Geometric control theory and sub-Riemannian geometry
Boscain, Ugo; Gauthier, Jean-Paul; Sarychev, Andrey; Sigalotti, Mario
2014-01-01
This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.
Riemannian geometry and geometric analysis
Jost, Jürgen
2017-01-01
This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research. The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the B...
Scattering theory for Riemannian Laplacians
DEFF Research Database (Denmark)
Ito, Kenichi; Skibsted, Erik
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another...... condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperbolic...
Geometric calculus: a new computational tool for Riemannian geometry
International Nuclear Information System (INIS)
Moussiaux, A.; Tombal, P.
1988-01-01
We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus
Riemannian geometry of Hamiltonian chaos: hints for a general theory.
Cerruti-Sola, Monica; Ciraolo, Guido; Franzosi, Roberto; Pettini, Marco
2008-10-01
We aim at assessing the validity limits of some simplifying hypotheses that, within a Riemmannian geometric framework, have provided an explanation of the origin of Hamiltonian chaos and have made it possible to develop a method of analytically computing the largest Lyapunov exponent of Hamiltonian systems with many degrees of freedom. Therefore, a numerical hypotheses testing has been performed for the Fermi-Pasta-Ulam beta model and for a chain of coupled rotators. These models, for which analytic computations of the largest Lyapunov exponents have been carried out in the mentioned Riemannian geometric framework, appear as paradigmatic examples to unveil the reason why the main hypothesis of quasi-isotropy of the mechanical manifolds sometimes breaks down. The breakdown is expected whenever the topology of the mechanical manifolds is nontrivial. This is an important step forward in view of developing a geometric theory of Hamiltonian chaos of general validity.
Aspects of quasi-Riemannian Kaluza-Klein theory
International Nuclear Information System (INIS)
Viswanathan, K.S.; Wong, B.
1985-01-01
We consider the applications of quasi-Riemannian geometry in Kaluza-Klein theories. We find that such theories cannot be implemented for all choices of the tangent group G/sub T/ and internal space G/H for reasons of gauge invariance. Coupling of fermions to gravity poses further problems in these theories
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...
The Riemannian geometry is not sufficient for the geometrization of the Maxwell's equations
Kulyabov, Dmitry S.; Korolkova, Anna V.; Velieva, Tatyana R.
2018-04-01
The transformation optics uses geometrized Maxwell's constitutive equations to solve the inverse problem of optics, namely to solve the problem of finding the parameters of the medium along the paths of propagation of the electromagnetic field. For the geometrization of Maxwell's constitutive equations, the quadratic Riemannian geometry is usually used. This is due to the use of the approaches of the general relativity. However, there arises the question of the insufficiency of the Riemannian structure for describing the constitutive tensor of the Maxwell's equations. The authors analyze the structure of the constitutive tensor and correlate it with the structure of the metric tensor of Riemannian geometry. It is concluded that the use of the quadratic metric for the geometrization of Maxwell's equations is insufficient, since the number of components of the metric tensor is less than the number of components of the constitutive tensor. A possible solution to this problem may be a transition to Finslerian geometry, in particular, the use of the Berwald-Moor metric to establish the structural correspondence between the field tensors of the electromagnetic field.
Chaos based on Riemannian geometric approach to Abelian-Higgs dynamical system
International Nuclear Information System (INIS)
Kawabe, Tetsuji
2003-01-01
Based on the Riemannian geometric approach, we study chaos of the Abelian-Higgs dynamical system derived from a classical field equation consisting of a spatially homogeneous Abelian gauge field and Higgs field. Using the global indicator of chaos formulated by the sectional curvature of the ambient manifold, we show that this approach brings the same qualitative and quantitative information about order and chaos as has been provided by the Lyapunov exponents in the conventional and phenomenological approach. We confirm that the mechanism of chaos is a parametric instability of the system. By analyzing a close relation between the sectional curvature and the Gaussian curvature, we point out that the Toda-Brumer criterion becomes a sufficient condition to the criterion based on this geometric approach as to the stability condition
Riemannian theory of Hamiltonian chaos and Lyapunov exponents
Casetti, Lapo; Clementi, Cecilia; Pettini, Marco
1996-12-01
A nonvanishing Lyapunov exponent λ1 provides the very definition of deterministic chaos in the solutions of a dynamical system; however, no theoretical mean of predicting its value exists. This paper copes with the problem of analytically computing the largest Lyapunov exponent λ1 for many degrees of freedom Hamiltonian systems as a function of ɛ=E/N, the energy per degree of freedom. The functional dependence λ1(ɛ) is of great interest because, among other reasons, it detects the existence of weakly and strongly chaotic regimes. This aim, the analytic computation of λ1(ɛ), is successfully reached within a theoretical framework that makes use of a geometrization of Newtonian dynamics in the language of Riemannian differential geometry. An alternative point of view about the origin of chaos in these systems is obtained independently of the standard explanation based on homoclinic intersections. Dynamical instability (chaos) is here related to curvature fluctuations of the manifolds whose geodesics are natural motions and is described by means of the Jacobi-Levi-Civita equation (JLCE) for geodesic spread. In this paper it is shown how to derive from the JLCE an effective stability equation. Under general conditions, this effective equation formally describes a stochastic oscillator; an analytic formula for the instability growth rate of its solutions is worked out and applied to the Fermi-Pasta-Ulam β model and to a chain of coupled rotators. Excellent agreement is found between the theoretical prediction and numeric values of λ1(ɛ) for both models.
On geometrized gravitation theories
International Nuclear Information System (INIS)
Logunov, A.A.; Folomeshkin, V.N.
