Limit Algebras of Differential Forms in Non-Commutative Geometry
Indian Academy of Sciences (India)
S J Bhatt; A Inoue
2008-08-01
Given a C∗-normed algebra A which is either a Banach ∗-algebra or a Frechet ∗-algebra, we study the algebras ∞A and A obtained by taking respectively the projective limit and the inductive limit of Banach ∗-algebras obtained by completing the universal graded differential algebra ∗A of abstract non-commutative differential forms over A. Various quantized integrals on ∞A induced by a K-cycle on A are considered. The GNS-representation of ∞A defined by a d-dimensional non-commutative volume integral on a d+-summable K-cycle on A is realized as the representation induced by the left action of A on ∗A. This supplements the representation A on the space of forms discussed by Connes (Ch. VI.1, Prop. 5, p. 550 of [C]).
Gravity in Non-Commutative Geometry
Chamseddine, A H; Fröhlich, J
1993-01-01
We study general relativity in the framework of non-commutative differential geometry. In particular, we introduce a gravity action for a space-time which is the product of a four dimensional manifold by a two-point space. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.
Two Approaches to Non-Commutative Geometry
Kisil, V V
1997-01-01
Looking to the history of mathematics one could find out two outer approaches to Geometry. First one (algebraic) is due to Descartes and second one (group-theoretic)--to Klein. We will see that they are not rivalling but are tied (by Galois). We also examine their modern life as philosophies of non-commutative geometry. Connections between different objects (see keywords) are discussed. Keywords: Heisenberg group, Weyl commutation relation, Manin plain, quantum groups, SL(2, R), Hardy space, Bergman space, Segal-Bargmann space, Szeg"o projection, Bergman projection, Clifford analysis, Moebius transformations, functional calculus, Weyl calculus (quantization), Berezin quantization, Wick ordering, quantum mechanics.
An introduction to non-commutative differential geometry on quantum groups
Aschieri, Paolo
1993-01-01
We give a pedagogical introduction to the differential calculus on quantum groups by stressing at all stages the connection with the classical case ($q \\rightarrow 1$ limit). The Lie derivative and the contraction operator on forms and tensor fields are found. A new, explicit form of the Cartan--Maurer equations is presented. The example of a bicovariant differential calculus on the quantum group $GL_q(2)$ is given in detail. The softening of a quantum group is considered, and we introduce $q$-curvatures satisfying q-Bianchi identities, a basic ingredient for the construction of $q$-gravity and $q$-gauge theories.
Geometry of time-spaces non-commutative algebraic geometry, applied to quantum theory
Landau, Olav Arnfinn
2011-01-01
This is a monograph about non-commutative algebraic geometry, and its application to physics. The main mathematical inputs are the non-commutative deformation theory, moduli theory of representations of associative algebras, a new non-commutative theory o
Grand Unification in Non-Commutative Geometry
Chamseddine, A H; Fröhlich, J
1993-01-01
The formalism of non-commutative geometry of A. Connes is used to construct models in particle physics. The physical space-time is taken to be a product of a continuous four-manifold by a discrete set of points. The treatment of Connes is modified in such a way that the basic algebra is defined over the space of matrices, and the breaking mechanism is planted in the Dirac operator. This mechanism is then applied to three examples. In the first example the discrete space consists of two points, and the two algebras are taken respectively to be those of $2\\times 2$ and $1\\times 1$ matrices. With the Dirac operator containing the vacuum breaking $SU(2)\\times U(1)$ to $U(1)$, the model is shown to correspond to the standard model. In the second example the discrete space has three points, two of the algebras are identical and consist of $5\\times 5$ complex matrices, and the third algebra consists of functions. With an appropriate Dirac operator this model is almost identical to the minimal $SU(5)$ model of Georgi...
Non-Commutative Geometry, Categories and Quantum Physics
Bertozzini, Paolo; Lewkeeratiyutkul, Wicharn
2008-01-01
After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of A.Connes' non-commutative geometry: morphisms/categories of spectral triples, categorification of Gel'fand duality. We conclude with a summary of the expected applications of "categorical non-commutative geometry" to structural questions in relativistic quantum physics: (hyper)covariance, quantum space-time, (algebraic) quantum gravity.
Computational commutative and non-commutative algebraic geometry
Cojocaru, S; Ufnarovski, V
2005-01-01
This publication gives a good insight in the interplay between commutative and non-commutative algebraic geometry. The theoretical and computational aspects are the central theme in this study. The topic is looked at from different perspectives in over 20 lecture reports. It emphasizes the current trends in commutative and non-commutative algebraic geometry and algebra. The contributors to this publication present the most recent and state-of-the-art progresses which reflect the topic discussed in this publication. Both researchers and graduate students will find this book a good source of information on commutative and non-commutative algebraic geometry.
Modular Theory, Non-Commutative Geometry and Quantum Gravity
Directory of Open Access Journals (Sweden)
Wicharn Lewkeeratiyutkul
2010-08-01
Full Text Available This paper contains the first written exposition of some ideas (announced in a previous survey on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.
Kastler, Daniel
We describe in detail Alain Connes’ last presentation of the (classical level of the) standard model in noncommutative differential geometry, now free of the cumbersome adynamical fields which parasited the initial treatment. Accessorily, the theory is presented in a more transparent way by systematic use of the skew tensor-product structure, and of 2×2 matrices with 2×2 matrix-entries instead of the previous 4×4 matrices.
Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach
Bertram, Wolfgang
2013-01-01
By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via exercises using dynamical software (such as geogebra), hopefully accessible to a wide mathematical audience, from undergraduate students and high school teachers to researchers, proceeding in three steps: (1) experimental geometry, (2) algebra (linear algebr...
Wormhole inspired by non-commutative geometry
Energy Technology Data Exchange (ETDEWEB)
Rahaman, Farook, E-mail: rahaman@iucaa.ernet.in [Department of Mathematics, Jadavpur University, Kolkata 700032, West Bengal (India); Karmakar, Sreya, E-mail: sreya.karmakar@gmail.com [Department of Physics, Calcutta Institute of Engineering and Management, Kolkata 700040, West Bengal (India); Karar, Indrani, E-mail: indrani.karar08@gmail.com [Department of Mathematics, Saroj Mohan Institute of Technology, Guptipara, West Bengal (India); Ray, Saibal, E-mail: saibal@iucaa.ernet.in [Department of Physics, Government College of Engineering & Ceramic Technology, Kolkata 700010, West Bengal (India)
2015-06-30
In the present Letter we search for a new wormhole solution inspired by noncommutative geometry with the additional condition of allowing conformal Killing vectors (CKV). A special aspect of noncommutative geometry is that it replaces point-like structures of gravitational sources with smeared objects under Gaussian distribution. However, the purpose of this letter is to obtain wormhole solutions with noncommutative geometry as a background where we consider a point-like structure of gravitational object without smearing effect. It is found through this investigation that wormhole solutions exist in this Lorentzian distribution with viable physical properties.
Wormhole inspired by non-commutative geometry
Directory of Open Access Journals (Sweden)
Farook Rahaman
2015-06-01
Full Text Available In the present Letter we search for a new wormhole solution inspired by noncommutative geometry with the additional condition of allowing conformal Killing vectors (CKV. A special aspect of noncommutative geometry is that it replaces point-like structures of gravitational sources with smeared objects under Gaussian distribution. However, the purpose of this letter is to obtain wormhole solutions with noncommutative geometry as a background where we consider a point-like structure of gravitational object without smearing effect. It is found through this investigation that wormhole solutions exist in this Lorentzian distribution with viable physical properties.
A Partial Unification Model in Non-commutative Geometry
Hanlon, B E
1994-01-01
We consider the construction of $SU(2)_{L}\\otimes SU(2)_{R}\\otimes SU(4)$ partial unification models as an example of phenomenologically acceptable unification models in the absence of supersymmetry in non-commutative geometry. We exploit the Chamseddine, Felder and Fr\\"ohlich generalization of the Connes and Lott model building prescription. By introducing a bi-module structure and appropriate permutation symmetries we construct a model with triplet Higgs fields in the $SU(2)$ sectors and spontaneous breaking of $SU(4)$.
Non-commutative tachyon action and D-brane geometry
Herbst, Manfred; Kreuzer, M; Herbst, Manfred; Kling, Alexander; Kreuzer, Maximilian
2002-01-01
We analyse open string correlators in non-constant background fields, including the metric $g$, the antisymmetric $B$-field, and the gauge field $A$. Working with a derivative expansion for the background fields, but exact in their constant parts, we obtain a tachyonic on-shell condition for the inserted functions and extract the kinetic term for the tachyon action. The 3-point correlator yields a non-commutative tachyon potential. We also find a remarkable feature of the differential structure on the D-brane: Although the boundary metric $G$ plays an essential role in the action, the natural connection on the D-brane is the same as in closed string theory, i.e. it is compatible with the bulk metric and has torsion $H=dB$. This means, in particular, that the parallel transport on the brane is independent of the gauge field $A$.
Notes on "quantum gravity" and non-commutative geometry
Gracia-Bondia, Jose M
2010-01-01
I hesitated for a long time before giving shape to these notes, originally intended for preliminary reading by the attendees to the Summer School "New paths towards quantum gravity" (Holbaek Bay, Denmark, May 2008). At the end, I decide against just selling my mathematical wares, and for a survey, necessarily very selective, but taking a global phenomenological approach to its subject matter. After all, non-commutative geometry does not purport yet to solve the riddle of quantum gravity; it is more of an insurance policy against the probable failure of the other approaches. The plan is as follows: the introduction invites students to the fruitful doubts and conundrums besetting the application of even classical gravity. Next, the first experiments detecting quantum gravitational states inoculate us a healthy dose of skepticism on some of the current ideologies. In Section 3 we look at the action for general relativity as a consequence of gauge theory for quantum tensor fields. Section 4 briefly deals with the...
Quantum Analysis - Non-Commutative Differential and Integral Calculi
Suzuki, Masuo
1997-01-01
A new scheme of quantum analysis, namely a non-commutative calculus of operator derivatives and integrals is introduced. This treats differentiation of an operator-valued function with respect to the relevant operator in a Banach space. In this new scheme, operator derivatives are expressed in terms of the relevant operator and its inner derivation explicitly. Derivatives of hyperoperators are also defined. Some possible applications of the present calculus to quantum statistical physics are briefly discussed. Acknowledgements The author would like to thank Professor H. Araki, Professor K. Aomoto, Professor H. Hiai, Professor N. Obata and Dr. R.I. McLachlan for useful comments. Added in proof. Recently it has been proven that the quantum derivatives {dn f(A)/ dAn} are invariant for any choice of definitions of the differential df(A) satisfying the Leibniz rule and the linearity (M. Suzuki, J. Math. Phys.).->
Non-commutative geometry, the Bohm interpretation and the mind-matter relationship
Hiley, B. J.
2001-06-01
It is argued that in order to address the mind/matter relationship, we will have to radically change the conceptual structure normally assumed in physics. Rather than fields and/or particles-in-interaction described in the traditional Cartesian order based a local evolution in spacetime, we need to introduce a more general notion of process described by a non-commutative algebra. This will have radical implications for both for physical processes and for geometry. By showing how the Bohm interpretation of quantum mechanics can be understood within a non-commutative structure, we can give a much clearer meaning to the implicate order introduced by Bohm. It is through this implicate order that mind and matter can be seen as different aspects of the same general process.
A computational non-commutative geometry program for disordered topological insulators
Prodan, Emil
2017-01-01
This work presents a computational program based on the principles of non-commutative geometry and showcases several applications to topological insulators. Noncommutative geometry has been originally proposed by Jean Bellissard as a theoretical framework for the investigation of homogeneous condensed matter systems. Recently, this approach has been successfully applied to topological insulators, where it facilitated many rigorous results concerning the stability of the topological invariants against disorder. In the first part of the book the notion of a homogeneous material is introduced and the class of disordered crystals defined together with the classification table, which conjectures all topological phases from this class. The manuscript continues with a discussion of electrons’ dynamics in disordered crystals and the theory of topological invariants in the presence of strong disorder is briefly reviewed. It is shown how all this can be captured in the language of noncommutative geometry using the co...
Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry
Boyle, Latham; Farnsworth, Shane
2015-04-01
Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. Recently, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra U(1) B - L gauge symmetry, and a single extra complex scalar field σ, which is a singlet under SU(3)C × SU(2)L × U(1)Y , but has B - L = 2 . This field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction. We acknowledge support from an NSERC Discovery Grant.
Rethinking Connes’ Approach to the Standard Model of Particle Physics Via Non-Commutative Geometry
Farnsworth, Shane; Boyle, Latham
2015-02-01
Connes’ non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. In a previous paper, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra U{{(1)}B-L} gauge symmetry, and a single extra complex scalar field σ, which is a singlet under SU{{(3)}C}× SU{{(2)}L}× U{{(1)}Y}, but has B-L=2. This field has cosmological implications, and offers a new solution to the discrepancy between the observed Higgs mass and the NCG prediction.
Rethinking Connes' approach to the standard model of particle physics via non-commutative geometry
Farnsworth, Shane
2015-01-01
Connes' non-commutative geometry (NCG) is a generalization of Riemannian geometry that is particularly apt for expressing the standard model of particle physics coupled to Einstein gravity. In a previous paper, we suggested a reformulation of this framework that is: (i) simpler and more unified in its axioms, and (ii) allows the Lagrangian for the standard model of particle physics (coupled to Einstein gravity) to be specified in a way that is tighter and more explanatory than the traditional algorithm based on effective field theory. Here we explain how this same reformulation yields a new perspective on the symmetries of a given NCG. Applying this perspective to the NCG traditionally used to describe the standard model we find, instead, an extension of the standard model by an extra $U(1)_{B-L}$ gauge symmetry, and a single extra complex scalar field $\\sigma$, which is a singlet under $SU(3)_{C}\\times SU(2)_{L}\\times U(1)_{Y}$, but has $B-L=2$. This field has cosmological implications, and offers a new solu...
Twisted rings and moduli stacks of "fat" point modules in non-commutative projective geometry
Chan, Daniel
2010-01-01
The Hilbert scheme of point modules was introduced by Artin-Tate-Van den Bergh to study non-commutative graded algebras. The key tool is the construction of a map from the algebra to a twisted ring on this Hilbert scheme. In this paper, we study moduli stacks of more general "fat" point modules, and show that there is a similar map to a twisted ring associated to the stack. This is used to provide a sufficient criterion for a non-commutative projective surface to be birationally PI. It is hoped that such a criterion will be useful in understanding Mike Artin's conjecture on the birational classification of non-commutative surfaces.
Pati-Salam Unification from Non-commutative Geometry and the TeV-scale W_R boson
Aydemir, Ufuk; Sun, Chen; Takeuchi, Tatsu
2016-01-01
We analyze the compatibility of the unified left-right symmetric Pati-Salam models motivated by non-commutative geometry and the TeV scale right-handed W boson suggested by recent LHC data. We find that the unification/matching conditions place conflicting demands on the symmetry breaking scales and that generating the required W_R mass and coupling is non-trivial.
Non-commutative Differential Calculus and the Axial Anomaly in Abelian Lattice Gauge Theories
Fujiwara, T; Wu, K; Fujiwara, Takanori; Suzuki, Hiroshi; Wu, Ke
2000-01-01
The axial anomaly in lattice gauge theories has topological nature when the Dirac operator satisfies the Ginsparg-Wilson relation. We study the axial anomaly in Abelian gauge theories on an infinite hypercubic lattice by utilizing cohomological techniques. The crucial tool in our approach is the non-commutative differential calculus (NCDC) which validates the Leibniz rule of exterior derivatives on the lattice. The topological nature of the ``Chern character'' on the lattice becomes manifest with NCDC. Our result provides an algebraic proof of Lüscher's theorem for a four-dimensional lattice and its generalization to arbitrary dimensions.
Muon $g-2$ measurements and non-commutative geometry of quantum beams
Indian Academy of Sciences (India)
Y Srivastava; A Widom
2004-03-01
We discuss a completely quantum mechanical treatment of the measurement of the anomalous magnetic moment of the muon. A beam of muons move in a strong uniform magnetic field and a weak focusing electrostatic field. Errors in the classical beam analysis are exposed. In the Dirac quantum beam analysis, an important role is played by non-commutative muon beam coordinates leading to a discrepancy between the classical and quantum theories. We obtain a quantum limit to the accuracy achievable in BNL type experiments. Some implications of the quantum corrected data analysis for supersymmetry are briefly mentioned.
Non-commutative geometry as a realization of varying speed of light cosmology
Alexander, S H S; Alexander, Stephon H.S.; Magueijo, Jo\\~ao
2001-01-01
We examine the cosmological implications of space-time non-commutativity, discovering yet another realization of the varying speed of light model. Our starting point is the well-known fact that non-commutativity leads to deformed dispersion relations, relating energy and momentum, implying a frequency dependent speed of light. A Hot Big Bang Universe therefore experiences a higher speed of light as it gets hotter. We study the statistical physics of this "deformed radiation", recovering standard results at low temperatures, but a number of novelties at high temperatures: a deformed Planck's spectrum, a temperature dependent equation of state $w=p/\\rho$ (ranging from 1/3 to infinity), a new Stephan-Boltzmann law, and a new entropy relation. These new photon properties closely mimic those of phonons in crystals, hardly a surprising analogy. They combine to solve the horizon and flatness problems, explaining also the large entropy of the Universe. We also show how one would find a direct imprint of non-commutati...
Left-right symmetric gauge theory in non-commutative geometry on M{sub 4} x Z{sub N}
Energy Technology Data Exchange (ETDEWEB)
Okumura, Yoshitaka [Chubu Univ., Kasugai, Aichi (Japan)
1995-10-01
The left-right symmetric gauge model (LRSM) is reconstructed using the previously proposed formalism based on the non-commutative differential geometry extended on the discrete space M{sub 4} x Z{sub N}. This formalism is so flexible and applicable that not only the standard model but also the SU(5) grand unified model have already been reformulated in this formalism, which presents many attractive points such as the unified picture of the gauge field and Higgs field as the generalized connection in non-commutative geometry. LRSM is still alive as a model with the intermediate symmetry of the spontaneously broken SO(10) grand unified theory (GUT). Six sheets are prepared for LRSM (N=6), one is for SU(3){sub c} color symmetry and the rest of five are for SU(2){sub L} x SU(2){sub R} x U(1) symmetry. We can achieve the reformulation of LRSM with the quite different configurations of Higgs particles from the ordinary one. Namely, the left-right symmetric gauge groups are broken owing to two (2, 1) and two (1, 2) doublet Higgs fields with hypercharge 1, one (2, 2{sup *}) Higgs field, and one (1, 3) Higgs field with hypercharge -2. The fermion sectors are nicely incorporated so that the seesaw mechanism works well to make the right-handed neutrino super heavy and the left-handed neutrino super light. (author).
Guggenheimer, Heinrich W
1977-01-01
This is a text of local differential geometry considered as an application of advanced calculus and linear algebra. The discussion is designed for advanced undergraduate or beginning graduate study, and presumes of readers only a fair knowledge of matrix algebra and of advanced calculus of functions of several real variables. The author, who is a Professor of Mathematics at the Polytechnic Institute of New York, begins with a discussion of plane geometry and then treats the local theory of Lie groups and transformation groups, solid differential geometry, and Riemannian geometry, leading to a
Graustein, William C
2006-01-01
This first course in differential geometry presents the fundamentals of the metric differential geometry of curves and surfaces in a Euclidean space of three dimensions. Written by an outstanding teacher and mathematician, it explains the material in the most effective way, using vector notation and technique. It also provides an introduction to the study of Riemannian geometry.Suitable for advanced undergraduates and graduate students, the text presupposes a knowledge of calculus. The first nine chapters focus on the theory, treating the basic properties of curves and surfaces, the mapping of
Kreyszig, Erwin
1991-01-01
An introductory textbook on the differential geometry of curves and surfaces in three-dimensional Euclidean space, presented in its simplest, most essential form, but with many explanatory details, figures and examples, and in a manner that conveys the theoretical and practical importance of the different concepts, methods and results involved. With problems at the end of each section, and solutions listed at the end of the book. Includes 99 illustrations.
Ciarlet, Philippe G
2007-01-01
This book gives the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory in a selfcontained and accessible manner. Although the field is often considered a classical one, it has recently been rejuvenated, thanks to the manifold applications where it plays an essential role. The book presents some important applications to shells, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and
Stoker, J J
2011-01-01
This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis.
Non-commuting variations in mathematics and physics a survey
Preston, Serge
2016-01-01
This text presents and studies the method of so –called noncommuting variations in Variational Calculus. This method was pioneered by Vito Volterra who noticed that the conventional Euler-Lagrange (EL-) equations are not applicable in Non-Holonomic Mechanics and suggested to modify the basic rule used in Variational Calculus. This book presents a survey of Variational Calculus with non-commutative variations and shows that most basic properties of conventional Euler-Lagrange Equations are, with some modifications, preserved for EL-equations with K-twisted (defined by K)-variations. Most of the book can be understood by readers without strong mathematical preparation (some knowledge of Differential Geometry is necessary). In order to make the text more accessible the definitions and several necessary results in Geometry are presented separately in Appendices I and II Furthermore in Appendix III a short presentation of the Noether Theorem describing the relation between the symmetries of the differential equa...
Bicovariant differential geometry of the quantum group GL$_{q}$(3)
Aschieri, Paolo; Aschieri, Paolo; Castellani, Leonardo
1992-01-01
We construct a bicovariant differential calculus on the quantum group $GL_q(3)$, and discuss its restriction to $[SU(3) \\otimes U(1)]_q$. The $q$-algebra of Lie derivatives is found, as well as the Cartan-Maurer equations. All the quantities characterizing the non-commutative geometry of $GL_q(3)$ are given explicitly.
Bär, Christian; Schwarz, Matthias
2012-01-01
This volume contains a collection of well-written surveys provided by experts in Global Differential Geometry to give an overview over recent developments in Riemannian Geometry, Geometric Analysis and Symplectic Geometry. The papers are written for graduate students and researchers with a general interest in geometry, who want to get acquainted with the current trends in these central fields of modern mathematics.
Optimization of polynomials in non-commuting variables
Burgdorf, Sabine; Povh, Janez
2016-01-01
This book presents recent results on positivity and optimization of polynomials in non-commuting variables. Researchers in non-commutative algebraic geometry, control theory, system engineering, optimization, quantum physics and information science will find the unified notation and mixture of algebraic geometry and mathematical programming useful. Theoretical results are matched with algorithmic considerations; several examples and information on how to use NCSOStools open source package to obtain the results provided. Results are presented on detecting the eigenvalue and trace positivity of polynomials in non-commuting variables using Newton chip method and Newton cyclic chip method, relaxations for constrained and unconstrained optimization problems, semidefinite programming formulations of the relaxations and finite convergence of the hierarchies of these relaxations, and the practical efficiency of algorithms.
Non-commutative Nash inequalities
Energy Technology Data Exchange (ETDEWEB)
Kastoryano, Michael [NBIA, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen (Denmark); Temme, Kristan [Institute for Quantum Information and Matter, California Institute of Technology, Pasadena California 91125 (United States)
2016-01-15
A set of functional inequalities—called Nash inequalities—are introduced and analyzed in the context of quantum Markov process mixing. The basic theory of Nash inequalities is extended to the setting of non-commutative L{sub p} spaces, where their relationship to Poincaré and log-Sobolev inequalities is fleshed out. We prove Nash inequalities for a number of unital reversible semigroups.
Digital Differential Geometry Processing
Institute of Scientific and Technical Information of China (English)
Xin-Guo Liu; Hu-Jun Bao; Qun-Sheng Peng
2006-01-01
The theory and methods of digital geometry processing has been a hot research area in computer graphics, as geometric models serves as the core data for 3D graphics applications. The purpose of this paper is to introduce some recent advances in digital geometry processing, particularly mesh fairing, surface parameterization and mesh editing, that heavily use differential geometry quantities. Some related concepts from differential geometry, such as normal, curvature, gradient,Laplacian and their counterparts on digital geometry are also reviewed for understanding the strength and weakness of various digital geometry processing methods.
A non-commutative framework for topological insulators
Bourne, C.; Carey, A. L.; Rennie, A.
2016-04-01
We study topological insulators, regarded as physical systems giving rise to topological invariants determined by symmetries both linear and anti-linear. Our perspective is that of non-commutative index theory of operator algebras. In particular, we formulate the index problems using Kasparov theory, both complex and real. We show that the periodic table of topological insulators and superconductors can be realized as a real or complex index pairing of a Kasparov module capturing internal symmetries of the Hamiltonian with a spectral triple encoding the geometry of the sample’s (possibly non-commutative) Brillouin zone.
Non-commutative black holes in D dimensions
Klimcík, C; Pompos, A
1994-01-01
Recently introduced classical theory of gravity in non-commutative geometry is studied. The most general (four parametric) family of D dibensional static spherically symmetric spacetimes is identified and its properties are studied in detail. For wide class of the choices of parameters, the corresponding spacetimes have the structure of asymptotically flat black holes with a smooth event horizon hiding the curvature singularity. A specific attention is devoted to the behavior of components of the metric in non-commutative direction, which are interpreted as the black hole hair.
Non-topological non-commutativity in string theory
Guttenberg, Sebastian; Kreuzer, Maximilian; Rashkov, Radoslav
2007-01-01
Quantization of coordinates leads to the non-commutative product of deformation quantization, but is also at the roots of string theory, for which space-time coordinates become the dynamical fields of a two-dimensional conformal quantum field theory. Appositely, open string diagrams provided the inspiration for Kontsevich's solution of the long-standing problem of quantization of Poisson geometry by virtue of his formality theorem. In the context of D-brane physics non-commutativity is not limited, however, to the topolocial sector. We show that non-commutative effective actions still make sense when associativity is lost and establish a generalized Connes-Flato-Sternheimer condition through second order in a derivative expansion. The measure in general curved backgrounds is naturally provided by the Born--Infeld action and reduces to the symplectic measure in the topological limit, but remains non-singular even for degenerate Poisson structures. Analogous superspace deformations by RR--fields are also discus...
Non commutative quantum spacetime with topological vortex states, and dark matter in the universe
Patwardhan, A
2003-01-01
Non commutative geometry is creating new possibilities for physics. Quantum spacetime geometry and post inflationary models of the universe with matter creation have an enormous range of scales of time, distance and energy in between. There is a variety of physics possible till the nucleosynthesis epoch is reached. The use of topology and non commutative geometry in cosmology is a recent approach. This paper considers the possibility of topological solutions of a vortex kind given by non commutative structures. These are interpreted as dark matter, with the grand unified Yang-Mills field theory energy scale used to describe its properties. The relation of the model with other existing theories is discussed.
Elementary differential geometry
Pressley, Andrew
2001-01-01
Curves and surfaces are objects that everyone can see, and many of the questions that can be asked about them are natural and easily understood Differential geometry is concerned with the precise mathematical formulation of some of these questions, and with trying to answer them using calculus techniques It is a subject that contains some of the most beautiful and profound results in mathematics yet many of these are accessible to higher-level undergraduates Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces while keeping the prerequisites to an absolute minimum Nothing more than first courses in linear algebra and multivariate calculus are required, and the most direct and straightforward approach is used at all times Numerous diagrams illustrate both the ideas in the text and the examples of curves and surfaces discussed there The book will provide an invaluable resource to all those taking a first course in differential geometry, for their lecture...
Geometry of differential equations
Khovanskiĭ, A; Vassiliev, V
1998-01-01
This volume contains articles written by V. I. Arnold's colleagues on the occasion of his 60th birthday. The articles are mostly devoted to various aspects of geometry of differential equations and relations to global analysis and Hamiltonian mechanics.
On Non-commutative Geodesic Motion
Ulhoa, S C; Santos, A F
2013-01-01
In this work we study the geodesic motion on a noncommutative space-time. As a result we find a non-commutative geodesic equation and then we derive corrections of the deviation angle per revolution in terms of the non-commutative parameter when we specify the problem of Mercury's perihelion. In this way, we estimate the noncommutative parameter based in experimental data.
On non-commutative geodesic motion
Ulhoa, S. C.; Amorim, R. G. G.; Santos, A. F.
2014-07-01
In this work we study the geodesic motion on a noncommutative space-time. As a result we find a non-commutative geodesic equation and then we derive corrections of the deviation angle per revolution in terms of the non-commutative parameter when we specify the problem of Mercury's perihelion. In this way, we estimate the noncommutative parameter based in experimental data.
Zheng, Fangyang
2002-01-01
The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds. The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classifi...
Differential geometry and thermodynamics
Quevedo, H
2003-01-01
In this work we present the first steps of a new approach to the study of thermodynamics in the context of differential geometry. We introduce a fundamental differential 1-form and a metric on a pseudo-Euclidean manifold coordinatized by means of the extensive thermodynamic variables. The study of the connection and the curvature of these objects is initialized in this work by using Cartan structure equations. (Author)
Multi linear formulation of differential geometry and matrix regularizations
Arnlind, Joakim; Huisken, Gerhard
2010-01-01
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations. For matrix analogues of embedded surfaces we define discrete curvatures and Euler characteristics, and a non-commutative Gauss--Bonnet theorem is shown to follow. We derive simple expressions for the discrete Gauss curvature in terms of matrices representing the embedding coordinates, and a large class of explicit examples is provided. Furthermore, we illustrate the fact that techniques from differential geometry can carry over to matrix analogues by proving that a bound on the discrete Gauss curvature implies a bound on the eigenvalues of the discrete Laplace operator.
Non-Commutative Geometry from Strings
Chu, Chong-Sun
2005-01-01
Comment: 26 pages, no figures. To appear in the Elsevier Encyclopedia of Mathematical Physics. This web version has a more comprehensive list of references. Comments and corrections welcome. v2: Typos corrected, references and some comments added. v3: 27 pages. more references and Comments added. v4: references added. Final version
Non-commutative Complex Projective Spaces and the Standard Model
Dolan, Brian P
2003-01-01
The standard model fermion spectrum, including a right handed neutrino, can be obtained as a zero-mode of the Dirac operator on a space which is the product of complex projective spaces of complex dimension two and three. The construction requires the introduction of topologically non-trivial background gauge fields. By borrowing from ideas in Connes' non-commutative geometry and making the complex spaces `fuzzy' a matrix approximation to the fuzzy space allows for three generations to emerge...
Chiral bosonization for non-commutative fields
Das, A; Méndez, F; López-Sarrion, J; Das, Ashok; Gamboa, Jorge; M\\'endez, Fernando; L\\'opez-Sarri\\'on, Justo
2004-01-01
A model of chiral bosons on a non-commutative field space is constructed and new generalized bosonization (fermionization) rules for these fields are given. The conformal structure of the theory is characterized by a level of the Kac-Moody algebra equal to $(1+ \\theta^2)$ where $\\theta$ is the non-commutativity parameter and chiral bosons living in a non-commutative fields space are described by a rational conformal field theory with the central charge of the Virasoro algebra equal to 1. The non-commutative chiral bosons are shown to correspond to a free fermion moving with a speed equal to $ c^{\\prime} = c \\sqrt{1+\\theta^2} $ where $c$ is the speed of light. Lorentz invariance remains intact if $c$ is rescaled by $c \\to c^{\\prime}$. The dispersion relation for bosons and fermions, in this case, is given by $\\omega = c^{\\prime} | k|$.
Covariant non-commutative space–time
Directory of Open Access Journals (Sweden)
Jonathan J. Heckman
2015-05-01
Full Text Available We introduce a covariant non-commutative deformation of 3+1-dimensional conformal field theory. The deformation introduces a short-distance scale ℓp, and thus breaks scale invariance, but preserves all space–time isometries. The non-commutative algebra is defined on space–times with non-zero constant curvature, i.e. dS4 or AdS4. The construction makes essential use of the representation of CFT tensor operators as polynomials in an auxiliary polarization tensor. The polarization tensor takes active part in the non-commutative algebra, which for dS4 takes the form of so(5,1, while for AdS4 it assembles into so(4,2. The structure of the non-commutative correlation functions hints that the deformed theory contains gravitational interactions and a Regge-like trajectory of higher spin excitations.
Non-commutative standard model: model building
Chaichian, Masud; Presnajder, P
2003-01-01
A non-commutative version of the usual electro-weak theory is constructed. We discuss how to overcome the two major problems: (1) although we can have non-commutative U(n) (which we denote by U sub * (n)) gauge theory we cannot have non-commutative SU(n) and (2) the charges in non-commutative QED are quantized to just 0,+-1. We show how the latter problem with charge quantization, as well as with the gauge group, can be resolved by taking the U sub * (3) x U sub * (2) x U sub * (1) gauge group and reducing the extra U(1) factors in an appropriate way. Then we proceed with building the non-commutative version of the standard model by specifying the proper representations for the entire particle content of the theory, the gauge bosons, the fermions and Higgs. We also present the full action for the non-commutative standard model (NCSM). In addition, among several peculiar features of our model, we address the inherentCP violation and new neutrino interactions. (orig.)
Phase space quantization, non-commutativity and the gravitational field
Chatzistavrakidis, Athanasios
2014-01-01
In this paper we study the structure of the phase space in non-commutative geometry in the presence of a non-trivial frame. Our basic assumptions are that the underlying space is a symplectic and parallelizable manifold. Furthermore, we assume the validity of the Leibniz rule and the Jacobi identities. We consider non-commutative spaces due to the quantization of the symplectic structure and determine the momentum operators that guarantee a set of canonical commutation relations, appropriately extended to include the non-trivial frame. We stress the important role of left vs. right acting operators and of symplectic duality. This enables us to write down the form of the full phase space algebra on these non-commutative spaces, both in the non-compact and in the compact case. We test our results against the class of 4D and 6D symplectic nilmanifolds, thus presenting a large set of non-trivial examples that realize the general formalism.
Non-topological non-commutativity in string theory
Energy Technology Data Exchange (ETDEWEB)
Guttenberg, S. [NCSR Demokritos, INP, Patriarchou Gregoriou and Neapoleos Str., 15310 Agia Paraskevi Attikis (Greece); Herbst, M. [CERN, 1211 Geneva 23 (Switzerland); Kreuzer, M. [Institute for Theoretical Physics, TU Wien, Wiedner Hauptstr. 8-10, 1040 Vienna (Austria); Rashkov, R. [Erwin Schroedinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090 Vienna (Austria)
2008-04-15
Quantization of coordinates leads to the non-commutative product of deformation quantization, but is also at the roots of string theory, for which space-time coordinates become the dynamical fields of a two-dimensional conformal quantum field theory. Appositely, open string diagrams provided the inspiration for Kontsevich's solution of the long-standing problem of quantization of Poisson geometry by virtue of his formality theorem. In the context of D-brane physics non-commutativity is not limited, however, to the topological sector. We show that non-commutative effective actions still make sense when associativity is lost and establish a generalized Connes-Flato-Sternheimer condition through second order in a derivative expansion. The measure in general curved backgrounds is naturally provided by the Born-Infeld action and reduces to the symplectic measure in the topological limit, but remains non-singular even for degenerate Poisson structures. Analogous superspace deformations by RR-fields are also discussed. (Abstract Copyright [2008], Wiley Periodicals, Inc.)
Non-commutativity in polar coordinates
Energy Technology Data Exchange (ETDEWEB)
Edwards, James P. [Universidad Michoacana de San Nicolas de Hidalgo, Ciudad Universitaria, Instituto de Fisica y Matematicas, Morelia, Michoacan (Mexico)
2017-05-15
We reconsider the fundamental commutation relations for non-commutative R{sup 2} described in polar coordinates with non-commutativity parameter θ. Previous analysis found that the natural transition from Cartesian coordinates to the traditional polar system led to a representation of [r, φ] as an everywhere diverging series. In this article we compute the Borel resummation of this series, showing that it can subsequently be extended throughout parameter space and hence provide an interpretation of this commutator. Our analysis provides a complete solution for arbitrary r and θ that reproduces the earlier calculations at lowest order and benefits from being generally applicable to problems in a two-dimensional non-commutative space. We compare our results to previous literature in the (pseudo-)commuting limit, finding a surprising spatial dependence for the coordinate commutator when θ >> r{sup 2}. Finally, we raise some questions for future study in light of this progress. (orig.)
Non-commutative multi-dimensional cosmology
Khosravi, N; Sepangi, H R
2006-01-01
A non-commutative multi-dimensional cosmological model is introduced and used to address the issues of compactification and stabilization of extra dimensions and the cosmological constant problem. We show that in such a scenario these problems find natural solutions in a universe described by an increasing time parameter.
Chiral bosonization for non-commutative fields
Energy Technology Data Exchange (ETDEWEB)
Das, Ashok [Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627-0171 (United States)]. E-mail: das@pas.rochester.edu; Gamboa, Jorge [Departamento de Fisica, Universidad de Santiago de Chile, Casilla 307, Santiago 2 (Chile); Mendez, Fernando [INFN, Laboratorio Nazionali del Gran Sasso, SS, 17bis, 67010 Asergi, L' Aquila (Italy); Lopez-Sarrion, Justo [Departamento de Fisica Teorica, Universidad de Zaragoza, Zaragoza 50009 (Spain)
2004-05-01
A model of chiral bosons on a non-commutative field space is constructed and new generalized bosonization (fermionization) rules for these fields are given. The conformal structure of the theory is characterized by a level of the Kac-Moody algebra equal to (1+{theta}{sup 2}) where {theta} is the non-commutativity parameter and chiral bosons living in a non-commutative fields space are described by a rational conformal field theory with the central charge of the Virasoro algebra equal to 1. The non-commutative chiral bosons are shown to correspond to a free fermion moving with a speed equal to c' = c(1+{theta}{sup 2}){sup 1/2} where c is the speed of light. Lorentz invariance remains intact if c is rescaled by c{yields}c'. The dispersion relation for bosons and fermions, in this case, is given by {omega} = c' vertical bar k vertical bar. (author)
Discrete Symmetries In Lorentz-Invariant Non-Commutative QED
Morita, K
2003-01-01
It is pointed out that the usual $\\theta$-algebra assumed for non-commuting coordinates is not $P$- and $T$-invariant, unless one {\\it formally} transforms the non-commutativity parameter $\\theta^{\\mu\
Non-commutative time-frequency tomography
Man'ko, V I
1999-01-01
The characterization of non-stationary signals requires joint time and frequency information. However, time (t) and frequency (omega) being non-commuting variables there cannot be a joint probability density in the (t,omega) plane and the time-frequency distributions, that have been proposed, have difficult interpretation problems arising from negative or complex values and spurious components. As an alternative we propose to obtain time-frequency information by looking at the marginal distributions along rotated directions in the (t,omega) plane. The rigorous probability interpretation of the marginal distributions avoids all interpretation ambiguities. Applications to signal analysis and signal detection are discussed as well as an extension of the method to other pairs of non-commuting variables.
Non-commutativity in polar coordinates
Edwards, James P
2016-01-01
We reconsider the fundamental commutation relations for non-commutative $\\mathbb{R}^{2}$ described in polar coordinates with non-commutativity parameter $\\theta$. Previous analysis found that the natural transition from Cartesian coordinates to polars led to a representation of $\\left[\\hat{r}, \\hat{\\varphi}\\right]$ as an everywhere diverging series. We compute the Borel resummation of this series, showing that it can subsequently be extended throughout parameter space and hence provide an interpretation of this commutator. Our analysis provides a complete solution for arbitrary $r$ and $\\theta$ that reproduces the earlier calculations at lowest order. We compare our results to previous literature in the (pseudo-)commuting limit, finding a surprising spatial dependence for the coordinate commutator when $\\theta \\gg r^{2}$. We raise some questions for future study in light of this progress.
Symposium on Differential Geometry and Differential Equations
Berger, Marcel; Bryant, Robert
1987-01-01
The DD6 Symposium was, like its predecessors DD1 to DD5 both a research symposium and a summer seminar and concentrated on differential geometry. This volume contains a selection of the invited papers and some additional contributions. They cover recent advances and principal trends in current research in differential geometry.
An introduction to differential geometry
Willmore, T J
2012-01-01
This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.
Advances in discrete differential geometry
2016-01-01
This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, ...
Discretising differential geometry via a new product on the space of chains
de Beauce, V; Beauce, Vivien de; Sen, Siddhartha
2006-01-01
A discretisation of differential geometry using the Whitney forms of algebraic topology is consistently extended via the introduction of a pairing on the space of chains. This pairing of chains enables us to give a definition of the discrete interior product and thus provides a solution to a notorious puzzle in discretisation techniques. Further prescriptions are made to introduce metric data, as a discrete substitute for the continuum vielbein, or Cartan formulation. The original topological data of the de Rham complex is then recovered as a discrete version of the Pontryagin class, a sketch of a few examples of the technique is also provided. A map of discrete differential geometry into the non-commutative geometry of graphs is constructed which shows in a precise way the difference between them.
Shadow of a charged rotating non-commutative black hole
Energy Technology Data Exchange (ETDEWEB)
Sharif, M. [University of the Punjab, Department of Mathematics, Lahore (Pakistan); Pakistan Academy of Sciences, Islamabad (Pakistan); Iftikhar, Sehrish [University of the Punjab, Department of Mathematics, Lahore (Pakistan)
2016-11-15
This paper investigates the shadow of a charged rotating non-commutative black hole. For this purpose, we first formulate the null geodesics and study the effects of a non-commutative charge on the photon orbit. We then explore the effect of spin, angle of inclination as well as non-commutative charge on the silhouette of the shadow. It is found that shape of the shadow deviates from the circle with the decrease in the non-commutative charge. We also discuss observable quantities to study the deformation and distortion in the shadow cast by the black hole which decreases for small values of a non-commutative charge. Finally, we study the shadows in the presence of plasma. We conclude that the non-commutativity has a great impact on the black hole shadow. (orig.)
