possibility of applying external pressure is implemented. The constant-pressure optimization offers an alternative optimization method in addition to the already implemented optimization at constant volume.
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
Proposal for the proper gravitational energy-momentum tensor
Shimizu, Katsutaro
2016-08-01
We propose a gravitational energy-momentum (GEMT) tensor of the general relativity obtained using Noether’s theorem. It transforms as a tensor under general coordinate transformations. One of the two indices of the GEMT labels a local Lorentz frame that satisfies the energy-momentum conservation law. The energies for a gravitational wave, a Schwarzschild black hole and a Friedmann-Lemaitre-Robertson-Walker (FLRW) universe are calculated as examples. The gravitational energy of the Schwarzschild black hole exists only outside the horizon, its value being the negative of the black hole mass.
Extended tensor products and generalization of the notion of entanglement
Khrennikov, Andrei; Rosinger, Elemer E.
2012-03-01
Motivated by the novel applications of the mathematical formalism of quantum theory and its generalizations in cognitive science, psychology, social and political sciences, and economics, we extend the notion of the tensor product and entanglement. We also study the relation between conventional entanglement of complex qubits and our generalized entanglement. Our construction can also be used to describe entanglement in the framework of non-Archimedean physics. It is also possible to construct tensor products of non-Archimedean (e.g., p-adic) and complex Hilbert spaces.
New 1/N expansions in random tensor models
Bonzom, Valentin
2012-01-01
Although random tensor models were introduced twenty years ago, it is only in 2011 that Gurau proved the existence of a 1/N expansion. Here we show that there actually is more than a single 1/N expansion, depending on the dimension. In the large N limit, these new expansions retain more than the melonic graphs. Still, in most cases, the large N limit is found to be Gaussian, and therefore extends the scope of the universality theorem for large random tensors. Nevertheless, a scaling which leads to non-Gaussian large N limits, in even dimensions, is identified for the first time.
Dipole modulation in tensor modes: signatures in CMB polarization
Zarei, Moslem [Isfahan University of Technology, Department of Physics, Isfahan (Iran, Islamic Republic of); Institute for Research in Fundamental Sciences (IPM), School of Astronomy, P. O. Box 19395-5531, Tehran (Iran, Islamic Republic of)
2015-06-15
In this work we consider a dipole asymmetry in tensor modes and study the effects of this asymmetry on the angular power spectra of CMB. We derive analytical expressions for the C{sub l}{sup TT} and C{sub l}{sup BB} in the presence of such dipole modulation in tensor modes for l < 100. We also discuss on the amplitude of modulation term and show that the C{sub l}{sup BB} is considerably modified due to this term. (orig.) 3.
Intermediate Field Representation for Positive Matrix and Tensor Interactions
Lionni, Luca
2016-01-01
In this paper we introduce an intermediate field representation for random matrices and random tensors with positive (stable) interactions of degree higher than 4. This representation respects the symmetry axis responsible for positivity. It is non-perturbative and allows to prove that such models are Borel-Le Roy summable of the appropriate order in their coupling constant. However we have not been able yet to associate a convergent Loop Vertex Expansion to this representation, hence our Borel summability result is not of the optimal expected form when the size N of the matrix or of the tensor tends to infinity.
Multicritical tensor models and hard dimers on spherical random lattices
Bonzom, Valentin
2012-01-01
Random tensor models which display multicritical behaviors in a remarkably simple fashion are presented. They come with entropy exponents \\gamma = (m-1)/m, similarly to multicritical random branched polymers. Moreover, they are interpreted as models of hard dimers on a set of random lattices for the sphere in dimension three and higher. Dimers with their exclusion rules are generated by the different interactions between tensors, whose coupling constants are dimer activities. As an illustration, we describe one multicritical point, which is interpreted as a transition between the dilute phase and a crystallized phase, though with negative activities.
Abelian tensor hierarchy in 4D, N = 1 superspace
Becker, Katrin; Becker, Melanie; Linch, William D.; Robbins, Daniel
2016-03-01
With the goal of constructing the supersymmetric action for all fields, massless and massive, obtained by Kaluza-Klein compactification from type II theory or M-theory in a closed form, we embed the (Abelian) tensor hierarchy of p-forms in four-dimensional, N =1superspaceandconstructitsChern-Simons-likeinvariants. Whenspecializedtothe case in which the tensors arise from a higher-dimensional theory, the invariants may be interpreted as higher-dimensional Chern-Simons forms reduced to four dimensions. As an application of the formalism, we construct the eleven-dimensional Chern-Simons form in terms of four-dimensional, N = 1 superfields.
Making Tensor Factorizations Robust to Non-Gaussian Noise
Chi, Eric C
2010-01-01
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 i.i.d. Gaussian noise. We demonstrate that this loss function can actually 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 algorithm for fitting a CP model using our proposed loss function.