1977-01-01
General properties of the geometrized gravitation theories have been considered. Geometrization of the theory is realized only to the extent that by necessity follows from an experiment (geometrization of the density of the matter Lagrangian only). Aor a general case the gravitation field equations and the equations of motion for matter are formulated in the different Riemann spaces. A covariant formulation of the energy-momentum conservation laws is given in an arbitrary geometrized theory. The noncovariant notion of ''pseudotensor'' is not required in formulating the conservation laws. It is shown that in the general case (i.e., when there is an explicit dependence of the matter Lagrangian density on the covariant derivatives) a symmetric energy-momentum tensor of the matter is explicitly dependent on the curvature tensor. There are enlisted different geometrized theories that describe a known set of the experimental facts. The properties of one of the versions of the quasilinear geometrized theory that describes the experimental facts are considered. In such a theory the fundamental static spherically symmetrical solution has a singularity only in the coordinate origin. The theory permits to create a satisfactory model of the homogeneous nonstationary Universe
Curvature of fluctuation geometry and its implications on Riemannian fluctuation theory
International Nuclear Information System (INIS)
Velazquez, L
2013-01-01
Fluctuation geometry was recently proposed as a counterpart approach of the Riemannian geometry of inference theory (widely known as information geometry). This theory describes the geometric features of the statistical manifold M of random events that are described by a family of continuous distributions dp(x|θ). A main goal of this work is to clarify the statistical relevance of the Levi-Civita curvature tensor R ijkl (x|θ) of the statistical manifold M. For this purpose, the notion of irreducible statistical correlations is introduced. Specifically, a distribution dp(x|θ) exhibits irreducible statistical correlations if every distribution dp(x-check|θ) obtained from dp(x|θ) by considering a coordinate change x-check = φ(x) cannot be factorized into independent distributions as dp(x-check|θ) = prod i dp (i) (x-check i |θ). It is shown that the curvature tensor R ijkl (x|θ) arises as a direct indicator about the existence of irreducible statistical correlations. Moreover, the curvature scalar R(x|θ) allows us to introduce a criterium for the applicability of the Gaussian approximation of a given distribution function. This type of asymptotic result is obtained in the framework of the second-order geometric expansion of the distribution family dp(x|θ), which appears as a counterpart development of the high-order asymptotic theory of statistical estimation. In physics, fluctuation geometry represents the mathematical apparatus of a Riemannian extension for Einstein’s fluctuation theory of statistical mechanics. Some exact results of fluctuation geometry are now employed to derive the invariant fluctuation theorems. Moreover, the curvature scalar allows us to express some asymptotic formulae that account for the system fluctuating behavior beyond the Gaussian approximation, e.g.: it appears as a second-order correction of the Legendre transformation between thermodynamic potentials, P(θ)=θ i x-bar i -s( x-bar |θ)+k 2 R(x|θ)/6. (paper)
Druţu, Cornelia
2018-01-01
The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamental groups of manifolds, groups of matrices with integer coefficients, etc. The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls. The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the f...
Quantum theory of spinor field in four-dimensional Riemannian space-time
International Nuclear Information System (INIS)
Shavokhina, N.S.
1996-01-01
The review deals with the spinor field in the four-dimensional Riemannian space-time. The field beys the Dirac-Fock-Ivanenko equation. Principles of quantization of the spinor field in the Riemannian space-time are formulated which in a particular case of the plane space-time are equivalent to the canonical rules of quantization. The formulated principles are exemplified by the De Sitter space-time. The study of quantum field theory in the De Sitter space-time is interesting because it itself leads to a method of an invariant well for plane space-time. However, the study of the quantum spinor field theory in an arbitrary Riemannian space-time allows one to take into account the influence of the external gravitational field on the quantized spinor field. 60 refs
Waerden, B
1996-01-01
From the reviews: "... Federer's timely and beautiful book indeed fills the need for a comprehensive treatise on geometric measure theory, and his detailed exposition leads from the foundations of the theory to the most recent discoveries. ... The author writes with a distinctive style which is both natural and powerfully economical in treating a complicated subject. This book is a major treatise in mathematics and is essential in the working library of the modern analyst." Bulletin of the London Mathematical Society.
Bestvina, Mladen; Vogtmann, Karen
2014-01-01
Geometric group theory refers to the study of discrete groups using tools from topology, geometry, dynamics and analysis. The field is evolving very rapidly and the present volume provides an introduction to and overview of various topics which have played critical roles in this evolution. The book contains lecture notes from courses given at the Park City Math Institute on Geometric Group Theory. The institute consists of a set of intensive short courses offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The courses begin at an introductory level suitable for graduate students and lead up to currently active topics of research. The articles in this volume include introductions to CAT(0) cube complexes and groups, to modern small cancellation theory, to isometry groups of general CAT(0) spaces, and a discussion of nilpotent genus in the context of mapping class groups and CAT(0) gro...
Convex functions and optimization methods on Riemannian manifolds
Udrişte, Constantin
1994-01-01
This unique monograph discusses the interaction between Riemannian geometry, convex programming, numerical analysis, dynamical systems and mathematical modelling. The book is the first account of the development of this subject as it emerged at the beginning of the 'seventies. A unified theory of convexity of functions, dynamical systems and optimization methods on Riemannian manifolds is also presented. Topics covered include geodesics and completeness of Riemannian manifolds, variations of the p-energy of a curve and Jacobi fields, convex programs on Riemannian manifolds, geometrical constructions of convex functions, flows and energies, applications of convexity, descent algorithms on Riemannian manifolds, TC and TP programs for calculations and plots, all allowing the user to explore and experiment interactively with real life problems in the language of Riemannian geometry. An appendix is devoted to convexity and completeness in Finsler manifolds. For students and researchers in such diverse fields as pu...
A geometrical foundation of a unified field theory
International Nuclear Information System (INIS)
Tauber, G.E.
1983-01-01
In a series of two little known papers Einstein and Mayer proposed a formalism by which they were able to obtain a theory of gravitation and electromagnetism similar to that of Kaluza and Klein. Instead of assuming, as these authors did, the existence of a five-dimensional continuum they assumed that at each point of space-time, regarded as a Riemannian space there exists a five-dimensional vector space. The purpose of this work is to generalize the approach of Einstein and Mayer to N dimensions and to lay the geometrical foundation of a possible unified field theory of gravitation with other fields. (Auth.)
Riemannian computing in computer vision
Srivastava, Anuj
2016-01-01
This book presents a comprehensive treatise on Riemannian geometric computations and related statistical inferences in several computer vision problems. This edited volume includes chapter contributions from leading figures in the field of computer vision who are applying Riemannian geometric approaches in problems such as face recognition, activity recognition, object detection, biomedical image analysis, and structure-from-motion. Some of the mathematical entities that necessitate a geometric analysis include rotation matrices (e.g. in modeling camera motion), stick figures (e.g. for activity recognition), subspace comparisons (e.g. in face recognition), symmetric positive-definite matrices (e.g. in diffusion tensor imaging), and function-spaces (e.g. in studying shapes of closed contours). · Illustrates Riemannian computing theory on applications in computer vision, machine learning, and robotics · Emphasis on algorithmic advances that will allow re-application in other...
Geometric theory of information
2014-01-01
This book brings together geometric tools and their applications for Information analysis. It collects current and many uses of in the interdisciplinary fields of Information Geometry Manifolds in Advanced Signal, Image & Video Processing, Complex Data Modeling and Analysis, Information Ranking and Retrieval, Coding, Cognitive Systems, Optimal Control, Statistics on Manifolds, Machine Learning, Speech/sound recognition, and natural language treatment which are also substantially relevant for the industry.
Geometric group theory an introduction
Löh, Clara
2017-01-01
Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.
A new geometrical gravitational theory
International Nuclear Information System (INIS)
Obata, T.; Chiba, J.; Oshima, H.