Exploring the thermodynamics of non-commutative scalar fields
Brito, Francisco A
2015-01-01
We study the thermodynamic properties of the Bose-Einstein condensate (BEC) in the context of the quantum field theory with non-commutative target space. Our main goal is to investigate in which temperature and/or energy regimes the non-commutativity can characterize some influence in the BEC properties described by a relativistic massive non-commutative boson gas. The non-commutative parameters play a key role in the modified dispersion relations of the non-commutative fields, leading to a new phenomenology. We have obtained the condensate fraction, internal energy, pressure and specific heat of the system and taken ultra-relativistic (UR) and non-relativistic limits (NR). The non-commutative effects in the thermodynamic properties of the system are discussed. We found that there appear interesting signatures around the critical temperature.
Shadow of a Charged Rotating Non-Commutative Black Hole
Sharif, M
2016-01-01
This paper investigates the shadow of a charged rotating non-commutative black hole. For this purpose, we first formulate the null geodesics and study the effects of non-commutative charge on the photon orbit. We then explore the effect of spin, angle of inclination as well as non-commutative charge on the silhouette of the shadow. It is found that shape of the shadow deviates from the circle with the decrease in the non-commutative charge. We also discuss observable quantities to study the deformation and distortion in the shadow cast by the black hole which decreases for small values of non-commutative charge. Finally, we study the shadows in the presence of plasma. We conclude that the non-commutativity has a great impact on the black hole shadow.
Axiomatic differential geometry II-2 - differential forms
Nishimura, Hirokazu
2013-01-01
We refurbish our axiomatics of differential geometry introduced in [Mathematics for Applications,, 1 (2012), 171-182]. Then the notion of Euclideaness can naturally be formulated. The principal objective in this paper is to present an adaptation of our theory of differential forms developed in [International Journal of Pure and Applied Mathematics, 64 (2010), 85-102] to our present axiomatic framework.
Axiomatic Differential Geometry Ⅱ-2: Differential Forms
Nishimura, Hirokazu
2013-01-01
We refurbish our axiomatics of differential geometry introduced in [arXiv 1203.3911]. Then the notion of Euclideaness can naturally be formulated. The principal objective in this paper is to present an adaptation of our theory of differential forms developed in [International Journal of Pure and Applied Mathematics, 64 (2010), 85-102] to our present axiomatic framework.
Non-commutative covering spaces and their symmetries
DEFF Research Database (Denmark)
Canlubo, Clarisson
dened and its corresponding Galois theory. Using this and basic concepts from algebraic geometryand spectral theory, we will give a full description of the general structure of non-centralcoverings. Examples of coverings of the rational and irrational non-commutative tori will alsobe studied. Using......-commutative covering space using Galois theory of Hopfalgebroids. We will look at basic properties of classical covering spaces that generalize to thenon-commutative framework. Afterwards, we will explore a series of examples. We will startwith coverings of a point and central coverings of commutative spaces and see...... how these areclosely tied up. Coupled Hopf algebras will be presented to give a general description of coveringsof a point. We will give a complete description of the geometry of the central coverings ofcommutative spaces using the coverings of a point. A topologized version of Hopf categories willbe...
A review of non-commutative gauge theories
Indian Academy of Sciences (India)
N G Deshpande
2003-02-01
Construction of quantum ﬁeld theory based on operators that are functions of non-commutative space-time operators is reviewed. Examples of 4 theory and QED are then discussed. Problems of extending the theories to () gauge theories and arbitrary charges in QED are considered. Construction of standard model on non-commutative space is then brieﬂy discussed. The phenomenological implications are then considered. Limits on non-commutativity from atomic physics as well as accelerator experiments are presented.
The topological AC effect on non-commutative phase space
Energy Technology Data Exchange (ETDEWEB)
Li, Kang [Hangzhou Teachers College, Department of Physics, Hangzhou (China); The Abdus Salam International Center for Theoretical Physics, Trieste (Italy); Wang, Jianhua [Shaanxi University of Technology, Department of Physics, Hanzhong (China); The Abdus Salam International Center for Theoretical Physics, Trieste (Italy)
2007-05-15
The Aharonov-Casher (AC) effect in non-commutative (NC) quantum mechanics is studied. Instead of using the star product method, we use a generalization of Bopp's shift method. After solving the Dirac equations both on non-commutative space and non-commutative phase space by the new method, we obtain corrections to the AC phase on NC space and NC phase space, respectively. (orig.)
Classical mechanics in non-commutative phase space
Institute of Scientific and Technical Information of China (English)
WEI Gao-Feng; LONG Chao-Yun; LONG Zheng-Wen; QIN Shui-Jie; Fu Qiang
2008-01-01
In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space.The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity.First,new Poisson brackets have been defined in non-commutative phase space.They contain corrections due to the non-commutativity of coordinates and momenta.On the basis of this new Poisson brackets,a new modified second law of Newton has been obtained.For two cases,the free particle and the harmonic oscillator,the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys.Rev.D,2005,72:025010).The consistency between both methods is demonstrated.It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space.but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.
Differential geometry meets the cell.
Marshall, Wallace F
2013-07-18
A new study by Terasaki et al. highlights the role of physical forces in biological form by showing that connections between stacked endoplasmic reticulum cisternae have a shape well known in classical differential geometry, the helicoid, and that this shape is a predictable consequence of membrane physics.
Parabosonic string and space-time non-commutativity
Energy Technology Data Exchange (ETDEWEB)
Seridi, M. A.; Belaloui, N. [Laboratoire de Physique Mathematique et Subatomique, Universite Mentouri Constantine (Algeria)
2012-06-27
We investigate the para-quantum extension of the bosonic strings in a non-commutative space-time. We calculate the trilinear relations between the mass-center variables and the modes and we derive the Virasoro algebra where a new anomaly term due to the non-commutativity is obtained.
Parity-dependent non-commutative quantum mechanics
Chung, Won Sang
2017-01-01
In this paper, we consider the non-commutative quantum mechanics (NCQM) with parity (or space reflection) in two dimensions. Using the parity operators Ri, we construct the deformed Heisenberg algebra with parity in the non-commutative plane. We use this algebra to discuss the isotropic harmonic Hamiltonian with parity.
High-Energy Scattering in Non-Commutative Field Theory
Kumar, J; Kumar, Jason; Rajaraman, Arvind
2005-01-01
We analyze high energy scattering for non-commutative field theories using the dual gravity description. We find that the Froissart-Martin bound still holds, but that cross-sections stretch in the non-commutative directions in a way dependent on the infrared cutoff. This puzzling behavior suggests new aspects of UV/IR mixing.
Gaussian processes in non-commutative probability theory
Guţǎ, M.I.
2002-01-01
The generalisation of the notion of Gaussian processes from probability theory is investigated in the context of non-commutative probability theory. A non-commutative Gaussian process is viewed as a linear map from an infinite dimensional (real) Hilbert space into an algebra with involution and a po
Brownian Motion in Non-Commutative Super-Yang-Mills
Fischler, Willy; Garcia, Walter Tangarife
2012-01-01
Using the gauge/gravity correspondence, we study the dynamics of a heavy quark in strongly-coupled non-commutative Super-Yang-Mills at finite temperature. We propose a Langevin equation that accounts for the effects of non-commutativity and resembles the structure of Brownian motion in the presence of a magnetic field. As expected, fluctuations along non-commutative directions are generically correlated. Our results show that the viscosity of the plasma is smaller than the commutative case and that the diffusion properties of the quark are unaffected by non-commutativity. Finally, we compute the random force autocorrelator and verify that the fluctuation-dissipation theorem holds in the presence of non-commutativity.
Inflation on a non-commutative space–time
Directory of Open Access Journals (Sweden)
Xavier Calmet
2015-07-01
Full Text Available We study inflation on a non-commutative space–time within the framework of enveloping algebra approach which allows for a consistent formulation of general relativity and of the standard model of particle physics. We show that within this framework, the effects of the non-commutativity of spacetime are very subtle. The dominant effect comes from contributions to the process of structure formation. We describe the bound relevant to this class of non-commutative theories and derive the tightest bound to date of the value of the non-commutative scale within this framework. Assuming that inflation took place, we get a model independent bound on the scale of space–time non-commutativity of the order of 19 TeV.
Inflation on a non-commutative space–time
Energy Technology Data Exchange (ETDEWEB)
Calmet, Xavier, E-mail: x.calmet@sussex.ac.uk; Fritz, Christopher, E-mail: c.fritz@sussex.ac.uk
2015-07-30
We study inflation on a non-commutative space–time within the framework of enveloping algebra approach which allows for a consistent formulation of general relativity and of the standard model of particle physics. We show that within this framework, the effects of the non-commutativity of spacetime are very subtle. The dominant effect comes from contributions to the process of structure formation. We describe the bound relevant to this class of non-commutative theories and derive the tightest bound to date of the value of the non-commutative scale within this framework. Assuming that inflation took place, we get a model independent bound on the scale of space–time non-commutativity of the order of 19 TeV.
Lectures on classical differential geometry
Struik, Dirk J
1988-01-01
Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student.Writ
Differential geometry and mathematical physics
Rudolph, Gerd
Starting from an undergraduate level, this book systematically develops the basics of • Calculus on manifolds, vector bundles, vector fields and differential forms, • Lie groups and Lie group actions, • Linear symplectic algebra and symplectic geometry, • Hamiltonian systems, symmetries and reduction, integrable systems and Hamilton-Jacobi theory. The topics listed under the first item are relevant for virtually all areas of mathematical physics. The second and third items constitute the link between abstract calculus and the theory of Hamiltonian systems. The last item provides an introduction to various aspects of this theory, including Morse families, the Maslov class and caustics. The book guides the reader from elementary differential geometry to advanced topics in the theory of Hamiltonian systems with the aim of making current research literature accessible. The style is that of a mathematical textbook,with full proofs given in the text or as exercises. The material is illustrated by numerous d...
Non-Commutative Mechanics in Mathematical & in Condensed Matter Physics
Directory of Open Access Journals (Sweden)
Peter A. Horváthy
2006-12-01
Full Text Available Non-commutative structures were introduced, independently and around the same time, in mathematical and in condensed matter physics (see Table 1. Souriau's construction applied to the two-parameter central extension of the planar Galilei group leads to the ''exotic'' particle, which has non-commuting position coordinates. A Berry-phase argument applied to the Bloch electron yields in turn a semiclassical model that has been used to explain the anomalous/spin/optical Hall effects. The non-commutative parameter is momentum-dependent in this case, and can take the form of a monopole in momentum space.
Dimensional regularization and renormalization of non-commutative QFT
Gurau, R
2007-01-01
Using the recently introduced parametric representation of non-commutative quantum field theory, we implement here the dimensional regularization and renormalization of the vulcanized $\\Phi^{\\star 4}_4$ model on the Moyal space.
On the renormalization of non-commutative field theories
Blaschke, Daniel N.; Garschall, Thomas; Gieres, François; Heindl, Franz; Schweda, Manfred; Wohlgenannt, Michael
2013-01-01
This paper addresses three topics concerning the quantization of non-commutative field theories (as defined in terms of the Moyal star product involving a constant tensor describing the non-commutativity of coordinates in Euclidean space). To start with, we discuss the Quantum Action Principle and provide evidence for its validity for non-commutative quantum field theories by showing that the equation of motion considered as insertion in the generating functional Z c [ j] of connected Green functions makes sense (at least at one-loop level). Second, we consider the generalization of the BPHZ renormalization scheme to non-commutative field theories and apply it to the case of a self-interacting real scalar field: Explicit computations are performed at one-loop order and the generalization to higher loops is commented upon. Finally, we discuss the renormalizability of various models for a self-interacting complex scalar field by using the approach of algebraic renormalization.
Strong Planck constraints on braneworld and non-commutative inflation
Energy Technology Data Exchange (ETDEWEB)
Calcagni, Gianluca [Instituto de Estructura de la Materia, CSIC, Serrano 121, 28006 Madrid (Spain); Kuroyanagi, Sachiko; Ohashi, Junko; Tsujikawa, Shinji, E-mail: calcagni@iem.cfmac.csic.es, E-mail: skuro@rs.tus.ac.jp, E-mail: j1211703@ed.tus.ac.jp, E-mail: shinji@rs.kagu.tus.ac.jp [Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601 (Japan)
2014-03-01
We place observational likelihood constraints on braneworld and non-commutative inflation for a number of inflaton potentials, using Planck, WMAP polarization and BAO data. Both braneworld and non-commutative scenarios of the kind considered here are limited by the most recent data even more severely than standard general-relativity models. At more than 95 % confidence level, the monomial potential V(φ)∝φ{sup p} is ruled out for p ≥ 2 in the Randall-Sundrum (RS) braneworld cosmology and, for p > 0, also in the high-curvature limit of the Gauss-Bonnet (GB) braneworld and in the infrared limit of non-commutative inflation, due to a large scalar spectral index. Some parameter values for natural inflation, small-varying inflaton models and Starobinsky inflation are allowed in all scenarios, although some tuning is required for natural inflation in a non-commutative spacetime.
Non-commutative computer algebra and molecular computing
Directory of Open Access Journals (Sweden)
Svetlana Cojocaru
2001-12-01
Full Text Available Non-commutative calculations are considered from the molecular computing point of view. The main idea is that one can get more advantage in using molecular computing for non-commutative computer algebra compared with a commutative one. The restrictions, connected with the coefficient handling in Grobner basis calculations are investigated. Semigroup and group cases are considered as more appropriate. SAGBI basis constructions and possible implementations are discussed.
Non-commutative computer algebra and molecular computing
2001-01-01
Non-commutative calculations are considered from the molecular computing point of view. The main idea is that one can get more advantage in using molecular computing for non-commutative computer algebra compared with a commutative one. The restrictions, connected with the coefficient handling in Grobner basis calculations are investigated. Semigroup and group cases are considered as more appropriate. SAGBI basis constructions and possible implementations are discussed.
Relations between Non-Commutative and Commutative Spacetime
Tezuka, K I
2001-01-01
Spacetime non-commutativity appears in string theory. In this paper, the non-commutativity in string theory is reviewed. At first we review that a Dp-brane is equivalent to a configuration of infinitely many D($p-2$)-branes. If we consider the worldvolume as that of the Dp-brane, coordinates of the Dp-brane is commutative. On the other hand if we deal with the worldvolume as that of the D($p-2$)-branes, since coordinates of many D-branes are promoted to matrices the worldvolume theory is non-commutative one. Next we see that using a point splitting reguralization gives a non-commutative D-brane, and a non-commutative gauge field can be rewritten in terms of an ordinary gauge field. The transformation is called the Seiberg-Witten map. And we introduce second class constraints as boundary conditions of an open string. Since Neumann and Dirichlet boundary conditions are mixed in the constraints when the open string is coupled to a NS B field, the end points of the open string is non-commutative.
Non-commutative dynamics of spinning D0 branes
Loh, D; Sahakian, V V; Loh, Duane; Rudolfa, Kit; Sahakian, Vatche
2004-01-01
Rotational dynamics is known to polarize D0 branes into higher dimensional fuzzy Dp-branes: the tension forces between D0 branes provide the centripetal acceleration, and a puffed up spinning configuration stabilizes. In this work, we consider a rotating cylindrical formation of finite height, wrapping a compact cycle of the background space along the axis of rotation. We find a myriad of interesting results: an intriguing relation between the angular speed, the geometry of the cylinder, and the scale of non-commutativity; instabilities for small radii in relation to the height of the cylinder - reminiscent of the Gregory-LaFlamme phenomenon; a critical radius corresponding to the case where the area of the cylinder is proportional to the number of D0 branes - reminiscent of Matrix black holes; and no power radiated away through D0 brane charge. The instabilities appear to lead to the lateral collapse of the cylinder into possibly a slinky configuration, akin to the Matrix string.
Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory
Molina, Mercedes
2016-01-01
Presenting the collaborations of over thirty international experts in the latest developments in pure and applied mathematics, this volume serves as an anthology of research with a common basis in algebra, functional analysis and their applications. Special attention is devoted to non-commutative algebras, non-associative algebras, operator theory and ring and module theory. These themes are relevant in research and development in coding theory, cryptography and quantum mechanics. The topics in this volume were presented at the Workshop on Non-Associative & Non-Commutative Algebra and Operator Theory, held May 23—25, 2014 at Cheikh Anta Diop University in Dakar, Senegal in honor of Professor Amin Kaidi. The workshop was hosted by the university's Laboratory of Algebra, Cryptology, Algebraic Geometry and Applications, in cooperation with the University of Almería and the University of Málaga. Dr. Kaidi's work focuses on non-associative rings and algebras, operator theory and functional analysis, and he...
Differential geometry based multiscale models.
Wei, Guo-Wei
2010-08-01
Large chemical and biological systems such as fuel cells, ion channels, molecular motors, and viruses are of great importance to the scientific community and public health. Typically, these complex systems in conjunction with their aquatic environment pose a fabulous challenge to theoretical description, simulation, and prediction. In this work, we propose a differential geometry based multiscale paradigm to model complex macromolecular systems, and to put macroscopic and microscopic descriptions on an equal footing. In our approach, the differential geometry theory of surfaces and geometric measure theory are employed as a natural means to couple the macroscopic continuum mechanical description of the aquatic environment with the microscopic discrete atomistic description of the macromolecule. Multiscale free energy functionals, or multiscale action functionals are constructed as a unified framework to derive the governing equations for the dynamics of different scales and different descriptions. Two types of aqueous macromolecular complexes, ones that are near equilibrium and others that are far from equilibrium, are considered in our formulations. We show that generalized Navier-Stokes equations for the fluid dynamics, generalized Poisson equations or generalized Poisson-Boltzmann equations for electrostatic interactions, and Newton's equation for the molecular dynamics can be derived by the least action principle. These equations are coupled through the continuum-discrete interface whose dynamics is governed by potential driven geometric flows. Comparison is given to classical descriptions of the fluid and electrostatic interactions without geometric flow based micro-macro interfaces. The detailed balance of forces is emphasized in the present work. We further extend the proposed multiscale paradigm to micro-macro analysis of electrohydrodynamics, electrophoresis, fuel cells, and ion channels. We derive generalized Poisson-Nernst-Planck equations that are
Foundations of arithmetic differential geometry
Buium, Alexandru
2017-01-01
The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is "intrinsically curved"; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.
Functional approach to squeezed states in non commutative theories
Lubo, Musongela
2004-05-01
We review here the quantum mechanics of some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose positions and momenta mean values are not strictly equal to the ones predicted by classical mechanics. This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the absolute values of the expressions associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory i.e. we recover the known gaussian functions. Besides them, we find other states which can be expressed as products of gaussians with specific hyper geometrics. We illustrate our construction in two models defined on a four dimensional phase space: a model endowed with a minimal length uncertainty and the non commutative plane. Our proposal leads to second order partial differential equations. We find analytical solutions in specific cases. We briefly discuss how our proposal may be applied to the fuzzy sphere and analyze its shortcomings.
Klein geometries, parabolic geometries and differential equations of finite type
Abadoglu, Ender
2009-01-01
We define the infinitesimal and geometric orders of an effective Klein geometry G/H. Using these concepts, we prove i) For any integer m>1, there exists an effective Klein geometry G/H of infinitesimal order m such that G/H is a projective variety (Corollary 9). ii) An effective Klein geometry G/H of geometric order M defines a differential equation of order M+1 on G/H whose global solution space is G (Proposition 18).
Non-commutativity from coarse grained classical probabilities
Wetterich, C
2010-01-01
Non-commutative quantum physics at the atom scale can arise from coarse graining of a classical statistical ensemble at the Planck scale. Position and momentum of an isolated particle are classical observables which remain computable in terms of the coarse grained information. However, the commuting classical product of position and momentum observables is no longer defined in the coarse grained system, which is therefore described by incomplete statistics. The microphysical classical statistical ensemble at the Planck scale admits an alternative non-commuting product structure for position and momentum observables which is compatible with the coarse graining. Measurement correlations for isolated atoms are based on this non-commutative product structure. We present an explicit example for these ideas. It also realizes the discreteness of the spin observable within a microphysical classical statistical ensemble.
Non-linear Vacuum Phenomena in Non-commutative QED
Alvarez-Gaumé, Luís
2001-01-01
We show that the classic results of Schwinger on the exact propagation of particles in the background of constant field-strengths and plane waves can be readily extended to the case of non-commutative QED. It is shown that non-perturbative effects on constant backgrounds are the same as their commutative counterparts, provided the on-shell gauge invariant dynamics is referred to a non-perturbatively related space-time frame. For the case of the plane wave background, we find evidence of the effective extended nature of non-commutative particles, producing retarded and advanced effects in scattering. Besides the known `dipolar' character of non-commutative neutral particles, we find that charged particles are also effectively extended, but they behave instead as `half-dipoles'.
Rotation of D-brane and Non-commutative Geometry
Wang, P; Wang, Pei; Yue, Ruihong
1999-01-01
Our motivation is to find the relationship between the commutator of coordinates and uncertainty relation involving only the coordinates. The boundary condition with constant background field is connected with the rotation of D-brane at general angle. And the mode expansions of D-brane we found is more reasonable than those appeared in literature. The partition functions and scattering amplitudes are also discussed.
Quantum dynamics of simultaneously measured non-commuting observables
Hacohen-Gourgy, Shay; Martin, Leigh S.; Flurin, Emmanuel; Ramasesh, Vinay V.; Whaley, K. Birgitta; Siddiqi, Irfan
2016-10-01
In quantum mechanics, measurements cause wavefunction collapse that yields precise outcomes, whereas for non-commuting observables such as position and momentum Heisenberg’s uncertainty principle limits the intrinsic precision of a state. Although theoretical work has demonstrated that it should be possible to perform simultaneous non-commuting measurements and has revealed the limits on measurement outcomes, only recently has the dynamics of the quantum state been discussed. To realize this unexplored regime, we simultaneously apply two continuous quantum non-demolition probes of non-commuting observables to a superconducting qubit. We implement multiple readout channels by coupling the qubit to multiple modes of a cavity. To control the measurement observables, we implement a ‘single quadrature’ measurement by driving the qubit and applying cavity sidebands with a relative phase that sets the observable. Here, we use this approach to show that the uncertainty principle governs the dynamics of the wavefunction by enforcing a lower bound on the measurement-induced disturbance. Consequently, as we transition from measuring identical to measuring non-commuting observables, the dynamics make a smooth transition from standard wavefunction collapse to localized persistent diffusion and then to isotropic persistent diffusion. Although the evolution of the state differs markedly from that of a conventional measurement, information about both non-commuting observables is extracted by keeping track of the time ordering of the measurement record, enabling quantum state tomography without alternating measurements. Our work creates novel capabilities for quantum control, including rapid state purification, adaptive measurement, measurement-based state steering and continuous quantum error correction. As physical systems often interact continuously with their environment via non-commuting degrees of freedom, our work offers a way to study how notions of contemporary
Exotic Galilean Symmetry and Non-Commutative Mechanics
Directory of Open Access Journals (Sweden)
Peter A. Horváthy
2010-07-01
Full Text Available Some aspects of the ''exotic'' particle, associated with the two-parameter central extension of the planar Galilei group are reviewed. A fundamental property is that it has non-commuting position coordinates. Other and generalized non-commutative models are also discussed. Minimal as well as anomalous coupling to an external electromagnetic field is presented. Supersymmetric extension is also considered. Exotic Galilean symmetry is also found in Moyal field theory. Similar equations arise for a semiclassical Bloch electron, used to explain the anomalous/spin/optical Hall effects.
Directory of Open Access Journals (Sweden)
FENG Chaojun
2014-08-01
Full Text Available Inflation could be also driven by kinetic terms of the inflaton field,which is called the K-inflation model.During the inflation epoch,one could not neglect gravitational effect since the energy was so much high.According to general relativity,gravity is described by space-time geometry.By considering the space-time uncertainty,it is found that all the modes were created inside the Hubble horizon,and it contributes a linear term in the spectral index of the scalar and tensor power spectral.
Seiberg-Witten equations and non-commutative spectral curves in Liouville theory
Energy Technology Data Exchange (ETDEWEB)
Chekhov, Leonid [Department of Theoretical Physics, Steklov Mathematical Institute, Moscow, 119991 Russia and School of Mathematics, Loughborough University, LE11 3TU Leicestershire (United Kingdom); Eynard, Bertrand [Institut de Physique Theorique, IPhT, CNRS, URA 2306, F-91191 Gif-sur-Yvette (France); Ribault, Sylvain [Institut de Physique Theorique, IPhT, CNRS, URA 2306, F-91191 Gif-sur-Yvette (France); Laboratoire Charles Coulomb UMR 5221 CNRS-UM2, Universite Montpellier 2, Place Eugene Bataillon, F-34095 Montpellier Cedex 5 (France)
2013-02-15
We propose that there exist generalized Seiberg-Witten equations in the Liouville conformal field theory, which allow the computation of correlation functions from the resolution of certain Ward identities. These identities involve a multivalued spin one chiral field, which is built from the energy-momentum tensor. We solve the Ward identities perturbatively in an expansion around the heavy asymptotic limit, and check that the first two terms of the Liouville three-point function agree with the known result of Dorn, Otto, Zamolodchikov, and Zamolodchikov. We argue that such calculations can be interpreted in terms of the geometry of non-commutative spectral curves.
Topics in modern differential geometry
Verstraelen, Leopold
2017-01-01
A variety of introductory articles is provided on a wide range of topics, including variational problems on curves and surfaces with anisotropic curvature. Experts in the fields of Riemannian, Lorentzian and contact geometry present state-of-the-art reviews of their topics. The contributions are written on a graduate level and contain extended bibliographies. The ten chapters are the result of various doctoral courses which were held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.
Chiral anomalies and differential geometry
Energy Technology Data Exchange (ETDEWEB)
Zumino, B.
1983-10-01
Some properties of chiral anomalies are described from a geometric point of view. Topics include chiral anomalies and differential forms, transformation properties of the anomalies, identification and use of the anomalies, and normalization of the anomalies. 22 references. (WHK)
The Z-> gamma gamma,gg decays in the non-commutative standard model
Behr, W; Duplancic, G; Schupp, P; Trampetic, J; Wess, J
2003-01-01
On non-commutative spacetime, the standard model (SM) allows new, usually SM forbidden, triple gauge boson interactions to occur. In this letter we propose the SM strictly forbidden Z-> gamma gamma and Z->gg decay modes coming from the gauge sector of the non-commutative standard model (NCSM) as a place where non-commutativity could be experimentally discovered. (orig.)
Differential geometry and topology of curves
Animov, Yu
2001-01-01
Differential geometry is an actively developing area of modern mathematics. This volume presents a classical approach to the general topics of the geometry of curves, including the theory of curves in n-dimensional Euclidean space. The author investigates problems for special classes of curves and gives the working method used to obtain the conditions for closed polygonal curves. The proof of the Bakel-Werner theorem in conditions of boundedness for curves with periodic curvature and torsion is also presented. This volume also highlights the contributions made by great geometers. past and present, to differential geometry and the topology of curves.
Some operator ideals in non-commutative functional analysis
Fidaleo, F
1997-01-01
We characterize classes of linear maps between operator spaces $E$, $F$ which factorize through maps arising in a natural manner via the Pisier vector-valued non-commutative $L^p$ spaces $S_p[E^*]$ based on the Schatten classes on the separable Hilbert space $l^2$. These classes of maps can be viewed as quasi-normed operator ideals in the category of operator spaces, that is in non-commutative (quantized) functional analysis. The case $p=2$ provides a Banach operator ideal and allows us to characterize the split property for inclusions of $W^*$-algebras by the 2-factorable maps. The various characterizations of the split property have interesting applications in Quantum Field Theory.
Quantum Mechanics: Harbinger of a Non-Commutative Probability Theory?
Hiley, Basil J.
2014-01-01
In this paper we discuss the relevance of the algebraic approach to quantum phenomena first introduced by von Neumann before he confessed to Birkoff that he no longer believed in Hilbert space. This approach is more general and allows us to see the structure of quantum processes in terms of non-commutative probability theory, a non-Boolean structure of the implicate order which contains Boolean sub-structures which accommodates the explicate classical world. We move away from mechanical `wave...
Non-commutative Field Theory on S^4
Nakayama, R; Nakayama, Ryuichi; Shimono, Yusuke
2004-01-01
In the previous paper (hep-th/0402010) we proposed a matrix configuration for a non-commutative S^4 (NC4S) and constructed a non-commutative (star) product for field theories on NC4S. This star product and the functions on NC4S turned out to be singular (ambiguous) on a circle on S^4. In the present paper we will show that any matrix can be expanded in terms of the matrix configuration representing NC4S just like any matrix can be expanded into symmetrized products of the matrix configuration for non-commutative S^2. Then we will show that the singularities of the functions on S^4 and the star product can be removed by covering the (commutative) manifold by coordinate neighborhoods and performing appropriate coordinate transformations. Finally a scalar field theory on NC4S is constructed. Our matrix configuration describes two S^4's joined at the circle and the Matrix theory action contains a projection matrix inside the trace to restrict the space of matrices to that for one S^4.
The entropy of dense non-commutative fermion gases
Kriel, Johannes N
2011-01-01
We investigate the properties of two- and three-dimensional non-commutative fermion gases with fixed total z-component of angular momentum, J_z, and at high density for the simplest form of non-commutativity involving constant spatial commutators. Analytic expressions for the entropy and pressure are found. The entropy exhibits non-extensive behaviour while the pressure reveals the presence of incompressibility in two, but not in three dimensions. Remarkably, for two-dimensional systems close to the incompressible density, the entropy is proportional to the square root of the system size, i.e., for such systems the number of microscopic degrees of freedom is determined by the circumference, rather than the area (size) of the system. The absence of incompressibility in three dimensions, and subsequently also the absence of a scaling law for the entropy analogous to the one found in two dimensions, is attributed to the form of the non-commutativity used here, the breaking of the rotational symmetry it implies a...
Symbolic computations in applied differential geometry
Gragert, P.K.H.; Kersten, P.H.M.; Martini, R.
1983-01-01
The main aim of this paper is to contribute to the automatic calculations in differential geometry and its applications, with emphasis on the prolongation theory of Estabrook and Wahlquist, and the calculation of invariance groups of exterior differential systems. A large number of worked examples h
Non-commutative and commutative vacua effects in a scalar torsion scenario
Energy Technology Data Exchange (ETDEWEB)
Sheikhahmadi, Haidar, E-mail: h.sh.ahmadi@gmail.com [Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj (Iran, Islamic Republic of); Aghamohammadi, Ali, E-mail: a.aghamohamadi@iausdj.ac.ir [Sanandaj Branch, Islamic Azad University, Sanandaj (Iran, Islamic Republic of); Saaidi, Khaled, E-mail: ksaaidi@uok.ac.ir [Department of Physics, Faculty of Science, University of Kurdistan, Sanandaj (Iran, Islamic Republic of)
2015-10-07
In this work, the effects of non-commutative and commutative vacua on the phase space generated by a scalar field in a scalar torsion scenario are investigated. For both classical and quantum regimes, the commutative and non-commutative cases are compared. To take account the effects of non-commutativity, two well known non-commutative parameters, θ and β, are introduced. It should be emphasized, the effects of β which is related to momentum sector has more key role in comparison to θ which is related to space sector. Also the different boundary conditions and mathematical interpretations of non-commutativity are explored.
Non-commutative and commutative vacua effects in a scalar torsion scenario
Directory of Open Access Journals (Sweden)
Haidar Sheikhahmadi
2015-10-01
Full Text Available In this work, the effects of non-commutative and commutative vacua on the phase space generated by a scalar field in a scalar torsion scenario are investigated. For both classical and quantum regimes, the commutative and non-commutative cases are compared. To take account the effects of non-commutativity, two well known non-commutative parameters, θ and β, are introduced. It should be emphasized, the effects of β which is related to momentum sector has more key role in comparison to θ which is related to space sector. Also the different boundary conditions and mathematical interpretations of non-commutativity are explored.
LAMB SHIFT IN HYDROGEN-LIKE ATOM INDUCED FROM NON-COMMUTATIVE QUANTUM SPACE-TIME
Directory of Open Access Journals (Sweden)
S Zaim
2015-06-01
Full Text Available In this work we present an important contribution to the non-commutative approach to the hydrogen atom to deal with lamb shift corrections. This can be done by studying the Klein-Gordon equation in a non-commutative space-time as applied to the Hydrogen atom to extract the energy levels, by considering the second-order corrections in the non commutativity parameter and by comparing with the result of the current experimental results on the Lamb shift of the 2P level to extract a bound on the parameter of non-commutativity. Phenomenologically we show that the non-commutativity effects induce lamb shift corrections.
Ordinary differential equations in affine geometry
Directory of Open Access Journals (Sweden)
Salvador Gigena
1996-05-01
Full Text Available The method of qualitative analysis is used, as applied to a class of fourth order, nonlinear ordinary differential equations, in order to classify, both locally and globally, two classes of hypersurfaces of decomposable type in affine geometry: those with constant unimodular affine mean curvature L , and those with constant Riemannian scalar curvature R. This allows to provide a large number of new examples of hypersurfaces in affine geometry.
Ordinary differential equations in affine geometry
Salvador Gigena
1996-01-01
The method of qualitative analysis is used, as applied to a class of fourth order, nonlinear ordinary differential equations, in order to classify, both locally and globally, two classes of hypersurfaces of decomposable type in affine geometry: those with constant unimodular affine mean curvature L , and those with constant Riemannian scalar curvature R. This allows to provide a large number of new examples of hypersurfaces in affine geometry.
Fat Triangulations and Differential Geometry
Saucan, Emil
2011-01-01
We study the differential geometric consequences of our previous result on the existence of fat triangulations, in conjunction with a result of Cheeger, M\\"{u}ller and Schrader, regarding the convergence of Lipschitz-Killing curvatures of piecewise-flat approximations of smooth Riemannian manifolds. A further application to the existence of quasiconformal mappings between manifolds, as well as an extension of the triangulation result to the case of almost Riemannian manifolds, are also given. In addition, the notion of fatness of triangulations and its relation to metric curvature and to excess is explored. Moreover, applications of the main results, and in particular a purely metric approach to Regge calculus, are also investigated.
On Discrete Differential Geometry in Twistor Space
2011-01-01
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Pl\\"ucker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Pl\\"ucker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete differential geometry aims to develop disc...
Charge and/or spin limits for black holes at a non-commutative scale
Paik, Biplab
2017-08-01
In the commutative geometrical background, one finds the total charge ( Q) and/or the total angular momentum ( J) of a generalized black hole of mass M to be bounded by the condition Q^2+( J{/}M) ^2≤ M^2, whereas the inclusion of the concept of non-commutativity in geometry leads to a much more richer result. It predicts that the upper limit to Q and/or J is not fixed but depends on the mass/length scale of black holes; it (the upper limit to Q and/or J) goes towards a `commutative limit' when {M≫ √{θ}} (√{θ} characterizes the minimal length scale) and rapidly diminishes towards zero with M decreasing in the strongly non-commutative regime, until approaching a perfect zero value for {M˜eq 1.904√{θ}}. We have performed separate calculations for a pure Kerr or a pure Reissner-Nordström black hole, and briefly done it for a generalized black hole.
Charge and/or spin limits for black holes at a non-commutative scale
Indian Academy of Sciences (India)
BIPLAB PAIK
2017-08-01
In the commutative geometrical background, one finds the total charge $\\mathcal{(Q)}$ and/or the total angular momentum $\\mathcal{(J)}$ of a generalized black hole of mass $M$ to be bounded by the condition $\\mathcal{Q^{2} + (J/M)^{2} \\leq M^{2}}$, whereas the inclusion of the concept of non-commutativity in geometry leads to a much more richer result. It predicts that the upper limit to $\\mathcal{Q}$ and/or $\\mathcal{J}$ is not fixed but depends on the mass/length scale of black holes; it (the upper limit to $\\mathcal{Q}$ and/or $\\mathcal{J}$ ) goes towards a ‘commutative limit’ when $M \\gg \\sqrt{\\vartheta} (\\sqrt{\\vartheta}$ characterizes the minimal length scale) and rapidly diminishes towards zero with $M$ decreasing in the strongly non-commutative regime, until approaching a perfect zero value for $M \\simeq 1.904\\sqrt{\\vartheta}$. We have performed separate calculations for a pure Kerr or a pure Reissner–Nordström black hole, and briefly done it for a generalized black hole.
Recent topics in differential and analytic geometry
Ochiai, T
1990-01-01
Advanced Studies in Pure Mathematics, Volume 18-I: Recent Topics in Differential and Analytic Geometry presents the developments in the field of analytical and differential geometry. This book provides some generalities about bounded symmetric domains.Organized into two parts encompassing 12 chapters, this volume begins with an overview of harmonic mappings and holomorphic foliations. This text then discusses the global structures of a compact Kähler manifold that is locally decomposable as an isometric product of Ricci-positive, Ricci-negative, and Ricci-flat parts. Other chapters con
Differential Geometry of Microlinear Frolicher Spaces I
Nishimura, Hirokazu
2010-01-01
The central object of synthetic differential geometry is microlinear spaces. In our previous paper [Microlinearity in Frolicher spaces -beyond the regnant philosophy of manifolds-, to appear in International Journal of Pure and Applied Mathematics] we have emancipated microlinearity from within well-adapted models to Frolicher spaces. Therein we have shown that Frolicher spaces which are microlinear as well as Weil exponentiable form a cartesian closed category. To make sure that such Frolicher spaces are the central object of infinite-dimensional differential geometry, we develop the theory of vector fields on them in this paper. The central result is that all vector fields on such a Frolicher space form a Lie algebra.
Differential geometry basic notions and physical examples
Epstein, Marcelo
2014-01-01
Differential Geometry offers a concise introduction to some basic notions of modern differential geometry and their applications to solid mechanics and physics. Concepts such as manifolds, groups, fibre bundles and groupoids are first introduced within a purely topological framework. They are shown to be relevant to the description of space-time, configuration spaces of mechanical systems, symmetries in general, microstructure and local and distant symmetries of the constitutive response of continuous media. Once these ideas have been grasped at the topological level, the differential structure needed for the description of physical fields is introduced in terms of differentiable manifolds and principal frame bundles. These mathematical concepts are then illustrated with examples from continuum kinematics, Lagrangian and Hamiltonian mechanics, Cauchy fluxes and dislocation theory. This book will be useful for researchers and graduate students in science and engineering.
Symbolic computations in applied differential geometry
Gragert, P.K.H.; Kersten, P. H. M.; Martini, R.
1983-01-01
The main aim of this paper is to contribute to the automatic calculations in differential geometry and its applications, with emphasis on the prolongation theory of Estabrook and Wahlquist, and the calculation of invariance groups of exterior differential systems. A large number of worked examples have been included in the text to demonstrate the concrete manipulations in practice. In the appendix, a list of programs discussed in the paper is added.
Can non-commutativity resolve the big-bang singularity?
Energy Technology Data Exchange (ETDEWEB)
Maceda, M.; Madore, J. [Laboratoire de Physique Theorique, Universite de Paris-Sud, Batiment 211, 91405, Orsay (France); Manousselis, P. [Department of Engineering Sciences, University of Patras, 26110, Patras (Greece); Physics Department, National Technical University, Zografou Campus, 157 80, Zografou, Athens (Greece); Zoupanos, G. [Physics Department, National Technical University, Zografou Campus, 157 80, Zografou, Athens (Greece); Theory Division, CERN, 1211, Geneva 23 (Switzerland)
2004-08-01
A possible way to resolve the singularities of general relativity is proposed based on the assumption that the description of space-time using commuting coordinates is not valid above a certain fundamental scale. Beyond that scale it is assumed that the space-time has non-commutative structure leading in turn to a resolution of the singularity. As a first attempt towards realizing the above programme a modification of the Kasner metric is constructed which is commutative only at large time scales. At small time scales, near the singularity, the commutation relations among the space coordinates diverge. We interpret this result as meaning that the singularity has been completely delocalized. (orig.)
Bethe-Salpeter equation in non-commutative space
Directory of Open Access Journals (Sweden)
M. Haghighat
2005-06-01
Full Text Available We consider Bethe-Salpeter (BS equation for the bound state of two point particles in the non-commutative space-time. We subsequently explore the BS equation for spin0-spin0, spin1/2-spin1/2 and spin0-spin1/2 bound states. we show that the lowest order spin independent correction to energy spectrum in each case is of the order θ a 4 while the spin dependent one in NC space, is started at the order θ a 6.
Scalar fields in a non-commutative space
Bietenholz, Wolfgang; Mejía-Díaz, Héctor; Panero, Marco
2014-01-01
We discuss the lambda phi**4 model in 2- and 3-dimensional non-commutative spaces. The mapping onto a Hermitian matrix model enables its non-perturbative investigation by Monte Carlo simulations. The numerical results reveal a phase where stripe patterns dominate. In d=3 we show that in this phase the dispersion relation is deformed in the IR regime, in agreement with the property of UV/IR mixing. This "striped phase" also occurs in d=2. For both dimensions we provide evidence that it persists in the simultaneous limit to the continuum and to infinite volume ("Double Scaling Limit"). This implies the spontaneous breaking of translation symmetry.
Supergravity and Light-Like Non-commutativity
Alishahiha, M; Russo, Jorge G; Alishahiha, Mohsen; Oz, Yaron; Russo, Jorge G.
2000-01-01
We construct dual supergravity descriptions of field theories and little string theories with light-like non-commutativity. The field theories are realized on the world-volume of Dp branes with light-like NS $B$ field and M5 branes with light-like $C$ field. The little string theories are realized on the world-volume of NS5 branes with light-like RR $A$ fields. The supergravity backgrounds are closely related to the $A=0,B=0,C=0$ backgrounds. We discuss the implications of these results. We also construct dual supergravity descriptions of ODp theories realized on the worldvolume of NS5 branes with RR backgrounds.
Applications of Differential Geometry to Cartography
Benitez, Julio; Thome, Nestor
2004-01-01
This work introduces an application of differential geometry to cartography. The mathematical aspects of some geographical projections of Earth surface are revealed together with some of its more important properties. An important problem since the discovery of the 'spherical' form of the Earth is how to compose a reliable map of the surface of…
Advances in differential geometry and topology
Institute for Scientific Interchange. Turin
1990-01-01
The aim of this volume is to offer a set of high quality contributions on recent advances in Differential Geometry and Topology, with some emphasis on their application in physics.A broad range of themes is covered, including convex sets, Kaehler manifolds and moment map, combinatorial Morse theory and 3-manifolds, knot theory and statistical mechanics.