Primordial power spectrum of tensor perturbations in Finsler spacetime
Li, Xin
2015-01-01
We first investigate the gravitational wave in the flat Finsler spacetime. In the Finslerian universe, we derive the perturbed gravitational field equation with tensor perturbations. The Finslerian background spacetime breaks rotational symmetry and induces parity violation. Then we obtain the modified primordial power spectrum of tensor perturbations. The parity violation feature requires that the anisotropic effect contributes to $TT,TE,EE,BB$ angular correlation coefficients with $l'=l+1$ and $TB,EB$ with $l'=l$. The numerical results show that the anisotropic contributions to angular correlation coefficients depend on $m$, and $TE$ and $ET$ angular correlation coefficients are different.
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
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....
Labeling spherically symmetric spacetimes with the Ricci tensor
Ferrando, Joan Josep; Sáez, Juan Antonio
2017-02-01
We complete the intrinsic characterization of spherically symmetric solutions partially accomplished in a previous paper (Ferrando and Sáez 2010 Class. Quantum Grav. 27 205024). In this approach we consider every compatible algebraic type of the Ricci tensor, and we analyze specifically the conformally flat case for perfect fluid and Einstein–Maxwell solutions. As a direct application we obtain the ideal labeling (exclusively involving explicit concomitants of the metric tensor) of the Schwarzschild interior metric and the Vaidya solution. The Stephani universes and some significative subfamilies are also characterized.
Symmetry-imposed shape of linear response tensors
Seemann, M.; Ködderitzsch, D.; Wimmer, S.; Ebert, H.
2015-10-01
A scheme suggested in the literature to determine the symmetry-imposed shape of linear response tensors is revised and extended to allow for the treatment of more complex situations. The extended scheme is applied to discuss the shape of the spin conductivity tensor for all magnetic space groups. This allows in particular investigating the character of longitudinal as well as transverse spin transport for arbitrary crystal structure and magnetic order that give rise, e.g., to the spin Hall, Nernst, and the spin-dependent Seebeck effects.
Analytical effective tensor for flow-through composites
Sviercoski, Rosangela De Fatima
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.
Quantum group symmetry and q-tensor algebras
Biedenharn, Lawrence Christian
1995-01-01
Quantum groups are a generalization of the classical Lie groups and Lie algebras and provide a natural extension of the concept of symmetry fundamental to physics. This monograph is a survey of the major developments in quantum groups, using an original approach based on the fundamental concept of a tensor operator. Using this concept, properties of both the algebra and co-algebra are developed from a single uniform point of view, which is especially helpful for understanding the noncommuting co-ordinates of the quantum plane, which we interpret as elementary tensor operators. Representations
Product numerical range in a space with tensor product structure
Puchała, Zbigniew; Miszczak, Jarosław Adam; Skowronek, Łukasz; Choi, Man-Duen; Zyczkowski, Karol \\
2010-01-01
We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.
Effect of change in head position on diffusion tensor analysis
Sawamoto, Megumi; Takaba, Junko; Kushima, Toshio [Hiroshima Univ. (Japan). Hospital; Kabasawa, Hiroyuki
2001-06-01
It is important to understand the motion probing gradient direction and nerve fiber tract structure to assess diffusion-weighted imaging, which is affected by the position of subjects. However, diffusion tensor imaging may be able to reproduce exact anisotropy regardless of the position of the subject. Therefore, we examined the effect of the position of subjects on diffusion tensor imaging. The value of fractional anisotropy and eigenvalue 1 in standard position were equal to the values of other documented records. Change in head position caused no significant difference in eigenvector imaging, the value of fractional anisotropy, or eigenvalue 1. (author)
Dynamical system approach to scalar-vector-tensor cosmology
Ghaffarnejad, H
2016-01-01
We use scalar-vector-tensor gravity [1] which is obtained by generalizing Brans Dicke (BD) gravity model [2] via dynamical vector field. We study flat Friedmann Robertson Walker (FRW) cosmology by using dynamical system approach in the presence of self interaction BD potential and perfect fluid matter stress tensor. We obtained 3 critical points for $\\Lambda CDM$ vacuum de Sitter era which one of them is spiral attractor absolutely independent of particular values of the BD parameter $\\omega$ but not two other critical points. The latter take real values only for $-0.54-0.54.$ Even if the eigne values become complex imaginary where $\\omega\
Tensor renormalization group analysis of CP(N-1) model
Kawauchi, Hikaru
2016-01-01
We apply the higher order tensor renormalization group to lattice CP($N-1$) model in two dimensions. A tensor network representation of the CP($N-1$) model in the presence of the $\\theta$-term is derived. We confirm that the numerical results of the CP(1) model without the $\\theta$-term using this method are consistent with that of the O(3) model which is analyzed by the same method in the region $\\beta \\gg 1$ and that obtained by Monte Carlo simulation in a wider range of $\\beta$. The numerical computation including the $\\theta$-term is left for future challenges.
Tensor network quotient takes the vacuum to the thermal state
Czech, Bartłomiej; Evenbly, Glen; Lamprou, Lampros; McCandlish, Samuel; Qi, Xiao-liang; Sully, James; Vidal, Guifré
2016-08-01
In 1+1-dimensional conformal-field theory, the thermal state on a circle is related to a certain quotient of the vacuum on a line. We explain how to take this quotient in the MERA tensor network representation of the vacuum and confirm the validity of the construction in the critical Ising model. This result suggests that the tensors comprising MERA can be interpreted as performing local scale transformations, so that adding or removing them emulates conformal maps. In this sense, the optimized MERA recovers local conformal invariance that is broken by the choice of lattice.