1981-01-01
A geometrical gravitational theory is developed. The field equations are uniquely determined apart from one unknown dimensionless parameter ω 2 . It is based on an extension of the Weyl geometry, and by the extension the gravitational coupling constant and the gravitational mass are made to be dynamical and geometrical. The fundamental geometrical objects in the theory are a metric gsub(μν) and two gauge scalars phi and psi. The theory satisfies the weak equivalence principle, but breaks the strong one generally. u(phi, psi) = phi is found out on the assumption that the strong one keeps holding good at least for bosons of low spins. Thus there is the simple correspondence between the geometrical objects and the gravitational objects. Since the theory satisfies the weak one, the inertial mass is also dynamical and geometrical in the same way as is the gravitational mass. Moreover, the cosmological term in the theory is a coscalar of power -4 algebraically made of psi and u(phi, psi), so it is dynamical, too. Finally spherically symmetric exact solutions are given. The permissible range of the unknown parameter ω 2 is experimentally determined by applying the solutions to the solar system. (author)
Geometric Topology and Shape Theory
Segal, Jack
1987-01-01
The aim of this international conference the third of its type was to survey recent developments in Geometric Topology and Shape Theory with an emphasis on their interaction. The volume contains original research papers and carefully selected survey of currently active areas. The main topics and themes represented by the papers of this volume include decomposition theory, cell-like mappings and CE-equivalent compacta, covering dimension versus cohomological dimension, ANR's and LCn-compacta, homology manifolds, embeddings of continua into manifolds, complement theorems in shape theory, approximate fibrations and shape fibrations, fibered shape, exact homologies and strong shape theory.
Geometrical methods in learning theory
International Nuclear Information System (INIS)
Burdet, G.; Combe, Ph.; Nencka, H.
2001-01-01
The methods of information theory provide natural approaches to learning algorithms in the case of stochastic formal neural networks. Most of the classical techniques are based on some extremization principle. A geometrical interpretation of the associated algorithms provides a powerful tool for understanding the learning process and its stability and offers a framework for discussing possible new learning rules. An illustration is given using sequential and parallel learning in the Boltzmann machine
Minimal Webs in Riemannian Manifolds
DEFF Research Database (Denmark)
Markvorsen, Steen
2008-01-01
For a given combinatorial graph $G$ a {\\it geometrization} $(G, g)$ of the graph is obtained by considering each edge of the graph as a $1-$dimensional manifold with an associated metric $g$. In this paper we are concerned with {\\it minimal isometric immersions} of geometrized graphs $(G, g......)$ into Riemannian manifolds $(N^{n}, h)$. Such immersions we call {\\em{minimal webs}}. They admit a natural 'geometric' extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which...... are of instrumental importance for the applications. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence...
Geometric function theory in higher dimension
2017-01-01
The book collects the most relevant outcomes from the INdAM Workshop “Geometric Function Theory in Higher Dimension” held in Cortona on September 5-9, 2016. The Workshop was mainly devoted to discussions of basic open problems in the area, and this volume follows the same line. In particular, it offers a selection of original contributions on Loewner theory in one and higher dimensions, semigroups theory, iteration theory and related topics. Written by experts in geometric function theory in one and several complex variables, it focuses on new research frontiers in this area and on challenging open problems. The book is intended for graduate students and researchers working in complex analysis, several complex variables and geometric function theory.
Yang, Paul; Gambino, Nicola; Kock, Joachim
2015-01-01
The two parts of the present volume contain extended conference abstracts corresponding to selected talks given by participants at the "Conference on Geometric Analysis" (thirteen abstracts) and at the "Conference on Type Theory, Homotopy Theory and Univalent Foundations" (seven abstracts), both held at the Centre de Recerca Matemàtica (CRM) in Barcelona from July 1st to 5th, 2013, and from September 23th to 27th, 2013, respectively. Most of them are brief articles, containing preliminary presentations of new results not yet published in regular research journals. The articles are the result of a direct collaboration between active researchers in the area after working in a dynamic and productive atmosphere. The first part is about Geometric Analysis and Conformal Geometry; this modern field lies at the intersection of many branches of mathematics (Riemannian, Conformal, Complex or Algebraic Geometry, Calculus of Variations, PDE's, etc) and relates directly to the physical world, since many natural phenomena...
Geometrical theory of spin motion
International Nuclear Information System (INIS)
Halpern, L.
1983-01-01
A discussion of the fundamental interrelation of geometry and physical laws with Lie groups leads to a reformulation and heuristic modification of the principle of inertia and the principle of equivalence, which is based on the simple De Sitter group instead of the Poincare group. The resulting law of motion allows a unified formulation for structureless and spinning test particles. A metrical theory of gravitation is constructed with the modified principle, which is structured after the geometry of the manifold of the De Sitter group. The theory is equivalent to a particular Kaluza-Klein theory in ten dimensions with the Lorentz group as gauge group. A restricted version of this theory excludes torsion. It is shown by a reformulation of the energy momentum complex that this version is equivalent to general relativity with a cosmologic term quadratic in the curvature tensor and in which the existence of spinning particle fields is inherent from first principles. The equations of the general theory with torsion are presented and it is shown in a special case how the boundary conditions for the torsion degree of freedom have to be chosen such as to treat orbital and spin angular momenta on an equal footing. The possibility of verification of the resulting anomalous spin-spin interaction is mentioned and a model imposed by the group topology of SO(3, 2) is outlined in which the unexplained discrepancy between the magnitude of the discrete valued coupling constants and the gravitational constant in Kaluza-Klein theories is resolved by the identification of identical fermions as one orbit. The mathematical structure can be adapted to larger groups to include other degrees of freedom. 41 references
Initial singularity and pure geometric field theories
Wanas, M. I.; Kamal, Mona M.; Dabash, Tahia F.
2018-01-01
In the present article we use a modified version of the geodesic equation, together with a modified version of the Raychaudhuri equation, to study initial singularities. These modified equations are used to account for the effect of the spin-torsion interaction on the existence of initial singularities in cosmological models. Such models are the results of solutions of the field equations of a class of field theories termed pure geometric. The geometric structure used in this study is an absolute parallelism structure satisfying the cosmological principle. It is shown that the existence of initial singularities is subject to some mathematical (geometric) conditions. The scheme suggested for this study can be easily generalized.
A Geometrical View of Higgs Effective Theory
CERN. Geneva
2016-01-01
A geometric formulation of Higgs Effective Field Theory (HEFT) is presented. Experimental observables are given in terms of geometric invariants of the scalar sigma model sector such as the curvature of the scalar field manifold M. We show how the curvature can be measured experimentally via Higgs cross-sections, W_L scattering, and the S parameter. The one-loop action of HEFT is given in terms of geometric invariants of M. The distinction between the Standard Model (SM) and HEFT is whether M is flat or curved, with the curvature a signal of the scale of new physics.