Methods from Differential Geometry in Polytope Theory
Adiprasito, Karim Alexander
2014-01-01
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-dimensional topology.
Applications of Differential Geometry to Cartography
Benitez, Julio; Thome, Nestor
2004-01-01
This work introduces an application of differential geometry to cartography. The mathematical aspects of some geographical projections of Earth surface are revealed together with some of its more important properties. An important problem since the discovery of the 'spherical' form of the Earth is how to compose a reliable map of the surface of…
Differential geometry connections, curvature, and characteristic classes
Tu, Loring W
2017-01-01
This text presents a graduate-level introduction to differential geometry for mathematics and physics students. The exposition follows the historical development of the concepts of connection and curvature with the goal of explaining the Chern–Weil theory of characteristic classes on a principal bundle. Along the way we encounter some of the high points in the history of differential geometry, for example, Gauss' Theorema Egregium and the Gauss–Bonnet theorem. Exercises throughout the book test the reader’s understanding of the material and sometimes illustrate extensions of the theory. Initially, the prerequisites for the reader include a passing familiarity with manifolds. After the first chapter, it becomes necessary to understand and manipulate differential forms. A knowledge of de Rham cohomology is required for the last third of the text. Prerequisite material is contained in author's text An Introduction to Manifolds, and can be learned in one semester. For the benefit of the reader and to establ...
Non-commutative multiple-valued logic algebras
Ciungu, Lavinia Corina
2014-01-01
This monograph provides a self-contained and easy-to-read introduction to non-commutative multiple-valued logic algebras; a subject which has attracted much interest in the past few years because of its impact on information science, artificial intelligence and other subjects. A study of the newest results in the field, the monograph includes treatment of pseudo-BCK algebras, pseudo-hoops, residuated lattices, bounded divisible residuated lattices, pseudo-MTL algebras, pseudo-BL algebras and pseudo-MV algebras. It provides a fresh perspective on new trends in logic and algebras in that algebraic structures can be developed into fuzzy logics which connect quantum mechanics, mathematical logic, probability theory, algebra and soft computing. Written in a clear, concise and direct manner, Non-Commutative Multiple-Valued Logic Algebras will be of interest to masters and PhD students, as well as researchers in mathematical logic and theoretical computer science.
A black hole cast on a non-commutative background
Mbonye, Manasse R
2010-01-01
In this work we describe a black hole, set on a non-commutative background. The model, which is relatively simple, is an exact solution of the Einstein Field Equations. Based on a proposition we put forward, we argue that introducing a matter density field on a non-commutative background sets up a mechanism that deforms the field into two distinct fields, one residing dominantly on the lattice tops (hereafter, on-cell) and the other residing dominantly in the inter-lattice regions (hereafter, off-cell). The two fields have different physical and themodynamic characterics which we describe, and some of which play a role in halting collpse to a singularity. For example, not surprisingly the on-cell (off-cell) fields manifest standard on-shell (off-shell) characteristics, respectively. Both the density and the net mass-energy are unchanged by the deformation mechanism. In our treatment the mass of a black hole defines its own size scale L of the interior region it occupies. Moreover, such a length is quantized, ...
Non-commutative U(1) Gauge Theory on R**4 with Oscillator Term
Blaschke, Daniel N; Schweda, Manfred
2007-01-01
Inspired by the renormalizability of the non-commutative $\\Phi^4$ model with added oscillator term, we formulate a non-commutative gauge theory, where the oscillator enters as a gauge fixing term. All propagators turn out to be essentially given by the Mehler kernel and the bilinear part of the action is invariant under the Langmann-Szabo duality. The model is a promising candidate for a renormalizable non-commutative U(1) gauge theory.
The Aharonov-Casher effect for spin-1 particles in non-commutative quantum mechanics
Dulat, Sayipjamal
2008-01-01
By using a generalized Bopp's shift formulation, instead of star product method, we investigate the Aharonov-Casher(AC) effect for a spin-1 neutral particle in non-commutative(NC) quantum mechanics. After solving the Kemmer equations both on a non-commutative space and a non-commutative phase space, we obtain the corrections to the topological phase of the AC effect for a spin-1 neutral particle both on a NC space and a NC phase space.
Differential geometry and scalar gravitational waves
Corda, Christian
2013-01-01
Following some strong argumentations of differential geometry in the Landau's book, some corrections about errors in the old literature on scalar gravitational waves (SGWs) are given and discussed. In the analysis of the response ofi nterferometers the computation is first performed in the low frequencies approximation, then the analysis is applied to all SGWs in the full frequency and angular dependences. The presented results are in agreement with the more recent literature on SGWs.
A non-commuting twist in the partition function
Govindarajan, Suresh
2012-01-01
We compute a twisted index for an orbifold theory when the twist generating group does not commute with the orbifold group. The twisted index requires the theory to be defined on moduli spaces that are compatible with the twist. This is carried out for CHL models at special points in the moduli space where they admit dihedral symmetries. The commutator subgroup of the dihedral groups are cyclic groups that are used to construct the CHL orbifolds. The residual reflection symmetry is chosen to act as a `twist' on the partition function. The reflection symmetries do not commute with the orbifolding group and hence we refer to this as a non-commuting twist. We count the degeneracy of half-BPS states using the twisted partition function and find that the contribution comes mainly from the untwisted sector. We show that the generating function for these twisted BPS states are related to the Mathieu group M_{24}.
Functional approach to squeezed states in non commutative theories
Lubo, M
2004-01-01
We review some noncommutative theories in which no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. We show how the difference of structure between the Poisson brackets and the commutators in these theories generically leads to a harmonic oscillator whose position mean value is not strictly equal to the one predicted by classical mechanics. This raises the question of the nature of quasi classical states in these models. We propose an extension based on a variational principle. The action considered is the sum of the squares of the terms associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory: we recover the known gaussian functions and, besides them, other states which can be expressed as products of gaussians with specific hypergeometrics. We illustrate our construction in three models defined on a four dimensional phase space: two models endowed with a minimal length uncertainty and the non commutative p...
Non-commutative Iwasawa theory for modular forms
Coates, John; Liang, Zhibin; Stein, William; Sujatha, Ramdorai
2012-01-01
The aim of the present paper is to give evidence, largely numerical, in support of the non-commutative main conjecture of Iwasawa theory for the motive of a primitive modular form of weight k>2 over the Galois extension of Q obtained by adjoining to Q all p-power roots of unity, and all p-power roots of a fixed integer m>1. The predictions of the main conjecture are rather intricate in this case because there is more than one critical point, and also there is no canonical choice of periods. Nevertheless, our numerical data agrees perfectly with all aspects of the main conjecture, including Kato's mysterious congruence between the cyclotomic Manin p-adic L-function, and the cyclotomic p-adic L-function of a twist of the motive by a certain non-abelian Artin character of the Galois group of this extension.
Canonical approach to the closed string non-commutativity
Energy Technology Data Exchange (ETDEWEB)
Davidovic, Lj.; Nikolic, B.; Sazdovic, B. [University of Belgrade, Institute of Physics, P.O.Box 57, Belgrade (Serbia)
2014-01-15
We consider the closed stringmoving in a weakly curved background and its totally T-dualized background. Using T-duality transformation laws, we find the structure of the Poisson brackets in the T-dual space corresponding to the fundamental Poisson brackets in the original theory. From this structure we see that the commutative original theory is equivalent to the non-commutative T-dual theory, whose Poisson brackets are proportional to the background fluxes times winding and momentum numbers. The noncommutative theory of the present article is more nongeometrical than T-folds and in the case of three space-time dimensions corresponds to the nongeometric space-time with R-flux. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Privault, N. [Universite d`Evry, 91 (France)
1996-05-20
Using two different constructions of the chaotic and variational calculus on Poisson space, we show that the Wiener and Poisson processes have a non-commutative representation which is different from the one obtained by transfer of the Fock space creation and annihilation operators. We obtain in this way an extension of the non-commutative It calculus. The associated commutation relations show a link between the geometric and exponential distributions. (author). 11 refs.
Non-commutative Poisson Algebra Structures on the Lie Algebra son(CQ)
Institute of Scientific and Technical Information of China (English)
Jie Tong; Quanqin Jin
2007-01-01
Non-commutative Poisson algebras are the algebras having both an associativealgebra structure and a Lie algebra structure together with the Leibniz law.In this paper,the non-commutative poisson algebra structures on son(CQ) are determined.
The numerical approach to quantum field theory in a non-commutative space
Panero, Marco
2016-01-01
Numerical simulation is an important non-perturbative tool to study quantum field theories defined in non-commutative spaces. In this contribution, a selection of results from Monte Carlo calculations for non-commutative models is presented, and their implications are reviewed. In addition, we also discuss how related numerical techniques have been recently applied in computer simulations of dimensionally reduced supersymmetric theories.
Infinite divisibility and a non-commutative Boolean-to-free Bercovici-Pata bijection
Belinschi, Serban T; Vinnikov, Victor
2010-01-01
We use the theory of fully matricial, or non-commutative, functions to investigate infinite divisibility and limit theorems in operator-valued non-commutative probability. Our main result is an operator-valued analogue of the Bercovici-Pata bijection. An important tool is Voiculescu's subordination property for operator-valued free convolution.
NON-COMMUTATIVE POISSON ALGEBRA STRUCTURES ON LIE ALGEBRA sln(fCq) WITH NULLITY M
Institute of Scientific and Technical Information of China (English)
Jie TONG; Quanqin JIN
2013-01-01
Non-commutative Poisson algebras are the algebras having both an associa-tive algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(fCq) are determined.
Introduction to differential geometry for engineers
Doolin, Brian F
2013-01-01
This outstanding guide supplies important mathematical tools for diverse engineering applications, offering engineers the basic concepts and terminology of modern global differential geometry. Suitable for independent study as well as a supplementary text for advanced undergraduate and graduate courses, this volume also constitutes a valuable reference for control, systems, aeronautical, electrical, and mechanical engineers.The treatment's ideas are applied mainly as an introduction to the Lie theory of differential equations and to examine the role of Grassmannians in control systems analysis. Additional topics include the fundamental notions of manifolds, tangent spaces, vector fields, exterior algebra, and Lie algebras. An appendix reviews concepts related to vector calculus, including open and closed sets, compactness, continuity, and derivative.
Noncommutative Differential Geometry of Generalized Weyl Algebras
Brzeziński, Tomasz
2016-06-01
Elements of noncommutative differential geometry of Z-graded generalized Weyl algebras A(p;q) over the ring of polynomials in two variables and their zero-degree subalgebras B(p;q), which themselves are generalized Weyl algebras over the ring of polynomials in one variable, are discussed. In particular, three classes of skew derivations of A(p;q) are constructed, and three-dimensional first-order differential calculi induced by these derivations are described. The associated integrals are computed and it is shown that the dimension of the integral space coincides with the order of the defining polynomial p(z). It is proven that the restriction of these first-order differential calculi to the calculi on B(p;q) is isomorphic to the direct sum of degree 2 and degree -2 components of A(p;q). A Dirac operator for B(p;q) is constructed from a (strong) connection with respect to this differential calculus on the (free) spinor bimodule defined as the direct sum of degree 1 and degree -1 components of A(p;q). The real structure of KO-dimension two for this Dirac operator is also described.
Non-commutative solitons and strong-weak duality
Blas, H; Rojas, M
2005-01-01
Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM) associated to the group GL(2), in which the Toda field belongs to certain representations of either $U(1){x} U(1)$ or $U(1)_{C}$ corresponding to the Lechtenfeld et al. (NCSG$_{1}$) or Grisaru-Penati (NCSG$_{2}$) proposals for the NC versions of the sine-Gordon model, respectively. Besides, the relevant NCMT$_{1, 2}$ models are written for two (four) types of Dirac fields corresponding to the Moyal product extension of one (two) copy(ies) of the ordinary massive Thirring model. The NCATM$_{1,2}$ models share the same one-soliton (real Toda field sector of model 2) exact solutions, which are found without expansion in the NC ...
Non-commutative solitons and strong-weak duality
Energy Technology Data Exchange (ETDEWEB)
Blas, Harold [Departamento de Matematica - ICET, Universidade Federal de Mato Grosso, Av. Fernando Correa, s/n, Coxipo, 78060-900, Cuiaba - MT (Brazil)]. E-mail: blas@cpd.ufmt.br; Carrion, Hector L. [Instituto de Fisica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro (Brazil); Rojas, Moises [Centro Brasileiro de Pesquisas Fisicas, Rua Dr. Xavier Sigaud, 150 CEP 22290-180, Rio de Janeiro-RJ (Brazil)
2005-03-01
Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian approach to deal with dual theories. The master lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM) associated to the group GL(2), in which the Toda field belongs to certain representations of either U(1)xU(1) or U(1){sub C} corresponding to the Lechtenfeld et al. (NCSG{sub 1}) or Grisaru-Penati (NCSG{sub 2}) proposals for the NC versions of the sine-Gordon model, respectively. Besides, the relevant NCMT{sub 1,2} models are written for two (four) types of Dirac fields corresponding to the Moyal product extension of one (two) copy(ies) of the ordinary massive Thirring model. The NCATM{sub 1,2} models share the same one-soliton (real Toda field sector of model 2) exact solutions, which are found without expansion in the NC parameter {theta} for the corresponding Toda and matter fields describing the strong-weak phases, respectively. The correspondence NCSG{sub 1} {r_reversible} NCMT{sub 1} is promising since it is expected to hold on the quantum level. (author)
Non-commutative solitons and strong-weak duality
Energy Technology Data Exchange (ETDEWEB)
Blas, H. [Univerdidade Federal de Mato Grosso, Cuiaba, MT (Brazil). Dept. de Matematica]. E-mail: blas@cpd.ufmt.br; Carrion, H.L. [Universidade Federal, Rio de Janeiro, RJ (Brazil). Inst. de Fisica]. E-mail: mlm@if.ufrj.br; Rojas, M. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil)]. E-mail: mrojas@cbpf.br
2004-07-01
Some properties of the non-commutative versions of the sine-Gordon model (NCSG) and the corresponding massive Thirring theories (NCMT) are studied. Our method relies on the NC extension of integrable models and the master Lagrangian approach to deal with dual theories. The master Lagrangians turn out to be the NC versions of the so-called affine Toda model coupled to matter fields (NCATM) associated to the group GL(2), in which the Toda field belongs to certain representations of either U(1)xU(1) or U(1){sub C} corresponding to the Lechtenfeld et al. (NCSG{sub 1}) or Grisaru- Penati (NCSG{sub 2}) proposals for the NC versions of the sine-Gordon model, respectively. Besides, the relevant NCMT{sub 1,2} models are written for two (four) types of Dirac fields corresponding to the Moyal product extension of one (two) copy(ies) of the ordinary massive Thirring model. The NCATM{sub 1,2} models share the same one-soliton (real Toda field sector of model 2) exact solutions, which are found without expansion in the NC parameter {theta} for the corresponding Toda and matter fields describing the strong-weak phases, respectively. The correspondence NCSG{sub 1} {r_reversible} NCMT{sub 1} is promising since it is expected to hold on the quantum level (author)
Differential geometry of curves and surfaces
Tapp, Kristopher
2016-01-01
This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging. Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships. Applications abound! The study of conformal and equiareal functions is grounded in its application to carto...
Differential geometry of curves and surfaces
Banchoff, Thomas F
2010-01-01
Students and professors of an undergraduate course in differential geometry will appreciate the clear exposition and comprehensive exercises in this book that focuses on the geometric properties of curves and surfaces, one- and two-dimensional objects in Euclidean space. The problems generally relate to questions of local properties (the properties observed at a point on the curve or surface) or global properties (the properties of the object as a whole). Some of the more interesting theorems explore relationships between local and global properties. A special feature is the availability of accompanying online interactive java applets coordinated with each section. The applets allow students to investigate and manipulate curves and surfaces to develop intuition and to help analyze geometric phenomena.
The Aharonov-Casher effect for spin-1 particles in non-commutative quantum mechanics
Energy Technology Data Exchange (ETDEWEB)
Dulat, S. [Xinjiang University, School of Physics Science and Technology, Urumqi (China); The Abdus Salam International Center for Theoretical Physics, Trieste (Italy); Li, Kang [Hangzhou Normal University, Department of Physics, Hangzhou (China); The Abdus Salam International Center for Theoretical Physics, Trieste (Italy)
2008-03-15
By using a generalized Bopp's shift formulation, instead of the star product method, we investigate the Aharonov-Casher (AC) effect for a spin-1 neutral particle in non-commutative (NC) quantum mechanics. After solving the Kemmer equations both on a non-commutative space and a non-commutative phase space, we obtain the corrections to the topological phase of the AC effect for a spin-1 neutral particle both on a NC space and a NC phase space. (orig.)
Aspects of perturbative quantum field theory on non-commutative spaces
Blaschke, Daniel N
2016-01-01
In this contribution to the proceedings of the Corfu Summer Institute 2015, I give an overview over quantum field theories on non-commutative Moyal space and renormalization. In particular, I review the new features and challenges one faces when constructing various scalar, fermionic and gauge field theories on Moyal space, and especially how the UV/IR mixing problem was solved for certain models. Finally, I outline more recent progress in constructing a renormalizable gauge field model on non-commutative space, and how one might attempt to prove renormalizability of such a model using a generalized renormalization scheme adapted to the non-commutative (and hence non-local) setting.
Note On The Extended Non Commutativity of Coordinates
Boulahoual, A
2001-01-01
We present in this short note an idea about a possible extension of the standard noncommutative algebra to the formal differential operators framework. In this sense, we develop an analysis and derive an extended noncommutative algebra given by $[x_{a}, x_{b}]_{\\star} = i(\\theta + \\chi)_{ab}$ where $\\theta_{ab}$ is the standard noncommutative parameter and $\\chi_{ab}(x)\\equiv \\chi^{\\mu}_{ab}(x)\\partial_{\\mu} ={1/2}(x_a \\theta^{\\mu}_{b} - x_b \\theta^{\\mu}_{a})\\partial_\\mu$ is an antisymmetric non-constant vector-field shown to play the role of the extended deformation parameter. This idea was motivated by the importance of noncommutative geometry framework in the current subject of D-brane and matrix theory physics.
Discrete differential geometry: the nonplanar quadrilateral mesh.
Twining, Carole J; Marsland, Stephen
2012-06-01
We consider the problem of constructing a discrete differential geometry defined on nonplanar quadrilateral meshes. Physical models on discrete nonflat spaces are of inherent interest, as well as being used in applications such as computation for electromagnetism, fluid mechanics, and image analysis. However, the majority of analysis has focused on triangulated meshes. We consider two approaches: discretizing the tensor calculus, and a discrete mesh version of differential forms. While these two approaches are equivalent in the continuum, we show that this is not true in the discrete case. Nevertheless, we show that it is possible to construct mesh versions of the Levi-Civita connection (and hence the tensorial covariant derivative and the associated covariant exterior derivative), the torsion, and the curvature. We show how discrete analogs of the usual vector integral theorems are constructed in such a way that the appropriate conservation laws hold exactly on the mesh, rather than only as approximations to the continuum limit. We demonstrate the success of our method by constructing a mesh version of classical electromagnetism and discuss how our formalism could be used to deal with other physical models, such as fluids.
Differential geometry of groups in string theory
Energy Technology Data Exchange (ETDEWEB)
Schmidke, W.B. Jr.
1990-09-01
Techniques from differential geometry and group theory are applied to two topics from string theory. The first topic studied is quantum groups, with the example of GL (1{vert bar}1). The quantum group GL{sub q}(1{vert bar}1) is introduced, and an exponential description is derived. The algebra and coproduct are determined using the invariant differential calculus method introduced by Woronowicz and generalized by Wess and Zumino. An invariant calculus is also introduced on the quantum superplane, and a representation of the algebra of GL{sub q}(1{vert bar}1) in terms of the super-plane coordinates is constructed. The second topic follows the approach to string theory introduced by Bowick and Rajeev. Here the ghost contribution to the anomaly of the energy-momentum tensor is calculated as the Ricci curvature of the Kaehler quotient space Diff(S{sup 1})/S{sup 1}. We discuss general Kaehler quotient spaces and derive an expression for their Ricci curvatures. Application is made to the string and superstring diffeomorphism groups, considering all possible choices of subgroup. The formalism is extended to associated holomorphic vector bundles, where the Ricci curvature corresponds to the anomaly for different ghost sea levels. 26 refs.
Energy Technology Data Exchange (ETDEWEB)
Amini, Nina H. [Stanford University, Edward L. Ginzton Laboratory, Stanford, CA (United States); CNRS, Laboratoire des Signaux et Systemes (L2S) CentraleSupelec, Gif-sur-Yvette (France); Miao, Zibo; Pan, Yu; James, Matthew R. [Australian National University, ARC Centre for Quantum Computation and Communication Technology, Research School of Engineering, Canberra, ACT (Australia); Mabuchi, Hideo [Stanford University, Edward L. Ginzton Laboratory, Stanford, CA (United States)
2015-12-15
The purpose of this paper is to study the problem of generalizing the Belavkin-Kalman filter to the case where the classical measurement signal is replaced by a fully quantum non-commutative output signal. We formulate a least mean squares estimation problem that involves a non-commutative system as the filter processing the non-commutative output signal. We solve this estimation problem within the framework of non-commutative probability. Also, we find the necessary and sufficient conditions which make these non-commutative estimators physically realizable. These conditions are restrictive in practice. (orig.)
Landau-like Atomic Problem on a Non-commutative Phase Space
Mamat, Jumakari; Dulat, Sayipjamal; Mamatabdulla, Hekim
2016-06-01
We study the motion of a neutral particle in symmetric gauge and in the framework of non-commutative Quantum Mechanics. Starting from the corresponding Hamiltonian we derive the eigenfunction and eigenvalues.
Determinants of self-employment among commuters and non-commuters
DEFF Research Database (Denmark)
Backman, M.; Karlsson, C.
2016-01-01
We analyse the determinants of self-employment and focus on the contextual environment. By distinguishing between commuters and non-commuters we are able to analyse the influence from the work and home environment, respectively. Our results indicate a significant difference between non-commuters ......We analyse the determinants of self-employment and focus on the contextual environment. By distinguishing between commuters and non-commuters we are able to analyse the influence from the work and home environment, respectively. Our results indicate a significant difference between non......-commuters and commuters in terms of the role of networks for becoming self-employed. Our results indicate that it is the business networks where people work, rather than where they live that exerts a positive influence on the probability of becoming self-employed. These effects are further robust over educational...
Dirac equation, hydrogen atom spectrum and the Lamb shift in dynamical non-commutative spaces
Indian Academy of Sciences (India)
S A ALAVI; N REZAEI
2017-05-01
We derive the relativistic Hamiltonian of hydrogen atom in dynamical non-commutative spaces (DNCS or $\\tau$ -space). Using this Hamiltonian we calculate the energy shift of the ground state as well the $2P_{1/2}$, $2S_{1/2}$levels. In all the cases, the energy shift depends on the dynamical non-commutative parameter $\\tau$. Using the accuracy of the energy measurement, we obtain an upper bound for $\\tau$. We also study the Lamb shift in DNCS. Both $2P_{1/2}$ and $2S_{1/2}$ levels receive corrections due to dynamical non-commutativity of space which is in contrast with the non-dynamical non-commutative spaces (NDNCS or $\\theta$-space) in which the $2S_{1/2}$ level receives no correction.
2015-01-01
In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over $\\mathbb{Q}$ is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (whether or not randomization is ...
Energy-momentum tensors for non-commutative Abelian Proca field
Darabi, F
2014-01-01
We study two different possibilities of constructing the energy-momentum tensors for non-commutative Abelian Proca field, by using (i) general Noether theorem and (ii) coupling to a weak external gravitational field. Both energy-momentum tensors are not traceless due to the violation of Lorentz invariance in non-commutative spaces. In particular, we show that the obtained energy density of the latter case coincides exactly with that of obtained by Dirac quantization method.
On Non-Commutative Correction of the G\\"odel-type Metric
Ulhoa, S C; Amorim, R G G
2015-01-01
In this paper, we will study non-commutative corrections in the metric tensor for the G\\"{o}del-type universe, a model that has as its main characteristic the possibility of violation of causality, allowing therefore time travel. We also find that the critical radius in such a model, which eventually will determine the time travel possibility, is modified due to the non commutativity of spatial coordinates.
Singlet particles as cold dark matter in θ-exact non-commutative space-time
Directory of Open Access Journals (Sweden)
S A A Alavi
2017-02-01
Full Text Available First, singlet dark matter annihilation into pair charged fermions and pair bosons was studied to the first order of non-commutativity parameter in perturbative model. Our results are different from the results reported in some previous studies. Then the problem is formulated in -exact non-commutative space-time and non-perturbative model, then the exact results are presented
Quantization maps, algebra representation, and non-commutative Fourier transform for Lie groups
Energy Technology Data Exchange (ETDEWEB)
Guedes, Carlos; Oriti, Daniele [Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam (Germany); Raasakka, Matti [Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam (Germany); LIPN, Institut Galilée, Université Paris-Nord, 99, av. Clement, 93430 Villetaneuse (France)
2013-08-15
The phase space given by the cotangent bundle of a Lie group appears in the context of several models for physical systems. A representation for the quantum system in terms of non-commutative functions on the (dual) Lie algebra, and a generalized notion of (non-commutative) Fourier transform, different from standard harmonic analysis, has been recently developed, and found several applications, especially in the quantum gravity literature. We show that this algebra representation can be defined on the sole basis of a quantization map of the classical Poisson algebra, and identify the conditions for its existence. In particular, the corresponding non-commutative star-product carried by this representation is obtained directly from the quantization map via deformation quantization. We then clarify under which conditions a unitary intertwiner between such algebra representation and the usual group representation can be constructed giving rise to the non-commutative plane waves and consequently, the non-commutative Fourier transform. The compact groups U(1) and SU(2) are considered for different choices of quantization maps, such as the symmetric and the Duflo map, and we exhibit the corresponding star-products, algebra representations, and non-commutative plane waves.
A non-commutative n-particle 3D Wigner quantum oscillator
King, R C; Stoilova, N I; Van der Jeugt, J
2003-01-01
An n-particle 3-dimensional Wigner quantum oscillator model is constructed explicitly. It is non-canonical in that the usual coordinate and linear momentum commutation relations are abandoned in favour of Wigner's suggestion that Hamilton's equations and the Heisenberg equations are identical as operator equations. The construction is based on the use of Fock states corresponding to a family of irreducible representations of the Lie superalgebra sl(1|3n) indexed by an A-superstatistics parameter p. These representations are typical for p\\geq 3n but atypical for p<3n. The branching rules for the restriction from sl(1|3n) to gl(1) \\oplus so(3) \\oplus sl(n) are used to enumerate energy and angular momentum eigenstates. These are constructed explicitly and tabulated for n\\leq 2. It is shown that measurements of the coordinates of the individual particles gives rise to a set of discrete values defining nests in the 3-dimensional configuration space. The fact that the underlying geometry is non-commutative is sh...
Differential geometry of proteins. Helical approximations.
Louie, A H; Somorjai, R L
1983-07-25
We regard a protein molecule as a geometric object, and in a first approximation represent it as a regular parametrized space curve passing through its alpha-carbon atoms (the backbone). In an earlier paper we argued that the regular patterns of secondary structures of proteins (morphons) correspond to geodesics on minimal surfaces. In this paper we discuss methods of recognizing these morphons on space curves that represent the protein backbone conformation. The mathematical tool we employ is the differential geometry of curves and surfaces. We introduce a natural approximation of backbone space curves in terms of helical approximating elements and present a computer algorithm to implement the approximation. Simple recognition criteria are given for the various morphons of proteins. These are incorporated into our helical approximation algorithm, together with more non-local criteria for the recognition of beta-sheet topologies. The method and the algorithm are illustrated with several examples of representative proteins. Generalizations of the helical approximation method are considered and their possible implications for protein energetics are sketched.
T-duality with H-flux. Non-commutativity, T-folds and G x G structure
Energy Technology Data Exchange (ETDEWEB)
Grange, P. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik]|[Hamburg Univ. (Germany). Zentrum fuer Mathematische Physik; Schaefer-Nameki, S. [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik]|[Hamburg Univ. (Germany). Zentrum fuer Mathematische Physik]|[California Inst. of Tech., Pasadena (United States)
2006-09-15
Various approaches to T-duality with NSNS three-form flux are reconciled. Non-commutative torus fibrations are shown to be the open-string version of T-folds. The non-geometric T-dual of a three-torus with uniform flux is embedded into a generalized complex six-torus, and the non-geometry is probed by D0-branes regarded as generalized complex submanifolds. The non-commutativity scale, which is present in these compactifications, is given by a holomorphic Poisson bivector that also encodes the variation of the dimension of the world-volume of D-branes under monodromy. This bivector is shown to exist in SU(3) x SU(3) structure compactifications, which have been proposed as mirrors to NSNS-flux backgrounds. The two SU(3)-invariant spinors are generically not parallel, thereby giving rise to a non-trivial Poisson bivector. Furthermore we show that for non-geometric T-duals, the Poisson bivector may not be decomposable into the tensor product of vectors. (orig.)
Botnan, Magnus Bakke
2011-01-01
We study persistent homology, methods in discrete differential geometry and discrete Morse theory. Persistent homology is applied to computational biology and range image analysis. Theory from differential geometry is used to define curvature estimates of triangulated hypersurfaces. In particular, a well-known method for triangulated surfacesis generalised to hypersurfaces of any dimension. The thesis concludesby discussing a discrete analogue of Morse theory.
Nonlinear partial differential equations: Integrability, geometry and related topics
Krasil'shchik, Joseph; Rubtsov, Volodya
2017-03-01
Geometry and Differential Equations became inextricably entwined during the last one hundred fifty years after S. Lie and F. Klein's fundamental insights. The two subjects go hand in hand and they mutually enrich each other, especially after the "Soliton Revolution" and the glorious streak of Symplectic and Poisson Geometry methods in the context of Integrability and Solvability problems for Non-linear Differential Equations.
Lectures on Differential Geometry of Modules and Rings
Sardanashvily, G
2009-01-01
Generalizing differential geometry of smooth vector bundles formulated in algebraic terms of the ring of smooth functions, its derivations and the Koszul connection, one can define differential operators, differential calculus and connections on modules over arbitrary commutative, graded commutative and noncommutative rings. For instance, this is the case of quantum theory, SUSY theory and noncommutative geometry, respectively. The relevant material on this subject is summarized.
Matrix Configurations for Spherical 4-branes and Non-commutative Structures on S^4
Nakayama, R; Nakayama, Ryuichi; Shimono, Yusuke
2004-01-01
We present a Matrix theory action and Matrix configurations for spherical 4-branes. The dimension of the representations is given by N=2(2j+1) (j=1/2,1,3/2,...). The algebra which defines these configurations is not invariant under SO(5) rotations but under SO(3) \\otimes SO(2). We also construct a non-commutative product for field theories on S^4 in terms of that on S^2. An explicit formula of the non-commutative product which corresponds to the N=4 dim representation of the non-commutative S^4 algebra is worked out. Because we use S^2 \\otimes S^2 parametrization of S^4, our S^4 is doubled and the non-commutative product and functions on S^4 are indeterminate on a great circle (S^1) on S^4. We will however, show that despite this mild singularity it is possible to write down a finite action integral of the non-commutative field thoery on S^4. NS-NS B field background on S^4 which is associated with our Matrix S^4 configurations is also constructed.
Non-commutative field theory and the parameters of Lorentz violation in QED
Directory of Open Access Journals (Sweden)
S Aghababaei
2011-09-01
Full Text Available Non-commutative field theory as a theory including the Lorentz violation can be constructed in two different ways. In the first method, the non-commutative fields are the same as the ordinary ones while the gauge group is restricted to U(n. For example, the symmetry group of standard model in non-commutative space is U(3×(2×U(1 which can be reduced to SU(3×SU(2×U(1 by two appropriate spontaneous symmetry breaking. In contrast, in the second method, the non-commutative gauge theory can be constructed for SU(n gauge group via Seiberg- Witten map. In this work, we want to find the relation between the NC-parameter and the Lorentz violation parameters for the first method and compare our results with what is already found in the second one. At the end, we obtain new limits on non-commutative parameter by using the existing bounds on the Lorentz Violation parameters.
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...
Blaschke, D. N.; Grosse, H.; Schweda, M.
2007-09-01
Inspired by the renormalizability of the non-commutative Φ4 model with added oscillator term, we formulate a non-commutative gauge theory, where the oscillator enters as a gauge fixing term in a BRST invariant manner. All propagators turn out to be essentially given by the Mehler kernel and the bilinear part of the action is invariant under the Langmann-Szabo duality. The model is a promising candidate for a renormalizable non-commutative U(1) gauge theory.
Energy Technology Data Exchange (ETDEWEB)
Varshovi, Amir Abbass [School of Mathematics, Institute for Research in Fundamental Sciences (IPM) and School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran (Iran, Islamic Republic of)
2013-07-15
The theory of α*-cohomology is studied thoroughly and it is shown that in each cohomology class there exists a unique 2-cocycle, the harmonic form, which generates a particular Groenewold-Moyal star product. This leads to an algebraic classification of translation-invariant non-commutative structures and shows that any general translation-invariant non-commutative quantum field theory is physically equivalent to a Groenewold-Moyal non-commutative quantum field theory.
Differential and complex geometry origins, abstractions and embeddings
Wells, Jr , Raymond O
2017-01-01
Differential and complex geometry are two central areas of mathematics with a long and intertwined history. This book, the first to provide a unified historical perspective of both subjects, explores their origins and developments from the sixteenth to the twentieth century. Providing a detailed examination of the seminal contributions to differential and complex geometry up to the twentieth century embedding theorems, this monograph includes valuable excerpts from the original documents, including works of Descartes, Fermat, Newton, Euler, Huygens, Gauss, Riemann, Abel, and Nash. Suitable for beginning graduate students interested in differential, algebraic or complex geometry, this book will also appeal to more experienced readers.
A non-perturbative study of non-commutative U(1) gauge theory
Energy Technology Data Exchange (ETDEWEB)
Nishimura, J. [High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki (Japan)]|[Graduate Univ. for Advanced Studies (SOKENDAI), Tsukuba (Japan). Dept. of Particle and Nuclear Physics; Bietenholz, W. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Susaki, Y. [High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki (Japan)]|[Tsukuba Univ. (Japan). Graduate School of Pure and Applied Science; Volkholz, J. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik
2007-06-15
We study U(1) gauge theory on a 4d non-commutative torus, where two directions are non-commutative. Monte Carlo simulations are performed after mapping the regularized theory onto a U(N) lattice gauge theory in d=2. At intermediate coupling strength, we find a phase in which open Wilson lines acquire non-zero vacuum expectation values, which implies the spontaneous breakdown of translational invariance. In this phase, various physical quantities obey clear scaling behaviors in the continuum limit with a fixed non- commutativity parameter {theta}, which provides evidence for a possible continuum theory. In the weak coupling symmetric phase, the dispersion relation involves a negative IR-singular term, which is responsible for the observed phase transition. (orig.)
Non-Commutative Integration, Zeta Functions and the Haar State for SU{sub q}(2)
Energy Technology Data Exchange (ETDEWEB)
Matassa, Marco, E-mail: marco.matassa@gmail.com [SISSA (Italy)
2015-12-15
We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group SU{sub q}(2). In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of SU{sub q}(2). We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of SU{sub q}(2). Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.
Simulation Results for U(1) Gauge Theory on Non-Commutative Spaces
Bietenholz, W; Nishimura, J; Susaki, Y; Torrielli, A; Volkholz, J
2007-01-01
We present numerical results for U(1) gauge theory in 2d and 4d spaces involving a non-commutative plane. Simulations are feasible thanks to a mapping of the non-commutative plane onto a twisted matrix model. In d=2 it was a long-standing issue if Wilson loops are (partially) invariant under area-preserving diffeomorphisms. We show that non-perturbatively this invariance breaks, including the subgroup SL(2,R). In both cases, d=2 and d=4, we extrapolate our results to the continuum and infinite volume by means of a Double Scaling Limit. In d=4 this limit leads to a phase with broken translation symmetry, which is not affected by the perturbatively known IR instability. Therefore the photon may survive in a non-commutative world.
Non-Commutative Integration, Zeta Functions and the Haar State for SU q (2)
Matassa, Marco
2015-12-01
We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group SU q (2). In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of SU q (2). We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of SU q (2). Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.
Relativistic Spectrum of Hydrogen Atom in Space-Time Non-Commutativity
Moumni, Mustafa; Zaim, Slimane; 10.1063/1.4715429
2012-01-01
We study space-time non-commutativity applied to the hydrogen atom via the Seiberg-Witten map and its phenomenological effects. We find that it modifies the Coulomb potential in the Hamiltonian and add an r-3 part. By calculating the energies from Dirac equation using perturbation theory, we study the modifications to the hydrogen spectrum. We find that it removes the degeneracy with respect to the total angular momentum quantum number and acts like a Lamb shift. Comparing the results with experimental values from spectroscopy, we get a new bound for the space-time non-commutative parameter. N.B: In precedent works (arXiv:0907.1904, arXiv:1003.5732 and arXiv:1006.4590), we have used the Bopp Shift formulation of non-commutativity but here use it \\`a la Seiberg-Witten in the Relativistic case.
Double Compactified d = 11 Supermembrane Dual as a Non-Commutative Super-Maxwell Theory
Martin, I; Restuccia, A
2000-01-01
The physical hamiltonian of the double compactified D=11 supermembrane dual with non trivial wrapping is explicitly obtained. It contains cubic and quartic interacting terms. It exactly agrees with the hamiltonian formulation of non-commutative super-Maxwell theory on the world volume, minimally coupled to seven scalars fields corresponding to the transverse coordinates to the brane. The non commutative star product is intrinsically obtained from the simplectic 2-form defined by the minimal configuration of the hamiltonian, that is by the pull-back to the world volume of the canonical conection 1-form on the Hopf fibring over $CP_n$. The constraint generating the area preserving diffeomorphism is reformulated as the Gauss constraint of the non-commutative super-Maxwell theory.
Rohwer, CM
2012-01-01
In this thesis we shall demonstrate that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the formalism used entails a description that is non-local in that it requires all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary to specify quantum states completely. The remainder of the thesis, will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended...
Duality and gauge invariance of non-commutative spacetime Podolsky electromagnetic theory
Abreu, Everton M. C.; Fernandes, Rafael L.; Mendes, Albert C. R.; Neto, Jorge Ananias; Neves, Mario, Jr.
2017-01-01
The interest in higher derivative field theories has its origin mainly in their influence concerning the renormalization properties of physical models and to remove ultraviolet divergences. In this paper, we have introduced the non-commutative (NC) version of the Podolsky theory and we investigated the effect of the non-commutativity over its original gauge invariance property. We have demonstrated precisely that the non-commutativity spoiled the primary gauge invariance of the original action under this primary gauge transformation. After that we have used the Noether dualization technique to obtain a dual and gauge invariant action. We have demonstrated that through the introduction of a Stueckelberg field in this NC model, we can also recover the primary gauge invariance. In this way, we have accomplished a comparison between both methods.
An alternative way to explain how non-commutativity arises in the bosonic string theory
De Andrade, M A
2015-01-01
In this work we will investigate how the non-commutativity arises into the string theory, \\textit{i.e.}, how the bosonic string theory attaches to a D3-brane in the presence of magnetic fields. In order to accomplish the proposal, we departure from the commutative two-dimensional harmonic oscillator, which after the application of the general Bopp's shifts Matrix Method, the non-commutative version of the two-dimensional harmonic oscillator is obtained. After that, this non-commutative harmonic oscillator will be mapped into the bosonic string theory in the light cone frame, which it now appears as a bosonic string theory attached to a D3-brane.
On Non-commuting Sets in a Finite p-group with Derived Subgroup of Prime Order
Institute of Scientific and Technical Information of China (English)
Wang Yu-lei; Liu He-guo
2016-01-01
Let G be a finite group. A nonempty subset X of G is said to be non-commuting if xy = yx for any x, y ∈ X with x = y. If |X| ≥ |Y| for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite p-group with derived subgroup of prime order.
Some Penrose transforms in complex differential geometry
Institute of Scientific and Technical Information of China (English)
ANCO; Stephen; BLAND; John; EASTWOOD; Michael
2006-01-01
In this article, we review a construction in the complex geometry often known as the Penrose transform. We then present two new applications of this transform. One concerns the construction of symmetries of the massless field equations from mathematical physics. The otherconcerns obstructions to the embedding of CR structures on the three-sphere.
Area-preserving diffeomorphisms in gauge theory on a non-commutative plane. A lattice study
Energy Technology Data Exchange (ETDEWEB)
Bietenholz, W. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Bigarini, A. [Univ. degli Studi di Perugia (Italy). Dipt. di Fisica]|[INFN, Sezione di Perugia (Italy)]|[Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik; Torrielli, A. [Massachusetts Institute of Technology, Cambridge, MA (United States). Center for Theoretical Physics, Lab. for Nuclear Sciences
2007-06-15
We consider Yang-Mills theory with the U(1) gauge group on a non-commutative plane. Perturbatively it was observed that the invariance of this theory under area-preserving diffeomorphisms (APDs) breaks down to a rigid subgroup SL(2,R). Here we present explicit results for the APD symmetry breaking at finite gauge coupling and finite non-commutativity. They are based on lattice simulations and measurements of Wilson loops with the same area but with a variety of different shapes. Our results confirm the expected loss of invariance under APDs. Moreover, they strongly suggest that non-perturbatively the SL(2,R) symmetry does not persist either. (orig.)