Tensor renormalization group analysis of CP (N -1 ) model
Kawauchi, Hikaru; Takeda, Shinji
2016-06-01
We apply the higher-order tensor renormalization group to the lattice CP (N -1 ) model in two dimensions. A tensor network representation of the CP (N -1 ) model in the presence of the θ term is derived. We confirm that the numerical results of the CP(1) model without the θ term using this method are consistent with that of the O(3) model which is analyzed by the same method in the region β ≫1 and that obtained by the Monte Carlo simulation in a wider range of β . The numerical computation including the θ term is left for future challenges.
Rapid determination of global moment-tensor solutions
Sipkin, S.A.
1994-01-01
In an effort to improve data services, the National Earthquake Information Center has begun a program, in cooperation with the Incorporated Research Institutions for Seismology Data Management Center (IRIS DMC), to produce rapid estimates of the seismic moment tensor for most earthquakes with a bodywave magnitude of 5.8 or greater. An estimate of the moment tensor can usually be produced within 20 minutes of the arrival of the broadband P-waveform data from the IRIS DMC. The solutions do not vary significantly from the final solutions determined using the entire network. -from Author
Elasticity tensor and ultrasonic velocities for anisotropic cubic polycrystal
2008-01-01
The orientation distribution of crystallites in a polycrystal can be described by the orientation distribution function(ODF) . The ODF can be expanded under the Wigner D-bases. The expanded coefficients in the ODF are called the texture coefficients. In this paper,we use the Clebsch-Gordan expression to derive an explicit expression of the elasticity tensor for an anisotropic cubic polycrystal. The elasticity tensor contains three material constants and nine texture coefficients. In order to measure the nine texture coefficients by ultrasonic wave,we give relations between the nine texture coefficients and ultrasonic propagation velocities. We also give a numerical example to check the relations.
The Energy-Momentum Tensor for Cosmological Perturbations
Abramo, L R W; Mukhanov, V M
1997-01-01
We study the effective energy-momentum tensor (EMT) for cosmological perturbations and formulate the gravitational back-reaction problem in a gauge invariant manner. We analyze the explicit expressions for the EMT in the cases of scalar metric fluctuations and of gravitational waves and derive the resulting equations of state. The formalism is applied to investigate the back-reaction effects in chaotic inflation. We find that for long wavelength scalar and tensor perturbations, the effective energy density is negative and thus counteracts any pre-existing cosmological constant. For scalar perturbations during an epoch of inflation, the equation of state is de Sitter-like.
On the origin and the calculation of the force in electrostatic actuators
Jakoby, Bernhard
2016-07-01
This paper reviews fundamental ways to calculate the forces between charged electrodes as they appear, e.g., in electrostatic drives. In particular the consideration of the force acting on the surface charge layers on the electrodes, the principle of virtual displacement, and the Maxwell stress tensor are considered for two examples: a parallel plate capacitor and an electrostatic comb drive featuring interdigitated electrodes.
Kemp, Brandon A.; Sheppard, Cheyenne J.
2016-09-01
The momentum of light in media has been one of the most debated topics in physics over the past one hundred years. Originally a theoretical debate over the electrodynamics of moving media, practical applications have emerged over the past few decades due to interest in optical manipulation and nanotechnology. Resolution of the debate identifies a kinetic momentum as the momentum of the fields responsible for center of mass translations and a canonical momentum related to the coupled field and material system. The optical momentum resolution has been considered incomplete because it did not uniquely identify the full stress-energy-momentum (SEM) tensor of the field-kinetic subsystem. A consequence of this partial resolution is that the field-kinetic momentum could be described by three of the leading formulations found in the literature. The Abraham, Einstein-Laub, and Chu SEM tensors share the field-kinetic momentum, but their SEM tensors differ resulting in competing force densities. We can show now that the Abraham and Einstein-Laub formulations are invalid since their SEM tensors are not frame invariant, whereas the Chu SEM tensor satisfies relativistic principles as the field-kinetic formulation. However, a number of reports indicate that the force distribution in matter may not accurately represent experimental observations. In this correspondence, we show that the field-kinetic SEM tensor can be used along with the corresponding material subsystem to accurately predict experimental force and stress distributions. We model experimental examples from optical and static manipulation of particles and fluids.
崔洁; 江权; 冯夏庭; 李邵军; 高红; 李帅军
2016-01-01
Discontinuities constitute an integral part of rock mass and inherently affect its anisotropic deformation behavior. This work focuses on the equivalent elastic deformation of rock mass with multiple persistent joint sets. A new method based on the space geometric and mechanical properties of the modified crack tensor is proposed, providing an analytical solution for the equivalent elastic compliance tensor of rock mass. A series of experiments validate the capability of the compliance tensor to accurately represent the deformation of rock mass with multiple persistent joint sets, based on conditions set by the basic hypothesis. The spatially varying rules of the equivalent elastic parameters of rock mass with a single joint set are analyzed to reveal the universal law of the stratified rock mass.