Towards a theory of geometric graphs
Pach, Janos
2004-01-01
The early development of graph theory was heavily motivated and influenced by topological and geometric themes, such as the Konigsberg Bridge Problem, Euler's Polyhedral Formula, or Kuratowski's characterization of planar graphs. In 1936, when Denes Konig published his classical Theory of Finite and Infinite Graphs, the first book ever written on the subject, he stressed this connection by adding the subtitle Combinatorial Topology of Systems of Segments. He wanted to emphasize that the subject of his investigations was very concrete: planar figures consisting of points connected by straight-line segments. However, in the second half of the twentieth century, graph theoretical research took an interesting turn. In the most popular and most rapidly growing areas (the theory of random graphs, Ramsey theory, extremal graph theory, algebraic graph theory, etc.), graphs were considered as abstract binary relations rather than geometric objects. Many of the powerful techniques developed in these fields have been su...
Understanding geometric algebra for electromagnetic theory
Arthur, John W
2011-01-01
"This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison"--Provided by publisher.
Riemannian and Lorentzian flow-cut theorems
Headrick, Matthew; Hubeny, Veronika E.
2018-05-01
We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut (MFMC) theorem for boundary regions, applied recently to develop a ‘bit-thread’ interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous MFMC theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth’s theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.
Workshop on Topology and Geometric Group Theory
Fowler, James; Lafont, Jean-Francois; Leary, Ian
2016-01-01
This book presents articles at the interface of two active areas of research: classical topology and the relatively new field of geometric group theory. It includes two long survey articles, one on proofs of the Farrell–Jones conjectures, and the other on ends of spaces and groups. In 2010–2011, Ohio State University (OSU) hosted a special year in topology and geometric group theory. Over the course of the year, there were seminars, workshops, short weekend conferences, and a major conference out of which this book resulted. Four other research articles complement these surveys, making this book ideal for graduate students and established mathematicians interested in entering this area of research.
The geometric Hopf invariant and surgery theory
Crabb, Michael
2017-01-01
Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds. Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists. Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, with many results old and new. .
International Nuclear Information System (INIS)
Ezin, J.P.
1988-08-01
The lectures given at the ''5th Symposium of Mathematics in Abidjan: Differential Geometry and Mechanics'' are presented. They are divided into four chapters: Riemannian metric on a differential manifold, curvature tensor fields on a Riemannian manifold, some classical functionals on Riemannian manifolds and questions. 11 refs
Comparison theorems in Riemannian geometry
Cheeger, Jeff
2008-01-01
The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most re
Geometric continuum regularization of quantum field theory
International Nuclear Information System (INIS)
Halpern, M.B.
1989-01-01
An overview of the continuum regularization program is given. The program is traced from its roots in stochastic quantization, with emphasis on the examples of regularized gauge theory, the regularized general nonlinear sigma model and regularized quantum gravity. In its coordinate-invariant form, the regularization is seen as entirely geometric: only the supermetric on field deformations is regularized, and the prescription provides universal nonperturbative invariant continuum regularization across all quantum field theory. 54 refs
Geometric measure theory a beginner's guide
Morgan, Frank
1995-01-01
Geometric measure theory is the mathematical framework for the study of crystal growth, clusters of soap bubbles, and similar structures involving minimization of energy. Morgan emphasizes geometry over proofs and technicalities, and includes a bibliography and abundant illustrations and examples. This Second Edition features a new chapter on soap bubbles as well as updated sections addressing volume constraints, surfaces in manifolds, free boundaries, and Besicovitch constant results. The text will introduce newcomers to the field and appeal to mathematicians working in the field.
Geometric Measure Theory and Minimal Surfaces
Bombieri, Enrico
2011-01-01
W.K. ALLARD: On the first variation of area and generalized mean curvature.- F.J. ALMGREN Jr.: Geometric measure theory and elliptic variational problems.- E. GIUSTI: Minimal surfaces with obstacles.- J. GUCKENHEIMER: Singularities in soap-bubble-like and soap-film-like surfaces.- D. KINDERLEHRER: The analyticity of the coincidence set in variational inequalities.- M. MIRANDA: Boundaries of Caciopoli sets in the calculus of variations.- L. PICCININI: De Giorgi's measure and thin obstacles.
Austerity and geometric structure of field theories
International Nuclear Information System (INIS)
Kheyfets, A.
1986-01-01
The relation between the austerity idea and the geometric structure of the three basic field theories - electrodynamics, Yang-Mills theory, and general relativity - is studied. One of the most significant manifestations of the austerity idea in field theories is thought to be expressed by the boundary of a boundary principle (BBP). The BBP says that almost all content of the field theories can be deduced from the topological identity of delta dot produced with delta = 0 used twice, at the 1-2-3-dimensional level (providing the homogeneous field equations), and at the 2-3-4-dimensional level (providing the conservation laws for the source currents). There are some difficulties in this line of thought due to the apparent lack of universality in application of the BBP to the three basic modern field theories above. This dissertation: (a) analyzes the difficulties by means of algebraic topology, integration theory, and modern differential geometry based on the concepts of principal bundles and Ehresmann connections: (b) extends the BBP to the unified Kaluza-Klein theory; (c) reformulates the inhomogeneous field equations and the BBP in terms of E. Cartan moment of rotation, in the way universal for the three theories and compatible with the original austerity idea; and (d) underlines the important role of the soldering structure on spacetime, and indicates that the future development of the austerity idea would involve the generalized theories
Geometric perturbation theory and plasma physics
International Nuclear Information System (INIS)
Omohundro, S.M.