Encoding Phases using Commutativity and Non-commutativity in a Logical Framework
Amblard, Maxime
2011-01-01
This article presents an extension of Minimalist Categorial Gram- mars (MCG) to encode Chomsky's phases. These grammars are based on Par- tially Commutative Logic (PCL) and encode properties of Minimalist Grammars (MG) of Stabler. The first implementation of MCG were using both non- commutative properties (to respect the linear word order in an utterance) and commutative ones (to model features of different constituents). Here, we pro- pose to adding Chomsky's phases with the non-commutative tensor product of the logic. Then we could give account of the PIC just by using logical prop- erties of the framework.
Relativistic spectrum of hydrogen atom in the space-time non-commutativity
Energy Technology Data Exchange (ETDEWEB)
Moumni, Mustafa; BenSlama, Achour; Zaim, Slimane [Matter Sciences Department, Faculty of SE and SNV, University of Biskra (Algeria); Laboratoire de Physique Mathematique et Subatomique, Mentouri University, Constantine (Algeria); Matter Sciences Department, Faculty of Sciences, University of Batna (Algeria)
2012-06-27
We study space-time non-commutativity applied to the hydrogen atom and its phenomenological effects. We find that it modifies the Coulomb potential in the Hamiltonian and add an r{sup -3} part. By calculating the energies from Dirac equation using perturbation theory, we study the modifications to the hydrogen spectrum. We find that it removes the degeneracy with respect to the total angular momentum quantum number and acts like a Lamb shift. Comparing the results with experimental values from spectroscopy, we get a new bound for the space-time non-commutative parameter.
Radiative corrections to the non commutative photon propagator at one-loop order
Energy Technology Data Exchange (ETDEWEB)
Boutalbi, E.; Kouadik, S. [Laboratory of Particle Physics and Statistical Physics, Ecole Normale Superieure BP 92 Vieux kouba (Algeria); Faculty of Technologies Sciences,University of Medea (Algeria)
2012-06-27
We study the non-commutative gauge theory on the Moyal space. We add the harmonic potential introduced by Grosse and Wulkenhaar to the Maxwell Lagrange as well as the Gauge fixation. We determine the non-commutative U{sub *}(1) Gauge action which is invariant under the BRST transformations in the matrix basis. We determine in this basis the quadratic parts and the vertex of the Gauge field A{sub {mu}} and of the Faddeev-Popov ghost fields c(bar sign)andc. Finally, we study the perturbative correction to one loop order of the one point function in the matrix basis.
Aharonov-Casher effect for spin-1 particles in a non-commutative space
Energy Technology Data Exchange (ETDEWEB)
Mirza, B.; Narimani, R.; Zarei, M. [Isfahan University of Technology, Department of Physics, Isfahan (Iran)
2006-11-15
In this work, the Aharonov-Casher (AC) phase is calculated for spin-1 particles in a non-commutative space. The AC phase has previously been calculated from the Dirac equation in a non-commutative space using a gauge-like technique. In the spin-1 case, we use the Kemmer equation to calculate the phase in a similar manner. It is shown that the holonomy receives non-trivial kinematical corrections. By comparing the new result with the already known spin-1/2 case, one may conjecture a generalized formula for the corrections to holonomy for higher spins. (orig.)
On the non-commutative Local Main Conjecture for elliptic curves with complex multiplication
Venjakob, Otmar
2012-01-01
This paper is a natural continuation of the joint work [6] on non-commutative Main Conjectures for CM elliptic curves: now we concentrate on the local Main Conjecture or more precisely on the epsilon-isomorphism conjecture by Fukaya and Kato in [20]. Our results rely heavily on Kato's unpublished proof of (commutative) epsilon-isomorphisms for one dimensional representations of G_{Q_p} in [24]. For the convenience of the reader we give a slight modification or rather reformulation of it in the language of [20] and extend it to the (slightly non-commutative) semi-global setting.
Bastos, Catarina; Dias, Nuno Costa; Prata, João Nuno
2012-01-01
We show that a possible violation of the Robertson-Schr\\"odinger uncertainty principle may signal the existence of a deformation of the Heisenberg-Weyl algebra. More precisely, we prove that any Gaussian in phase-space (even if it violates the Robertson-Schr\\"odinger uncertainty principle) is always a quantum state of an appropriate non-commutative extension of quantum mechanics. Conversely, all canonical non-commutative extensions of quantum mechanics display states that violate the Robertson-Schr\\"odinger uncertainty principle.
First Simulation Results for the Photon in a Non-Commutative Space
Bietenholz, W; Nishimura, J; Susaki, Y; Volkholz, J
2005-01-01
We present preliminary simulation results for QED in a non-commutative 4d space-time, which is discretized to a fuzzy lattice. Its numerical treatment becomes feasible after its mapping onto a dimensionally reduced twisted Eguchi-Kawai matrix model. In this formulation we investigate the Wilson loops and in particular the Creutz ratios. This is an ongoing project which aims at non-perturbative predictions for the photon, which can be confronted with phenomenology in order to verify the possible existence of non-commutativity in nature.
Invertibility of random submatrices via the Non-Commutative Bernstein Inequality
Chrétien, Stéphane
2011-01-01
Let $X$ be a $n\\times p$ matrix. We provide a detailed study of the quasi isometry property for random submatrices of $X$ obtained by uniform column sampling. The analysis relies on a tail decoupling argument with explicit constants and a recent version of the Non-Commutative Bernstein inequality (NCBI) [14]. Our results complement those obtained in [13] for the moments of submatrices. They also generalize and improve on those in [2], which are based on a Non-Commutative Kahane- Kintchine inequality (NCKI).
Compactified D=11 Supermembranes and Symplectic Non-Commutative Gauge Theories
Martin, I; Restuccia, A
2001-01-01
It is shown that a double compactified D=11 supermembrane with non trivial wrapping may be formulated as a symplectic non-commutative gauge theory on the world volume. The symplectic non commutative structure is intrinsically obtained from the symplectic 2-form on the world volume defined by the minimal configuration of its hamiltonian. The gauge transformations on the symplectic fibration are generated by the area preserving diffeomorphisms on the world volume. Geometrically, this gauge theory corresponds to a symplectic fibration over a compact Riemman surface with a symplectic connection.
A treatise on the differential geometry of curves and surfaces
Eisenhart, Luther Pfahler
2004-01-01
Created especially for graduate students, this introductory treatise on differential geometry has been a highly successful textbook for many years. Its unusually detailed and concrete approach includes a thorough explanation of the geometry of curves and surfaces, concentrating on problems that will be most helpful to students. 1909 edition.
Noncommutative geometry with graded differential Lie algebras
Wulkenhaar, Raimar
1997-06-01
Starting with a Hilbert space endowed with a representation of a unitary Lie algebra and an action of a generalized Dirac operator, we develop a mathematical concept towards gauge field theories. This concept shares common features with the Connes-Lott prescription of noncommutative geometry, differs from that, however, by the implementation of unitary Lie algebras instead of associative * -algebras. The general scheme is presented in detail and is applied to functions ⊗ matrices.
ICMS Workshop on Differential Geometry and Continuum Mechanics
Grinfeld, Michael; Knops, R
2015-01-01
This book examines the exciting interface between differential geometry and continuum mechanics, now recognised as being of increasing technological significance. Topics discussed include isometric embeddings in differential geometry and the relation with microstructure in nonlinear elasticity, the use of manifolds in the description of microstructure in continuum mechanics, experimental measurement of microstructure, defects, dislocations, surface energies, and nematic liquid crystals. Compensated compactness in partial differential equations is also treated. The volume is intended for specialists and non-specialists in pure and applied geometry, continuum mechanics, theoretical physics, materials and engineering sciences, and partial differential equations. It will also be of interest to postdoctoral scientists and advanced postgraduate research students. These proceedings include revised written versions of the majority of papers presented by leading experts at the ICMS Edinburgh Workshop on Differential G...
Global Differential Geometry and Global Analysis
Pinkall, Ulrich; Simon, Udo; Wegner, Berd
1991-01-01
All papers appearing in this volume are original research articles and have not been published elsewhere. They meet the requirements that are necessary for publication in a good quality primary journal. E.Belchev, S.Hineva: On the minimal hypersurfaces of a locally symmetric manifold. -N.Blasic, N.Bokan, P.Gilkey: The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary. -J.Bolton, W.M.Oxbury, L.Vrancken, L.M. Woodward: Minimal immersions of RP2 into CPn. -W.Cieslak, A. Miernowski, W.Mozgawa: Isoptics of a strictly convex curve. -F.Dillen, L.Vrancken: Generalized Cayley surfaces. -A.Ferrandez, O.J.Garay, P.Lucas: On a certain class of conformally flat Euclidean hypersurfaces. -P.Gauduchon: Self-dual manifolds with non-negative Ricci operator. -B.Hajduk: On the obstruction group toexistence of Riemannian metrics of positive scalar curvature. -U.Hammenstaedt: Compact manifolds with 1/4-pinched negative curvature. -J.Jost, Xiaowei Peng: The geometry of moduli spaces of stabl...
FINAL REPORT: GEOMETRY AND ELEMENTARY PARTICLE PHYSICS
Energy Technology Data Exchange (ETDEWEB)
Singer, Isadore M.
2008-03-04
The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists’ quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energy for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.
On Some Isomorphisms between Bounded Linear Maps and Non-Commutative Lp-Spaces
Directory of Open Access Journals (Sweden)
E. J. Atto
2014-04-01
Full Text Available We define a particular space of bounded linear maps using a Von Neumann algebra and some operator spaces. By this, we prove some isomorphisms, and using interpolation in some particular cases, we get analogue of non-commutative Lp spaces.
Coincidence and ﬁxed point results for non-commuting maps
Directory of Open Access Journals (Sweden)
Abdul Latif
2008-06-01
Full Text Available In the setting of Banach spaces, some results on the existence of coincidence and common fixed points for single-valued and multivalued non-commuting maps with and without contractive type conditions are obtained.
The He-McKellar-Wilkens effect for spin-1 particles on non-commutative space
Institute of Scientific and Technical Information of China (English)
Li Kang; Sayipjamal Dulat; Wang Jian-Hua
2008-01-01
By using star product method,the He-McKellar-Wilkeus (HMW) effect for spin-one neutral particle on noncommutative (NC) space is studied.After solving the Kemmer-like equations on NC space,we obtain the topological HMW phase on NC space where the additional terms related to the space-space non-commutativity are given explicitly.
A Generalized Rule For Non-Commuting Operators in Extended Phase Space
Nasiri, S; Khademi, S.; Bahrami, S; Taati, F.
2005-01-01
A generalized quantum distribution function is introduced. The corresponding ordering rule for non-commuting operators is given in terms of a single parameter. The origin of this parameter is in the extended canonical transformations that guarantees the equivalence of different distribution functions obtained by assuming appropriate values for this parameter.
Some Aspects of Production Functions Differential Geometry
Directory of Open Access Journals (Sweden)
Cătălin Angelo Ioan
2017-04-01
Full Text Available The article deals with some aspects of differential production functions with examples for Cobb-Douglas function in two or three variables. There are studied in each case, the conditions of the parameters in order that the sectional curvature be constant.
The geometry of differential difference equations
Helminck, G.F.; Post, G.F.
1994-01-01
To each maximal commuting subalgebra h of glm(C) is associated a system of differential difference equations, generalizing several known systems. Starting from a Grassmann manifold, solutions are constructed, their properties are discussed and the relation with other systems is given. Finally it is shown how to express these solutions in T-functions.
New symbolic tools for differential geometry, gravitation, and field theory
Anderson, I. M.; Torre, C. G.
2012-01-01
DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, spinor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors, algebraic classification of solutions of the Einstein equations, and symmetry reduction of field equations.
Cartan for beginners differential geometry via moving frames and exterior differential systems
Ivey, Thomas A
2016-01-01
Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explici...
Differential geometry of complex vector bundles
Kobayashi, Shoshichi
2014-01-01
Holomorphic vector bundles have become objects of interest not only to algebraic and differential geometers and complex analysts but also to low dimensional topologists and mathematical physicists working on gauge theory. This book, which grew out of the author's lectures and seminars in Berkeley and Japan, is written for researchers and graduate students in these various fields of mathematics. Originally published in 1987. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeto
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.
Differential geometry the mathematical works of J. H. C. Whitehead
James, I M
1962-01-01
The Mathematical Works of J. H. C. Whitehead, Volume 1: Differential Geometry contains all of Whitehead's published work on differential geometry, along with some papers on algebras. Most of these were written in the period 1929-1937, but a few later articles are included. The book begins with a list of Whitehead's works, in chronological order of writing as well as a biographical note by M. H. A. Newman and Barbara Whitehead, and a mathematical appreciation by John Milnor. This is followed by separate chapters on topics such as linear connections; a method of obtaining normal representations
Einstein Geometrization Philosophy and Differential Identities in PAP-Geometry
Wanas, M I; Hanafy, H El; Osman, S N
2016-01-01
The importance of Einstein's geometrization philosophy, as an alternative to the least action principle, in constructing general relativity (GR), is illuminated. The role of differential identities in this philosophy is clarified. The use of Bianchi identity to write the field equations of GR is shown. Another similar identity in the absolute parallelism geometry is given. A more general differential identity in the parameterized absolute parallelism geometry is derived. Comparison and interrelationships between the above mentioned identities and their role in constructing field theories are discussed.
Differential forms and the geometry of general relativity
Dray, Tevian
2015-01-01
Differential Forms and the Geometry of General Relativity provides readers with a coherent path to understanding relativity. Requiring little more than calculus and some linear algebra, it helps readers learn just enough differential geometry to grasp the basics of general relativity.The book contains two intertwined but distinct halves. Designed for advanced undergraduate or beginning graduate students in mathematics or physics, most of the text requires little more than familiarity with calculus and linear algebra. The first half presents an introduction to general relativity that describes
Simulations results for U(1) gauge theory on non-commutative spaces
Energy Technology Data Exchange (ETDEWEB)
Bietenholz, W. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC; Bigarini, A. [Univ. degli Studi di Perugia (Italy). Dipt. di Fisica; Nishimura, J. [High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki (Japan)]|[Graduate Univ. for Advanced Studies Tsukuba (Japan). Dept. of Particle and Nuclear Physics; Susaki, Y. [High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki (Japan)]|[Tsukuba Univ. (Japan). Graduate School of Pure and Applied Science; Torrielli, A. [Massachusetts Institute of Technology (MIT), Cambridge, MA (United States). Center for Theoretical Physics, Lab. for Nuclear Sciences and Dept. of Physics; Volkholz, J. [Humboldt-Universitaet, Berlin (Germany). Inst. fuer Physik
2007-11-15
We present numerical results for U(1) gauge theory in 2d and 4d spaces involving a noncommutative plane. Simulations are feasible thanks to a mapping of the non-commutative plane onto a twisted matrix model. In d=2 it was a long-standing issue if Wilson loops are (partially) invariant under area-preserving diffeomorphisms. We show that non-perturbatively this invariance breaks, including the subgroup SL(2,R). In both cases, d=2 and d=4, we extrapolate our results to the continuum and infinite volume by means of a Double Scaling Limit. In d=4 this limit leads to a phase with broken translation symmetry, which is not affected by the perturbatively known IR instability. Therefore the photon may survive in a non-commutative world. (orig.)
A Dream of Yukawa — Non-Local Fields out of Non-Commutative Spacetime —
Naka, Shigefumi; Toyoda, Haruki; Takanashi, Takahiro; Umezawa, Eizo
The coordinates of κ-Minkowski spacetime form Lie algebraic elements, in which time and space coordinates do not commute in spite of that space coordinates commute each other. The non-commutativity is realized by a Planck-length-scale constant κ - 1( ne 0), which is a universal constant other than the light velocity under the κ-Poincare transformation. Such a non-commutative structure can be realized by SO(1,4) generators in dS4 spacetime. In this work, we try to construct a κ-Minkowski like spacetime with commutative 4-dimensional spacetime based on Adsn+1 spacetime. Another aim of this work is to study invariant wave equations in this spacetime from the viewpoint of non-local field theory by H. Yukawa, who expected to realize elementary particle theories without divergence according to this viewpoint.
Einstein–Podolski–Rosen paradox, non-commuting operator, complete wavefunction and entanglement
Indian Academy of Sciences (India)
Andrew Das Arulsamy
2014-03-01
Einstein, Podolski and Rosen (EPR) have shown that any wavefunction (subject to the Schrödinger equation) can describe the physical reality completely, and any two observables associated with two non-commuting operators can have simultaneous reality. In contrast, quantum theory claims that the wavefunction can capture the physical reality completely, and the physical quantities associated with two non-commuting operators cannot have simultaneous reality. The above contradiction is known as the EPR paradox. Here, we unambiguously expose that there is a hidden assumption made by EPR, which gives rise to this famous paradox. Putting the assumption right this time leads us not to the paradox, but only reinforces the correctness of the quantum theory. However, it is shown here that the entanglement phenomenon between two physically separated particles (they were entangled prior to separation) can only be proven to exist with a `proper’ measurement.
Late time acceleration in a non-commutative model of modified cosmology
Energy Technology Data Exchange (ETDEWEB)
Malekolkalami, B., E-mail: b.malakolkalami@uok.ac.ir [Department of Physics, University of Kurdistan, Pasdaran St., Sanandaj (Iran, Islamic Republic of); Atazadeh, K., E-mail: atazadeh@azaruniv.ac.ir [Department of Physics, Azarbaijan Shahid Madani University, 53714-161, Tabriz (Iran, Islamic Republic of); Vakili, B., E-mail: b-vakili@iauc.ac.ir [Department of Physics, Central Tehran Branch, Islamic Azad University, Tehran (Iran, Islamic Republic of)
2014-12-12
We investigate the effects of non-commutativity between the position–position, position–momentum and momentum–momentum of a phase space corresponding to a modified cosmological model. We show that the existence of such non-commutativity results in a Moyal Poisson algebra between the phase space variables in which the product law between the functions is of the kind of an α-deformed product. We then transform the variables in such a way that the Poisson brackets between the dynamical variables take the form of a usual Poisson bracket but this time with a noncommutative structure. For a power law expression for the function of the Ricci scalar with which the action of the gravity model is modified, the exact solutions in the commutative and noncommutative cases are presented and compared. In terms of these solutions we address the issue of the late time acceleration in cosmic evolution.
Late time acceleration in a non-commutative model of modified cosmology
Malekolkalami, B.; Atazadeh, K.; Vakili, B.
2014-12-01
We investigate the effects of non-commutativity between the position-position, position-momentum and momentum-momentum of a phase space corresponding to a modified cosmological model. We show that the existence of such non-commutativity results in a Moyal Poisson algebra between the phase space variables in which the product law between the functions is of the kind of an α-deformed product. We then transform the variables in such a way that the Poisson brackets between the dynamical variables take the form of a usual Poisson bracket but this time with a noncommutative structure. For a power law expression for the function of the Ricci scalar with which the action of the gravity model is modified, the exact solutions in the commutative and noncommutative cases are presented and compared. In terms of these solutions we address the issue of the late time acceleration in cosmic evolution.
Late time acceleration in a non-commutative model of modified cosmology
Directory of Open Access Journals (Sweden)
B. Malekolkalami
2014-12-01
Full Text Available We investigate the effects of non-commutativity between the position–position, position–momentum and momentum–momentum of a phase space corresponding to a modified cosmological model. We show that the existence of such non-commutativity results in a Moyal Poisson algebra between the phase space variables in which the product law between the functions is of the kind of an α-deformed product. We then transform the variables in such a way that the Poisson brackets between the dynamical variables take the form of a usual Poisson bracket but this time with a noncommutative structure. For a power law expression for the function of the Ricci scalar with which the action of the gravity model is modified, the exact solutions in the commutative and noncommutative cases are presented and compared. In terms of these solutions we address the issue of the late time acceleration in cosmic evolution.
Zaim, Slimane
2015-01-01
We study the effect of the non-commutativity on the creation of scalar particles from vacuum in the anisotropic universe space-time. We derive the deformed Klein-Gordon equation up to second order in the non-commutativity parameter using the general modified field equation. Then the canonical method based on Bogoliubov transformation is applied to calculate the probability of particle creation in vacuum and the corresponding number density in the $k$ mode. We deduce that the non-commutative space-time introduces a new source of particle creation.
Marginal and non-commutative deformations via non-abelian T-duality
Hoare, Ben; Thompson, Daniel C.
2017-02-01
In this short article we develop recent proposals to relate Yang-Baxter sigmamodels and non-abelian T-duality. We demonstrate explicitly that the holographic spacetimes associated to both (multi-parameter)- β-deformations and non-commutative deformations of N = 4 super Yang-Mills gauge theory including the RR fluxes can be obtained via the machinery of non-abelian T-duality in Type II supergravity.
Marginal and non-commutative deformations via non-abelian T-duality
Hoare, Ben
2016-01-01
In this short article we develop recent proposals to relate Yang-Baxter sigma-models and non-abelian T-duality. We demonstrate explicitly that the holographic space-times associated to both (multi-parameter)-$\\beta$-deformations and non-commutative deformations of ${\\cal N}=4$ super Yang-Mills gauge theory including the RR fluxes can be obtained via the machinery of non-abelian T-duality in Type II supergravity.
Non-commutative residue of projections in Boutet de Monvel's calculus
DEFF Research Database (Denmark)
Gaarde, Anders
2007-01-01
Using results by Melo, Nest, Schick, and Schrohe on the K-theory of Boutet de Monvel's calculus of boundary value problems, we show that the non-commutative residue introduced by Fedosov, Golse, Leichtnam, and Schrohe vanishes on projections in the calculus. This partially answers a question raised...... in a recent collaboration with Grubb, namely whether the residue is zero on sectorial projections for boundary value problems: This is confirmed to be true when the sectorial projections is in the calculus....
Non-commutative residue of projections in Boutet de Monvel's calculus
DEFF Research Database (Denmark)
Gaarde, Anders
2007-01-01
Using results by Melo, Nest, Schick, and Schrohe on the K-theory of Boutet de Monvel's calculus of boundary value problems, we show that the non-commutative residue introduced by Fedosov, Golse, Leichtnam, and Schrohe vanishes on projections in the calculus. This partially answers a question raised...... in a recent collaboration with Grubb, namely whether the residue is zero on sectorial projections for boundary value problems: This is confirmed to be true when the sectorial projections is in the calculus....
Toeplitz Quantization for Non-commutating Symbol Spaces such as SUq(2
Directory of Open Access Journals (Sweden)
Sontz Stephen Bruce
2016-08-01
Full Text Available Toeplitz quantization is defined in a general setting in which the symbols are the elements of a possibly non-commutative algebra with a conjugation and a possibly degenerate inner product. We show that the quantum group SUq(2 is such an algebra. Unlike many quantization schemes, this Toeplitz quantization does not require a measure. The theory is based on the mathematical structures defined and studied in several recent papers of the author; those papers dealt with some specific examples of this new Toeplitz quantization. Annihilation and creation operators are defined as densely defined Toeplitz operators acting in a quantum Hilbert space, and their commutation relations are discussed. At this point Planck’s constant is introduced into the theory. Due to the possibility of non-commuting symbols, there are now two definitions for anti-Wick quantization; these two definitions are equivalent in the commutative case. The Toeplitz quantization introduced here satisfies one of these definitions, but not necessarily the other. This theory should be considered as a second quantization, since it quantizes non-commutative (that is, already quantum objects. The quantization theory presented here has two essential features of a physically useful quantization: Planck’s constant and a Hilbert space where natural, densely defined operators act.
Differential geometry techniques for sets of nonlinear partial differential equations
Estabrook, Frank B.
1990-01-01
An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.
Sica, Louis
2012-01-01
The usual interpretation of the Greenberger, Horne, Zeilinger (GHZ) theorem is that only nonlocal hidden variables are consistent with quantum mechanics. This conclusion is reasoned from the fact that combinations of results of unperformed non-commutative measurement procedures (counterfactuals) do not agree with quantum mechanical predictions taking non-commutation into account. However, it is shown from simple counter-examples, that combinations of such counterfactuals are inconsistent with classical non-commutative measurement sequences as well. There is thus no regime in which the validity of combined non-commutative counterfactuals may be depended upon. As a consequence, negative conclusions regarding local hidden variables do not follow from the GHZ and Bell theorems as historically reasoned.
Pseudo-differential operators groups, geometry and applications
Zhu, Hongmei
2017-01-01
This volume consists of papers inspired by the special session on pseudo-differential operators at the 10th ISAAC Congress held at the University of Macau, August 3-8, 2015 and the mini-symposium on pseudo-differential operators in industries and technologies at the 8th ICIAM held at the National Convention Center in Beijing, August 10-14, 2015. The twelve papers included present cutting-edge trends in pseudo-differential operators and applications from the perspectives of Lie groups (Chapters 1-2), geometry (Chapters 3-5) and applications (Chapters 6-12). Many contributions cover applications in probability, differential equations and time-frequency analysis. A focus on the synergies of pseudo-differential operators with applications, especially real-life applications, enhances understanding of the analysis and the usefulness of these operators.
Institute of Scientific and Technical Information of China (English)
WEI Gao-Feng; LONG Chao-Yun; LONG Zheng-Wen; QIN Shui-Jie
2008-01-01
In this paper,the isotropic charged harmonic oscillator in uniform magnetic field is researched in the non-commutative phase space;the corresponding exact energy is obtained,and the analytic eigenfunction is presented in terms of the confluent hypergeometric function.It is shown that in the non-commutative space,the isotropic charged harmonic oscillator in uniform magnetic field has the similar behaviors to the Landau problem.
Coordinated standoff tracking of moving targets using differential geometry
Institute of Scientific and Technical Information of China (English)
Zhi-qiang SONG; Hua-xiong LI; Chun-lin CHEN; Xian-zhong ZHOU; Feng XU
2014-01-01
This research is concerned with coordinated standoff tracking, and a guidance law against a moving target is proposed by using differential geometry. We first present the geometry between the unmanned aircraft (UA) and the target to obtain the convergent solution of standoff tracking when the speed ratio of the UA to the target is larger than one. Then, the convergent solution is used to guide the UA onto the standoff tracking geometry. We propose an improved guidance law by adding a derivative term to the relevant algorithm. To keep the phase angle difference of multiple UAs, we add a second derivative term to the relevant control law. Simulations are done to demonstrate the feasibility and performance of the proposed approach. The proposed algo-rithm can achieve coordinated control of multiple UAs with its simplicity and stability in terms of the standoff distance and phase angle difference.
Application of Noncommutative Differential Geometry on Lattice to Anomaly
Fujiwara, T; Wu, K; Fujiwara, Takanori; Suzuki, Hiroshi; Wu, Ke
1999-01-01
The chiral anomaly in lattice abelian gauge theory is investigated by applying the geometric and topological method in noncommutative differential geometry(NCDG). A new kind of double complex and descent equation are proposed on infinite hypercubic lattice in arbitrary even dimensional Euclidean space, in the framework of NCDG. Using the general solutions to proposed descent equation, we derive the chiral anomaly in Abelian lattice gauge theory. The topological origin of anomaly is nothing but the Chern classes in NCDG.
System theory as applied differential geometry. [linear system
Hermann, R.
1979-01-01
The invariants of input-output systems under the action of the feedback group was examined. The approach used the theory of Lie groups and concepts of modern differential geometry, and illustrated how the latter provides a basis for the discussion of the analytic structure of systems. Finite dimensional linear systems in a single independent variable are considered. Lessons of more general situations (e.g., distributed parameter and multidimensional systems) which are increasingly encountered as technology advances are presented.
Quantum κ-deformed differential geometry and field theory
Mercati, Flavio
2016-03-01
I introduce in κ-Minkowski noncommutative spacetime the basic tools of quantum differential geometry, namely bicovariant differential calculus, Lie and inner derivatives, the integral, the Hodge-∗ and the metric. I show the relevance of these tools for field theory with an application to complex scalar field, for which I am able to identify a vector-valued four-form which generalizes the energy-momentum tensor. Its closedness is proved, expressing in a covariant form the conservation of energy-momentum.
Global differential geometry: An introduction for control engineers
Doolin, B. F.; Martin, C. F.
1982-01-01
The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.
Classification of 5-Dimensional MD-Algebras Having Non-Commutative Derived Ideals
Vu, Le Anh; Nghia, Tran Thi Hieu
2011-01-01
The paper presents a subclass of the class of MD5-algebras and MD5-groups, i.e. five dimensional solvable Lie algebras and Lie groups such that their orbits in the co-adjoint representation (K-orbits) are orbits of zero or maximal dimension. The main result of the paper is the classification up to an isomorphism of all MD5-algebras with the non-commutative derived ideal. With this result, we have the complete classification of 5-dimensional solvable Lie algebras.
On prime and semiprime rings with generalized derivations and non-commutative Banach algebras
Indian Academy of Sciences (India)
MOHD ARIF RAZA; NADEEM UR REHMAN
2016-08-01
Let $R$ be a prime ring of characteristic different from 2 and $m$ a fixed positive integer. If $R$ admits a generalized derivation associated with a nonzero deviation $d$ such that $[F(x), d(y)]_m = [x, y]$ for all $x$, $y$ in some appropriate subset of $R$, then $R$ is commutative. Moreover, we also examine the case $R$ is a semiprime ring. Finally, we apply the above result to Banach algebras, and we obtain a non-commutative version of the Singer--Werner theorem.
a Note on the Non-Commutative Laplace-Varadhan Integral Lemma
de Roeck, W.; Maes, Christian; Netočný, Karel; Rey-Bellet, Luc
We continue the study of the free energy of quantum lattice spin systems where to the local Hamiltonian H an arbitrary mean field term is added, a polynomial function of the arithmetic mean of some local observables X and Y that do not necessarily commute. By slightly extending a recent paper by Hiai, Mosonyi, Ohno and Petz [10], we prove in general that the free energy is given by a variational principle over the range of the operators X and Y. As in [10], the result is a non-commutative extension of the Laplace-Varadhan asymptotic formula.
Differential and Twistor Geometry of the Quantum Hopf Fibration
Brain, Simon
2011-01-01
We study a quantum version of the SU(2) Hopf fibration $S^7 \\to S^4$ and its associated twistor geometry. Our quantum sphere $S^7_q$ arises as the unit sphere inside a q-deformed quaternion space $\\mathbb{H}^2_q$. The resulting four-sphere $S^4_q$ is a quantum analogue of the quaternionic projective space $\\mathbb{HP}^1$. The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space $\\mathbb{CP}^3_q$ and use it to study a system of anti-self-duality equations on $S^4_q$, for which we find an `instanton' solution coming from the natural projection defining the tautological bundle over $S^4_q$.
Differential Geometry Applied to Rings and Möbius Nanostructures
DEFF Research Database (Denmark)
Lassen, Benny; Willatzen, Morten; Gravesen, Jens
2014-01-01
Nanostructure shape effects have become a topic of increasing interest due to advancements in fabrication technology. In order to pursue novel physics and better devices by tailoring the shape and size of nanostructures, effective analytical and computational tools are indispensable....... In this chapter, we present analytical and computational differential geometry methods to examine particle quantum eigenstates and eigenenergies in curved and strained nanostructures. Example studies are carried out for a set of ring structures with different radii and it is shown that eigenstate and eigenenergy...... at bending radii above 50 nm. In the second part of the chapter, a more complicated topological structure, the Möbius nanostructure, is analyzed and geometry effects for eigenstate properties are discussed including dependencies on the Möbius nanostructure width, length, thickness, and strain....
Noncommutative differential geometry, and the matrix representations of generalised algebras
Gratus, J.
1998-05-01
The underly ing algebra I or a noncummutative geometry is taken to be a matrix algebra, and the set of derivatives the ad joint of a subset of traceless matrices. This is sufficient to calculate the dual 1-forms, and show that the space of 1-firms is at free module over the algebra of matrices. The concept of a generalised algebra is delined and it is shown that this is required in order for the space of 2-forms to exist, The exterior derivative is generalised for higher-order forms and these are also shown to he free modules over the matrix algebra. Examples of mappings that preserve the differential Structure are peen, Also giken are four examples of matrix generalised algebras, and the corresponding noncommutntive geometries, including the cases where the generalised algebra corresponds to a representation of a Lie algebra or a q-deformed algebra.
Interpretation of the prominence differential emissions measure for 3 geometries
Schmahl, E. J.; Orrall, F. Q.
1986-01-01
Researchers have used prominence extreme ultraviolet line intensities observed from Skylab to derive the differential emission measure Q(T) in the prominence-corona (PC) interface from 3 x 10,000 to 3 times 1 million K, including the effects of Lyman Continuum absorption. Using lines both shortward and longward of the Lyman limit, researchers have estimated the importance of absorption as function of temperature. The magnitude of the absorption, as well as its rate of increase as a function of temperature, place limits on the thread scales and the character of the interfilar medium. Researchers have calculated models based on three assumed geometries: (1) threads with hot sheaths and cool cores; (2) isothermal threads; and (3) threads with longitudinal temperature gradients along the magnetic field. Comparison of the absorption computed from these models with the observed absorption in prominences shows that none of the geometries is totally satisfactory.
DiracQ: A Package for Algebraic Manipulation of Non-Commuting Quantum Variables
Directory of Open Access Journals (Sweden)
John G. Wright
2015-11-01
Full Text Available In several problems of quantum many body physics, one is required to handle complex expressions originating in the non-commutative nature of quantum operators. Their manipulation requires precise ordering and application of simplification rules. This can be cumbersome, tedious and error prone, and often a challenge to the most expert researcher. In this paper we present a software package DiracQ to facilitate such manipulations. The package DiracQ consists of functions based upon and extending considerably the symbolic capabilities of 'Mathematica'. With DiracQ, one can proceed with large scale algebraic manipulations of expressions containing combinations of ordinary numbers or symbols (the c-numbers and arbitrary sets of non-commuting variables (the q-numbers with user defined properties. The DiracQ package is user extendable and comes encoded with the algebraic properties of several standard operators in popular usage. These include Fermionic and Bosonic creation and annihilation operators, spin operators, and canonical position and momentum operators. An example book is provided with some suggestive calculations of large-scale algebraic manipulations.
M-theory in the Omega-background and 5-dimensional non-commutative gauge theory
Costello, Kevin
2016-01-01
The $\\Omega$-background is defined for supergravity, and a very general class of such backgrounds is constructed for 11-dimensional supergravity. 11-dimensional supergravity in a certain $\\Omega$-background is shown to be equivalent to a 5-dimensional non-commutative gauge theory of Chern-Simons type. M2 and M5 branes are expressed as 1 and 2-dimensional extended objects in the 5-dimensional gauge theory. This 5-dimensional gauge theory is shown to admit a consistent quantization with two coupling constants, despite being formally non-renormalizable. A check of a twisted version of AdS/CFT is performed relating this 5-dimensional non-commutative gauge theory to the theory on N M5 branes, wrapping an $A_{k-1}$ singularity and placed in an $\\Omega$-background. The operators on the M5 branes, in the $\\Omega$-background, are described by a certain chiral algebra which in the large N limit becomes a $W_{k+\\infty}$ algebra. This chiral algebra is recovered from the 5-dimensional gauge theory. This argument also pro...
Cosmic microwave background polarization in non-commutative space-time
Tizchang, S.; Batebi, S.; Haghighat, M.; Mohammadi, R.
2016-09-01
In the standard model of cosmology (SMC) the B-mode polarization of the CMB can be explained by the gravitational effects in the inflation epoch. However, this is not the only way to explain the B-mode polarization for the CMB. It can be shown that the Compton scattering in the presence of a background, besides generating a circularly polarized microwave, can lead to a B-mode polarization for the CMB. Here we consider the non-commutative (NC) space-time as a background to explore the CMB polarization at the last scattering surface. We obtain the B-mode spectrum of the CMB radiation by scalar perturbation of metric via a correction on the Compton scattering in NC-space-time in terms of the circular polarization power spectrum and the non-commutative energy scale. It can be shown that even for the NC scale as large as 20 TeV the NC-effects on the CMB polarization and the r parameter are significant. We show that the V-mode power spectrum can be obtained in terms of linearly polarized power spectrum in the range of micro- to nano-kelvin squared for the NC scale of about 1-20 TeV, respectively.
Cosmic microwave background polarization in non-commutative space-time
Energy Technology Data Exchange (ETDEWEB)
Tizchang, S.; Batebi, S. [Isfahan University of Technology, Department of Physics, Isfahan (Iran, Islamic Republic of); Haghighat, M. [Shiraz University, Department of Physics, Shiraz (Iran, Islamic Republic of); Mohammadi, R. [Iran Science and Technology Museum (IRSTM), Tehran (Iran, Islamic Republic of)
2016-09-15
In the standard model of cosmology (SMC) the B-mode polarization of the CMB can be explained by the gravitational effects in the inflation epoch. However, this is not the only way to explain the B-mode polarization for the CMB. It can be shown that the Compton scattering in the presence of a background, besides generating a circularly polarized microwave, can lead to a B-mode polarization for the CMB. Here we consider the non-commutative (NC) space-time as a background to explore the CMB polarization at the last scattering surface. We obtain the B-mode spectrum of the CMB radiation by scalar perturbation of metric via a correction on the Compton scattering in NC-space-time in terms of the circular polarization power spectrum and the non-commutative energy scale. It can be shown that even for the NC scale as large as 20 TeV the NC-effects on the CMB polarization and the r parameter are significant. We show that the V-mode power spectrum can be obtained in terms of linearly polarized power spectrum in the range of micro- to nano-kelvin squared for the NC scale of about 1-20 TeV, respectively. (orig.)
Lie groups, differential equations, and geometry advances and surveys
2017-01-01
This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.
Differential geometry based solvation model I: Eulerian formulation.
Chen, Zhan; Baker, Nathan A; Wei, G W
2010-11-01
This paper presents a differential geometry based model for the analysis and computation of the equilibrium property of solvation. Differential geometry theory of surfaces is utilized to define and construct smooth interfaces with good stability and differentiability for use in characterizing the solvent-solute boundaries and in generating continuous dielectric functions across the computational domain. A total free energy functional is constructed to couple polar and nonpolar contributions to the salvation process. Geometric measure theory is employed to rigorously convert a Lagrangian formulation of the surface energy into an Eulerian formulation so as to bring all energy terms into an equal footing. By minimizing the total free energy functional, we derive coupled generalized Poisson-Boltzmann equation (GPBE) and generalized geometric flow equation (GGFE) for the electrostatic potential and the construction of realistic solvent-solute boundaries, respectively. By solving the coupled GPBE and GGFE, we obtain the electrostatic potential, the solvent-solute boundary profile, and the smooth dielectric function, and thereby improve the accuracy and stability of implicit solvation calculations. We also design efficient second order numerical schemes for the solution of the GPBE and GGFE. Matrix resulted from the discretization of the GPBE is accelerated with appropriate preconditioners. An alternative direct implicit (ADI) scheme is designed to improve the stability of solving the GGFE. Two iterative approaches are designed to solve the coupled system of nonlinear partial differential equations. Extensive numerical experiments are designed to validate the present theoretical model, test computational methods, and optimize numerical algorithms. Example solvation analysis of both small compounds and proteins are carried out to further demonstrate the accuracy, stability, efficiency and robustness of the present new model and numerical approaches. Comparison is given to
Bassetto, A; Torrielli, A
2002-01-01
Commutative Yang-Mills theories in 1+1 dimensions exhibit an interesting interplay between geometrical properties and U(N) gauge structures: in the exact expression of a Wilson loop with $n$ windings a non trivial scaling intertwines $n$ and $N$. In the non-commutative case the interplay becomes tighter owing to the merging of space-time and ``internal'' symmetries in a larger group $U(\\infty)$. We perform an explicit perturbative calculation of such a loop up to ${\\cal O}(g^6)$; rather surprisingly, we find that in the contribution from the crossed graphs (the genuine non-commutative terms) the scaling we mentioned occurs for large $n$ and $N$ in the limit of maximal non-commutativity $\\theta=\\infty$. We present arguments in favour of the persistence of such a scaling at any perturbative order and succeed in summing the related perturbative series.
Directory of Open Access Journals (Sweden)
Marco Panero
2006-11-01
Full Text Available We review some recent progress in quantum field theory in non-commutative space, focusing onto the fuzzy sphere as a non-perturbative regularisation scheme. We first introduce the basic formalism, and discuss the limits corresponding to different commutative or non-commutative spaces. We present some of the theories which have been investigated in this framework, with a particular attention to the scalar model. Then we comment on the results recently obtained from Monte Carlo simulations, and show a preview of new numerical data, which are consistent with the expected transition between two phases characterised by the topology of the support of a matrix eigenvalue distribution.
De quelques questions relatives à la (co)homologie et à la descente en algèbre non commutative.
Nuss, P
2005-01-01
Le présent mémoire d'habilitation a pour objectif de donner une vue d'ensemble du contexte et de la teneur de nos travaux, ainsi qu'une présentation succincte de quelques directions de recherches ultérieures. Sa subdivision en trois chapitres reflète nos orientations thématiques : 1) (co)homologie de Hochschild et cyclique des algèbres ; 2) descente fidèlement plate non commutative ; 3) extensions (Hopf-)galoisiennes non commutatives
Lagrangian for Frenkel electron and position's non-commutativity due to spin
Deriglazov, Alexei A
2014-01-01
We construct relativistic-invariant spinning-particle Lagrangian without auxiliary variables. Spin is considered as a composed quantity constructed on the base of non-Grassmann vector-like variable. The variational problem guarantees both fixed value of spin and Frenkel condition on spin-tensor. Taking into account the Frenkel condition, we obtain, inevitably, relativistic corrections to the algebra of position variables: their classical brackets became noncommutative, with the "parameter of non-commutativity" proportional to the spin-tensor. This leads to a number of interesting consequences in quantum theory. We construct the relativistic quantum mechanics in canonical formalism (in physical-time parametrization) and in covariant formalism (in arbitrary parametrization). We show how state-vectors and operators of covariant formulation can be used to compute mean values of physical operators of position and spin. This proves relativistic covariance of canonical formalism. Various candidates for position and ...
Differential geometry based solvation model. III. Quantum formulation.
Chen, Zhan; Wei, Guo-Wei
2011-11-21
Solvation is of fundamental importance to biomolecular systems. Implicit solvent models, particularly those based on the Poisson-Boltzmann equation for electrostatic analysis, are established approaches for solvation analysis. However, ad hoc solvent-solute interfaces are commonly used in the implicit solvent theory. Recently, we have introduced differential geometry based solvation models which allow the solvent-solute interface to be determined by the variation of a total free energy functional. Atomic fixed partial charges (point charges) are used in our earlier models, which depends on existing molecular mechanical force field software packages for partial charge assignments. As most force field models are parameterized for a certain class of molecules or materials, the use of partial charges limits the accuracy and applicability of our earlier models. Moreover, fixed partial charges do not account for the charge rearrangement during the solvation process. The present work proposes a differential geometry based multiscale solvation model which makes use of the electron density computed directly from the quantum mechanical principle. To this end, we construct a new multiscale total energy functional which consists of not only polar and nonpolar solvation contributions, but also the electronic kinetic and potential energies. By using the Euler-Lagrange variation, we derive a system of three coupled governing equations, i.e., the generalized Poisson-Boltzmann equation for the electrostatic potential, the generalized Laplace-Beltrami equation for the solvent-solute boundary, and the Kohn-Sham equations for the electronic structure. We develop an iterative procedure to solve three coupled equations and to minimize the solvation free energy. The present multiscale model is numerically validated for its stability, consistency and accuracy, and is applied to a few sets of molecules, including a case which is difficult for existing solvation models. Comparison is made
Self quartic interaction for a scalar field in a non-commutative space with Lorentz invariance
Energy Technology Data Exchange (ETDEWEB)
Neves, M.J.; Abreu, Everton M.C. [UFRRJ, Seropedica, RJ (Brazil)
2013-07-01
Full text: The framework Doplicher-Fredenhagen-Roberts (DFR) of a noncommutative (NC) space-time is considered as alternative approach to study the NC space-time of the early Universe. In this formalism, the parameter of noncommutative θ{sup μν} is promoted to a coordinate of the space-time. The consequence of this statement is that we are describing a NC field theory with Lorentz invariance in a space-time with extra-dimension. The addition of a canonical momentum associated to θ-coordinate is a extension of the NC DFR, in which the effects of a new physics can emerge in the propagation of the fields along the extra-dimension. This extension is called Doplicher-Fredenhagen-Roberts-Amorim (DFRA) NC space-time. The main concept that we would like to emphasize from the outset is that the formalism demonstrated here will not be constructed introducing a NC parameter in the system, as usual. It will be generated naturally from an already NC space. We study a scalar field with self-quartic interaction ϕ{sup 4} ∗ in the approach of non-commutative space with Lorentz invariance. We compare the two frameworks, DFR and DFRA NC space-time. We obtain the Feynman rules in the Fourier space for the scalar propagator and vertex. The divergences are analyzed at the one loop approximation, in which the non-commutativity scale can improve the ultraviolet behavior for the mass correction in the propagator. (author)
Fluid lipid membranes: from differential geometry to curvature stresses.
Deserno, Markus
2015-01-01
A fluid lipid membrane transmits stresses and torques that are fully determined by its geometry. They can be described by a stress- and torque-tensor, respectively, which yield the force or torque per length through any curve drawn on the membrane's surface. In the absence of external forces or torques the surface divergence of these tensors vanishes, revealing them as conserved quantities of the underlying Euler-Lagrange equation for the membrane's shape. This review provides a comprehensive introduction into these concepts without assuming the reader's familiarity with differential geometry, which instead will be developed as needed, relying on little more than vector calculus. The Helfrich Hamiltonian is then introduced and discussed in some depth. By expressing the quest for the energy-minimizing shape as a functional variation problem subject to geometric constraints, as proposed by Guven (2004), stress- and torque-tensors naturally emerge, and their connection to the shape equation becomes evident. How to reason with both tensors is then illustrated with a number of simple examples, after which this review concludes with four more sophisticated applications: boundary conditions for adhering membranes, corrections to the classical micropipette aspiration equation, membrane buckling, and membrane mediated interactions.
Bearing diagnostics: A method based on differential geometry
Tian, Ye; Wang, Zili; Lu, Chen; Wang, Zhipeng
2016-12-01
The structures around bearings are complex, and the working environment is variable. These conditions cause the collected vibration signals to become nonlinear, non-stationary, and chaotic characteristics that make noise reduction, feature extraction, fault diagnosis, and health assessment significantly challenging. Thus, a set of differential geometry-based methods with superiorities in nonlinear analysis is presented in this study. For noise reduction, the Local Projection method is modified by both selecting the neighborhood radius based on empirical mode decomposition and determining noise subspace constrained by neighborhood distribution information. For feature extraction, Hessian locally linear embedding is introduced to acquire manifold features from the manifold topological structures, and singular values of eigenmatrices as well as several specific frequency amplitudes in spectrograms are extracted subsequently to reduce the complexity of the manifold features. For fault diagnosis, information geometry-based support vector machine is applied to classify the fault states. For health assessment, the manifold distance is employed to represent the health information; the Gaussian mixture model is utilized to calculate the confidence values, which directly reflect the health status. Case studies on Lorenz signals and vibration datasets of bearings demonstrate the effectiveness of the proposed methods.
An application of differential geometry to SSC magnet end winding
Energy Technology Data Exchange (ETDEWEB)
Cook, J.M. (Argonne National Lab., IL (USA))
1990-04-01
It is expected that a large fraction of the total cost of the proposed Superconducting Supercollider will be spent on magnets, and, as Leon Lederman has remarked, most of the cost of making a magnet is in the ends.'' Among the mechanical problems to be solved there is the construction of an end-configuration for the superconducting cables which will minimize their strain energy. The purpose of this paper is to promote the use of differential geometry in this minimization. The use will be illustrated by a specific application to the winding of dipole ends. The cables are assumed to be clamped so firmly that their strain is not altered by Lorentz stresses. 15 refs.
Triple differential cross sections of magnesium in doubly symmetric geometry
S, Y. Sun; X, Y. Miao; Xiang-Fu, Jia
2016-01-01
A dynamically screened three-Coulomb-wave (DS3C) method is applied to study the single ionization of magnesium by electron impact. Triple differential cross sections (TDCS) are calculated in doubly symmetric geometry at incident energies of 13.65, 17.65, 22.65, 27.65, 37.65, 47.65, 57.65, and 67.65 eV. Comparisons are made with experimental data and theoretical predictions from a three-Coulomb-wave function (3C) approach and distorted-wave Born approximation (DWBA). The overall agreement between the predictions of the DS3C model and the DWBA approach with the experimental data is satisfactory. Project supported by the National Natural Science Foundation of China (Grant No. 11274215).
Differential geometry based solvation model II: Lagrangian formulation.
Chen, Zhan; Baker, Nathan A; Wei, G W
2011-12-01
Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and biomolecular processes. The understanding of solvation is an essential prerequisite for the quantitative description and analysis of biomolecular systems. This work presents a Lagrangian formulation of our differential geometry based solvation models. The Lagrangian representation of biomolecular surfaces has a few utilities/advantages. First, it provides an essential basis for biomolecular visualization, surface electrostatic potential map and visual perception of biomolecules. Additionally, it is consistent with the conventional setting of implicit solvent theories and thus, many existing theoretical algorithms and computational software packages can be directly employed. Finally, the Lagrangian representation does not need to resort to artificially enlarged van der Waals radii as often required by the Eulerian representation in solvation analysis. The main goal of the present work is to analyze the connection, similarity and difference between the Eulerian and Lagrangian formalisms of the solvation model. Such analysis is important to the understanding of the differential geometry based solvation model. The present model extends the scaled particle theory of nonpolar solvation model with a solvent-solute interaction potential. The nonpolar solvation model is completed with a Poisson-Boltzmann (PB) theory based polar solvation model. The differential geometry theory of surfaces is employed to provide a natural description of solvent-solute interfaces. The optimization of the total free energy functional, which encompasses the polar and nonpolar contributions, leads to coupled potential driven geometric flow and PB equations. Due to the development of singularities and nonsmooth manifolds in the Lagrangian representation, the resulting potential-driven geometric flow equation is embedded into the Eulerian representation for the purpose of
Tensors and Differential Geometry Applied to Analytic and Numerical Coordinate Generation
1981-01-01
Schild [9], Brand [10], Spain [111, Truesdell and Toupin [12], Struik [13], Sokolnikoff [14], Willmore [15], O’Neill [16], and Kreyszig [17], [18], on...Oxford, At The Clarendon Press (1959). [16] O’Neill, B., Elementary Differential Geometry, Academic Press, New York (1966). 193 [17) Kreyszig , E...Differential Geometry, Mathematical Exposition No. 11, University of Toronto Press (1959) [181 Kreyszig , E., Introduction to Differential Geometry and
Parameter optimization in differential geometry based solvation models.
Wang, Bao; Wei, G W
2015-10-01
Differential geometry (DG) based solvation models are a new class of variational implicit solvent approaches that are able to avoid unphysical solvent-solute boundary definitions and associated geometric singularities, and dynamically couple polar and non-polar interactions in a self-consistent framework. Our earlier study indicates that DG based non-polar solvation model outperforms other methods in non-polar solvation energy predictions. However, the DG based full solvation model has not shown its superiority in solvation analysis, due to its difficulty in parametrization, which must ensure the stability of the solution of strongly coupled nonlinear Laplace-Beltrami and Poisson-Boltzmann equations. In this work, we introduce new parameter learning algorithms based on perturbation and convex optimization theories to stabilize the numerical solution and thus achieve an optimal parametrization of the DG based solvation models. An interesting feature of the present DG based solvation model is that it provides accurate solvation free energy predictions for both polar and non-polar molecules in a unified formulation. Extensive numerical experiment demonstrates that the present DG based solvation model delivers some of the most accurate predictions of the solvation free energies for a large number of molecules.
Non-Commutative Worlds - Classical Constraints, Relativity and the Bianchi Identity
Kauffman, Louis H
2011-01-01
This paper shows how discrete measurement leads to commutators and how discrete derivatives are naturally represented by commutators in a non-commutative extension of the calculus in which they originally occurred. We show how the square root of minus one (i) arises naturally as a time-sensitive observable for an elementary oscillator. In this sense the square root of minus one is a clock and/or a clock/observer. This sheds new light on Wick rotation, which replaces t (temporal quantity) by it. In this view, the Wick rotation replaces numerical time with elementary temporal observation. The relationship of this remark with the Heisenberg commutator [P,Q]=ihbar is explained in the Introduction. After a review of previous work, the paper begins with a section of iterants - a generalization of the complex numbers as described above. This generalization includes all of matrix algebra in a temporal interpretation. We then give a generalization of the Feynman-Dyson derivation of electromagnetism in the context of n...
Row Sampling for Matrix Algorithms via a Non-Commutative Bernstein Bound
Magdon-Ismail, Malik
2010-01-01
We focus the use of \\emph{row sampling} for approximating matrix algorithms. We give applications to matrix multipication; sparse matrix reconstruction; and, \\math{\\ell_2} regression. For a matrix \\math{\\matA\\in\\R^{m\\times d}} which represents \\math{m} points in \\math{d\\ll m} dimensions, all of these tasks can be achieved in \\math{O(md^2)} via the singular value decomposition (SVD). For appropriate row-sampling probabilities (which typically depend on the norms of the rows of the \\math{m\\times d} left singular matrix of \\math{\\matA} (the \\emph{leverage scores}), we give row-sampling algorithms with linear (up to polylog factors) dependence on the stable rank of \\math{\\matA}. This result is achieved through the application of non-commutative Bernstein bounds. We then give, to our knowledge, the first algorithms for computing approximations to the appropriate row-sampling probabilities without going through the SVD of \\math{\\matA}. Thus, these are the first \\math{o(md^2)} algorithms for row-sampling based appro...
Lagrangian for Frenkel electron and position's non-commutativity due to spin
Energy Technology Data Exchange (ETDEWEB)
Deriglazov, Alexei A. [Universidade Federal de Juiz de Fora, Depto. de Matematica, ICE, Juiz de Fora, MG (Brazil); Tomsk Polytechnic University, Laboratory of Mathematical Physics, Tomsk (Russian Federation); Pupasov-Maksimov, Andrey M. [Universidade Federal de Juiz de Fora, Depto. de Matematica, ICE, Juiz de Fora, MG (Brazil)
2014-10-15
We construct a relativistic spinning-particle Lagrangian where spin is considered as a composite quantity constructed on the base of a non-Grassmann vector-like variable. The variational problem guarantees both a fixed value of the spin and the Frenkel condition on the spin-tensor. The Frenkel condition inevitably leads to relativistic corrections of the Poisson algebra of the position variables: their classical brackets became noncommutative. We construct the relativistic quantum mechanics in the canonical formalism (in the physical-time parametrization) and in the covariant formalism (in an arbitrary parametrization). We show how state vectors and operators of the covariant formulation can be used to compute the mean values of physical operators in the canonical formalism, thus proving its relativistic covariance. We establish relations between the Frenkel electron and positive-energy sector of the Dirac equation. Various candidates for the position and spin operators of an electron acquire clear meaning and interpretation in the Lagrangian model of the Frenkel electron. Our results argue in favor of Pryce's (d)-type operators as the spin and position operators of Dirac theory. This implies that the effects of non-commutativity could be expected already at the Compton wavelength. We also present the manifestly covariant form of the spin and position operators of the Dirac equation. (orig.)
Sica, Louis
2011-01-01
As discussed below, Bell's inequalities and experimental results rule out commutative hidden variable models as a basis for Bell correlations, but not necessarily non-commutative probability models. A local probability model is constructed for Bell correlations based on non-commutative operations involving polarizers. As in the entanglement model, the Bell correlation is obtained from a probability calculus without explicit use of deterministic hidden variables. The probability calculus used is associated with chaotic light. Joint wave intensity correlations at spatially separated polarization analyzers are computed using common information originating at the source. When interpreted as photon count rates, these yield quantum mechanical joint probabilities after the contribution of indeterminate numbers of photon pairs greater than one is subtracted out. The formalism appears to give a local account of Bell correlations.
Non-commutative U(1) gauge theory on R{sub {theta}}{sup 4} with oscillator term and BRST symmetry
Energy Technology Data Exchange (ETDEWEB)
Blaschke, D.N.; Schweda, M. [Vienna Univ. of Technology, Institute for Theoretical Physics (Austria); Grosse, H. [Vienna Univ., Faculty of Physics(Austria)
2007-09-15
Inspired by the renormalizability of the non-commutative {phi}{sup 4} model with added oscillator term, we formulate a non-commutative gauge theory, where the oscillator enters as a gauge fixing term in a BRST (Becchi-Rouet-Stora-Tyutin) invariant manner. All propagators turn out to be essentially given by the Mehler kernel and the bilinear part of the action is invariant under the Langmann-Szabo duality. The model is a promising candidate for a renormalizable non-commutative U(1) gauge theory. (authors)
Modern Differential Geometry For Physicists. 2nd Edn
Energy Technology Data Exchange (ETDEWEB)
Chrusciel, P T [Universite de Tours (France)
2006-06-21
Most of us sometimes have to face a student asking: 'What do I need to get started on this'. (In my case 'this' would typically be a topic in general relativity.) After thinking about it for quite a while, and consulting candidate texts again and again, a few days later I usually end up saying: read this chapter in book I (but without going too much detail), then that chapter in book II (but ignore all those comments), then the first few sections of this review paper (but do not try to work out equations NN to NNN), and then come back to see me. In the unlikely event that the student comes back without changing the topic, there follows quite a bit of explaining on a blackboard over the following weeks. From now on I will say: get acquainted with the material covered by this book. As far as Isham's book is concerned, 'this' in the student's question above can stand for any topic in theoretical physics which touches upon differential geometry (and I can only think of very few which do not). Said plainly: this book contains most of the introductory material necessary to get started in general relativity, or those branches of mathematical physics which require differential geometry. A student who has mastered the notions presented in the book will have a solid basis to continue into specialized topics. I am not aware of any other book which would be as useful as this one in terms of the spectrum of topics covered, stopping at the right place to get sufficient introductory insight. According to the publisher, these lecture notes are the content of an introductory course on differential geometry which is taken by first-year theoretical physics PhD students, or by students attending the one-year MSc course 'Quantum Fields and Fundamental Forces' at Imperial College, London. The volume is divided into six chapters: - An Introduction to Topology; - Differential Manifolds; - Vector Fields and n-Forms; - Lie Groups; - Fibre
Vargas, José G
2014-01-01
This is a book that the author wishes had been available to him when he was student. It reflects his interest in knowing (like expert mathematicians) the most relevant mathematics for theoretical physics, but in the style of physicists. This means that one is not facing the study of a collection of definitions, remarks, theorems, corollaries, lemmas, etc. but a narrative - almost like a story being told - that does not impede sophistication and deep results. It covers differential geometry far beyond what general relativists perceive they need to know. And it introduces readers to other areas
Institute of Scientific and Technical Information of China (English)
YIN Yajun; WU Jiye; YIN Jie
2008-01-01
To make the geometrical basis for soft matters with curved surfaces such as biomembranes as simple as possible, a symmetrical analytical system was developed in conventional differential geometry.The conventional second fundamental tensor is replaced by the so-called conjugate fundamental tensor.Because the differential properties of the conjugate fundamental tensor and the first fundamental tensor are symmetrical, the symmetrical analytical system including the symmetrical differential operators, symmetrical differential characteristics, and symmetrical integral theorems for tensor fields defined on curved surfaces can be constructed. From the symmetrical analytical system, the symmetrical integral theorems for mean curvature and Gauss curvature, with which the symmetrical Minkowski integral formulas are easily deduced just as special cases, can be derived. The applications of this symmetrical analytical system to biology not only display its simplicity and beauty, but also show its powers in depicting the symmetrical patterns of net-works of biomembrane nanotubes. All these symmetrical patterns in soft matters should be just the reason-able and natural results of the symmetrical analytical system.
Exact Random Walk Distributions using Noncommutative Geometry
Bellissard, J; Barelli, A; Claro, F; Bellissard, Jean; Camacho, Carlos J; Barelli, Armelle; Claro, Francisco
1997-01-01
Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length $ N $ on a two-dimensional square lattice for large $ N $, taking into account finite size contributions.
CutFEM : Discretizing geometry and partial differential equations
Burman, Erik; Claus, Susanne; Hansbo, Peter; Larson, Mats G.; Massing, Andre
2015-01-01
We discuss recent advances on robust unfitted finite element methods on cut meshes. These methods are designed to facilitate computations on complex geometries obtained, for example, from computer-aided design or image data from applied sciences. Both the treatment of boundaries and interfaces and the discretization of PDEs on surfaces are discussed and illustrated numerically.
Some questions of differential geometry in the large
Shikin, E V
1996-01-01
This collection contains articles that present recent results by geometers in Russia and the Ukraine. Papers in the collection deal with various questions related to the structure, symmetries, and embeddings of submanifolds in Euclidean and pseudo-Euclidian spaces. This collection offers a review of the challenges facing specialists in geometry in the large and features current research in the field.
2015-04-01
of dislocations in anisotropic crystals, Int. J. Eng. Sci. 5, 171–190 (1967). [92] A. Yavari and A. Goriely, Riemann -Cartan geometry of nonlinear...distributed point defects, Proc. R. Soc. Lond. A 468, 3902–3922 (2012). [94] A. Yavari and A. Goriely, Riemann -Cartan geometry of nonlinear disclination...ARL-RP-0522 ● APR 2015 US Army Research Laboratory Defects in Nonlinear Elastic Crystals: Differential Geometry , Finite
Curriculum, Translation, and Differential Functioning of Measurement and Geometry Items
Emenogu, Barnabas C.; Childs, Ruth A.
2005-01-01
A test item exhibits differential item functioning (DIF) if students with the same ability find it differentially difficult. When the item is administered in French and English, differences in language difficulty and meaning are the most likely explanations. However, curriculum differences may also contribute to DIF. The responses of Ontario…
Wei, Guo Wei; Baker, Nathan A.
2014-01-01
This chapter reviews the differential geometry-based solvation and electrolyte transport for biomolecular solvation that have been developed over the past decade. A key component of these methods is the differential geometry of surfaces theory, as applied to the solvent-solute boundary. In these approaches, the solvent-solute boundary is determined by a variational principle that determines the major physical observables of interest, for example, biomolecular surface area, enclosed volume, el...
Bożejko, Marek; Lytvynov, Eugene
2011-03-01
Let T be an underlying space with a non-atomic measure σ on it. In [ Comm. Math. Phys. 292, 99-129 (2009)] the Meixner class of non-commutative generalized stochastic processes with freely independent values, {ω=(ω(t))_{tin T}} , was characterized through the continuity of the corresponding orthogonal polynomials. In this paper, we derive a generating function for these orthogonal polynomials. The first question we have to answer is: What should serve as a generating function for a system of polynomials of infinitely many non-commuting variables? We construct a class of operator-valued functions {Z=(Z(t))_{tin T}} such that Z( t) commutes with ω( s) for any {s,tin T}. Then a generating function can be understood as {G(Z,ω)=sum_{n=0}^infty int_{T^n}P^{(n)}(ω(t_1),dots,ω(t_n))Z(t_1)dots Z(t_n)} {σ(dt_1) dots σ(dt_n)} , where {P^{(n)}(ω(t_1),dots,ω(t_n))} is (the kernel of the) n th orthogonal polynomial. We derive an explicit form of G( Z, ω), which has a resolvent form and resembles the generating function in the classical case, albeit it involves integrals of non-commuting operators. We finally discuss a related problem of the action of the annihilation operators {partial_t,t in T} . In contrast to the classical case, we prove that the operators ∂ t related to the free Gaussian and Poisson processes have a property of globality. This result is genuinely infinite-dimensional, since in one dimension one loses the notion of globality.
Pedoe, Dan
1988-01-01
""A lucid and masterly survey."" - Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to he
Wilson Line Correlators in N=4 Non-commutative Gauge Theory on S^2 x S^2
Kitazawa, Y; Tomino, D
2004-01-01
We investigate the Wilson line correlators dual to supergravity multiplets in N=4 non-commutative gauge theory on S^2 x S^2. We find additional non-analytic contributions to the correlators due to UV/IR mixing in comparison to ordinary gauge theory. Although they are no longer BPS off shell, their renormalization effects are finite as long as they carry finite momenta. We propose a renormalization procedure to obtain local operators with no anomalous dimensions in perturbation theory. We reflect on our results from dual supergravity point of view. We show that supergravity can account for both IR and UV/IR contributions.
Indian Academy of Sciences (India)
Barbara Jasiulis-Gołdyn; Anna Kula
2012-08-01
The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the -convolution and the (,1)-convolution satisfy the Urbanik’s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.
Residues of Logarithmic Differential Forms in Complex Analysis and Geometry
Institute of Scientific and Technical Information of China (English)
A.G.Aleksandrov
2014-01-01
In the article, we discuss basic concepts of the residue theory of logarithmic and multi-logarithmic differential forms, and describe some aspects of the theory, de-veloped by the author in the past few years. In particular, we introduce the notion of logarithmic differential forms with the use of the classical de Rham lemma and give an explicit description of regular meromorphic differential forms in terms of residues of logarithmic or multi-logarithmic differential forms with respect to hypersurfaces, com-plete intersections or pure-dimensional Cohen-Macaulay spaces. Among other things, several useful applications are considered, which are related with the theory of holo-nomic D-modules, the theory of Hodge structures, the theory of residual currents and others.
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
Tensors and Riemannian geometry with applications to differential equations
Ibragimov, Nail H
2015-01-01
This graduate textbook begins by introducing Tensors and Riemannian Spaces, and then elaborates their application in solving second-order differential equations, and ends with introducing theory of relativity and de Sitter space. Based on 40 years of teaching experience, the author compiles a well-developed collection of examples and exercises to facilitate the reader’s learning.
Differential geometry and topology with a view to dynamical systems
Burns, Keith
2005-01-01
MANIFOLDSIntroductionReview of topological conceptsSmooth manifoldsSmooth mapsTangent vectors and the tangent bundleTangent vectors as derivationsThe derivative of a smooth mapOrientationImmersions, embeddings and submersionsRegular and critical points and valuesManifolds with boundarySard's theoremTransversalityStabilityExercisesVECTOR FIELDS AND DYNAMICAL SYSTEMSIntroductionVector fieldsSmooth dynamical systemsLie derivative, Lie bracketDiscrete dynamical systemsHyperbolic fixed points and periodic orbitsExercisesRIEMANNIAN METRICSIntroductionRiemannian metricsStandard geometries on surfacesExercisesRIEMANNIAN CONNECTIONS AND GEODESICSIntroductionAffine connectionsRiemannian connectionsGeodesicsThe exponential mapMinimizing properties of geodesicsThe Riemannian distanceExercisesCURVATUREIntroductionThe curvature tensorThe second fundamental formSectional and Ricci curvaturesJacobi fieldsManifolds of constant curvatureConjugate pointsHorizontal and vertical sub-bundlesThe geodesic flowExercisesTENSORS AND DI...
Surfaces in 4-space from the affine differential geometry viewpoint
Luis Florial Espinoza Sánchez
2014-01-01
In this thesis, we study locally strictly convex surfaces from the affine differential viewpoint and generalize some tools for locally strictly submanifolds of codimension 2. We introduce a family of affine metrics on a locally strictly convex surface M in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if M is immersed in a locally atrictly convex hyperquadric, then the symmetric and the antisymmetric planes coincid...
Differential geometry of the space of Ising models
Machta, Benjamin; Chachra, Ricky; Transtrum, Mark; Sethna, James
2012-02-01
We use information geometry to understand the emergence of simple effective theories, using an Ising model perturbed with terms coupling non-nearest-neighbor spins as an example. The Fisher information is a natural metric of distinguishability for a parameterized space of probability distributions, applicable to models in statistical physics. Near critical points both the metric components (four-point susceptibilities) and the scalar curvature diverge with corresponding critical exponents. However, connections to Renormalization Group (RG) ideas have remained elusive. Here, rather than looking at RG flows of parameters, we consider the reparameterization-invariant flow of the manifold itself. To do this we numerically calculate the metric in the original parameters, taking care to use only information available after coarse-graining. We show that under coarse-graining the metric contracts very anisotropically, leading to a ``sloppy'' spectrum with the metric's Eigenvalues spanning many orders of magnitude. Our results give a qualitative explanation for the success of simple models: most directions in parameter space become fundamentally indistinguishable after coarse-graining.
Complex J-Symplectic Geometry With Application to Ordinary Differential Operators
Institute of Scientific and Technical Information of China (English)
王万义
2001-01-01
@@In this paper, we deal with complex J-symplectic geometry with application to ordinary differential operators. We define complex J-symplectic spaces and their J-Lagrangian subspaces and complete J-Lagrangian subspaces, and then we discuss their basic algebraic properties. Then we apply them to the theory of J-selfadjoint operators and give J-symplectic geometry complete characterizations of J-selfadjoint extensions of J-symmetric operators.
A Computational Differential Geometry Approach to Grid Generation
Liseikin, Vladimir D
2007-01-01
The process of breaking up a physical domain into smaller sub-domains, known as meshing, facilitates the numerical solution of partial differential equations used to simulate physical systems. This monograph gives a detailed treatment of applications of geometric methods to advanced grid technology. It focuses on and describes a comprehensive approach based on the numerical solution of inverted Beltramian and diffusion equations with respect to monitor metrics for generating both structured and unstructured grids in domains and on surfaces. In this second edition the author takes a more detailed and practice-oriented approach towards explaining how to implement the method by: Employing geometric and numerical analyses of monitor metrics as the basis for developing efficient tools for controlling grid properties. Describing new grid generation codes based on finite differences for generating both structured and unstructured surface and domain grids. Providing examples of applications of the codes to the genera...
Shape Morphing of Complex Geometries Using Partial Differential Equations
Directory of Open Access Journals (Sweden)
Gabriela González Castro
2007-11-01
Full Text Available An alternative technique for shape morphing using a surface generating method using partial differential equations is outlined throughout this work. The boundaryvalue nature that is inherent to this surface generation technique together with its mathematical properties are hereby exploited for creating intermediate shapes between an initial shape and a final one. Four alternative shape morphing techniques are proposed here. The first one is based on the use of a linear combination of the boundary conditions associated with the initial and final surfaces, the second one consists of varying the Fourier mode for which the PDE is solved whilst the third results from a combination of the first two. The fourth of these alternatives is based on the manipulation of the spine of the surfaces, which is computed as a by-product of the solution. Results of morphing sequences between two topologically nonequivalent surfaces are presented. Thus, it is shown that the PDE based approach for morphing is capable of obtaining smooth intermediate surfaces automatically in most of the methodologies presented in this work and the spine has been revealed as a powerful tool for morphing surfaces arising from the method proposed here.
Chern-Simons in the Seiberg-Witten map for non-commutative Abelian gauge theories in 4D
Picariello, M; Sorella, S P; Picariello, Marco; Quadri, Andrea; Sorella, Silvio P.
2002-01-01
A cohomological BRST characterization of the Seiberg-Witten (SW) map is given. We prove that the coefficients of the SW map can be identified with elements of the cohomology of the BRST operator modulo a total derivative. As an example, it will be illustrated how the first coefficients of the SW map can be written in terms of the Chern-Simons three form. This suggests a deep topological and geometrical origin of the SW map. The existence of the map for both Abelian and non-Abelian case is discussed. By using a recursive argument and the associativity of the $\\star$-product, we shall be able to prove that the Wess-Zumino consistency condition for non-commutative BRST transformations is fulfilled. The recipe of obtaining an explicit solution by use of the homotopy operator is briefly reviewed in the Abelian case.
One-Loop Calculations and Detailed Analysis of the Localized Non-Commutative p^{-2} U(1 Gauge Model
Directory of Open Access Journals (Sweden)
Daniel N. Blaschke
2010-05-01
Full Text Available This paper carries forward a series of articles describing our enterprise to construct a gauge equivalent for the θ-deformed non-commutative p^{-2} model originally introduced by Gurau et al. [Comm. Math. Phys. 287 (2009, 275-290]. It is shown that breaking terms of the form used by Vilar et al. [J. Phys. A: Math. Theor. 43 (2010, 135401, 13 pages] and ourselves [Eur. Phys. J. C: Part. Fields 62 (2009, 433-443] to localize the BRST covariant operator (D^2θ^2D^2^{-1} lead to difficulties concerning renormalization. The reason is that this dimensionless operator is invariant with respect to any symmetry of the model, and can be inserted to arbitrary power. In the present article we discuss explicit one-loop calculations, and analyze the mechanism the mentioned problems originate from.
Classical geometries defined by exterior differential systems on higher frame bundles
Estabrook, Frank B.; Wahlquist, Hugo D.
1989-01-01
Exterior differential ideals are discussed, and sets of invariant generators presented, for Reimannian, conformal and projective geometries, and for specializations such as Ricci-flat, self-dual and Einstein-Maxwell theories. The Cartan characteristic integers are explicitly calculated, and involutory basis forms found, for each of these (specialized to four dimensions), exposing their algebraic structure and showing how they generate well-posed sets of partial differential equations.
Synthetic Differential Geometry A Way to Intuitionistic Models of General Relativity in Toposes
Grinkevich, Y B
1996-01-01
W.Lawvere suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E, particular in some elementary topos. The synthetic differential geometry (SDG) is the theory developed by A.Kock in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a some nondegenerate commutative ring R of a line type in E. In this work we shall show that SDG allows to develop a Riemannian geometry and write the Einstein's equations of a field on pseudo-Riemannian formal manifold. This give a way for constructing a intuitionistic models of general relativity in suitable toposes.
Smooth spaces versus continuous spaces in models for synthetic differential geometry
Reyes, G.E.; Moerdijk, I.
1984-01-01
In topos models for synthetic differential geometry we study connections between smooth spaces (which interpret synthetic calculus) and continuous spaces (which interpret intuitionistic analysis). Our main tools are adjoint retractions of toposes and the standard map from the smooth reals to the con
Control of Differentiation of Human Mesenchymal Stem Cells by Altering the Geometry of Nanofibers
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Satoshi Fujita
2012-01-01
Full Text Available Effective differentiation of mesenchymal stem cells (MSCs is required for clinical applications. To control MSC differentiation, induction media containing different types of soluble factors have been used to date; however, it remains challenging to obtain a uniformly differentiated population of an appropriate quality for clinical application by this approach. We attempted to develop nanofiber scaffolds for effective MSC differentiation by mimicking anisotropy of the extracellular matrix structure, to assess whether differentiation of these cells can be controlled by using geometrically different scaffolds. We evaluated MSC differentiation on aligned and random nanofibers, fabricated by electrospinning. We found that induction of MSCs into adipocytes was markedly more inhibited on random nanofibers than on aligned nanofibers. In addition, adipoinduction on aligned nanofibers was also inhibited in the presence of mixed adipoinduction and osteoinduction medium, although osteoinduction was not affected by a change in scaffold geometry. Thus, we have achieved localized control over the direction of differentiation through changes in the alignment of the scaffold even in the presence of a mixed medium. These findings indicate that precise control of MSC differentiation can be attained by using scaffolds with different geometry, rather than by the conventional use of soluble factors in the medium.
Solution of the vacuum Einstein equations in Synthetic Differential Geometry of Kock-Lawvere
Guts, A K; Guts, Alexandr K.; Zvyagintsev, Artem A.
1999-01-01
The topos theory is a theory which is used for deciding a number of problems of theory of relativity, gravitation and quantum physics. It is known that topos-theoretic geometry can be successfully developed within the framework of Synthetic Differential Geometry of Kock-Lawvere (SDG), the models of which are serving the toposes, i.e. categories possessing many characteristics of traditional Theory of Sets. In the article by using ideas SDG, non-classical spherically symmetric solution of the vacuum Einstein equations is given.
Geometry, topology, and string theory
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Varadarajan, Uday
2003-07-10
A variety of scenarios are considered which shed light upon the uses and limitations of classical geometric and topological notions in string theory. The primary focus is on situations in which D-brane or string probes of a given classical space-time see the geometry quite differently than one might naively expect. In particular, situations in which extra dimensions, non-commutative geometries as well as other non-local structures emerge are explored in detail. Further, a preliminary exploration of such issues in Lorentzian space-times with non-trivial causal structures within string theory is initiated.
Ye, Han-Feng; Guo, Shu-Hai; Wu, Bo; Wang, Yan-Hu
2009-10-01
Based on the basic concepts of differential geometry in analyzing environmental data and establishing related models, the methodology for differential geometry expression and analysis of pollutants concentration in terrestrial environment was presented. As a kind of regionalized variables, the spatial distribution pattern of the pollutants concentration was transformed into 3-dimension form, and fitted with conicoid. This approach made it possible to analyze the quantitative relationships between the regionalized variables and their spatial structural attributes. For illustration purpose, several sorts of typical space fabrics, such as convexity, concavity, ridge, ravine, saddle, and slope, were calculated and characterized. It was suggested that this approach was feasible for analyzing the regionalized variables of pollutants concentration in terrestrial environment.
Nonlocality, No-Signalling and Bell's Theorem investigated by Weyl's Conformal Differential Geometry
De Martini, Francesco; Santamato, Enrico
2014-01-01
The principles and methods of the Conformal Quantum Geometrodynamics (CQG) based on the Weyl's differential geometry are presented. The theory applied to the case of the relativistic single quantum spin 1/2 leads a novel and unconventional derivation of Dirac's equation. The further extension of the theory to the case of two spins 1/2 in EPR entangled state and to the related violation of Bell's inequalities leads, by an exact albeit non relativistic analysis, to an insightful resolution of a...
Liao, Y
2003-01-01
A framework was recently proposed for doing perturbation theory on non-commutative (NC) spacetime. It preserves the unitarity of the S matrix and differs from the naive, popular approach already at the lowest order in perturbation when time does not commute with space. In this work, we investigate its phenomenological implications at linear colliders, especially the TESLA at DESY, through the processes of e sup + e sup --> mu sup +mu sup - ,H sup + H sup - ,H sup 0 H sup 0. We find that some NC effects computed previously are now modified and that there are new processes which now exhibit NC effects. Indeed, the first two processes get corrected at tree level as opposed to the null result in the naive approach, while the third one coincides with the naive result only in the low energy limit. The impact of the earth's rotation is incorporated. The NC signals are generally significant when the NC scale is comparable to the collider energy. If this is not the case, the non-trivial azimuthal angle distribution an...
The Geometry of Noncommutative Space-Time
Mendes, R. Vilela
2016-10-01
Stabilization, by deformation, of the Poincaré-Heisenberg algebra requires both the introduction of a fundamental lentgh and the noncommutativity of translations which is associated to the gravitational field. The noncommutative geometry structure that follows from the deformed algebra is studied both for the non-commutative tangent space and the full space with gravity. The contact points of this approach with the work of David Finkelstein are emphasized.
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Wei, Guowei; Baker, Nathan A.
2016-11-11
This chapter reviews the differential geometry-based solvation and electrolyte transport for biomolecular solvation that have been developed over the past decade. A key component of these methods is the differential geometry of surfaces theory, as applied to the solvent-solute boundary. In these approaches, the solvent-solute boundary is determined by a variational principle that determines the major physical observables of interest, for example, biomolecular surface area, enclosed volume, electrostatic potential, ion density, electron density, etc. Recently, differential geometry theory has been used to define the surfaces that separate the microscopic (solute) domains for biomolecules from the macroscopic (solvent) domains. In these approaches, the microscopic domains are modeled with atomistic or quantum mechanical descriptions, while continuum mechanics models (including fluid mechanics, elastic mechanics, and continuum electrostatics) are applied to the macroscopic domains. This multiphysics description is integrated through an energy functional formalism and the resulting Euler-Lagrange equation is employed to derive a variety of governing partial differential equations for different solvation and transport processes; e.g., the Laplace-Beltrami equation for the solvent-solute interface, Poisson or Poisson-Boltzmann equations for electrostatic potentials, the Nernst-Planck equation for ion densities, and the Kohn-Sham equation for solute electron density. Extensive validation of these models has been carried out over hundreds of molecules, including proteins and ion channels, and the experimental data have been compared in terms of solvation energies, voltage-current curves, and density distributions. We also propose a new quantum model for electrolyte transport.
The space of Penrose tilings and the non-commutative curve with homogeneous coordinate ring k
Smith, S Paul
2011-01-01
We construct a non-commutative scheme that behaves as if it is the space of Penrose tilings of the plane. Let k be a field and B=k(y^2). We consider B as the homogeneous coordinate ring of a non-commutative projective scheme. The category of "quasi-coherent sheaves" on it is, by fiat, the quotient category QGr(B):=Gr(B)/Fdim(B) and the category of coherent sheaves on it is qgr(B):=gr(B)/fdim(B), where gr(B) is the category of finitely presented graded modules and fdim(B) is the full subcategory of finite dimensional graded modules. We show that QGr B is equivalent to Mod S, the category of left modules over the ring S that is the direct limit of the directed system of finite dimensional semisimple algebras S_n=M_{f_n}(k) + M_{f_{n-1}}(k) where f_{n-1} and f_n$ are adjacent Fibonacci numbers and the maps S_n \\to S_{n+1} are (a,b)--->(diag(a,b),a). When k is the complex numbers, the norm closure of S is the C^*-algebra Connes uses to view the space of Penrose tilings as a non-commutative space. Objects in QGr B...
Quantum groups: Geometry and applications
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Chu, C.S. [Lawrence Berkeley Lab., CA (United States). Theoretical Physics Group
1996-05-13
The main theme of this thesis is a study of the geometry of quantum groups and quantum spaces, with the hope that they will be useful for the construction of quantum field theory with quantum group symmetry. The main tool used is the Faddeev-Reshetikhin-Takhtajan description of quantum groups. A few content-rich examples of quantum complex spaces with quantum group symmetry are treated in details. In chapter 1, the author reviews some of the basic concepts and notions for Hopf algebras and other background materials. In chapter 2, he studies the vector fields of quantum groups. A compact realization of these vector fields as pseudodifferential operators acting on the linear quantum spaces is given. In chapter 3, he describes the quantum sphere as a complex quantum manifold by means of a quantum stereographic projection. A covariant calculus is introduced. An interesting property of this calculus is the existence of a one-form realization of the exterior differential operator. The concept of a braided comodule is introduced and a braided algebra of quantum spheres is constructed. In chapter 4, the author considers the more general higher dimensional quantum complex projective spaces and the quantum Grassman manifolds. Differential calculus, integration and braiding can be introduced as in the one dimensional case. Finally, in chapter 5, he studies the framework of quantum principal bundle and construct the q-deformed Dirac monopole as a quantum principal bundle with a quantum sphere as the base and a U(1) with non-commutative calculus as the fiber. The first Chern class can be introduced and integrated to give the monopole charge.
Alexander, S; Magueijo, J; Alexander, Stephon; Brandenberger, Robert; Magueijo, Joao
2001-01-01
We show how a radiation dominated universe subject to space-time quantization may give rise to inflation as the radiation temperature exceeds the Planck temperature. We consider dispersion relations with a maximal momentum (i.e. a mimimum Compton wavelength, or quantum of space), noting that some of these lead to a trans-Planckian branch where energy increases with decreasing momenta. This feature translates into negative radiation pressure and, in well-defined circumstances, into an inflationary equation of state. We thus realize the inflationary scenario without the aid of an inflaton field. As the radiation cools down below the Planck temperature, inflation gracefully exits into a standard Big Bang universe, dispensing with a period of reheating. Thermal fluctuations in the radiation bath will in this case generate curvature fluctuations on cosmological scales whose amplitude and spectrum can be tuned to agree with observations.
Raptis, Ioannis
2007-12-01
We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide- cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of ‘poetic language’.