1985-01-01
Modern differential geometric techniques are used to unify the physical asymptotics underlying mechanics, wave theory, and statistical mechanics. The approach gives new insights into the structure of physical theories and is suited to the needs of modern large-scale computer simulation and symbol manipulation systems. A coordinate-free formulation of non-singular perturbation theory is given, from which a new Hamiltonian perturbation structure is derived and related to the unperturbed structure in five different ways. The theory of perturbations in the presence of symmetry is developed, and the method of averaging is related to reduction by a circle-group action. The pseudo-forces and magnetic Poisson bracket terms due to reduction are given a natural asymptotic interpretation. Similar terms due to changing reference frames are related to the method of variation of parameters, which is also given a Hamiltonian formulation. These methods are used to answer a long-standing question posed by Kruskal about nearly periodic systems. The answer leads to a new secular perturbation theory that contains no adhoc elements, which is then applied to gyromotion. Eikonal wave theory is given a Hamiltonian formulation that generalizes Whitham's Lagrangian approach. The evolution of wave action density on ray phase space is given a Hamiltonian structure using a Lie-Poisson bracket. The relationship between dissipative and Hamiltonian systems is discussed. A theory motivated by free electron lasers gives new restrictions on the change of area of projected parallelepipeds under canonical transformations
Bray, Hubert L; Mazzeo, Rafe; Sesum, Natasa
2015-01-01
This volume includes expanded versions of the lectures delivered in the Graduate Minicourse portion of the 2013 Park City Mathematics Institute session on Geometric Analysis. The papers give excellent high-level introductions, suitable for graduate students wishing to enter the field and experienced researchers alike, to a range of the most important areas of geometric analysis. These include: the general issue of geometric evolution, with more detailed lectures on Ricci flow and Kähler-Ricci flow, new progress on the analytic aspects of the Willmore equation as well as an introduction to the recent proof of the Willmore conjecture and new directions in min-max theory for geometric variational problems, the current state of the art regarding minimal surfaces in R^3, the role of critical metrics in Riemannian geometry, and the modern perspective on the study of eigenfunctions and eigenvalues for Laplace-Beltrami operators.
Geometric perturbation theory and plasma physics
Energy Technology Data Exchange (ETDEWEB)
Omohundro, S.M.
1985-04-04
Modern differential geometric techniques are used to unify the physical asymptotics underlying mechanics, wave theory and statistical mechanics. The approach gives new insights into the structure of physical theories and is suited to the needs of modern large-scale computer simulation and symbol manipulation systems. A coordinate-free formulation of non-singular perturbation theory is given, from which a new Hamiltonian perturbation structure is derived and related to the unperturbed structure. The theory of perturbations in the presence of symmetry is developed, and the method of averaging is related to reduction by a circle group action. The pseudo-forces and magnetic Poisson bracket terms due to reduction are given a natural asymptotic interpretation. Similar terms due to changing reference frames are related to the method of variation of parameters, which is also given a Hamiltonian formulation. These methods are used to answer a question about nearly periodic systems. The answer leads to a new secular perturbation theory that contains no ad hoc elements. Eikonal wave theory is given a Hamiltonian formulation that generalizes Whitham's Lagrangian approach. The evolution of wave action density on ray phase space is given a Hamiltonian structure using a Lie-Poisson bracket. The relationship between dissipative and Hamiltonian systems is discussed. A new type of attractor is defined which attracts both forward and backward in time and is shown to occur in infinite-dimensional Hamiltonian systems with dissipative behavior. The theory of Smale horseshoes is applied to gyromotion in the neighborhood of a magnetic field reversal and the phenomenon of reinsertion in area-preserving horseshoes is introduced. The central limit theorem is proved by renormalization group techniques. A natural symplectic structure for thermodynamics is shown to arise asymptotically from the maximum entropy formalism.
Geometric perturbation theory and plasma physics
International Nuclear Information System (INIS)
Omohundro, S.M.
1985-01-01
Modern differential geometric techniques are used to unify the physical asymptotics underlying mechanics, wave theory and statistical mechanics. The approach gives new insights into the structure of physical theories and is suited to the needs of modern large-scale computer simulation and symbol manipulation systems. A coordinate-free formulation of non-singular perturbation theory is given, from which a new Hamiltonian perturbation structure is derived and related to the unperturbed structure. The theory of perturbations in the presence of symmetry is developed, and the method of averaging is related to reduction by a circle group action. The pseudo-forces and magnetic Poisson bracket terms due to reduction are given a natural asymptotic interpretation. Similar terms due to changing reference frames are related to the method of variation of parameters, which is also given a Hamiltonian formulation. These methods are used to answer a question about nearly periodic systems. The answer leads to a new secular perturbation theory that contains no ad hoc elements. Eikonal wave theory is given a Hamiltonian formulation that generalizes Whitham's Lagrangian approach. The evolution of wave action density on ray phase space is given a Hamiltonian structure using a Lie-Poisson bracket. The relationship between dissipative and Hamiltonian systems is discussed. A new type of attractor is defined which attracts both forward and backward in time and is shown to occur in infinite-dimensional Hamiltonian systems with dissipative behavior. The theory of Smale horseshoes is applied to gyromotion in the neighborhood of a magnetic field reversal and the phenomenon of reinsertion in area-preserving horseshoes is introduced. The central limit theorem is proved by renormalization group techniques. A natural symplectic structure for thermodynamics is shown to arise asymptotically from the maximum entropy formalism
Absence of embedded eigenvalues for Riemannian Laplacians
DEFF Research Database (Denmark)
Ito, Kenichi; Skibsted, Erik
Schrödinger operators on non-compact connected Riemannian manifolds. A principal example is given by a manifold with an end (possibly more than one) in which geodesic coordinates are naturally defined. In this case one of our geometric conditions is a positive lower bound of the second fundamenta...
Key Notes on a Geometric Theory of Fields
Directory of Open Access Journals (Sweden)
Bruchholz U. E.
2009-04-01
Full Text Available The role of potentials and sources in electromagnetic and gravitational fields is investi- gated. A critical analysis leads to the result that sources have to be replaced by integra- tion constants. The existence of spatial boundaries gives reasons for this step. Potentials gain physical relevance first with it. The common view, that fields are “generated” by sources, appears as not tenable. Fields do exist by their own. These insights as well as results from numerical simulations force the conclusion that a Riemannian-geometrical background of electromagnetism and even quantum phenomena cannot be excluded. Nature could differ from abstract geometry in a way that distances and intervals never become infinitesimally small.
Conservation laws in quantum mechanics on a Riemannian manifold
International Nuclear Information System (INIS)
Chepilko, N.M.
1992-01-01
In Refs. 1-5 the quantum dynamics of a particle on a Riemannian manifold V n is considered. The advantage of Ref. 5, in comparison with Refs. 1-4, is the fact that in it the differential-geometric character of the theory and the covariant definition (via the known Lagrangian of the particle) of the algebra of quantum-mechanical operators on V n are mutually consistent. However, in Ref. 5 the procedure for calculating the expectation values of operators from the known wave function of the particle is not discussed. In the authors view, this question is problematical and requires special study. The essence of the problem is that integration on a Riemannian manifold V n , unlike that of a Euclidean manifold R n , is uniquely defined only for scalars. For this reason, the calculation of the expectation value of, e.g., the operator of the momentum or angular momentum of a particle on V n is not defined in the usual sense. However, this circumstance was not taken into account by the authors of Refs. 1-4, in which quantum mechanics on a Riemannian manifold V n was studied. In this paper the author considers the conservation laws and a procedure for calculating observable quantities in the classical mechanics (Sec. 2) and quantum mechanics (Sec. 3) of a particle on V n . It is found that a key role here is played by the Killing vectors of the Riemannian manifold V n . It is shown that the proposed approach to the problem satisfies the correspondence principle for both the classical and the quantum mechanics of a particle on a Euclidean manifold R n
Geometric Theory of Reduction of Nonlinear Control Systems
Elkin, V. I.
2018-02-01
The foundations of a differential geometric theory of nonlinear control systems are described on the basis of categorical concepts (isomorphism, factorization, restrictions) by analogy with classical mathematical theories (of linear spaces, groups, etc.).
From the geometric quantization to conformal field theory
International Nuclear Information System (INIS)
Alekseev, A.; Shatashvili, S.
1990-01-01
Investigation of 2d conformal field theory in terms of geometric quantization is given. We quantize the so-called model space of the compact Lie group, Virasoro group and Kac-Moody group. In particular, we give a geometrical interpretation of the Virasoro discrete series and explain that this type of geometric quantization reproduces the chiral part of CFT (minimal models, 2d-gravity, WZNW theory). In the appendix we discuss the relation between classical (constant) r-matrices and this geometrical approach. (orig.)
Noncritical String Liouville Theory and Geometric Bootstrap Hypothesis
Hadasz, Leszek; Jaskólski, Zbigniew
The applications of the existing Liouville theories for the description of the longitudinal dynamics of noncritical Nambu-Goto string are analyzed. We show that the recently developed DOZZ solution to the Liouville theory leads to the cut singularities in tree string amplitudes. We propose a new version of the Polyakov geometric approach to Liouville theory and formulate its basic consistency condition — the geometric bootstrap equation. Also in this approach the tree amplitudes develop cut singularities.
The geometrical theory of diffraction for axially symmetric reflectors
DEFF Research Database (Denmark)
Rusch, W.; Sørensen, O.
1975-01-01
The geometrical theory of diffraction (GTD) (cf. [1], for example) may be applied advantageously to many axially symmetric reflector antenna geometries. The material in this communication presents analytical, computational, and experimental results for commonly encountered reflector geometries...
Transformation optics, isotropic chiral media and non-Riemannian geometry
International Nuclear Information System (INIS)
Horsley, S A R
2011-01-01
The geometrical interpretation of electromagnetism in transparent media (transformation optics) is extended to include chiral media that are isotropic but inhomogeneous. It was found that such media may be described through introducing the non-Riemannian geometrical property of torsion into the Maxwell equations, and it is shown how such an interpretation may be applied to the design of optical devices.
Geometric Hamiltonian structures and perturbation theory
International Nuclear Information System (INIS)
Omohundro, S.
1984-08-01
We have been engaged in a program of investigating the Hamiltonian structure of the various perturbation theories used in practice. We describe the geometry of a Hamiltonian structure for non-singular perturbation theory applied to Hamiltonian systems on symplectic manifolds and the connection with singular perturbation techniques based on the method of averaging
Differential geometric methods in system theory.
Brockett, R. W.
1971-01-01
Discussion of certain problems in system theory which have been or might be solved using some basic concepts from differential geometry. The problems considered involve differential equations, controllability, optimal control, qualitative behavior, stochastic processes, and bilinear systems. The main goal is to extend the essentials of linear theory to some nonlinear classes of problems.
A geometrical introduction to screw theory
International Nuclear Information System (INIS)
Minguzzi, E
2013-01-01
This work introduces screw theory, a venerable but little known theory aimed at describing rigid body dynamics. This formulation of mechanics unifies in the concept of screw the translational and rotational degrees of freedom of the body. It captures a remarkable mathematical analogy between mechanical momenta and linear velocities, and between forces and angular velocities. For instance, it clarifies that angular velocities should be treated as applied vectors and that, under the composition of motions, they sum with the same rules of applied forces. This work provides a short and rigorous introduction to screw theory intended for an undergraduate and general readership. (paper)
Pseudo-Riemannian Novikov algebras
Energy Technology Data Exchange (ETDEWEB)
Chen Zhiqi; Zhu Fuhai [School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071 (China)], E-mail: chenzhiqi@nankai.edu.cn, E-mail: zhufuhai@nankai.edu.cn
2008-08-08
Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic-type and Hamiltonian operators in formal variational calculus. Pseudo-Riemannian Novikov algebras denote Novikov algebras with non-degenerate invariant symmetric bilinear forms. In this paper, we find that there is a remarkable geometry on pseudo-Riemannian Novikov algebras, and give a special class of pseudo-Riemannian Novikov algebras.
A geometric formulation of exceptional field theory
Energy Technology Data Exchange (ETDEWEB)
Bosque, Pascal du [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Hassler, Falk [Department of Physics and Astronomy, University of North Carolina, Phillips Hall, CB #3255, 120 E. Cameron Ave., Chapel Hill, NC 27599-3255 (United States); City University of New York, The Graduate Center, 365 Fifth Avenue, New York, NY 10016 (United States); Department of Physics, Columbia University, Pupin Hall, 550 West 120th St., New York, NY 10027 (United States); Lüst, Dieter [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Malek, Emanuel [Arnold Sommerfeld Center for Theoretical Physics,Department für Physik, Ludwig-Maximilians-Universität München,Theresienstraße 37, 80333 München (Germany)
2017-03-01
We formulate the full bosonic SL(5) exceptional field theory in a coordinate-invariant manner. Thereby we interpret the 10-dimensional extended space as a manifold with SL(5)×ℝ{sup +}-structure. We show that the algebra of generalised diffeomorphisms closes subject to a set of closure constraints which are reminiscent of the quadratic and linear constraints of maximal seven-dimensional gauged supergravities, as well as the section condition. We construct an action for the full bosonic SL(5) exceptional field theory, even when the SL(5)×ℝ{sup +}-structure is not locally flat.
Quantum effects from a purely geometrical relativity theory
International Nuclear Information System (INIS)
Ellis, Homer G
2005-01-01
A purely geometrical relativity theory results from a construction that produces from three-dimensional space a happy unification of Kaluza's five-dimensional theory and Weyl's conformal theory. The theory can provide geometrical explanations for the following observed phenomena, among others: (a) visibility lifetimes of elementary particles of lengths inversely proportional to their rest masses; (b) the equality of charge magnitude among all charged particles interacting at an event; (c) the propensity of electrons in atoms to be seen in discretely spaced orbits; and (d) 'quantum jumps' between those orbits. This suggests the possibility that the theory can provide a deterministic underpinning of quantum mechanics like that provided to thermodynamics by the molecular theory of gases
The geometric $\\beta$-function in curved space-time under operator regularization
Agarwala, Susama
2009-01-01
In this paper, I compare the generators of the renormalization group flow, or the geometric $\\beta$-functions for dimensional regularization and operator regularization. I then extend the analysis to show that the geometric $\\beta$-function for a scalar field theory on a closed compact Riemannian manifold is defined on the entire manifold. I then extend the analysis to find the generator of the renormalization group flow for a conformal scalar-field theories on the same manifolds. The geometr...