Differential Geometry applied to Acoustics : Non Linear Propagation in Reissner Beams
Bensoam, Joël
2013-01-01
Although acoustics is one of the disciplines of mechanics, its "geometrization" is still limited to a few areas. As shown in the work on nonlinear propagation in Reissner beams, it seems that an interpretation of the theories of acoustics through the concepts of differential geometry can help to address the non-linear phenomena in their intrinsic qualities. This results in a field of research aimed at establishing and solving dynamic models purged of any artificial nonlinearity by taking advantage of symmetry properties underlying the use of Lie groups. The geometric constructions needed for reduction are presented in the context of the "covariant" approach.
Raptis, I
2006-01-01
We summarize the twelve most important in our view novel concepts that have arisen, based on results that have been obtained, from various applications of Abstract Differential Geometry (ADG) to Quantum Gravity (QG). The present document may be used as a concise, yet informal, discursive and peripatetic conceptual guide-cum-terminological glossary to the voluminous technical research literature on the subject. In a bonus section at the end, we dwell on the significance of introducing new conceptual terminology in future QG research by means of `poetic language'
Discrete Differential Geometry Applied to the Coil-End Design of Superconducting Magnets
Auchmann, B; Schwerg, N
2007-01-01
Coil-end design for superconducting accelerator magnets, based on the continuous strip theory of differential geometry, has been introduced by Cook in 1991. A similar method has later been coupled to numerical field calculation and used in an integrated design process for LHC magnets within the CERN field computation program ROXIE. In this paper we present a discrete analog on to the continuous theory of strips. Its inherent simplicity enhances the computational performance, while reproducing the accuracy of the continuous model. The method has been applied to the design
An Approach to Differential Geometry of Fractional Order via Modified Riemann-Liouville Derivative
Institute of Scientific and Technical Information of China (English)
Guy JUMARIE
2012-01-01
In order to cope with some difficulties due to the fact that the derivative of a constant is not zero with the commonly accepted Riemann-Liouville definition of fractional derivative,one (Jumarie)has proposed recently an alternative referred to as (local) modified Riemann-Liouville definition,which directly,provides a Taylor's series of fractional order for non differentiable functions.We examine here in which way this calculus can be used as a framework for a differential geometry of fractional order.One will examine successively implicit function,manifold,length of curves,radius of curvature,Christoffel coefficients,velocity,acceleration.One outlines the application of this framework to Lagrange optimization in mechanics,and one concludes with some considerations on a possible fractional extension of the pseudo-geodesic of thespecial relativity and of the Lorentz transformation.
Differential geometry on the space of connections via graphs and projective limits
Ashtekar, Abhay; Ashtekar, Abhay; Lewandowski, Jerzy
1994-01-01
In a quantum mechanical treatment of gauge theories(including general relativity), one is led to consider a certain completion, \\agb, of the space \\ag of gauge equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory. \\agb is a very large space and serves as a ``universal home'' for measures in theories in which the Wilson loop observables are well-defined. In this paper, \\agb is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as ``floating lattices'' in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on \\agb: differential forms exterior derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and a...
Differential integrin expression regulates cell sensing of the matrix nanoscale geometry.
Di Cio, Stefania; Bøggild, Thea M L; Connelly, John; Sutherland, Duncan S; Gautrot, Julien E
2017-03-01
The nanoscale geometry and topography of the extra-cellular matrix (ECM) is an important parameter controlling cell adhesion and phenotype. Similarly, integrin expression and the geometrical maturation of adhesions they regulate have been correlated with important changes in cell spreading and phenotype. However, how integrin expression controls the nanoscale sensing of the ECM geometry is not clearly understood. Here we develop a new nanopatterning technique, electrospun nanofiber lithography (ENL), which allows the production of a quasi-2D fibrous nanopattern with controlled dimensions (250-1000nm) and densities. ENL relies on electrospun fibres to act as a mask for the controlled growth of protein-resistant polymer brushes. SEM, AFM and immunofluorescence imaging were used to characterise the resulting patterns and the adsorption of the extra-cellular matrix protein fibronectin to the patterned fibres. The control of adhesion formation was studied, as well as the remodelling and deposition of novel matrix. Cell spreading was found to be regulated by the size of fibres, similarly to previous observations made on circular nanopatterns. However, cell shape and polarity were more significantly affected. These changes correlated with important cytoskeleton reorganisation, with a gradual decrease in stress fibre formation as the pattern dimensions decrease. Finally, the differential expression of αvβ3 and α5β1 integrins in engineered cell lines was found to be an important mediator of cell sensing of the nanoscale geometry of the ECM. The novel nanofiber patterns developed in this study, via ENL, mimic the geometry and continuity of natural matrices found in the stroma of tissues, whilst preserving a quasi-2D character (to facilitate imaging and for comparison with other 2D systems such as micropatterned monolayers and circular nanopatches generated by colloidal lithography). These results demonstrate that the nanoscale geometry of the ECM plays an important role
Fröhlich, Steffen
2012-01-01
This book is intended for advanced students and young researchers interested in the analysis of partial differential equations and differential geometry. It discusses elementary concepts of surface geometry in higher-dimensional Euclidean spaces, in particular the differential equations of Gauss-Weingarten together with various integrability conditions and corresponding surface curvatures. It includes a chapter on curvature estimates for such surfaces, and, using results from potential theory and harmonic analysis, it addresses geometric and analytic methods to establish the existence and regularity of Coulomb frames in their normal bundles, which arise as critical points for a functional of total torsion.
Arteaga, J R
2011-01-01
This is the lecture 3 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\\mathbb{R}^{2}$. Here we use the Cartan method to classify the differential equations but not to resolve. This is a classification can be a weak classification in the sense of not involving all the structural invariants.
Arteaga, J R
2011-01-01
This is the lecture 4 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\\mathbb{R}^{2}$. Here we use the Cartan method to classify the differential equations but not to resolve. This is a classification can be a weak classification in the sense of not involving all the structural invariants.
Arteaga, J R
2011-01-01
This is the lecture 1 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\\mathbb{R}^{2}$. Here we use the Cartan method to classify the differential equations but not to resolve. This is a classification can be a weak classification in the sense of not involving all the structural invariants.
Arteaga, J R
2011-01-01
This is the lecture 2 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\\mathbb{R}^{2}$. Here we use the Cartan method to classify the differential equations but not to resolve. This is a classification can be a weak classification in the sense of not involving all the structural invariants.
A bicategory of reduced orbifolds from the point of view of differential geometry
Tommasini, Matteo
2016-10-01
We describe a bicategory (R ed O rb) of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to the notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a bicategory, we firstly define a 2-category (R ed A tl) whose objects are reduced orbifold atlases (on any paracompact, second countable, Hausdorff topological space). The definition of morphisms is obtained as a slight modification of a definition by A. Pohl, while the definitions of 2-morphisms and compositions of them are new in this setup. Using the bicalculus of fractions described by D. Pronk, we are able to construct the bicategory (R ed O rb) from the 2-category (R ed A tl) . We prove that (R ed O rb) is equivalent to the bicategory of reduced orbifolds described in terms of proper, effective, étale Lie groupoids by D. Pronk and I. Moerdijk and to the well-known 2-category of reduced orbifolds constructed from a suitable class of differentiable Deligne-Mumford stacks.
Algebra and geometry of Hamilton's quaternions
Krishnaswami, Govind S
2016-01-01
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.
Nonlinear Differential Geometry Method and Its Application in Induction Motor Decoupling Control
Directory of Open Access Journals (Sweden)
Linyuan Fan
2016-05-01
Full Text Available An alternating current induction motor is a nonlinear, multi-variable, and strong-coupled system that is difficult to control. To address this problem, a novel control strategy based on nonlinear differential geometry theory was proposed. First, a five-order affine mathematical model for an alternating current induction motor was provided. Then, the feedback linearization method was used to realize decoupling and full linearization of the system model. Moreover, a general and simplified control law was adopted to facilitate practical applications. Finally, a controller was designed using the pole assignment method. Simulation results show that the proposed method can decouple the system model into two independent subsystems, and that the closed-loop system exhibits good dynamic and static performances. The proposed decoupling control method is useful to reduce the system complexity of an induction motor and to improve its control performance, thereby providing a new and feasible dynamic decoupling control for an alternating current induction motor.
Directory of Open Access Journals (Sweden)
Xiang Lu
2015-01-01
Full Text Available Aiming at the nonlinear characteristics of VIENNA rectifier and using differential geometry theory, a dual closed-loop control strategy is proposed, that is, outer voltage loop using sliding mode control strategy and inner current loop using feedback linearization control strategy. On the basis of establishing the nonlinear mathematical model of VIENNA rectifier in d-q synchronous rotating coordinate system, an affine nonlinear model of VIENNA rectifier is established. The theory of feedback linearization is utilized to linearize the inner current loop so as to realize the d-q axis variable decoupling. The control law of outer voltage loop is deduced by utilizing sliding mode control and index reaching law. In order to verify the feasibility of the proposed control strategy, simulation model is built in simulation platform of Matlab/Simulink. Simulation results verify the validity of the proposed control strategy, and the controller has a strong robustness in the case of parameter variations or load disturbances.
Polyakov, Felix
2017-02-01
Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (e.g., direction, shape) or temporal (e.g., speed) parameters of trajectories rather than trajectory's representation as a whole. This work is about identifying geometric building blocks of movements by unifying different empirically supported mathematical descriptions that characterize relationship between geometric and temporal aspects of biological motion. Movement primitives supposedly facilitate the efficiency of movements' representation in the brain and comply with such criteria for biological movements as kinematic smoothness and geometric constraint. The minimum-jerk model formalizes criterion for trajectories' maximal smoothness of order 3. I derive a class of differential equations obeyed by movement paths whose nth-order maximally smooth trajectories accumulate path measurement with constant rate. Constant rate of accumulating equi-affine arc complies with the 2/3 power-law model. Candidate primitive shapes identified as equations' solutions for arcs in different geometries in plane and in space are presented. Connection between geometric invariance, motion smoothness, compositionality and performance of the compromised motor control system is proposed within single invariance-smoothness framework. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives.
Differential geometry on the space of connections via graphs and projective limits
Ashtekar, Abhay; Lewandowski, Jerzy
1995-11-01
In a quantum mechanical treatment of gauge theories (including general relativity), one is led to consider a certain completion overline{A}/{G} of the space overline{A}/{G} of guage equivalent connections. This space serves as the quantum configuration space, or, as the space of all Euclidean histories over which one must integrate in the quantum theory overline{A}/{G} is a very large is a very large space and serves as a "universal home" for measures in theories in which the Wilson loop observables are well defined. In this paper, overline{A}/{G} is considered as the projective limit of a projective family of compact Hausdorff manifolds, labelled by graphs (which can be regarded as "floating lattices" in the physics terminology). Using this characterization, differential geometry is developed through algebraic methods. In particular, we are able to introduce the following notions on overline{A}/{G}: differential forms, exterio derivatives, volume forms, vector fields and Lie brackets between them, divergence of a vector field with respect to a volume form, Laplacians and associated heat kernels and heat kernel measures. Thus, although overline{A}/{G} is very large, it is small enough to be mathematically interesting and physically useful. A key feature of this approach is that it does not require a background metric. The geometrical framework is therefore well suited for diffeomorphism invariant theories such as quantum general relativity.
Detecting curvatures in digital images using filters derived from differential geometry
Toro Giraldo, Juanita
2015-09-01
Detection of curvature in digital images is an important theoretical and practical problem in image processing. Many important features in an image are associated with curvature and the detection of such features is reduced to detection and characterization of curvatures. Differential geometry studies many kinds of curvature operators and from these curvature operators is possible to derive powerful filters for image processing which are able to detect curvature in digital images and videos. The curvature operators are formulated in terms of partial differential operators which can be applied to images via convolution with generalized kernels derived from the the Korteweg- de Vries soliton . We present an algorithm for detection of curvature in digital images which is implemented using the Maple package ImageTools. Some experiments were performed and the results were very good. In a future research will be interesting to compare the results using the Korteweg-de Vries soliton with the results obtained using Airy derivatives. It is claimed that the resulting curvature detectors could be incorporated in standard programs for image processing.
Special Geometries Emerging from Yang-Mills Type Matrix Models
Blaschke, Daniel N
2011-01-01
I review some recent results which demonstrate how various geometries, such as Schwarzschild and Reissner-Nordstroem, can emerge from Yang-Mills type matrix models with branes. Furthermore, explicit embeddings of these branes as well as appropriate Poisson structures and star-products which determine the non-commutativity of space-time are provided. These structures are motivated by higher order terms in the effective matrix model action which semi-classically lead to an Einstein-Hilbert type action.
van den Broek, P.M.
1984-01-01
The aim of this paper is to give a detailed exposition of the relation between the geometry of twistor space and the geometry of Minkowski space. The paper has a didactical purpose; no use has been made of differential geometry and cohomology.
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Nadler, Boaz [Department of Mathematics, Yale University, New-Haven, CT 06520 (United States); Schuss, Zeev [Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv (Israel); Singer, Amit [Department of Applied Mathematics, Tel-Aviv University, Ramat-Aviv 69978, Tel-Aviv (Israel); Eisenberg, R S [Department of Molecular Biophysics and Physiology, Rush Medical Center, 1750 Harrison Street, Chicago, IL 60612 (United States)
2004-06-09
Ionic diffusion through and near small domains is of considerable importance in molecular biophysics in applications such as permeation through protein channels and diffusion near the charged active sites of macromolecules. The motion of the ions in these settings depends on the specific nanoscale geometry and charge distribution in and near the domain, so standard continuum type approaches have obvious limitations. The standard machinery of equilibrium statistical mechanics includes microscopic details, but is also not applicable, because these systems are usually not in equilibrium due to concentration gradients and to the presence of an external applied potential, which drive a non-vanishing stationary current through the system. We present a stochastic molecular model for the diffusive motion of interacting particles in an external field of force and a derivation of effective partial differential equations and their boundary conditions that describe the stationary non-equilibrium system. The interactions can include electrostatic, Lennard-Jones and other pairwise forces. The analysis yields a new type of Poisson-Nernst-Planck equations, that involves conditional and unconditional charge densities and potentials. The conditional charge densities are the non-equilibrium analogues of the well studied pair correlation functions of equilibrium statistical physics. Our proposed theory is an extension of equilibrium statistical mechanics of simple fluids to stationary non-equilibrium problems. The proposed system of equations differs from the standard Poisson-Nernst-Planck system in two important aspects. First, the force term depends on conditional densities and thus on the finite size of ions, and second, it contains the dielectric boundary force on a discrete ion near dielectric interfaces. Recently, various authors have shown that both of these terms are important for diffusion through confined geometries in the context of ion channels.
Berberoglu, Giray
1995-01-01
Item characteristic curves were compared across gender and socioeconomic status (SES) groups for the university entrance mathematics examination in Turkey to see if any group had an advantage in solving computation, word-problem, or geometry questions. Differential item functioning was found, and patterns are discussed. (SLD)
Optimal Energy Measurement in Nonlinear Systems: An Application of Differential Geometry
Fixsen, Dale J.; Moseley, S. H.; Gerrits, T.; Lita, A.; Nam, S. W.
2014-01-01
Design of TES microcalorimeters requires a tradeoff between resolution and dynamic range. Often, experimenters will require linearity for the highest energy signals, which requires additional heat capacity be added to the detector. This results in a reduction of low energy resolution in the detector. We derive and demonstrate an algorithm that allows operation far into the nonlinear regime with little loss in spectral resolution. We use a least squares optimal filter that varies with photon energy to accommodate the nonlinearity of the detector and the non-stationarity of the noise. The fitting process we use can be seen as an application of differential geometry. This recognition provides a set of well-developed tools to extend our work to more complex situations. The proper calibration of a nonlinear microcalorimeter requires a source with densely spaced narrow lines. A pulsed laser multi-photon source is used here, and is seen to be a powerful tool for allowing us to develop practical systems with significant detector nonlinearity. The combination of our analysis techniques and the multi-photon laser source create a powerful tool for increasing the performance of future TES microcalorimeters.
Hansen, Ulrich; Maas, Christian
2017-04-01
About 4.5 billion years ago the early Earth experienced several giant impacts that lead to one or more deep terrestrial magma oceans of global extent. The crystallization of these vigorously convecting magma oceans is of key importance for the chemical structure of the Earth, the subsequent mantle evolution as well as for the initial conditions for the onset of plate tectonics. Due to the fast planetary rotation of the early Earth and the small magma viscosity, rotation probably had a profound effect on early differentiation processes and could for example influence the presence and distribution of chemical heterogeneities in the Earth's mantle [e.g. Matyska et al., 1994, Garnero and McNamara, 2008]. Previous work in Cartesian geometry revealed a strong influence of rotation as well as of latitude on the crystal settling in a terrestrial magma ocean [Maas and Hansen, 2015]. Based on the preceding study we developed a spherical shell model that allows to study crystal settling in-between pole and equator as well as the migration of crystals between these regions. Further we included centrifugal forces on the crystals, which significantly affect the lateral and radial distribution of the crystals. Depending on the strength of rotation the particles accumulate at mid-latitude or at the equator. At high rotation rates the dynamics of fluid and particles are dominated by jet-like motions in longitudinal direction that have different directions on northern and southern hemisphere. All in all the first numerical experiments in spherical geometry agree with Maas and Hansen [2015] that the crystal distribution crucially depends on latitude, rotational strength and crystal density. References E. J. Garnero and A. K. McNamara. Structure and dynamics of earth's lower mantle. Science, 320(5876):626-628, 2008. C. Maas and U. Hansen. Eff ects of earth's rotation on the early di erentiation of a terrestrial magma ocean. Journal of Geophysical Research: Solid Earth, 120
Directory of Open Access Journals (Sweden)
Abdelmadjid MAIRECHE
2015-09-01
Full Text Available We obtain here the modified bound-states solutions for central fraction power singular potential (C.F.P.S. in noncommutative 3-dimensional non relativistic quantum mechanics (NC-3D NRQM. It has been observed that the commutative energy spectra was changed, and replaced degenerate new states, depending on four quantum numbers: j, l and sz=±1/2 corresponding to the two spins states of electron by (up and down and the deformed Hamiltonian formed by two new operators: the first describes the spin-orbit interaction , while the second obtained Hamiltonian describes the modified Zeeman effect (containing ordinary Zeeman effect in addition to the usual commutative Hamiltonian. We showed that the isotropic commutative Hamiltonian HCFPS will be in non commutative space anisotropic Hamiltonian HNC-CFPS.
Unitary theories in the work of Mira Fernandes (beyond general relativity and differential geometry)
Lemos, José P S
2010-01-01
An analysis of the work of Mira Fernandes on unitary theories is presented. First it is briefly mentioned the Portuguese scientific context of the 1920s. A short analysis of the extension of Riemann geometries to new generalized geometries with new affine connections, such as those of Weyl and Cartan, is given. Based on these new geometries, the unitary theories of the gravitational and electromagnetic fields, proposed by Weyl, Eddington, Einstein, and others are then explained. Finally, the book and one paper on connections and two papers on unitary theories, all written by Mira Fernandes, are analyzed and put in context.
Piao, Daqing; Barbour, Randall L; Graber, Harry L; Lee, Daniel C
2015-10-01
This work analytically examines some dependences of the differential pathlength factor (DPF) for steady-state photon diffusion in a homogeneous medium on the shape, dimension, and absorption and reduced scattering coefficients of the medium. The medium geometries considered include a semi-infinite geometry, an infinite-length cylinder evaluated along the azimuthal direction, and a sphere. Steady-state photon fluence rate in the cylinder and sphere geometries is represented by a form involving the physical source, its image with respect to the associated extrapolated half-plane, and a radius-dependent term, leading to simplified formula for estimating the DPFs. With the source-detector distance and medium optical properties held fixed across all three geometries, and equal radii for the cylinder and sphere, the DPF is the greatest in the semi-infinite and the smallest in the sphere geometry. When compared to the results from finite-element method, the DPFs analytically estimated for 10 to 25 mm source–detector separations on a sphere of 50 mm radius with μa=0.01 mm(−1) and μ′s=1.0 mm(−1) are on average less than 5% different. The approximation for sphere, generally valid for a diameter≥20 times of the effective attenuation pathlength, may be useful for rapid estimation of DPFs in near-infrared spectroscopy of an infant head and for short source–detector separation.
Piao, Daqing; Barbour, Randall L.; Graber, Harry L.; Lee, Daniel C.
2015-01-01
Abstract. This work analytically examines some dependences of the differential pathlength factor (DPF) for steady-state photon diffusion in a homogeneous medium on the shape, dimension, and absorption and reduced scattering coefficients of the medium. The medium geometries considered include a semi-infinite geometry, an infinite-length cylinder evaluated along the azimuthal direction, and a sphere. Steady-state photon fluence rate in the cylinder and sphere geometries is represented by a form involving the physical source, its image with respect to the associated extrapolated half-plane, and a radius-dependent term, leading to simplified formula for estimating the DPFs. With the source-detector distance and medium optical properties held fixed across all three geometries, and equal radii for the cylinder and sphere, the DPF is the greatest in the semi-infinite and the smallest in the sphere geometry. When compared to the results from finite-element method, the DPFs analytically estimated for 10 to 25 mm source–detector separations on a sphere of 50 mm radius with μa=0.01 mm−1 and μs′=1.0 mm−1 are on average less than 5% different. The approximation for sphere, generally valid for a diameter ≥20 times of the effective attenuation pathlength, may be useful for rapid estimation of DPFs in near-infrared spectroscopy of an infant head and for short source–detector separation. PMID:26465613
Li, Yajun
2011-06-01
The theory developed in Part I of this study [Y. Li, "Differential geometry of the ruled surfaces optically generated by mirror-scanning devices. I. Intrinsic and extrinsic properties of the scan field," J. Opt. Soc. Am. A28, 667 (2011)] for the ruled surfaces optically generated by single-mirror scanning devices is extended to multimirror scanning systems for an investigation of optical generation of the well-known ruled surfaces, such as helicoid, Plücker's conoid, and hyperbolic paraboloid.
离差在微分几何中的应用%Role of Deviation in Differential Geometry
Institute of Scientific and Technical Information of China (English)
洪涛清
2014-01-01
将微分几何课程中的主要概念通过＂离差＂这一桥梁统一起来，指出相对曲率、挠率、法曲率、测地曲率等都是曲线或曲面上点与某平面间的离差的不同表现形式。其次，利用离差推导出与这些概念相关的许多经典结论。在将数学概念系统化的同时，沟通解析几何与微分几何两门课程的教学。%Deviation can be used to unify main concepts in the course of differential geometry . Relative curvature ,torsion ,normal curvature ,and geodesic curvature are all deviations between a point and a plane .Many classical results associated with these concepts are deducted through deviation . Analytic Geometry and Differential Geometry can be linked well when the mathematical concepts are systemized as above .
Directory of Open Access Journals (Sweden)
Emmanuel E Achor
2010-06-01
Full Text Available This study investigated the effect of games and simulations on the gender related differences in mathematics achievement and interest of students in geometry. The sample group consisted of 287 senior secondary school (SSS I students comprising 158 boys and 129 girls from six out of the 46 secondary schools in Gwer-West LGA of Benue state, Nigeria. The study adopted a pre-test and post-test quasi-experimental design, where intact classes were assigned to experimental and control groups. Data generated using Geometry Achievement Test (GAT and Geometry Interest Inventory (GII were analyzed using descriptive statistics to answer research questions and Analysis of Covariance (ANCOVA to test the hypotheses. Findings reveal that male and female students taught using games, and simulations did not differ significantly both in achievement and in interest. It was recommended among others that mathematics teacher should always use relevant games and simulations in teaching mathematics concepts but paying equal attention to the learning needs of both male and female students, and that school administrators should be encouraged to provide local games that could facilitate meaningful learning of mathematics.
(e, 2e) triple-differential cross sections for Ag+(4p, 4s) in coplanar symmetric geometry
Institute of Scientific and Technical Information of China (English)
Zhou Li-Xia; Yan You-Guo
2012-01-01
The (e,2e) triple-differential cross sections of Ag+ (4p,4s) are calculated based on the three-body distorted-wave Born approximation considering post-collision interaction in coplanar symmetric geometry.The energy of the outgoing electron is set to be 50,70,100,200,300,500,700,and 1000 eV,and the intensity and splitting of forward and backward peaks are discussed in detail.Some new structures are observed around 15° and 85° for 4p and 4s orbitals.Structures in triple-differential cross sections at 15° are reported for the first time.A double-binary collision is proposed to explain the formation of such structures.The structures at 85° are also considered as the result of one kind of double-binary collision.
Sakaamini, Ahmad; Amami, Sadek; Murray, Andrew James; Ning, Chuangang; Madison, Don
2016-10-01
Ionisation triple differential cross sections have been determined experimentally and theoretically for the neutral molecule N2 over a range of geometries from coplanar to the perpendicular plane. Data were obtained at incident electron energies ∼10 and ∼20 eV above the ionisation potential of the 3σ g, 1π u and 2σ g states, using both equal and non-equal outgoing electron energies. The data were taken with the incident electron beam in the scattering plane (ψ = 0°), at 45° to this plane and orthogonal to the plane (ψ = 90°). The set of nine measured differential cross sections at a given energy were then inter-normalised to each other. The data are compared to new calculations using various distorted wave methods, and differences between theory and experiment are discussed.
Winger Function for Spin Half Non-commutativeLandau Problem%自旋1/2非对易朗道问题的Wigner函数（英文）
Institute of Scientific and Technical Information of China (English)
王亚辉; 剡江峰; 袁毅
2011-01-01
With great significance in describing the state of quantum system,the Wigner function of the spin half non-commutative Landau problem is studied in this paper.On the basis of the review of the Wigner function in the commutative space,which is subject to the ＊-eigenvalue equation,Hamiltonian of the spin half Landau problem in the non-commutative phase space is given.Then,energy levels and Wigner functions in the form of a matrix of the spin half Landau problem in the non-commutative phase space are obtained by means of the ＊-eigenvalue equation（or Moyal equation）.%Wigner函数在对量子体系状态的描述方面具有重要的意义。讨论了自旋1/2非对易朗道问题的Wigner函数。首先回顾了对易空间中Wigner函数所服从的星本征方程,然后给出了非对易相空间中自旋1/2朗道问题的Hamiltonian,最后利用星本征方程（Moyal方程）计算了非对易相空间中自旋1/2朗道问题具有矩阵表示形式的Wigner函数及其能级。
Differential geometry and mathematical physics part II fibre bundles, topology and gauge fields
Rudolph, Gerd
2017-01-01
The book is devoted to the study of the geometrical and topological structure of gauge theories. It consists of the following three building blocks:- Geometry and topology of fibre bundles,- Clifford algebras, spin structures and Dirac operators,- Gauge theory.Written in the style of a mathematical textbook, it combines a comprehensive presentation of the mathematical foundations with a discussion of a variety of advanced topics in gauge theory.The first building block includes a number of specific topics, like invariant connections, universal connections, H-structures and the Postnikov approximation of classifying spaces.Given the great importance of Dirac operators in gauge theory, a complete proof of the Atiyah-Singer Index Theorem is presented. The gauge theory part contains the study of Yang-Mills equations (including the theory of instantons and the classical stability analysis), the discussion of various models with matter fields (including magnetic monopoles, the Seiberg-Witten model and dimensional r...
López-Permouth, Sergio
1990-01-01
The papers of this volume share as a common goal the structure and classi- fication of noncommutative rings and their modules, and deal with topics of current research including: localization, serial rings, perfect endomorphism rings, quantum groups, Morita contexts, generalizations of injectivitiy, and Cartan matrices.
Non-commutative Hardy inequalities
DEFF Research Database (Denmark)
Hansen, Frank
2009-01-01
We extend Hardy's inequality from sequences of non-negative numbers to sequences of positive semi-definite operators if the parameter p satisfies 1 1. Applications to trace functions are given. We introduce the tracial geometric mean...... and generalize Carleman's inequality....
67 A Study of Cobb-Douglas Production Function with Differential Geometry
Directory of Open Access Journals (Sweden)
Alin Cristian Ioan
2014-12-01
Full Text Available In this paper we shall made an analysis of Cobb-Douglas production function from the differential point of view. We shall obtain some interesting results about the nature of the points of the surface, the total curvature, the conditions when a production function is minimal and finally we give the equations of the geodesics on the surface i.e. the curves of minimal length between two points.
Differential Age-related Changes in Bone Geometry between the Humerus and the Femur in Healthy Men.
Allen, Matti D; McMillan, S Jared; Klein, Cliff S; Rice, Charles L; Marsh, Greg D
2012-04-01
Muscle pull and weight-bearing are key mechanical determinants of bone geometry which is an important feature of bone strength that declines with adult aging. However, the relative importance of these determinants in young and old adults has not been evaluated systematically. To differentiate the influence of each type of mechanical loading we compared humeral and femoral bone shaft geometry and cross-sectional area (CSA) of the arm and thigh muscles in young and old men. Contiguous transverse MRI (Siemens 1.5T) scans of the arm and thigh were made in 10 young men (21.9 ± 1.0 years) and 10 old men (78.1 ± 4.9 years). Image analysis yielded total (TA), cortical (CA) and medullary (MA) CSA of the humeral and femoral shafts, as well as muscle CSA of the corresponding regions of the arm and thigh. Humeral CA was significantly greater in the young, whereas humeral and femoral MA were significantly greater in the older group. Significant correlations were found between arm muscle CSA and humeral CA (r = 0.73); between thigh muscle CSA and femoral CA (r = 0.69); and between body mass and femoral CA (r = 0.63) and TA (r = 0.55). Moderate correlations between muscle CSA and CA suggest that muscle pull is an important determinant of bone geometry. The significant difference observed between young and old in humeral, but not femoral CA, and the correlation between body mass and femoral, but not humeral cortical area, suggests that weight-bearing attenuates bone loss associated with adult aging.
Mukherjee, Saptarshi; Rosell, Anders; Udpa, Lalita; Udpa, Satish; Tamburrino, Antonello
2017-02-01
The modeling of U-Bend segment in steam generator tubes for predicting eddy current probe signals from cracks, wear and pitting in this region poses challenges and is non-trivial. Meshing the geometry in the cartesian coordinate system might require a large number of elements to model the U-bend region. Also, since the lift-off distance between the probe and tube wall is usually very small, a very fine mesh is required near the probe region to accurately describe the eddy current field. This paper presents a U-bend model using differential geometry principles that exploit the result that Maxwell's equations are covariant with respect to changes of coordinates and independent of metrics. The equations remain unaltered in their form, regardless of the choice of the coordinates system, provided the field quantities are represented in the proper covariant and contravariant form. The complex shapes are mapped into simple straight sections, while small lift-off is mapped to larger values, thus reducing the intrinsic dimension of the mesh and stiffness matrix. In this contribution, the numerical implementation of the above approach will be discussed with regard to field and current distributions within the U-bend tube wall. For the sake of simplicity, a two dimensional test case will be considered. The approach is evaluated in terms of efficiency and accuracy by comparing the results with that obtained using a conventional FE model in cartesian coordinates.
Institute of Scientific and Technical Information of China (English)
Chen Xiaocen; Chen Maoyin
2013-01-01
Precise control of a magnetically suspended double-gimbal control moment gyroscope (MSDGCMG) is of vital importance and challenge to the attitude positioning of spacecraft owing to its multivariable,nonlinear and strong coupled properties.This paper proposes a novel linearization and decoupling method based on differential geometry theory and combines it with the internal model controller (IMC) to guarantee the system robustness to the external disturbance and parameter uncertainty.Furthermore,by introducing the dynamic compensation for the inner-gimbal rate-servo system and the magnetically suspended rotor (MSR) system only,we can eliminate the influence of the unmodeled dynamics to the decoupling control accuracy as well as save costs and inhibit noises effectively.The simulation results verify the nice decoupling and robustness performance of the system using the proposed method.
Yazdanbakhsh, Arash; Gori, Simone
2011-12-01
When an observer moves towards a square-wave grating display, a non-rigid distortion of the pattern occurs in which the stripes bulge and expand perpendicularly to their orientation; these effects reverse when the observer moves away. Such distortions present a new problem beyond the classical aperture problem faced by visual motion detectors, one we describe as a 3D aperture problem as it incorporates depth signals. We applied differential geometry to obtain a closed form solution to characterize the fluid distortion of the stripes. Our solution replicates the perceptual distortions and enabled us to design a nulling experiment to distinguish our 3D aperture solution from other candidate mechanisms (see Gori et al. (in this issue)). We suggest that our approach may generalize to other motion illusions visible in 2D displays.
Besse, Nicolas; Coulette, David
2016-08-01
Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov-Poisson and Vlasov-Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, "Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry" (submitted)] and were found to be surprisingly close to those for the original gyrokinetic
Pottmann, Helmut
2014-11-26
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.
Geometry from Information Geometry
Caticha, Ariel
2015-01-01
We use the method of maximum entropy to model physical space as a curved statistical manifold. It is then natural to use information geometry to explain the geometry of space. We find that the resultant information metric does not describe the full geometry of space but only its conformal geometry -- the geometry up to local changes of scale. Remarkably, this is precisely what is needed to model "physical" space in general relativity.
Solving differential equations for 3-loop diagrams relation to hyperbolic geometry and knot theory
Broadhurst, D J
1998-01-01
In hep-th/9805025, a result for the symmetric 3-loop massive tetrahedron in 3 dimensions was found, using the lattice algorithm PSLQ. Here we give a more general formula, involving 3 distinct masses. A proof is devised, though it cannot be accounted as a derivation; rather it certifies that an Ansatz found by PSLQ satisfies a more easily derived pair of partial differential equations. The result is similar to Schläfli's formula for the volume of a bi-rectangular hyperbolic tetrahedron, revealing a novel connection between 3-loop diagrams and 1-loop boxes. We show that each reduces to a common basis: volumes of ideal tetrahedra, corresponding to 1-loop massless triangle diagrams. Ideal tetrahedra are also obtained when evaluating the volume complementary to a hyperbolic knot. In the case that the knot is positive, and hence implicated in field theory, ease of ideal reduction correlates with likely appearance in counterterms. Volumes of knots relevant to the number content of multi-loop diagrams are evaluated;...
Leo, Marco; Cazzato, Dario; De Marco, Tommaso; Distante, Cosimo
2014-01-01
's shape that is obtained through a differential analysis of image intensities and the subsequent combination with the local variability of the appearance represented by self-similarity coefficients. The experimental evidence of the effectiveness of the method was demonstrated on challenging databases containing facial images. Moreover, its capabilities to accurately detect the centers of the eyes were also favourably compared with those of the leading state-of-the-art methods.
Directory of Open Access Journals (Sweden)
Marco Leo
representation of the eye's shape that is obtained through a differential analysis of image intensities and the subsequent combination with the local variability of the appearance represented by self-similarity coefficients. The experimental evidence of the effectiveness of the method was demonstrated on challenging databases containing facial images. Moreover, its capabilities to accurately detect the centers of the eyes were also favourably compared with those of the leading state-of-the-art methods.
Institute of Scientific and Technical Information of China (English)
李怀兵; 丑武胜; 冯震
2011-01-01
Torque ripple is an important factor affecting the performance of Brushless DC motor (BLDCM).In this paper, the non-commutation torque ripple was analyzed, focusing on the torque ripple caused by PWM_ON modulation.The main part of the torque ripple was caused by the PWM_OFF, and the diode freewheeling of inactive phase can compensate the torque ripple to some extent.Meanwhile, hysteresis current control was proposed to suppress the non-commutation torque ripple.The simulation and experiment show the difference between the method and the traditional method, and prove the validity of the method suppressing the torque ripple during non-commutation.%无刷直流电机的转矩脉动是影响其性能的重要因素.该文针对非换相期间的转矩脉动进行了分析,重点研究了PWM_ON调制方式时的转矩脉动.PWM信号为"OFF"时续流引起的转矩脉动占主要部分,非导通相续流在一定程度上可以补偿PWM关断时造成的转矩脉动.同时,提出使用滞环电流控制方法,采用该方法可以抑制非换相期间转矩脉动.通过仿真分析和实验验证了该方法和传统方法的差异,证明了该方法在非换相期间抑制转矩脉动的有效性.
Maireche Abdelmadjid
2015-01-01
In present search, we have studied the effect of the both non commutativity of three dimensional space and phase on the Schrödinger equation with companied Harmonic oscillator potential and it’s inverse, know by isotopic Harmonic oscillator plus inverse quadratic (h.p.i.) potential, we shown that the Hermitian NC Hamiltonian formed anisotropic operator and described many physics phenomena’s, we have also derived the exact degenerated spectrum for studied potential in the first order of two in...
Berger, Marcel
2010-01-01
Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces,
Algebra and Geometry of Hamilton's Quaternions: 'Well, Papa, Can You Multiply Triplets?'
Indian Academy of Sciences (India)
2016-06-01
Inspired by the relation between the algebra ofcomplex numbers and plane geometry, WilliamRowan Hamilton sought an algebra of triples forapplication to three-dimensional geometry. Unableto multiply and divide triples, he inventeda non-commutative division algebra of quadruples,in what he considered his most significantwork, generalizing the real and complex numbersystems. We give a motivated introduction toquaternions and discuss how they are related toPauli matrices, rotations in three dimensions, thethree sphere, the group SU(2) and the celebratedHopf fibrations.
Geometry and the Quantum: Basics
Chamseddine, Ali H; Mukhanov, Viatcheslav
2014-01-01
Motivated by the construction of spectral manifolds in noncommutative geometry, we introduce a higher degree Heisenberg commutation relation involving the Dirac operator and the Feynman slash of scalar fields. This commutation relation appears in two versions, one sided and two sided. It implies the quantization of the volume. In the one-sided case it implies that the manifold decomposes into a disconnected sum of spheres which will represent quanta of geometry. The two sided version in dimension 4 predicts the two algebras M_2(H) and M_4(C) which are the algebraic constituents of the Standard Model of particle physics. This taken together with the non-commutative algebra of functions allows one to reconstruct, using the spectral action, the Lagrangian of gravity coupled with the Standard Model. We show that any connected Riemannian Spin 4-manifold with quantized volume >4 (in suitable units) appears as an irreducible representation of the two-sided commutation relations in dimension 4 and that these represen...
Particle and Field Symmetries and Noncommutative Geometry
Patwardhan, A
2003-01-01
The development of Noncommutative Geometry is creating a reworking and new possibilities in physics. This paper identifies some of the commutation and derivation structures that arise in particle and field interactions and fundamental symmetries. The requirements of coexisting structures, and their consistency, produce a mathematical framework that underlies a fundamental physics theory. Among other developments in Quantum theory of particles and fields are the symmetries of gauge fields and the Fermi-Bose symmetry of particles. These involve a gauge covariant derivation and the action functionals; and commutation algebras and Bogoliubov transforms. The non commutative Theta form introduces an additional and fundamental structure. This paper obtains the interrelations of various structures, and the conditions for the symmetries of Fermionic/Bosonic particles interacting with Yang Mills gauge fields. Many example physical systems are being solved, and the mathematical formalism is being created to understand t...
Riemann-Finsler Geometry with Applications to Information Geometry
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
Information geometry is a new branch in mathematics, originated from the applications of differential geometry to statistics. In this paper we briefly introduce RiemannFinsler geometry, by which we establish Information Geometry on a much broader base,so that the potential applications of Information Geometry will be beyond statistics.
Linear Quaternion Differential Equations: Basic Theory and Fundamental Results
Kou, Kit Ian; Xia, Yong-Hui
2015-01-01
Quaternion-valued differential equations (QDEs) is a new kind of differential equations which have many applications in physics and life sciences. The largest difference between QDEs and ODEs is the algebraic structure. On the non-commutativity of the quaternion algebra, the algebraic structure of the solutions to the QDEs is completely different from ODEs. It is actually a left- or right- free module, not a linear vector space. This paper establishes a systematic frame work for the theory of...
Perkins, Daya I; Trudell, James R; Asatryan, Liana; Davies, Daryl L; Alkana, Ronald L
2012-05-01
Recent studies highlighted the importance of loop 2 of α1 glycine receptors (GlyRs) in the propagation of ligand-binding energy to the channel gate. Mutations that changed polarity at position 52 in the β hairpin of loop 2 significantly affected sensitivity to ethanol. The present study extends the investigation to charged residues. We found that substituting alanine with the negative glutamate at position 52 (A52E) significantly left-shifted the glycine concentration response curve and increased sensitivity to ethanol, whereas the negative aspartate substitution (A52D) significantly right-shifted the glycine EC₅₀ but did not affect ethanol sensitivity. It is noteworthy that the uncharged glutamine at position 52 (A52Q) caused only a small right shift of the glycine EC₅₀ while increasing ethanol sensitivity as much as A52E. In contrast, the shorter uncharged asparagine (A52N) caused the greatest right shift of glycine EC₅₀ and reduced ethanol sensitivity to half of wild type. Collectively, these findings suggest that charge interactions determined by the specific geometry of the amino acid at position 52 (e.g., the 1-Å chain length difference between aspartate and glutamate) play differential roles in receptor sensitivity to agonist and ethanol. We interpret these results in terms of a new homology model of GlyR based on a prokaryotic ion channel and propose that these mutations form salt bridges to residues across the β hairpin (A52E-R59 and A52N-D57). We hypothesize that these electrostatic interactions distort loop 2, thereby changing agonist activation and ethanol modulation. This knowledge will help to define the key physical-chemical parameters that cause the actions of ethanol in GlyRs.
Quantum Geometry of the ``Fuzzy-Lattice'' Hubbard Model and the Fractional Chern Insulator
Vijay, Sagar; Haldane, F. D. M.
2013-03-01
Recent studies of interacting particles on tight-binding lattices with broken time-reversal symmetry reveal ``zero-field'' fractional quantum Hall (FQH) phases (fractional Chern insulators, FCI). In a partially-filled Landau level, the non-commutative guiding-centers are the residual degrees of freedom, requiring a ``quantum geometry'' Hilbert-space description (a real-space Schrödinger description can only apply in the ``classical geometry'' of unprojected coordinates). The continuum description does not apply on a lattice, where we describe emergence of the FCI from a non-commutative quantum lattice geometry. We define a ``fuzzy lattice'' by projecting a one-particle bandstructure (with more than one orbital per unit cell) into a single band, and then renormalize the orbital on each site to unit weight. The resulting overcomplete basis of local states is analogous to a basis of more than one coherent state per flux quantum in a Landau level. The overlap matrix characterizes ``quantum geometry'' on the ``fuzzy lattice'', defining a ``quantum distance'' measure and Berry fluxes through elementary lattice triangles. We study quantum geometry at transitions between topologically-distinct instances of a fuzzy lattice, as well as N-body states with local Hubbard interactions. supported by NSF MRSEC Grant DMR-0819860
Artés, Joan C.; Rezende, Alex C.; Oliveira, Regilene D. S.
Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, 1902], are still open for this family. Our goal is to make a global study of the family QsnSN of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the collision of two infinite singular points. This family can be divided into three different subfamilies, all of them with the finite saddle-node in the origin of the plane with the eigenvectors on the axes and with the eigenvector associated with the zero eigenvalue on the horizontal axis and (A) with the infinite saddle-node in the horizontal axis, (B) with the infinite saddle-node in the vertical axis and (C) with the infinite saddle-node in the bisector of the first and third quadrants. These three subfamilies modulo the action of the affine group and time homotheties are three-dimensional and we give the bifurcation diagram of their closure with respect to specific normal forms, in the three-dimensional real projective space. The subfamilies (A) and (B) have already been studied [Artés et al., 2013b] and in this paper we provide the complete study of the geometry of the last family (C). The bifurcation diagram for the subfamily (C) yields 371 topologically distinct phase portraits with and without limit cycles for systems in the closure /line{QsnSN(C)} within the representatives of QsnSN(C) given by a chosen normal form. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set of /line{QsnSN(C)} is not only algebraic due to the presence of some surfaces found numerically. All points in these surfaces correspond to either connections of separatrices, or the
Directory of Open Access Journals (Sweden)
Basak KOK
2014-06-01
Full Text Available The purpose of this research is to evaluate the effects of teaching geometry which is differentiated based on the parallel curriculum for gifted/talented students on spatial ability. For this purpose; two units as “Polygons” and “Geometric Objects” of 5th grade mathematics book has been taken and formed a new differentiated geometry unit. In this study, pretest and posttest designs of experimental model were used. The study was conducted in Istanbul Science and Art Center, which offers differentiated program to those who are gifted and talented students after school, in the city of İstanbul and participants were 30 students being 15 of them are experimental group and the other 15 are control group. Experimental group students were underwent a differentiated program on “Polygons” and “Geometric Objects” whereas the other group continued their normal program without any differentiation. Spatial Ability Test developed by Talented Youth Center of John Hopkins University was used to collect data. Above mentioned test was presented to both groups of the study. Collected data was analyzed by Mann Whitney-U and Wilcoxon Signed Rank Test which is in the statistics program. It is presented as a result of the study that the program prepared for the gifted and talented students raised their spatial thinking ability.
Chiral topological insulator on Nambu 3-algebraic geometry
Directory of Open Access Journals (Sweden)
Kazuki Hasebe
2014-09-01
Full Text Available Chiral topological insulator (AIII-class with Landau levels is constructed based on the Nambu 3-algebraic geometry. We clarify the geometric origin of the chiral symmetry of the AIII-class topological insulator in the context of non-commutative geometry of 4D quantum Hall effect. The many-body groundstate wavefunction is explicitly derived as a (l,l,l−1 Laughlin–Halperin type wavefunction with unique K-matrix structure. Fundamental excitation is identified with anyonic string-like object with fractional charge 1/(2(l−12+1. The Hall effect of the chiral topological insulators turns out be a color version of Hall effect, which exhibits a dual property of the Hall and spin-Hall effects.
Symplectic geometries on supermanifolds
Lavrov, P M
2007-01-01
Extension of symplectic geometry on manifolds to the supersymmetric case is considered. In the even case it leads to the even symplectic geometry (or, equivalently, to the geometry on supermanifolds endowed with an non-degenerate Poisson bracket) or to the geometry on an even Fedosov supermanifolds. It is proven that in the odd case there are two different scalar symplectic structures (namely, an odd closed differential 2-form and the antibracket) which can be used for construction of different symplectic geometries on supermanifolds.
Indian Academy of Sciences (India)
S Dhar; M R Alam
2007-09-01
The triple differential cross-section for K-shell ionization of silver and copper atoms by relativistic electrons have been computed in the coplanar symmetric geometry with the inclusion of exchange effects following the multiple scattering theory of Das and Seal [1] multiplied by suitable spinors. Present computed results are marginally improved in some cases from the previous computed results [2]. Present results are compared with measured values [3] and with previous computation results [2]. Some other theoretical computational results are also presented here for comparison.
Busemann, Herbert
2005-01-01
A comprehensive approach to qualitative problems in intrinsic differential geometry, this text examines Desarguesian spaces, perpendiculars and parallels, covering spaces, the influence of the sign of the curvature on geodesics, more. 1955 edition. Includes 66 figures.
Li, Yajun
2011-04-01
Rectilinear propagation of light rays in homogeneous isotropic media makes it possible for optical generation of ruled surfaces as the ray is deflected by a rotatable mirror. Scan patterns on a plane or curved surface are merely curves on the ruled surface. Based on this understanding, structures of the scan fields produced by mirror-scanning devices of different configurations are investigated in terms of differential geometry. Expressions of the first and second fundamental coefficients and the first and second Gauss differential forms are given for an investigation of the intrinsic properties of the optically generated ruled surfaces. The Plücker ruled conoid is then generalized for mathematical modeling of the scan fields produced by single-mirror scanning devices of different configurations. Part II will be devoted to a study of multi-mirror scanning systems for optical generation of well-known ruled surfaces such as helicoids and hyperbolic paraboloids.
Moon, Jae Hoon; Jung, Kyong Yeun; Kim, Kyoung Min; Choi, Sung Hee; Lim, Soo; Park, Young Joo; Park, Do Joon; Jang, Hak Chul
2016-02-01
Subclinical hyperthyroidism has been reported to increase the fracture risk. However, the effect of thyroid stimulating hormone (TSH) suppressive therapy on bone geometry in the hip area of patients with differentiated thyroid carcinoma (DTC) is still unclear. The aim of this study was to investigate the effect of TSH suppression on bone geometry in the hip area of pre- and postmenopausal women with DTC. We conducted a retrospective cohort study including 99 women with DTC (25 pre- and 74 postmenopausal) who had received TSH suppressive therapy for at least 3years and 297 control subjects (75 and 222, respectively) matched for sex and age. Bone mineral density (BMD) in the spine and hip area and bone geometry at the femoral neck measured by dual energy X-ray absorptiometry (DXA) were compared between patients and controls. The association between thyroid hormone and bone parameters was investigated. All analyses of bone parameters were adjusted for age, body mass index, and serum calcium levels. In premenopausal subjects, TSH suppressive therapy was not associated with poor bone parameters. In postmenopausal subjects, patients with DTC undergoing TSH suppression showed lower cross-sectional moment of inertia (CSMI), cross-sectional area, and section modulus and thinner cortical thickness at the femoral neck than those of control subjects, whereas their femoral neck BMD was comparable with controls. Total hip BMD was lower in postmenopausal patients than in controls. CSMI and section modulus at the femoral neck were independently associated with serum free T4 levels in postmenopausal patients. The difference in femoral neck bone geometry between patients and controls was only apparent in postmenopausal DTC patients with free T4 >1.79ng/dL (23.04pmol/l), and not in those with free T4 levels ≤1.79ng/dL (23.04pmol/l). TSH suppression in postmenopausal DTC patients was associated with decreased bone strength by altering bone geometry rather than BMD in the hip area
Notte-Cuello, Eduardo A
2008-01-01
We reveal in a rigorous mathematical way using the theory of differential forms, here viewed as sections of a Clifford bundle over a Lorentzian manifold, the true meaning of Freud's identity of differential geometry discovered in 1939 (as a generalization of results already obtained by Einstein in 1916) and rediscovered in disguised forms by several people. We show moreover that contrary to some claims in the literature there is not a single (mathematical) inconsistency between Freud's identity (which is a decomposition of the Einstein indexed 3-forms in two gauge dependent objects) and the field equations of General Relativity. However, as we show there is an obvious inconsistency in the way that Freud's identity is usually applied in the formulation of energy-momentum "conservation laws" in GR. In order for this paper to be useful for a large class of readers (even those ones making a first contact with the theory of differential forms) all calculations are done with all details (disclosing some of the "tri...
Guide to Computational Geometry Processing
DEFF Research Database (Denmark)
Bærentzen, Jakob Andreas; Gravesen, Jens; Anton, François;
be processed before it is useful. This Guide to Computational Geometry Processing reviews the algorithms for processing geometric data, with a practical focus on important techniques not covered by traditional courses on computer vision and computer graphics. This is balanced with an introduction......, metric space, affine spaces, differential geometry, and finite difference methods for derivatives and differential equations Reviews geometry representations, including polygonal meshes, splines, and subdivision surfaces Examines techniques for computing curvature from polygonal meshes Describes...
Schreiber, Urs
2016-01-01
This is a survey of motivations, constructions and applications of higher prequantum geometry. In section 1 we highlight the open problem of prequantizing local field theory in a local and gauge invariant way, and we survey how a solution to this problem exists in higher differential geometry. In section 2 we survey examples and problems of interest. In section 3 we survey the abstract cohesive homotopy theory that serves to make all this precise and tractable.
Energy Technology Data Exchange (ETDEWEB)
Grotz, Andreas
2011-10-07
In this thesis, a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems is proposed. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to classical differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations in Minkowski space and on a globally hyperbolic Lorentzian manifold. When removing the regularization, the objects of our quantum geometry reduce to the common objects of spin geometry on Lorentzian manifolds, up to higher order curvature corrections.
A topological sigma model of biKaehler geometry
Energy Technology Data Exchange (ETDEWEB)
Zucchini, Roberto [Dipartimento di Fisica, Universita degli Studi di Bologna, V. Irnerio 46, I-40126 Bologna (Italy); I.N.F.N., sezione di Bologna (Italy)
2006-01-15
BiKaehler geometry is characterized by a riemannian metric g{sub ab} and two covariantly constant generally non commuting complex structures K{sub {+-}}{sup a}{sub b}, with respect to which g{sub ab} is hermitean. It is a particular case of the bihermitean geometry of Gates, Hull and Roceck, the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. We present a sigma model for biKaehler geometry that is topological in the following sense: i) the action is invariant under a fermionic symmetry {delta}; ii) {delta} is nilpotent on shell; iii) the action is {delta}-exact on shell up to a topological term; iv) the resulting field theory depends only on a subset of the target space geometrical data. The biKaehler sigma model is obtainable by gauge fixing the Hitchin model with generalized Kaehler target space. It further contains the customary A topological sigma model as a particular case. However, it is not seemingly related to the (2,2) supersymmetric biKaehler sigma model by twisting in general.
A topological sigma model of biKaehler geometry
Zucchini, R
2006-01-01
BiKaehler geometry is characterized by a Riemannian metric g_{ab} and two covariantly constant generally non commuting complex structures K_+^a_b, K_-^a_b, with respect to which g_{ab} is Hermitian. It is a particular case of the biHermitian geometry of Gates, Hull and Roceck, the most general sigma model target space geometry allowing for (2,2) world sheet supersymmetry. We present a sigma model for biKaehler geometry that is topological in the following sense: i) the action is invariant under a fermionic symmetry delta; ii) delta is nilpotent on shell; iii) the action is delta--exact on shell up to a topological term; iv) the resulting field theory depends only on a subset of the target space geometrical data. The biKaehler sigma model is obtainable by gauge fixing the Hitchin model with generalized Kaehler target space. It further contains the customary A topological sigma model as a particular case. However, it is not seemingly related to the (2,2) supersymmetric biKaehler sigma model by twisting in gener...
Black Hole Entropy and Finite Geometry
Lévay, Péter; Vrana, Péter; Pracna, Petr
2009-01-01
It is shown that the $E_{6(6)}$ symmetric entropy formula describing black holes and black strings in D=5 is intimately tied to the geometry of the generalized quadrangle GQ$(2,4)$ with automorphism group the Weyl group $W(E_6)$. The 27 charges correspond to the points and the 45 terms in the entropy formula to the lines of GQ$(2,4)$. Different truncations with $15, 11$ and 9 charges are represented by three distinguished subconfigurations of GQ$(2,4)$, well-known to finite geometers; these are the "doily" (i. e. GQ$(2,2)$) with 15, the "perp-set" of a point with 11, and the "grid" (i. e. GQ$(2,1)$) with 9 points, respectively. In order to obtain the correct signs for the terms in the entropy formula, we use a non- commutative labelling for the points of GQ$(2,4)$. For the 40 different possible truncations with 9 charges this labelling yields 120 Mermin squares -- objects well-known from studies concerning Bell-Kochen-Specker-like theorems. These results are connected to our previous ones obtained for the $E_...
The non-commutative Weil algebra
1999-01-01
Let G be a connected Lie group with Lie algebra g. The Duflo map is a vector space isomorphism between the symmetric algebra S(g) and the universal enveloping algebra U(g) which, as proved by Duflo, restricts to a ring isomorphism from invariant polynomials onto the center of the universal enveloping algebra. The Duflo map extends to a linear map from compactly supported distributions on the Lie algebra g to compactly supported distributions on the Lie group G, which is a ring homomorphism fo...
Connes distance function on fuzzy sphere and the connection between geometry and statistics
Energy Technology Data Exchange (ETDEWEB)
Devi, Yendrembam Chaoba, E-mail: chaoba@bose.res.in; Chakraborty, Biswajit, E-mail: biswajit@bose.res.in [S. N. Bose National Centre For Basic Sciences, Salt Lake, Kolkata 700098 (India); Prajapat, Shivraj, E-mail: shraprajapat@gmail.com [Technical Physics Division, Bhabha Atomic Research Centre (BARC), Mumbai 400085 (India); Mukhopadhyay, Aritra K., E-mail: aritra1910@gmail.com [Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg (Germany); Scholtz, Frederik G., E-mail: fgs@sun.ac.za [National Institute for Theoretical Physics (NITheP), Stellenbosch 7602 (South Africa)
2015-04-15
An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete and the Perelemov’s SU(2) coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ ℤ/2.
Connes distance function on fuzzy sphere and the connection between geometry and statistics
Prajapat, Shivraj; Mukhopadhyay, Aritra K; Chakraborty, Biswajit; Scholtz, Frederik G
2014-01-01
An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the $su(2)$ algebra. This has been computed for both the discrete, as well as for the Perelemov's $SU(2)$ coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by $n\\in\\mathbb{Z}/2$.
Connes distance function on fuzzy sphere and the connection between geometry and statistics
Devi, Yendrembam Chaoba; Prajapat, Shivraj; Mukhopadhyay, Aritra K.; Chakraborty, Biswajit; Scholtz, Frederik G.
2015-04-01
An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete and the Perelemov's SU(2) coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ ℤ/2.
Wang, Jian; Chen, Wei; Ruan, Litao; Toprak, Ahmet; Srinivasan, Sathanur R; Berenson, Gerald S
2011-03-01
Hypertension and left ventricular (LV) hypertrophy are both more common in blacks than in whites. The aim of the present study was to test the hypothesis that blood pressure (BP) has a differential effect on the LV geometry types in black versus white asymptomatic young adults. As a part of the Bogalusa Heart Study, echocardiography and cardiovascular risk factor measurements were performed in 780 white and 343 black subjects (aged 24 to 47 years). Four LV geometry types were identified as normal, concentric remodeling, eccentric, and concentric hypertrophy. Compared to the white subjects, the black subjects had a greater prevalence of eccentric (15.7% vs 9.1%, p <0.001) and concentric (9.3% vs 4.1%, p <0.001) hypertrophy. On multivariate logistic regression analyses, adjusting for age, gender, body mass index, lipids, and glucose, the black subjects showed a significantly stronger association of LV concentric hypertrophy with BP (systolic BP, odds ratio [OR] 3.74, p <0.001; diastolic BP, OR 2.86, p <0.001) than whites (systolic BP, OR 1.50, p = 0.037; and diastolic BP, OR 1.35, p = 0.167), with p values for the race difference of 0.007 for systolic BP and 0.026 for diastolic BP. LV eccentric hypertrophy showed similar trends for the race difference in the ORs; however, the association between eccentric hypertrophy and BP was not significant in the white subjects. With respect to LV concentric remodeling, its association with BP was not significant in either blacks or whites. In conclusion, elevated BP levels have a greater detrimental effect on LV hypertrophy patterns in the black versus white young adults. These findings suggest that blacks might be more susceptible than whites to BP-related adverse cardiac remodeling.
Eisenhart, L P
1927-01-01
The use of the differential geometry of a Riemannian space in the mathematical formulation of physical theories led to important developments in the geometry of such spaces. The concept of parallelism of vectors, as introduced by Levi-Civita, gave rise to a theory of the affine properties of a Riemannian space. Covariant differentiation, as developed by Christoffel and Ricci, is a fundamental process in this theory. Various writers, notably Eddington, Einstein and Weyl, in their efforts to formulate a combined theory of gravitation and electromagnetism, proposed a simultaneous generalization o
Rodger, Alison
1995-01-01
Molecular Geometry discusses topics relevant to the arrangement of atoms. The book is comprised of seven chapters that tackle several areas of molecular geometry. Chapter 1 reviews the definition and determination of molecular geometry, while Chapter 2 discusses the unified view of stereochemistry and stereochemical changes. Chapter 3 covers the geometry of molecules of second row atoms, and Chapter 4 deals with the main group elements beyond the second row. The book also talks about the complexes of transition metals and f-block elements, and then covers the organometallic compounds and trans
Institute of Scientific and Technical Information of China (English)
姜旭; 张量
2016-01-01
本文用微分几何的方法求解出椭球面上圆截线所在平面的一般方程。%This paper uses differential geometry method to obtain the general equations of the planes which contain the circular cross sections on an ellipsoid.
Institute of Scientific and Technical Information of China (English)
王万义; 孙炯
2003-01-01
本文利用J-辛几何,刻画了J-对称微分算子的J-对称扩张.%We give complex J-symplectic geometry characterizations for J-symmetric exten-sions of J-symmetric ordinary differential operators.
Thoughts on the Textbooks of Differential Geometry%《微分几何》教材的几点商榷
Institute of Scientific and Technical Information of China (English)
张先叶
2014-01-01
for the textbook , Differential Geometry , compiled by Mei Xiangming , Huang Jingzhi , the paper puts forward three different opinions about some knowledge in it .Firstly, surface of the equation is a net first order nonlinear differential equation .Sec-ondly, in developable surface r→ =a→( u) +v b→( u), parameters|v|is wires p( u,v) Point to the office of a bus straight distance . b→(u) can also just across the conductor on the straight a→(u) on the bus direction vector.Thirdly, when the fixed point P on the de-velopable surface along a straight moving bus , its normal vector is always collinear and cutting plane is unchanged .%对梅向明、黄敬之编写的《微分几何》教材中的3个知识点提出不同意见：曲面网的方程是一个一阶非线性微分方程；直纹面方程 r→＝a→（u ）＋vb→（u）中，参数｜v｜为导线上a→（u）点到直母线上任一点P（u，v）的距离，b→（u）也可以只是过导线上a→（u）点的直母线上的方向向量；可展曲面上动点P沿一条直母线移动时，它的法向量始终共线，切平面不变。
Lectures on Symplectic Geometry
Silva, Ana Cannas
2001-01-01
The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and cl...
Cecil, Thomas E
2015-01-01
This exposition provides the state-of-the art on the differential geometry of hypersurfaces in real, complex, and quaternionic space forms. Special emphasis is placed on isoparametric and Dupin hypersurfaces in real space forms as well as Hopf hypersurfaces in complex space forms. The book is accessible to a reader who has completed a one-year graduate course in differential geometry. The text, including open problems and an extensive list of references, is an excellent resource for researchers in this area. Geometry of Hypersurfaces begins with the basic theory of submanifolds in real space forms. Topics include shape operators, principal curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. The focus then turns to the theory of isoparametric hypersurfaces in spheres. Important examples and classification results are given, including the construction of isoparametric hypersurfaces based on representations of Clifford algebras. An in-depth treatment of Dupin hy...
An intrtoduction to differential geometry
2008-01-01
Resumo:A presente dissertação é um texto de Geometria Diferencial baseado nos principais textos editados em língua portuguesa sobre o assunto. A principal intenção ao redigir a dissertação foi compilar um material que possa ser utilizado em cursos introdutórios de Geometria Diferencial tanto em nível de licenciatura quanto de bacharelado. Para tornar o texto mais acessível, notas históricas sobre o desenvolvimento da Geometria Diferencial e seus principais personagens foram introduzidas logo ...
Differential geometry on Lie groups
2013-01-01
Resumo: Neste trabalho estudamos os aspectos geométricos dos grupos de Lie do ponto de vista da geometria Riemanniana, geometria Hermitiana e geometria Kähler, através das estruturas geométricas invariantes associadas. Exploramos resultados relacionados às curvaturas da variedade Riemanniana subjacente a um grupo de Lie através do estudo de sua álgebra de Lie correspondente. No contexto da geometria Hermitiana e geometria Kähler, para um caso concreto de grupo de Lie complexo, investigaram su...
Institute of Scientific and Technical Information of China (English)
王志敬; 李丽君
2011-01-01
研究了二阶奇型J-对称微分算子辛几何刻画问题,通过构造商空间,应用辛几何的方法讨论了二阶J-对称微分算子的自共轭扩张问题.给出了与二阶微分算子自共轭域相对应的完全J-Lagrangian子流型的分类与描述.%The symplectic geometry characterization of second order singular J - symmetric differential operators was investigated. By constructing different quotient spaces, self-adjoint extensions of second order J - symmetric differential operators were studied using the method of symplectic geometry. Then classification and description of complete J - Lagrangian submanifold corresponding with self-adjoint domains of second order differential operators were obtained.
Maor, Eli
2014-01-01
If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configur
Indian Academy of Sciences (India)
Cătălin Ciupală
2005-02-01
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: -algebras. We also define the Frölicher–Nijenhuis bracket in the non-commutative geometry on -algebras.
Lefschetz, Solomon
2005-01-01
An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.
Ay, Nihat; Lê, Hông Vân; Schwachhöfer, Lorenz
2017-01-01
The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated. This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, inf...
Saleem, Zain Hamid
In this thesis we study a special class of black hole geometries called subtracted geometries. Subtracted geometry black holes are obtained when one omits certain terms from the warp factor of the metric of general charged rotating black holes. The omission of these terms allows one to write the wave equation of the black hole in a completely separable way and one can explicitly see that the wave equation of a massless scalar field in this slightly altered background of a general multi-charged rotating black hole acquires an SL(2, R) x SL(2, R) x SO(3) symmetry. The "subtracted limit" is considered an appropriate limit for studying the internal structure of the non-subtracted black holes because new 'subtracted' black holes have the same horizon area and periodicity of the angular and time coordinates in the near horizon regions as the original black hole geometry it was constructed from. The new geometry is asymptotically conical and is physically similar to that of a black hole in an asymptotically confining box. We use the different nice properties of these geometries to understand various classically and quantum mechanically important features of general charged rotating black holes.
基于微分几何的隐式曲面交线跟踪方法%Tracing Implicit Surface Intersection Based on Differential Geometry
Institute of Scientific and Technical Information of China (English)
付明珠; 罗钟铉; 冯二宝
2016-01-01
Surface intersection is a fundamental problem in CAD applications. Instead of using Newton method to locate points on the curve for the marching method, a new method with dimidiate structure is proposed to trace implicit surface intersection in this paper. The starting and termination points are selected by solving constrained optimization problems. The tracing of intersection curve relies on differential geometry of the intersecting sur-faces. The curvature of intersection curve determines the adaptive step. A generalized tracing method is also pre-sented. Numerical examples show the effectiveness of both methods.%曲面求交是许多CAD应用的基本问题，针对目前在曲面交线跟踪方法中使用最广泛的行进方法要对估计点利用牛顿法进行校正的问题，提出一种二分方式的隐式曲面交线的跟踪方法。该方法通过求解约束优化问题选取起止点，根据相交曲面的微分几何结构跟踪2个隐式曲面的交线，在跟踪过程中使用由曲面交线的曲率确定的自适应步长，并给出此跟踪方法的一个拓展方法。最后通过数值算例验证文中方法的有效性。
The differential geometry of implicitly parametric curve and surface%隐参数曲线曲面的微分几何
Institute of Scientific and Technical Information of China (English)
张伟红; 李莹; 邓建松
2012-01-01
A new method named implicitly parametric curve and surface, which is a parametric form with an implicit domain, was presented. It inherits the advantages of parametric forms and implicit forms. Concretely, on the one hand, it is easy to computer the local information of curves or surfaces as well as parametric forms do. On the other hand, by an implicit domain, a curve and surface with complex topology was represented easily. And the relevant differential geometry concepts of the new representation form were described. Finally, several examples were given to show its application in representation of curve and surface with complex topology and the local control of the curve and surface.%基于参数表示和隐式表示的优点提出了一种新的曲线曲面表示方法——隐参数曲线曲面,并给出了新表示形式下曲线曲面的相关微分几何概念.这种表示在整体形式上为参数形式,但定义域为隐函数形式.从而不仅易于计算曲线、曲面上的点及其他信息,而且可以表示一些具有复杂拓扑的曲线曲面.最后通过实例展示了它在表示复杂拓扑形状和形状作局部控制等方面的应用.
Higher geometry an introduction to advanced methods in analytic geometry
Woods, Frederick S
2005-01-01
For students of mathematics with a sound background in analytic geometry and some knowledge of determinants, this volume has long been among the best available expositions of advanced work on projective and algebraic geometry. Developed from Professor Woods' lectures at the Massachusetts Institute of Technology, it bridges the gap between intermediate studies in the field and highly specialized works.With exceptional thoroughness, it presents the most important general concepts and methods of advanced algebraic geometry (as distinguished from differential geometry). It offers a thorough study
Burdette, A C
1971-01-01
Analytic Geometry covers several fundamental aspects of analytic geometry needed for advanced subjects, including calculus.This book is composed of 12 chapters that review the principles, concepts, and analytic proofs of geometric theorems, families of lines, the normal equation of the line, and related matters. Other chapters highlight the application of graphing, foci, directrices, eccentricity, and conic-related topics. The remaining chapters deal with the concept polar and rectangular coordinates, surfaces and curves, and planes.This book will prove useful to undergraduate trigonometric st
Robinson, Gilbert de B
2011-01-01
This brief undergraduate-level text by a prominent Cambridge-educated mathematician explores the relationship between algebra and geometry. An elementary course in plane geometry is the sole requirement for Gilbert de B. Robinson's text, which is the result of several years of teaching and learning the most effective methods from discussions with students. Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geom
The 750 GeV diphoton excess in unified Pati-Salam models from noncommutative geometry
Aydemir, Ufuk; Sun, Chen; Takeuchi, Tatsu
2016-01-01
We discuss a possible interpretation of the $750$ GeV diphoton resonance, recently reported at the LHC, within a class of Pati-Salam models with gauge coupling unification. The unification is imposed by the underlying non-commutative geometry (NCG), which in these models is extended to a left-right symmetric completion of the Standard Model (SM). Within such unified Pati-Salam models the Higgs content is restrictively determined from the underlying NCG, instead of being arbitrarily selected as in canonical, non-unified, Pati-Salam models. We show that the observed cross sections involving the $750$ GeV diphoton resonance could be realized through a SM singlet scalar field accompanied by colored scalars, present in these unified models. In view of this result we discuss the underlying rigidity of these models in the NCG framework and the wider implications of the NCG approach for physics beyond the SM.
Institute of Scientific and Technical Information of China (English)
陈维桓; 李海中
1999-01-01
用E.Cartan的等价方法,研究切触变换下四阶微分方程y(4)=f(x,y,y′,y″,y″′)的几何.%It is studied that the geometry of the differential equations of the fourth order y(4) = f(x, y,y′, y″, y′″) under contact transformations by E. Cartan's method of equivalence.
First order linear ordinary differential equations in associative algebras
Directory of Open Access Journals (Sweden)
Gordon Erlebacher
2004-01-01
Full Text Available In this paper, we study the linear differential equation $$ frac{dx}{dt}=sum_{i=1}^n a_i(t x b_i(t + f(t $$ in an associative but non-commutative algebra $mathcal{A}$, where the $b_i(t$ form a set of commuting $mathcal{A}$-valued functions expressed in a time-independent spectral basis consisting of mutually annihilating idempotents and nilpotents. Explicit new closed solutions are derived, and examples are presented to illustrate the theory.
Institute of Scientific and Technical Information of China (English)
潘前华; 危韧勇
2009-01-01
The torque ripple caused by diode freewheeling of the inactive phase during non-commutation state of PM brushless DC motor (BLDCM) when using the PWM_ON type of pulse width modulation was analyzed. And the characteris-tics of PWM_ON_PWM type of pulse width modulation was also discussed. Simulation with Matlab indicates that PWM_ON _PWM type is better than other traditional types of PWM in reducing torque ripple during non-commutation state of BLD-CM.%分析了传统的FWM_ON调制方式对无刷直流电动机在非换相期间由于截止相的反并联二极管续流而导致转矩脉动的原因.还分析了PWM_ON_PWM调制方式的运行特性.Matlab仿真表明此种调制方式应用在无刷直流电动机非换相转矩脉动抑制上比传统调制方式具有更好的效果.
Ochiai, T.; Nacher, J. C.
2011-09-01
Recently, the application of geometry and conformal mappings to artificial materials (metamaterials) has attracted the attention in various research communities. These materials, characterized by a unique man-made structure, have unusual optical properties, which materials found in nature do not exhibit. By applying the geometry and conformal mappings theory to metamaterial science, it may be possible to realize so-called "Harry Potter cloaking device". Although such a device is still in the science fiction realm, several works have shown that by using such metamaterials it may be possible to control the direction of the electromagnetic field at will. We could then make an object hidden inside of a cloaking device. Here, we will explain how to design invisibility device using differential geometry and conformal mappings.
Pottmann, Helmut; Eigensatz, Michael; Vaxman, A.; Wallner, Johannes
2015-01-01
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural
Pottmann, Helmut; Eigensatz, Michael; Vaxman, A.; Wallner, Johannes
2015-01-01
Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural
Institute of Scientific and Technical Information of China (English)
黄朝志; 肖发远
2011-01-01
This paper obtains the nonlinear decoupled control laws of 3-phase integrating magnetic VRM by differential geometry theory. The unified switch impulse function is given, and the three input and three output affine nonlinear model is built up;the state variable feedback linearization control law of 3-phase integrating magnetic VRM is given based on the differential geometry theory. At last, the simulation results show the performance on dynamic and steady state of integrating magnetic VRM is good based on differential geometry theory non-linearization control.%以三相磁集成VRM为研究对象,应用微分几何理论实现三相磁集成VRM的非线性解耦控制.在统一的开关脉冲函数下,基于微分几何理论得到三相磁集成VRM的状态反馈线性化解耦控制规律.建立三输入三输出仿射非线性模型,仿真实验表明,基于微分几何非线性控制的磁集成VRM具有良好的动态品质和稳态特性.
Petersen, Peter
2016-01-01
Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include: a substantial addition of unique and enriching exercises scattered throughout the text; inclusion of an increased number of coordinate calculations of connection and curvature; addition of general formulas for curvature on Lie Groups and submersions; integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger; incorporation of several recent results about manifolds with posit...
General Geometry and Geometry of Electromagnetism
Shahverdiyev, Shervgi S.
2002-01-01
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. And it is shown that equation of motion for a particle interacting with electromagnetic...
Recent Advances in Computational Conformal Geometry
Gu, Xianfeng David; Luo, Feng; Yau, Shing-Tung
2009-01-01
Computational conformal geometry focuses on developing the computational methodologies on discrete surfaces to discover conformal geometric invariants. In this work, we briefly summarize the recent developments for methods and related applications in computational conformal geometry. There are two major approaches, holomorphic differentials and curvature flow. Holomorphic differential method is a linear method, which is more efficient and robust to triangulations with lower qua...
Guide to Computational Geometry Processing
DEFF Research Database (Denmark)
Bærentzen, Jakob Andreas; Gravesen, Jens; Anton, François
be processed before it is useful. This Guide to Computational Geometry Processing reviews the algorithms for processing geometric data, with a practical focus on important techniques not covered by traditional courses on computer vision and computer graphics. This is balanced with an introduction......Optical scanning is rapidly becoming ubiquitous. From industrial laser scanners to medical CT, MR and 3D ultrasound scanners, numerous organizations now have easy access to optical acquisition devices that provide huge volumes of image data. However, the raw geometry data acquired must first......, metric space, affine spaces, differential geometry, and finite difference methods for derivatives and differential equations Reviews geometry representations, including polygonal meshes, splines, and subdivision surfaces Examines techniques for computing curvature from polygonal meshes Describes...
Topology and geometry for physicists
Nash, Charles
2011-01-01
Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. ""Thoroughly recommended"" by The Physics Bulletin, this volume's physics applications range fr
Index Theorems on Torsional Geometries
Kimura, Tetsuji
2007-01-01
We study various topological invariants on a differential geometry in the presence of a totally anti-symmetric torsion H under the closed condition dH=0. By using the identification between the Clifford algebra on a geometry and the canonical quantization condition of fermion in the quantum mechanics, we construct the N=1 quantum mechanical sigma model in the Hamiltonian formalism and extend this model to N=2 system, equipped with the totally anti-symmetric tensor associated with the torsion on the target space geometry. Next we construct transition elements in the Lagrangian path integral formalism and apply them to the analyses of the Witten indices in supersymmetric systems. We improve the formulation of the Dirac index on the torsional geometry which has already been studied. We also formulate the Euler characteristic and the Hirzebruch signature on the torsional geometry.
Institute of Scientific and Technical Information of China (English)
马良; 闫继宏; 赵杰; 陈志峰
2011-01-01
The differential geometry method was applied to coordinatod control of a nonholonomic mobile manipulator (NMM) system to solve the problems of local linearization and approximate linearization caused by using conventional linearization methods. The differential geometry method can realize the decoupling control of multi-input multi-output nonlinearization in a NMM system by diffeomorphism and nonlinear feedback, and transform accurately a multivariate, strong-coupling and nonlinear system into a linear-decoupled system. An affine nonlinear system was built up according to the state equations of the NMM system, and the decoupled conditions were validated. The linear-decoupled system of the NMM was obtained by the differential geometry method, and the PD trajectory tracking controller was designed for the linear-decoupled subsystem. The simulation results show the controller has the better tracking effect, and the linear system decoupled by differential geometry method has its validity.%针对在非完整移动操作臂(NMM)系统协调控制中传统解耦线性化方法所带来的局部线性化及近似线性化等问题,采用微分几何方法,通过适当的微分同胚和非线性反馈实现NMM系统多输入多输出非线性解耦控制,将多变量、强耦合、非线性的复杂系统精确转换为线性解耦系统.由NMM系统的状态方程建立其仿射非线性系统模型,并进行解耦条件验证,通过微分几何方法得到NMM的线性解耦系统,同时对解耦后的线性子系统设计PD轨迹跟踪控制器.仿真结果表明该控制器具有良好的跟踪效果,并验证了利用微分几何方法解耦后线性系统的正确性.
Directory of Open Access Journals (Sweden)
Charles-Michel Marle
2016-10-01
Full Text Available I present in this paper some tools in symplectic and Poisson geometry in view of their applications in geometric mechanics and mathematical physics. After a short discussion of the Lagrangian an Hamiltonian formalisms, including the use of symmetry groups, and a presentation of the Tulczyjew’s isomorphisms (which explain some aspects of the relations between these formalisms, I explain the concept of manifold of motions of a mechanical system and its use, due to J.-M. Souriau, in statistical mechanics and thermodynamics. The generalization of the notion of thermodynamic equilibrium in which the one-dimensional group of time translations is replaced by a multi-dimensional, maybe non-commutative Lie group, is fully discussed and examples of applications in physics are given.
Toward the classification of differential calculi on κ-Minkowski space and related field theories
Energy Technology Data Exchange (ETDEWEB)
Jurić, Tajron; Meljanac, Stjepan; Pikutić, Danijel [Ruđer Bošković Institute, Theoretical Physics Division,Bijenička c.54, HR-10002 Zagreb (Croatia); Štrajn, Rina [Dipartimento di Matematica e Informatica, Università di Cagliari,viale Merello 92, I-09123 Cagliari (Italy); INFN, Sezione di Cagliari,Cagliari (Italy)
2015-07-13
Classification of differential forms on κ-Minkowski space, particularly, the classification of all bicovariant differential calculi of classical dimension is presented. By imposing super-Jacobi identities we derive all possible differential algebras compatible with the κ-Minkowski algebra for time-like, space-like and light-like deformations. Embedding into the super-Heisenberg algebra is constructed using non-commutative (NC) coordinates and one-forms. Particularly, a class of differential calculi with an undeformed exterior derivative and one-forms is considered. Corresponding NC differential calculi are elaborated. Related class of new Drinfeld twists is proposed. It contains twist leading to κ-Poincaré Hopf algebra for light-like deformation. Corresponding super-algebra and deformed super-Hopf algebras, as well as the symmetries of differential algebras are presented and elaborated. Using the NC differential calculus, we analyze NC field theory, modified dispersion relations, and discuss further physical applications.
Energy Technology Data Exchange (ETDEWEB)
Byrd, M.
1997-10-01
The group SU(3) is parameterized in terms of generalized {open_quotes}Euler angles{close_quotes}. The differential operators of SU(3) corresponding to the Lie Algebra elements are obtained, the invariant forms are found, the group invariant volume element is found, and some relevant comments about the geometry of the group manifold are made.
The geometry of surfaces contact
Directory of Open Access Journals (Sweden)
Siegl J.
2007-11-01
Full Text Available This contribution deals with a geometrical exact description of contact between two given surfaces which are defined by the vector functions. These surfaces are substituted at a contact point by approximate surfaces of the second order in accordance with the Taylor series and consequently there is derived a differential surface of these second order surfaces. Knowledge of principal normal curvatures, their directions and the tensor (Dupin indicatrix of this differential surface are necessary for description of contact of these surfaces. For description of surface geometry the first and the second surface fundamental tensor and a further methods of the differential geometry are used. A geometrical visualisation of obtained results of this analysis is made. Method and results of this study will be applied to contact analysis of tooth screw surfaces of screw machines.
Foliation theory in algebraic geometry
McKernan, James; Pereira, Jorge
2016-01-01
Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013. Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...
Schmidt, Nathan W; Tai, Kenneth P; Kamdar, Karishma; Mishra, Abhijit; Lai, Ghee Hwee; Zhao, Kun; Ouellette, André J; Wong, Gerard C L
2012-06-22
The conserved tridisulfide array of the α-defensin family imposes a common triple-stranded β-sheet topology on peptides that may have highly diverse primary structures, resulting in differential outcomes after targeted mutagenesis. In mouse cryptdin-4 (Crp4) and rhesus myeloid α-defensin-4 (RMAD4), complete substitutions of Arg with Lys affect bactericidal peptide activity very differently. Lys-for-Arg mutagenesis attenuates Crp4, but RMAD4 activity remains mostly unchanged. Here, we show that the differential biological effect of Lys-for-Arg replacements can be understood by the distinct phase behavior of the experimental peptide-lipid system. In Crp4, small-angle x-ray scattering analyses showed that Arg-to-Lys replacements shifted the induced nanoporous phases to a different range of lipid compositions compared with the Arg-rich native peptide, consistent with the attenuation of bactericidal activity by Lys-for-Arg mutations. In contrast, such phases generated by RMAD4 were largely unchanged. The concordance between small-angle x-ray scattering measurements and biological activity provides evidence that specific types of α-defensin-induced membrane curvature-generating tendencies correspond directly to bactericidal activity via membrane destabilization.
The Geometry of Soft Materials: A Primer
2002-01-01
We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.
Wang, Zu-yong; Teo, Erin Yiling; Chong, Mark Seow Khoon; Zhang, Qin-yuan; Lim, Jing; Zhang, Zhi-yong; Hong, Ming-hui; Thian, Eng-san; Chan, Jerry Kok Yen; Teoh, Swee-hin
2013-07-01
Anisotropic geometries are critical for eliciting cell alignment to dictate tissue microarchitectures and biological functions. Current fabrication techniques are complex and utilize toxic solvents, hampering their applications for translational research. Here, we present a novel simple, solvent-free, and reproducible method via uniaxial stretching for incorporating anisotropic topographies on bioresorbable films with ambitions to realize stem cell alignment control. Uniaxial stretching of poly(ε-caprolactone) (PCL) films resulted in a three-dimensional micro-ridge/groove topography (inter-ridge-distance: ~6 μm; ridge-length: ~90 μm; ridge-depth: 200-900 nm) with uniform distribution and controllable orientation by the direction of stretch on the whole film surface. When stretch temperature (Ts) and draw ratio (DR) were increased, the inter-ridge-distance was reduced and ridge-length increased. Through modification of hydrolysis, increased surface hydrophilicity was achieved, while maintaining the morphology of PCL ridge/grooves. Upon seeding human mesenchymal stem cells (hMSCs) on uniaxial-stretched PCL (UX-PCL) films, aligned hMSC organization was obtained. Compared to unstretched films, hMSCs on UX-PCL had larger increase in cellular alignment (>85%) and elongation, without indication of cytotoxicity or reduction in cellular proliferation. This aligned hMSC organization was homogenous and stably maintained with controlled orientation along the ridges on the whole UX-PCL surface for over 2 weeks. Moreover, the hMSCs on UX-PCL had a higher level of myogenic genes' expression than that on the unstretched films. We conclude that uniaxial stretching has potential in patterning film topography with anisotropic structures. The UX-PCL in conjunction with hMSCs could be used as "basic units" to create tissue constructs with microscale control of cellular alignment and elongation for tissue engineering applications.