Geometrical identification of quantum and information theories
International Nuclear Information System (INIS)
Caianiello, E.R.
1983-01-01
The interrelation of quantum and information theories is investigation on the base of the conception of cross-entropy. It is assumed that ''complex information geometry'' may serve as a tool for ''technological transfer'' from one research field to the other which is not connected directly with the first one. It is pointed out that the ''infinitesimal distance'' ds 2 and ''infinitesimal cross-entropy'' dHsub(c) coincide
Geometric scalar theory of gravity beyond spherical symmetry
Moschella, U.; Novello, M.
2017-04-01
We construct several exact solutions for a recently proposed geometric scalar theory of gravity. We focus on a class of axisymmetric geometries and a big-bang-like geometry and discuss their Lorentzian character. The axisymmetric solutions are parametrized by an integer angular momentum l . The l =0 (spherical) case gives rise to the Schwarzschild geometry. The other solutions have naked singular surfaces. While not a priori obvious, all the solutions that we present here are globally Lorentzian. The Lorentzian signature appears to be a robust property of the disformal geometries solving the vacuum geometric scalar theory of gravity equations.
Comparisons between geometrical optics and Lorenz-Mie theory
Ungut, A.; Grehan, G.; Gouesbet, G.
1981-01-01
Both the Lorenz-Mie and geometrical optics theories are used in calculating the scattered light patterns produced by transparent spherical particles over a wide range of diameters, between 1.0 and 100 microns, and for the range of forward scattering angles from zero to 20 deg. A detailed comparison of the results shows the greater accuracy of the geometrical optics theory in the forward direction. Emphasis is given to the simultaneous sizing and velocimetry of particles by means of pedestal calibration methods.
Principal Curves on Riemannian Manifolds
DEFF Research Database (Denmark)
Hauberg, Søren
2015-01-01
Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only in Eucl...
Geometrical aspects of Kaluza-Klein theory
International Nuclear Information System (INIS)
Maia, M.D.
1983-08-01
The standard approaches to Kaluza-Klein theory assume the existence of a high dimensional space, from which the four-dimensional space-time would be recovered by a dimensional reduction procedure. In the present note the four-dimensional space-time is regarded as a hypersurface locally embedded in the ever-present (4+n)-dimensional space. In the simplest case where the high dimensional space is taken to be flat, the Kaluza-Klein metric is derived from the embedding assumption. However, this metric is invertible only if the range of the extra n coordinates is limited to an n - dimensional sphere with radius proportional to the local curvature radius of the space-time. Almost independently of this compactification the dimensional reduction, here described in terms of ''complete confinement'', is achieved by a symmetry breaking leading to P 4 x G and which is triggered by the limit of vanishing gravitation. The dynamics for an observer confined to the four-dimensional space-time is described by the embedding integrability conditions. On the other hand for a non-confined observer a unifying picture close to the Kaluza-Klein objectives is obtained. Finally a brief analysis of fermions and fermion masses is also included. (author)
On a Geometric Theory of Generalized Chiral Elasticity with Discontinuities
Directory of Open Access Journals (Sweden)
Suhendro I.
2008-01-01
Full Text Available In this work we develop, in a somewhat extensive manner, a geometric theory of chiral elasticity which in general is endowed with geometric discontinuities (sometimes referred to as defects. By itself, the present theory generalizes both Cosserat and void elasticity theories to a certain extent via geometrization as well as by taking intoaccount the action of the electromagnetic field, i.e., the incorporation of the electromagnetic field into the description of the so-called microspin (chirality also forms the underlying structure of this work. As we know, the description of the electromagnetic field as a unified phenomenon requires four-dimensional space-time rather than three-dimensional space as its background. For this reason we embed the three-dimensional material space in four-dimensional space-time. This way, the electromagnetic spin is coupled to the non-electromagnetic microspin, both being parts of the completemicrospin to be added to the macrospin in the full description of vorticity. In short, our objective is to generalize the existing continuum theories by especially describing microspin phenomena in a fully geometric way.
On a Geometric Theory of Generalized Chiral Elasticity with Discontinuities
Directory of Open Access Journals (Sweden)
Suhendro I.
2008-01-01
Full Text Available In this work we develop, in a somewhat extensive manner, a geometric theory of chiral elasticity which in general is endowed with geometric discontinuities (sometimes re- ferred to as defects . By itself, the present theory generalizes both Cosserat and void elasticity theories to a certain extent via geometrization as well as by taking into ac- count the action of the electromagnetic field, i.e., the incorporation of the electromag- netic field into the description of the so-called microspin ( chirality also forms the un- derlying structure of this work. As we know, the description of the electromagnetic field as a unified phenomenon requires four-dimensional space-time rather than three- dimensional space as its background. For this reason we embed the three-dimensional material space in four-dimensional space-time. This way, the electromagnetic spin is coupled to the non-electromagnetic microspin, both being parts of the complete mi- crospin to be added to the macrospin in the full description of vorticity. In short, our objective is to generalize the existing continuum theories by especially describing mi- crospin phenomena in a fully geometric way.
An existence result of energy minimizer maps between Riemannian polyhedra
International Nuclear Information System (INIS)
Bouziane, T.
2004-06-01
In this paper, we prove the existence of energy minimizers in each free homotopy class of maps between polyhedra with target space without focal points. Our proof involves a careful study of some geometric properties of Riemannian polyhedra without focal points. Among other things, we show that on the relevant polyhedra, there exists a convex supporting function. (author)
Norm of the Riemannian Curvature Tensor
Indian Academy of Sciences (India)
We consider the Riemannian functional R p ( g ) = ∫ M | R ( g ) | p d v g defined on the space of Riemannian metrics with unit volume on a closed smooth manifold where R ( g ) and d v g denote the corresponding Riemannian curvature tensor and volume form and p ∈ ( 0 , ∞ ) . First we prove that the Riemannian metrics ...
Geometric function theory: a modern view of a classical subject
International Nuclear Information System (INIS)
Crowdy, Darren
2008-01-01
Geometric function theory is a classical subject. Yet it continues to find new applications in an ever-growing variety of areas such as modern mathematical physics, more traditional fields of physics such as fluid dynamics, nonlinear integrable systems theory and the theory of partial differential equations. This paper surveys, with a view to modern applications, open problems and challenges in this subject. Here we advocate an approach based on the use of the Schottky–Klein prime function within a Schottky model of compact Riemann surfaces. (open problem)
Geometric model from microscopic theory for nuclear absorption
International Nuclear Information System (INIS)
John, S.; Townsend, L.W.; Wilson, J.W.; Tripathi, R.K.