An Elementary Account of Amari's Expected Geometry
1999-01-01
Differential geometry has found fruitful application in statistical inference.\\ud In particular, Amari’s (1990) expected geometry is used in higher order\\ud asymptotic analysis, and in the study of sufficiency and ancillarity. However,\\ud we can see three drawbacks to the use of a differential geometric approach in\\ud econometrics and statistics more generally. Firstly, the mathematics is unfamiliar\\ud and the terms involved can be difficult for the econometrician to fully\\ud appreciate. Seco...
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
Bochnak, Jacek; Roy, Marie-Françoise
1998-01-01
This book is a systematic treatment of real algebraic geometry, a subject that has strong interrelation with other areas of mathematics: singularity theory, differential topology, quadratic forms, commutative algebra, model theory, complexity theory etc. The careful and clearly written account covers both basic concepts and up-to-date research topics. It may be used as text for a graduate course. The present edition is a substantially revised and expanded English version of the book "Géometrie algébrique réelle" originally published in French, in 1987, as Volume 12 of ERGEBNISSE. Since the publication of the French version the theory has made advances in several directions. Many of these are included in this English version. Thus the English book may be regarded as a completely new treatment of the subject.
Vyas, Payal; Brown, David T
2012-04-01
Eukaryotic linker or H1 histones modulate DNA compaction and gene expression in vivo. In mammals, these proteins exist as multiple isotypes with distinct properties, suggesting a functional significance to the heterogeneity. Linker histones typically have a tripartite structure composed of a conserved central globular domain flanked by a highly variable short N-terminal domain and a longer highly basic C-terminal domain. We hypothesized that the variable terminal domains of individual subtypes contribute to their functional heterogeneity by influencing chromatin binding interactions. We developed a novel dual color fluorescence recovery after photobleaching assay system in which two H1 proteins fused to spectrally separable fluorescent proteins can be co-expressed and their independent binding kinetics simultaneously monitored in a single cell. This approach was combined with domain swap and point mutagenesis to determine the roles of the terminal domains in the differential binding characteristics of the linker histone isotypes, mouse H1(0) and H1c. Exchanging the N-terminal domains between H1(0) and H1c changed their overall binding affinity to that of the other variant. In contrast, switching the C-terminal domains altered the chromatin interaction surface of the globular domain. These results indicate that linker histone subtypes bind to chromatin in an intrinsically specific manner and that the highly variable terminal domains contribute to differences between subtypes. The methods developed in this study will have broad applications in studying dynamic properties of additional histone subtypes and other mobile proteins.
Institute of Scientific and Technical Information of China (English)
武唯强; 陈康; 符文星; 闫杰; 陈凯
2015-01-01
针对战术导弹的拦截问题，根据质点微分几何运动学在弧长系下及在时域内的关系，将弧长系下的微分几何制导律应用到实际的TBM拦截过程中，得到了空间中时域内的微分几何制导律以及相应的过载指令。根据拦截过程中目标的不同机动方式，采用微分几何制导与比例导引进行了仿真对比与分析，得到了两种导引律下的脱靶量与拦截时间。仿真结果表明，微分几何制导律能够在拦截过程中降低视线角速度并使其趋于稳定，在拦截开始其过载需求较大并逐渐降低至接近0，脱靶量及拦截时间都小于比例导引律，采用微分几何制导律能够在更短时的时间内进行精确拦截。%In this paper, the endo⁃atmosphere tactical ballistic missile interceptor is studied, especially on the terminal guidance law. According to the relationship between arc system and time domain, differential geometry guidance law will be applied to the process of TBM interception. Differential geometric guidance law and the relative overload command in the time domain of 3D space is derived. When it is compared with the classic proportion navigation law in the simulation toward a high speed maneuvering target, the designed differential geometry guidance law demonstrates its superiority by an obviously lower miss distance and lower line of sight rate.
Information geometry near randomness and near independence
Arwini, Khadiga A
2008-01-01
This volume will be useful to practising scientists and students working in the application of statistical models to real materials or to processes with perturbations of a Poisson process, a uniform process, or a state of independence for a bivariate process. We use information geometry to provide a common differential geometric framework for a wide range of illustrative applications including amino acid sequence spacings in protein chains, cryptology studies, clustering of communications and galaxies, cosmological voids, coupled spatial statistics in stochastic fibre networks and stochastic porous media, quantum chaology. Introduction sections are provided to mathematical statistics, differential geometry and the information geometry of spaces of probability density functions.
Cukier, Mimi; Asdourian, Tony; Thakker, Anand
2012-01-01
Geometry provides a natural window into what it is like to do mathematics. In the world of geometry, playful experimentation is often more fruitful than following a procedure, and logic plus a few axioms can open new worlds. Nonetheless, teaching a geometry course in a way that combines both rigor and play can be difficult. Many geometry courses…
Geometry of black hole spacetimes
Andersson, Lars; Blue, Pieter
2016-01-01
These notes, based on lectures given at the summer school on Asymptotic Analysis in General Relativity, collect material on the Einstein equations, the geometry of black hole spacetimes, and the analysis of fields on black hole backgrounds. The Kerr model of a rotating black hole in vacuum is expected to be unique and stable. The problem of proving these fundamental facts provides the background for the material presented in these notes. Among the many topics which are relevant for the uniqueness and stability problems are the theory of fields on black hole spacetimes, in particular for gravitational perturbations of the Kerr black hole, and more generally, the study of nonlinear field equations in the presence of trapping. The study of these questions requires tools from several different fields, including Lorentzian geometry, hyperbolic differential equations and spin geometry, which are all relevant to the black hole stability problem.
Second International workshop Geometry and Symbolic Computation
Walczak, Paweł; Geometry and its Applications
2014-01-01
This volume has been divided into two parts: Geometry and Applications. The geometry portion of the book relates primarily to geometric flows, laminations, integral formulae, geometry of vector fields on Lie groups, and osculation; the articles in the applications portion concern some particular problems of the theory of dynamical systems, including mathematical problems of liquid flows and a study of cycles for non-dynamical systems. This Work is based on the second international workshop entitled "Geometry and Symbolic Computations," held on May 15-18, 2013 at the University of Haifa and is dedicated to modeling (using symbolic calculations) in differential geometry and its applications in fields such as computer science, tomography, and mechanics. It is intended to create a forum for students and researchers in pure and applied geometry to promote discussion of modern state-of-the-art in geometric modeling using symbolic programs such as Maple™ and Mathematica®, as well as presentation of new results. ...
Directory of Open Access Journals (Sweden)
T. Wagner
2007-01-01
Full Text Available The results of a comparison exercise of radiative transfer models (RTM of various international research groups for Multiple AXis Differential Optical Absorption Spectroscopy (MAX-DOAS viewing geometry are presented. Besides the assessment of the agreement between the different models, a second focus of the comparison was the systematic investigation of the sensitivity of the MAX-DOAS technique under various viewing geometries and aerosol conditions. In contrast to previous comparison exercises, box-air-mass-factors (box-AMFs for different atmospheric height layers were modelled, which describe the sensitivity of the measurements as a function of altitude. In addition, radiances were calculated allowing the identification of potential errors, which might be overlooked if only AMFs are compared. Accurate modelling of radiances is also a prerequisite for the correct interpretation of satellite observations, for which the received radiance can strongly vary across the large ground pixels, and might be also important for the retrieval of aerosol properties as a future application of MAX-DOAS. The comparison exercises included different wavelengths and atmospheric scenarios (with and without aerosols. The strong and systematic influence of aerosol scattering indicates that from MAX-DOAS observations also information on atmospheric aerosols can be retrieved. During the various iterations of the exercises, the results from all models showed a substantial convergence, and the final data sets agreed for most cases within about 5%. Larger deviations were found for cases with low atmospheric optical depth, for which the photon path lengths along the line of sight of the instrument can become very large. The differences occurred between models including full spherical geometry and those using only plane parallel approximation indicating that the correct treatment of the Earth's sphericity becomes indispensable. The modelled box-AMFs constitute an
Mahé, Louis; Roy, Marie-Françoise
1992-01-01
Ten years after the first Rennes international meeting on real algebraic geometry, the second one looked at the developments in the subject during the intervening decade - see the 6 survey papers listed below. Further contributions from the participants on recent research covered real algebra and geometry, topology of real algebraic varieties and 16thHilbert problem, classical algebraic geometry, techniques in real algebraic geometry, algorithms in real algebraic geometry, semialgebraic geometry, real analytic geometry. CONTENTS: Survey papers: M. Knebusch: Semialgebraic topology in the last ten years.- R. Parimala: Algebraic and topological invariants of real algebraic varieties.- Polotovskii, G.M.: On the classification of decomposing plane algebraic curves.- Scheiderer, C.: Real algebra and its applications to geometry in the last ten years: some major developments and results.- Shustin, E.L.: Topology of real plane algebraic curves.- Silhol, R.: Moduli problems in real algebraic geometry. Further contribu...
Non-geometric fluxes and non-associative geometry
Plauschinn, Erik
2012-01-01
In these proceedings, we discuss non-commutativity in closed string theory. In analogy to the open-string sector, for closed strings we first motivate a cyclic double commutator to be evaluated for backgrounds with geometric or non-geometric fluxes. A non-trivial result for such an expression indicates a non-associative structure. Second, we define a conformal field theory at linear order in background fluxes and compute correlation functions therein. From these we motivate a tri-product which captures non-commutative and non-associative effects.
Introduction to geometry and relativity
2013-01-01
This book provides a lucid introduction to both modern differential geometry and relativity for advanced undergraduates and first-year graduate students of applied mathematics and physical sciences. This book meets an overwhelming need for a book on modern differential geometry and relativity that is student-friendly, and which is also suitable for self-study. The book presumes a minimal level of mathematical maturity so that any student who has completed the standard Calculus sequence should be able to read and understand the book. The key features of the book are: Detailed solutions are provided to the Exercises in each chapter; Many of the missing steps that are often omitted from standard mathematical derivations have been provided to make the book easier to read and understand; A detailed introduction to Electrodynamics is provided so that the book is accessible to students who have not had a formal course in this area; In its treatment of modern differential geometry, the book employs both a modern, c...
Institute of Scientific and Technical Information of China (English)
朱熀秋; 郝晓红; 张婷婷; 刁小燕
2011-01-01
Aiming at a bearingless permanent magnet slice motor, the radial suspension principle of this motor is introduced, and mathematical model of radial suspension forces are deduced. The decoupling control question has been investigated for radial suspension forces of the bearingless permanent magnet slice motor at load adopting the nonlinear differential geometry. The decoupling control has been realized among radical suspension forces and currents in radical suspension force's windings. The original coupling system is decoupled and linearized, and the neural network sliding mode variable structure (SMVS) controller is designed for the decoupled linear subsystems. Finally, the feasibility of the method is validated by the results of simulation. The simulation conclusions show that based on the nonlinear differential geometry neural SMVS controller method can achieve better stability of radial suspension force independent control.%以无轴承永磁薄片电机为研究对象,阐述了其径向力悬浮机理,推导了径向悬浮力数学模型.采用非线性微分几何的方法研究了无轴承永磁薄片电机在负载运行时径向悬浮力之间的解耦控制问题,实现电机径向悬浮力与悬浮力绕组中电流之间的解耦控制.将原耦合系统解耦和基本线性化成独立的伪线性系统,并对解耦后的伪线性子系统设计了神经滑模变结构控制器.最后对设计的控制系统进行仿真试验,验证了这种解耦控制方法的可行性.仿真结果表明,基于非线性微分几何的神经滑模变结构控制器方法能较好实现径向悬浮力的稳定独立控制.
Institute of Scientific and Technical Information of China (English)
赵韩; 邱明明; 黄康
2015-01-01
Aimed at nonlinearity,external disturbances and parameter uncertainty of the clutch control system,a sliding mode control was put forward based on differential geometry for speed track-ing during clutch engaging process.Considering the uncertainty of system parameters and external dis-turbances and other uncertain factors,a single clutch dynamic system model was established,feedback linearization was used based on differential geometry method,the control law was obtained,and then a sliding mode controller was designed based on reaching law control method for the clutch control sys-tem with disturbance.The stability of the system was proved by using Lyapunov theory.The simula-tion results show that the controller can make the process of clutch engagement speed tracking accura-cy and robustness.%针对离合器控制系统中存在的非线性、外部干扰和参数不确定问题，提出了基于微分几何的离合器接合过程速度跟踪滑模控制方法。考虑系统参数的不确定性和外界干扰等不确定因素，建立了单个离合器起步动力学模型；基于微分几何的反馈线性化方法，得出系统的控制律；采用基于趋近律的滑模控制方法，设计了存在不确定干扰的离合器控制系统滑模控制器。利用 Lyapunov 理论对系统的稳定性进行了证明。仿真结果表明该控制器使离合器接合过程的速度跟踪精度高，且鲁棒性好。
The geometry of ordinary variational equations
Krupková, Olga
1997-01-01
The book provides a comprehensive theory of ODE which come as Euler-Lagrange equations from generally higher-order Lagrangians. Emphasis is laid on applying methods from differential geometry (fibered manifolds and their jet-prolongations) and global analysis (distributions and exterior differential systems). Lagrangian and Hamiltonian dynamics, Hamilton-Jacobi theory, etc., for any Lagrangian system of any order are presented. The key idea - to build up these theories as related with the class of equivalent Lagrangians - distinguishes this book from other texts on higher-order mechanics. The reader should be familiar with elements of differential geometry, global analysis and the calculus of variations.
Liu, Chien-Hao
2014-01-01
In this Part II of D(11), we introduce new objects: super-$C^k$-schemes and Azumaya super-$C^k$-manifolds with a fundamental module (or, synonymously, matrix super-$C^k$-manifolds with a fundamental module), and extend the study in D(11.1) ([L-Y3], arXiv:1406.0929 [math.DG]) to define the notion of `differentiable maps from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion of `fermionic D-branes' in two different styles, one parallels Ramond-Neveu-Schwarz fermionic string and the other Green-Schwarz fermionic string. A more detailed discussion on the Higgs mechanism on dynamical D-branes in our setting, taking maps from the D-brane world-volume to the space-time in question and/or sections of the Chan-Paton bundle on the D-brane world-volume as Higgs fields, is also given for the first time in the D-project. Finally note that mathematically string theory begins with the notion of a differentiable map from a string world-sheet...
Institute of Scientific and Technical Information of China (English)
刘正权; 孙耀杰; 林燕丹
2012-01-01
A freeform reflector design method,which is mainly based on a first-order linear partial differential equation,is proposed for uniform rectangular illuminance distribution in the field of LED illumination. The interaction between the freeform surface and the light beam is depicted based on theory of the differential geometry and Snell's law. The energy topological relation between the Lambertian luminaire and the illuminated rectangular surface is established according to the LED luminous intensity distribution. The method deducts a first-order linear partial differential equation with some boundary conditions to represent the freeform reflector. The boundary conditions and the partial differential equation are solved by the Runge-Kutta method and finite difference method,respectively. The numerical results are validated in the form of raytracing,which reveal that the luminous flux efficiency is about 94 % ,the transverse uniformity of illuminance on the target surface is 0. 9 and the longitudinal uniformity of illuminance on the target surface is 0.8. The numerical computation time is less than 1 s.%在LED照明应用中为实现矩形均匀照度分布要求,提出了一种基于一阶线性偏微分方程的自由曲面反射器设计方法.基于微分几何理论和折射定律描述了光线与自由曲面的相互作用.根据LED光源特性建立了朗伯光源与矩形被照面之间的能量拓扑关系,推导了自由曲面反射器的一阶线性偏微分方程和边界条件.分别使用Runge-Kutta法和有限差分法对边界条件和偏微分方程进行数值计算,并对计算结果进行光线追迹仿真.仿真结果表明自由曲面反射器光通利用率达到了94％,矩形被照面横向照度均匀度达到了0.9,纵向照度均匀度达到了0.8.程序计算时间少于1 s.
Meyer, Walter J
2006-01-01
Meyer''s Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters. The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry''s usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers.* Realistic applications integrated throughout the text, including (but not limited to): - Symmetries of artistic patterns- Physics- Robotics- Computer vision- Computer graphics- Stability of architectural structures- Molecular biology- Medicine- Pattern recognition* Historical notes included in many chapters...
Institute of Scientific and Technical Information of China (English)
GUO Enli; MO Xiaohuan
2006-01-01
In this paper,a survey on Riemann-Finsler geometry is given.Non-trivial examples of Finsler metrics satisfying different curvature conditions are presented.Local and global results in Finsler geometry are analyzed.
Design of output stability control for BESS based on differential geometry%基于微分几何的电池储能系统输出稳定控制器设计
Institute of Scientific and Technical Information of China (English)
王忠勇
2011-01-01
为消除由于外界干扰引起的系统不稳定,通过对电池储能系统的数学模型进行分析,基于微分几何理论,采用非线性控制方法对系统输出控制器的设计,达到对系统输出量进行稳定控制的目的.为了消除控制偏差,对设计的控制器增加了抗干扰环节.仿真结果表明设计的非线性控制策略具有很好的动态性能,证明了控制方法的适用性.%To reduce the instability of system caused by interference, a proper control strategy is proposed based on the differential geometry by analyzing the established nonlinear model of battery energy storage system (BESS). This nonlinear control design method is effective for improving the output stability dynamic state. In order to eliminate the influence of deviation,anti - interference links are added to the proposed nonlinear controller. Simulation results verify the stability and the anticipant dynamic response of the control strategy.
Geometry of surfaces a practical guide for mechanical engineers
Radzevich, Stephen P
2012-01-01
Presents an in-depth analysis of geometry of part surfaces and provides the tools for solving complex engineering problems Geometry of Surfaces: A Practical Guide for Mechanical Engineers is a comprehensive guide to applied geometry of surfaces with focus on practical applications in various areas of mechanical engineering. The book is divided into three parts on Part Surfaces, Geometry of Contact of Part Surfaces and Mapping of the Contacting Part Surfaces. Geometry of Surfaces: A Practical Guide for Mechanical Engineers combines differential geometry and gearing theory and presents new developments in the elementary theory of enveloping surfaces. Written by a leading expert of the field, this book also provides the reader with the tools for solving complex engineering problems in the field of mechanical engineering. Presents an in-depth analysis of geometry of part surfaces Provides tools for solving complex engineering problems in the field of mechanical engineering Combines differential geometry an...
On Anholonomic Deformation, Geometry, and Differentiation
2013-02-01
twinning shear. In theories of porous or damaged media [18], F̃ represents volumetric expansion associated with voids. In theories of growth in...2009; 44: 675–688. [18] Bammann, DJ, and Solanki, KN. On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material...Clayton, JD, McDowell, DL, and Bammann, DJ. Modeling dislocations and disclinations with finite micropolar elastoplasticity . Int J Plasticity 2006; 22: 210
Introduction to differential geometry of plane curves
2015-01-01
A intenÃÃo desse trabalho serÃ de abordar de forma bÃsica e introdutÃria o estudo da Geometria Diferencial, que por sua vez tem seus estudos iniciados com as Curvas Planas. SerÃ necessÃrio um conhecimento de CÃlculo Diferencial, Integral e Geometria AnalÃtica para melhor compreensÃo desse trabalho, pois como seu prÃprio nome nos transparece Geometria Diferencial vem de uma junÃÃo do estudo da Geometria envolvendo CÃlculo. Assim abordaremos subtemas como curvas suaves, vetor tangente, co...
Differential Geometry of Time-Dependent Mechanics
Giachetta, G; Sardanashvily, G
1997-01-01
The usual formulations of time-dependent mechanics start from a given splitting $Y=R\\times M$ of the coordinate bundle $Y\\to R$. From physical viewpoint, this splitting means that a reference frame has been chosen. Obviously, such a splitting is broken under reference frame transformations and time-dependent canonical transformations. Our goal is to formulate time-dependent mechanics in gauge-invariant form, i.e., independently of any reference frame. The main ingredient in this formulation is a connection on the bundle $Y\\to R$ which describes an arbitrary reference frame. We emphasize the following peculiarities of this approach to time-dependent mechanics. A phase space does not admit any canonical contact or presymplectic structure which would be preserved under reference frame transformations, whereas the canonical Poisson structure is degenerate. A Hamiltonian fails to be a function on a phase space. In particular, it can not participate in a Poisson bracket so that the evolution equation is not reduced...
Differential geometry and the calculus of variations
Hermann, Robert
1968-01-01
In this book, we study theoretical and practical aspects of computing methods for mathematical modelling of nonlinear systems. A number of computing techniques are considered, such as methods of operator approximation with any given accuracy; operator interpolation techniques including a non-Lagrange interpolation; methods of system representation subject to constraints associated with concepts of causality, memory and stationarity; methods of system representation with an accuracy that is the best within a given class of models; methods of covariance matrix estimation;methods for low-rank mat
Kosinski, Antoni A
2007-01-01
The concepts of differential topology form the center of many mathematical disciplines such as differential geometry and Lie group theory. Differential Manifolds presents to advanced undergraduates and graduate students the systematic study of the topological structure of smooth manifolds. Author Antoni A. Kosinski, Professor Emeritus of Mathematics at Rutgers University, offers an accessible approach to both the h-cobordism theorem and the classification of differential structures on spheres.""How useful it is,"" noted the Bulletin of the American Mathematical Society, ""to have a single, sho
Geometry essentials for dummies
Ryan, Mark
2011-01-01
Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams. Get down to the basics - get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals Conque
Introduction to projective geometry
Wylie, C R
2008-01-01
This lucid introductory text offers both an analytic and an axiomatic approach to plane projective geometry. The analytic treatment builds and expands upon students' familiarity with elementary plane analytic geometry and provides a well-motivated approach to projective geometry. Subsequent chapters explore Euclidean and non-Euclidean geometry as specializations of the projective plane, revealing the existence of an infinite number of geometries, each Euclidean in nature but characterized by a different set of distance- and angle-measurement formulas. Outstanding pedagogical features include w
Affine and Projective Geometry
Bennett, M K
1995-01-01
An important new perspective on AFFINE AND PROJECTIVE GEOMETRY. This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view. Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and exercises, covering material for two semesters of upper-level undergraduate mathematics. The first part of the book deals with the correlation between synthetic geometry and linear algebra. In the second part, geometry is used to introduce lattice theory
Position dependent non-commutativity in two dimensions
López, Armand Idárraga
2015-01-01
Orientador: Prof. Dr. Vladislav Kupriyanov Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática , 2015. No presente trabalho estudamos as consequências físicas da não-comutatividade dependente da posição e rotacionalmente invariante em duas dimensões [x, y] = iq f (x2 + y2), usando a teoria de perturbações em mecânica quântica e considerando os modelos exatamente solúveis como o oscilador harmônico isotrópico e o problema de Landau. Nós ...
Bilangan Kromatik Grap Commuting dan Non Commuting Grup Dihedral
Directory of Open Access Journals (Sweden)
Handrini Rahayuningtyas
2015-11-01
Full Text Available Commuting graph is a graph that has a set of points X and two different vertices to be connected directly if each commutative in G. Let G non abelian group and Z(G is a center of G. Noncommuting graph is a graph which the the vertex is a set of G\\Z(G and two vertices x and y are adjacent if and only if xy≠yx. The vertex colouring of G is giving k colour at the vertex, two vertices that are adjacent not given the same colour. Edge colouring of G is two edges that have common vertex are coloured with different colour. The smallest number k so that a graph can be coloured by assigning k colours to the vertex and edge called chromatic number. In this article, it is available the general formula of chromatic number of commuting and noncommuting graph of dihedral group
Quasi-Normal Modes from Non-Commutative Matrix Dynamics
Aprile, Francesco
2016-01-01
We explore the connection between the process of relaxation in the BMN matrix model and the physics of black holes in AdS/CFT. Focusing on Dyson-fluid solutions of the matrix model, we perform numerical simulations of the real time dynamics of the system. By quenching the equilibrium distribution we study the quasi-normal oscillations of scalar single trace observables, we isolate the lowest quasi-normal mode, and we determine its frequencies as function of the energy. Considering the BMN matrix model as a truncation of $\\mathcal{N}=4$ SYM, we also compute the frequencies of the quasi-normal modes of the dual scalar fields in the AdS$_5$-Schwarzschild background. We compare the results of the black hole and the classical Dyson fluid, and we point out a correspondence between the two descriptions.
Functional approach to coherent states in non commutative theories
Lubo, M
2003-01-01
In many high dimensional noncommutative theories, no state saturates simultaneously all the non trivial Heisenberg uncertainty relations. This differs from the usual theory where the squeezed states possess this property. The important role played by these states when recovering classical mechanics as a limit of quantum theory makes necessary the investigation of the possible generalizations in the noncommutative context. We propose an extension based on a variational principle. The action considered is the sum of the squares of the terms associated to the non trivial Heisenberg uncertainty relations. We first verify that our proposal works in the usual theory: we find the known gaussian functions and, besides them, other states which can be expressed as products of gaussians with specific hypergeometrics. We illustrate our construction in three models defined on a four dimensional phase space: two models endowed with a minimal length uncertainty and the popular case in which the commutators of the positions ...
Authentication Schemes Using Polynomials Over Non-Commutative Rings
Valluri, Maheswara Rao
2013-01-01
Authentication is a process by which an entity,which could be a person or intended computer,establishes its identity to another entity.In private and public computer networks including the Internet,authentication is commonly done through the use of logon passwords. Knowledge of the password is assumed to guarantee that the user is authentic.Internet business and many other transactions require a more stringent authentication process. The aim of this paper is to propose two authentication sche...
Quantized equations of motion in non-commutative theories
Heslop, P; Heslop, Paul; Sibold, Klaus
2004-01-01
Quantum field theories based on interactions which contain the Moyal star product suffer, in the general case when time does not commute with space, from several diseases: quantum equation of motions contain unusual terms, conserved currents can not be defined and the residual spacetime symmetry is not maintained. All these problems have the same origin: time ordering does not commute with taking the star product. Here we show that these difficulties can be circumvented by a new definition of time ordering: namely with respect to a light-cone variable. In particular the original spacetime symmetries SO(1,1) x SO(2) and translation invariance turn out to be respected. Unitarity is guaranteed as well.
Non-commutative Modal Rings and Internalized Equality
Institute of Scientific and Technical Information of China (English)
D. Fearnley-Sander; A.V. Kelarev; T. Stokes
2002-01-01
We generalize the notion of modal Boolean rings to rings in general. The connection between Boolean rings with equality and modal Boolean rings provides a cue for the definition. The main motivation lies in the existence of examples such as matrix and polynomial rings over modal Boolean rings and Cartesian products of associative rings with identity, along with the desire that the class of "modal rings" be closed under formation of not only the usual homomorphic image, subalgebra and direct product constructions, but also the ring-theoretic constructions of forming matrices and polynomials.
Non Hermitian quantum mechanics in non commutative space
Giri, Pulak Ranjan
2008-01-01
We study non Hermitian quantum systems in noncommutative space as well as a \\cal{PT} symmetric deformation of this space. Specifically, a \\mathcal{PT}-symmetric harmonic oscillator together with iC(x_1+x_2) interaction is discussed in this space and solutions are obtained. It is shown that in the \\cal{PT} deformed noncommutative space the Hamiltonian may or may not possess real eigenvalues depending on the choice of the noncommutative parameters. However, it is shown that in standard noncommutative space, the iC(x_1+x_2) interaction generates only real eigenvalues despite the fact that the Hamiltonian is not \\mathcal{PT}-symmetric. A complex interacting anisotropic oscillator system has also been discussed.
Electrodynamics in Non-commutative Curved Space Time
Jafari, Abolfazl
2009-01-01
We study the issue of the electrodynamics theory in noncommutative curved space time (NCCST) with a new star-product. In this paper, the motion equation of electrodynamics and canonical energy-momentum tensor in noncommutative curved space time will be found. The most important point is the assumption of the noncommutative parameter ($\\theta$) be $x^{\\m}$-independent.
Causality in non-commutative quantum field theories
Energy Technology Data Exchange (ETDEWEB)
Haque, Asrarul; Joglekar, Satish D [Department of Physics, I.I.T. Kanpur, Kanpur 208 016 (India)], E-mail: ahaque@iitk.ac.in, E-mail: sdj@iitk.ac.in
2008-05-30
We study causality in noncommutative quantum field theory with a space-space noncommutativity. We employ the S operator approach of Bogoliubov-Shirkov (BS). We generalize the BS criterion of causality to the noncommutative theory. The criterion to test causality leads to a nonzero difference between the T* product and the T product as a condition of causality violation for a spacelike separation. We discuss two examples; one in a scalar theory and another in the Yukawa theory. In particular, in the context of a noncommutative Yukawa theory, with the interaction Lagrangian {psi}-bar(x)*{psi}(x)*{phi}(x), is observed to be causality violating even in the case of space-space noncommutativity for which {theta}{sup 0i} = 0.
Integrable systems, geometry, and topology
Terng, Chuu-Lian
2006-01-01
The articles in this volume are based on lectures from a program on integrable systems and differential geometry held at Taiwan's National Center for Theoretical Sciences. As is well-known, for many soliton equations, the solutions have interpretations as differential geometric objects, and thereby techniques of soliton equations have been successfully applied to the study of geometric problems. The article by Burstall gives a beautiful exposition on isothermic surfaces and their relations to integrable systems, and the two articles by Guest give an introduction to quantum cohomology, carry out explicit computations of the quantum cohomology of flag manifolds and Hirzebruch surfaces, and give a survey of Givental's quantum differential equations. The article by Heintze, Liu, and Olmos is on the theory of isoparametric submanifolds in an arbitrary Riemannian manifold, which is related to the n-wave equation when the ambient manifold is Euclidean. Mukai-Hidano and Ohnita present a survey on the moduli space of ...
Gualtieri, Marco
2010-01-01
Generalized Kahler geometry is the natural analogue of Kahler geometry, in the context of generalized complex geometry. Just as we may require a complex structure to be compatible with a Riemannian metric in a way which gives rise to a symplectic form, we may require a generalized complex structure to be compatible with a metric so that it defines a second generalized complex structure. We explore the fundamental aspects of this geometry, including its equivalence with the bi-Hermitian geometry on the target of a 2-dimensional sigma model with (2,2) supersymmetry, as well as the relation to holomorphic Dirac geometry and the resulting derived deformation theory. We also explore the analogy between pre-quantum line bundles and gerbes in the context of generalized Kahler geometry.
Methods for euclidean geometry
Byer, Owen; Smeltzer, Deirdre L
2010-01-01
Euclidean plane geometry is one of the oldest and most beautiful topics in mathematics. Instead of carefully building geometries from axiom sets, this book uses a wealth of methods to solve problems in Euclidean geometry. Many of these methods arose where existing techniques proved inadequate. In several cases, the new ideas used in solving specific problems later developed into independent areas of mathematics. This book is primarily a geometry textbook, but studying geometry in this way will also develop students' appreciation of the subject and of mathematics as a whole. For instance, despite the fact that the analytic method has been part of mathematics for four centuries, it is rarely a tool a student considers using when faced with a geometry problem. Methods for Euclidean Geometry explores the application of a broad range of mathematical topics to the solution of Euclidean problems.
Baratin, Aristide
2011-01-01
Using the non-commutative metric formulation of group field theories (GFT), we define a model of 4-dimensional quantum gravity as a constrained BF theory, without Immirzi parameter, encoding the quantum simplicial geometry of any triangulation used to define its quantum amplitudes. This involves a generalization of the usual GFT framework, where the usual field variables, associated to the four triangles of a tetrahedron, are supplemented by an S^3 vector playing the role of the normal to the tetrahedron. This leads naturally to projected spin network states. We give both a simplicial path integral and a spin foam formulation of the Feynman amplitudes, which correspond to a variant of the Barrett-Crane amplitudes. We then re-examin the arguments against the Barrett-Crane model(s), in light of our construction. We argue that it can still be considered a plausible quantization of 4d gravity, and that further work is needed to either confirm or refute its validity.
Geometry of Membrane Sigma Models
Vysoky, Jan
2015-01-01
String theory still remains one of the promising candidates for a unification of the theory of gravity and quantum field theory. One of its essential parts is relativistic description of moving multi-dimensional objects called membranes (or p-branes) in a curved spacetime. On the classical field theory level, they are described by an action functional extremalising the volume of a manifold swept by a propagating membrane. This and related field theories are collectively called membrane sigma models. Differential geometry is an important mathematical tool in the study of string theory. It turns out that string and membrane backgrounds can be conveniently described using objects defined on a direct sum of tangent and cotangent bundles of the spacetime manifold. Mathematical field studying such object is called generalized geometry. Its integral part is the theory of Leibniz algebroids, vector bundles with a Leibniz algebra bracket on its module of smooth sections. Special cases of Leibniz algebroids are better ...
Gear geometry of cycloid drives
Institute of Scientific and Technical Information of China (English)
CHEN BingKui; FANG TingTing; LI ChaoYang; WANG ShuYan
2008-01-01
According to differential geometry and gear geometry,the equation of meshing for small teeth difference planetary gearing and a universal equation of conjugated profile are established based on cylindrical pin tooth and given motion.The correct meshing condition,contact line,contact ratio,calculating method for pin tooth's maximum contact point are developed.Investigation on the theory of conjugated meshing is carried out when the tooth difference numbers between pin wheel and cycloidal gear are 1,2,3 and -1,respectively.A general method called enveloping method to generate hypocycloid and epicycloid is put forward.The correct mesh-ing condition for cycloid pin wheel gearing is provided,and the contact line and the contact ratio are also discussed.
Gear geometry of cycloid drives
Institute of Scientific and Technical Information of China (English)
2008-01-01
According to differential geometry and gear geometry, the equation of meshing for small teeth difference planetary gearing and a universal equation of conjugated profile are established based on cylindrical pin tooth and given motion. The correct meshing condition, contact line, contact ratio, calculating method for pin tooth’s maximum contact point are developed. Investigation on the theory of conjugated meshing is carried out when the tooth difference numbers between pin wheel and cycloidal gear are 1, 2, 3 and ?1, respectively. A general method called enveloping method to generate hypocycloid and epicycloid is put forward. The correct meshing condition for cycloid pin wheel gearing is provided, and the contact line and the contact ratio are also discussed.
Synchronization of Rossler chaotic system via the differential geometry method%基于微分几何方法和最优控制的混沌同步
Institute of Scientific and Technical Information of China (English)
李钢; 王水; 张吉泉; 姜明珠; 米佳; 姜珊; 王平
2015-01-01
The chaos synchronization between two same chaotic systems is investigated based on the differential geometry method and the optimal control theory .After nonlinear transformation accord‐ing to the Frobenius theorem by defining a proper output function of the error dynamical system for the case that the relative degree of the system is exactly equals to the order of the system ,the con‐troller can be designed by the exact feedback linearization of the error dynamical system .To demon‐strate the efficiency of the proposed scheme ,Rossler system is considered as the illustrative example .%基于微分几何方法和二次型性能指标最优控制原理研究了同结构混沌系统之间的同步问题。对于误差动力学系统的相对阶等于系统状态空间维数的情形，依据Frobe‐nius定理确定了用于非线性坐标变换的输出函数，对误差动力学系统进行了状态反馈精确线性化而得到了线性可控的正则形，从而确定了混沌同步的控制器。以Rossler系统为例的仿真模拟，验证了该方法的有效性。
Institute of Scientific and Technical Information of China (English)
沈娜; 韩凤琴; 久保田乔
2012-01-01
为了把时空非定常流动设计理念用于冲击式水轮机,旋转水斗自由曲面上复杂时空非定常水膜的数值可视化必不可少.文中采用局部非正交边界贴体网格(BFG)对水斗内表面进行描述,利用微分几何精确计算了网格处的自然基本矢量及其偏微分,得到了水斗表面的局部曲率、沿曲面最短距离以及微小曲面面积；成功地将流体的水膜移动网格点投影到了水斗内表面上.用建立的投影理论实现了从旋转水斗缺口及分水刃进入的非定常水膜移动网格的数值可视化,与模型试验照片的比较表明,该数值方法是有效的.%In order to apply the concept of unsteady flow in space and time domains to the design of Pelton buckets , a numerical visualization of the complicated unsteady water film flow on the free surface of a rotating bucket is indispensable. In this paper, the inner surface of a bucket is described by using the boundary-fitted grids with non-orthogonal curvilinear local coordinates. Then, the natural basic vectors and their partial differentials are precisely acquired based on the differential geometry, and the local curvature along the inner surface, the geodesic and the small surface area of the bucket are obtained. Moreover, the moving grids of the water film are successfully projected onto the bucket' s inner surface, and a projection algorithm is proposed to numerically visualize the moving grids of the unsteady water film flowing from the bucket cutout and the water separation edge. The visualization results are finally compared with the photos taken in the model test, which verifies the effectiveness of the proposed method.
Bárány, Imre; Vilcu, Costin
2016-01-01
This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems.
Algorithms in Algebraic Geometry
Dickenstein, Alicia; Sommese, Andrew J
2008-01-01
In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its
Fundamental concepts of geometry
Meserve, Bruce E
1983-01-01
Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.
O'Leary, Michael
2010-01-01
Guides readers through the development of geometry and basic proof writing using a historical approach to the topic. In an effort to fully appreciate the logic and structure of geometric proofs, Revolutions of Geometry places proofs into the context of geometry's history, helping readers to understand that proof writing is crucial to the job of a mathematician. Written for students and educators of mathematics alike, the book guides readers through the rich history and influential works, from ancient times to the present, behind the development of geometry. As a result, readers are successfull
Euclidean geometry and transformations
Dodge, Clayton W
1972-01-01
This introduction to Euclidean geometry emphasizes transformations, particularly isometries and similarities. Suitable for undergraduate courses, it includes numerous examples, many with detailed answers. 1972 edition.
Introduction to finite geometries
Kárteszi, F
1976-01-01
North-Holland Texts in Advanced Mathematics: Introduction to Finite Geometries focuses on the advancements in finite geometries, including mapping and combinatorics. The manuscript first offers information on the basic concepts on finite geometries and Galois geometries. Discussions focus on linear mapping of a given quadrangle onto another given quadrangle; point configurations of order 2 on a Galois plane of even order; canonical equation of curves of the second order on the Galois planes of even order; and set of collineations mapping a Galois plane onto itself. The text then ponders on geo
[Mathematics and string theory]. Progress report [August 1, 1992--July 31, 1993
Energy Technology Data Exchange (ETDEWEB)
Jaffe, A.; Yau, Shing-Tung
1993-07-01
Work on this grant was centered on connections between non- commutative geometry and physics. Topics covered included: cyclic cohomology, non-commutative manifolds, index theory, reflection positivity, space quantization, quantum groups, number theory, etc.
[Mathematics and string theory
Energy Technology Data Exchange (ETDEWEB)
Jaffe, A.; Yau, Shing-Tung.
1993-01-01
Work on this grant was centered on connections between non- commutative geometry and physics. Topics covered included: cyclic cohomology, non-commutative manifolds, index theory, reflection positivity, space quantization, quantum groups, number theory, etc.
Euclidean Geometry via Programming.
Filimonov, Rossen; Kreith, Kurt
1992-01-01
Describes the Plane Geometry System computer software developed at the Educational Computer Systems laboratory in Sofia, Bulgaria. The system enables students to use the concept of "algorithm" to correspond to the process of "deductive proof" in the development of plane geometry. Provides an example of the software's capability and compares it to…
Supersymmetric Sigma Model Geometry
Ulf Lindström
2012-01-01
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)k\\"ahler reduction; projective superspace; the generalized Legendre construction; generalized K\\"ahler geometry and constructions of hyperk\\"ahler metrics on Hermitean symmetric spaces.
Geometry of multihadron production
Energy Technology Data Exchange (ETDEWEB)
Bjorken, J.D.
1994-10-01
This summary talk only reviews a small sample of topics featured at this symposium: Introduction; The Geometry and Geography of Phase space; Space-Time Geometry and HBT; Multiplicities, Intermittency, Correlations; Disoriented Chiral Condensate; Deep Inelastic Scattering at HERA; and Other Contributions.
Supersymmetric Sigma Model geometry
Lindström, Ulf
2012-01-01
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)k\\"ahler reduction; projective superspace; the generalized Legendre construction; generalized K\\"ahler geometry and constructions of hyperk\\"ahler metrics on Hermitean symmetric spaces.
1996-01-01
Designs and Finite Geometries brings together in one place important contributions and up-to-date research results in this important area of mathematics. Designs and Finite Geometries serves as an excellent reference, providing insight into some of the most important research issues in the field.
Foundations of algebraic geometry
Weil, A
1946-01-01
This classic is one of the cornerstones of modern algebraic geometry. At the same time, it is entirely self-contained, assuming no knowledge whatsoever of algebraic geometry, and no knowledge of modern algebra beyond the simplest facts about abstract fields and their extensions, and the bare rudiments of the theory of ideals.
Bergshoeff, Eric A.; Riccioni, Fabio; Alvarez-Gaumé, L.
2011-01-01
We probe doubled geometry with dual fundamental branes. i.e. solitons. Restricting ourselves first to solitonic branes with more than two transverse directions we find that the doubled geometry requires an effective wrapping rule for the solitonic branes which is dual to the wrapping rule for fundam
Geometry and dynamics of integrable systems
Matveev, Vladimir
2016-01-01
Based on lectures given at an advanced course on integrable systems at the Centre de Recerca Matemàtica in Barcelona, these lecture notes address three major aspects of integrable systems: obstructions to integrability from differential Galois theory; the description of singularities of integrable systems on the basis of their relation to bi-Hamiltonian systems; and the generalization of integrable systems to the non-Hamiltonian settings. All three sections were written by top experts in their respective fields. Native to actual problem-solving challenges in mechanics, the topic of integrable systems is currently at the crossroads of several disciplines in pure and applied mathematics, and also has important interactions with physics. The study of integrable systems also actively employs methods from differential geometry. Moreover, it is extremely important in symplectic geometry and Hamiltonian dynamics, and has strong correlations with mathematical physics, Lie theory and algebraic geometry (including mir...