1993-07-01
A parameter-free geometric model for nuclear absorption is derived herein from microscopic theory. The expression for the absorption cross section in the eikonal approximation, taken in integral form, is separated into a geometric contribution that is described by an energy-dependent effective radius and two surface terms that cancel in an asymptotic series expansion. For collisions of light nuclei, an expression for the effective radius is derived from harmonic oscillator nuclear density functions. A direct extension to heavy nuclei with Woods-Saxon densities is made by identifying the equivalent half-density radius for the harmonic oscillator functions. Coulomb corrections are incorporated, and a simplified geometric form of the Bradt-Peters type is obtained. Results spanning the energy range from 1 MeV/nucleon to 1 GeV/nucleon are presented. Good agreement with experimental results is obtained
Geometric model for nuclear absorption from microscopic theory
International Nuclear Information System (INIS)
John, S.; Townsend, L.W.; Wilson, J.W.; Tripathi, R.K.
1993-01-01
A parameter-free geometric model for nuclear absorption is derived from microscopic theory. The expression for the absorption cross section in the eikonal approximation taken in integral form is separated into a geometric contribution, described by an energy-dependent effective radius, and two surface terms which are shown to cancel in an asymptotic series expansion. For collisions of light nuclei, an expression for the effective radius is derived using harmonic-oscillator nuclear density functions. A direct extension to heavy nuclei with Woods-Saxon densities is made by identifying the equivalent half density radius for the harmonic-oscillator functions. Coulomb corrections are incorporated and a simplified geometric form of the Bradt-Peters type obtained. Results spanning the energy range of 1 MeV/nucleon to 1 GeV/nucleon are presented. Good agreement with experimental results is obtained
Hoelder continuity of energy minimizer maps between Riemannian polyhedra
International Nuclear Information System (INIS)
Bouziane, Taoufik
2004-10-01
The goal of the present paper is to establish some kind of regularity of an energy minimizer map between Riemannian polyhedra. More precisely, we will show the Hoelder continuity of local energy minimizers between Riemannian polyhedra with the target spaces without focal points. With this new result, we also complete our existence theorem obtained elsewhere, and consequently we generalize completely, to the case of target polyhedra without focal points (which is a weaker geometric condition than the nonpositivity of the curvature), the Eells-Fuglede's existence and regularity theorem which is the new version of the famous Eells-Sampson's theorem. (author)
Local conformal symmetry in non-Riemannian geometry and the origin of physical scales
Energy Technology Data Exchange (ETDEWEB)
De Cesare, Marco [King' s College London, Theoretical Particle Physics and Cosmology Group, Department of Physics, London (United Kingdom); Moffat, John W. [Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada); Sakellariadou, Mairi [King' s College London, Theoretical Particle Physics and Cosmology Group, Department of Physics, London (United Kingdom); Perimeter Institute for Theoretical Physics, Waterloo, ON (Canada)
2017-09-15
We introduce an extension of the Standard Model and General Relativity built upon the principle of local conformal invariance, which represents a generalization of a previous work by Bars, Steinhardt and Turok. This is naturally realized by adopting as a geometric framework a particular class of non-Riemannian geometries, first studied by Weyl. The gravitational sector is enriched by a scalar and a vector field. The latter has a geometric origin and represents the novel feature of our approach. We argue that physical scales could emerge from a theory with no dimensionful parameters, as a result of the spontaneous breakdown of conformal and electroweak symmetries. We study the dynamics of matter fields in this modified gravity theory and show that test particles follow geodesics of the Levi-Civita connection, thus resolving an old criticism raised by Einstein against Weyl's original proposal. (orig.)
On the geometrical approach to the relativistic string theory
International Nuclear Information System (INIS)
Barbashov, B.M.; Nesterenko, V.V.
1978-01-01
In a geometrical approach to the string theory in the four-dimensional Minkowski space the relativistic invariant gauge proposed earlier for the string moving in three-dimensional space-time is used. In contrast to the results of previous paper the system of equations for the coefficients of the fundamental forms of the string model world sheet can be reduced now to one nonlinear Lionville equation again but for a complex valued function u. It is shown that in the case of space-time with arbitrary dimension there are such string motions which are described by one non-linear equation with a real function u. And as a consequence the soliton solutions investigated earlier take place in a geometrical approach to the string theory in any dimensional space-time
Geometric theory of functions of a complex variable
Goluzin, G M
1969-01-01
This book is based on lectures on geometric function theory given by the author at Leningrad State University. It studies univalent conformal mapping of simply and multiply connected domains, conformal mapping of multiply connected domains onto a disk, applications of conformal mapping to the study of interior and boundary properties of analytic functions, and general questions of a geometric nature dealing with analytic functions. The second Russian edition upon which this English translation is based differs from the first mainly in the expansion of two chapters and in the addition of a long survey of more recent developments. The book is intended for readers who are already familiar with the basics of the theory of functions of one complex variable.
A geometric theory for Lévy distributions
International Nuclear Information System (INIS)
Eliazar, Iddo
2014-01-01
Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT
A geometric theory for Lévy distributions
Eliazar, Iddo
2014-08-01
Lévy distributions are of prime importance in the physical sciences, and their universal emergence is commonly explained by the Generalized Central Limit Theorem (CLT). However, the Generalized CLT is a geometry-less probabilistic result, whereas physical processes usually take place in an embedding space whose spatial geometry is often of substantial significance. In this paper we introduce a model of random effects in random environments which, on the one hand, retains the underlying probabilistic structure of the Generalized CLT and, on the other hand, adds a general and versatile underlying geometric structure. Based on this model we obtain geometry-based counterparts of the Generalized CLT, thus establishing a geometric theory for Lévy distributions. The theory explains the universal emergence of Lévy distributions in physical settings which are well beyond the realm of the Generalized CLT.
Needle decompositions in Riemannian geometry
Klartag, Bo'az
2017-01-01
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.
Principal Curves on Riemannian Manifolds.
Hauberg, Soren
2016-09-01
Euclidean statistics are often generalized to Riemannian manifolds by replacing straight-line interpolations with geodesic ones. While these Riemannian models are familiar-looking, they are restricted by the inflexibility of geodesics, and they rely on constructions which are optimal only in Euclidean domains. We consider extensions of Principal Component Analysis (PCA) to Riemannian manifolds. Classic Riemannian approaches seek a geodesic curve passing through the mean that optimizes a criteria of interest. The requirements that the solution both is geodesic and must pass through the mean tend to imply that the methods only work well when the manifold is mostly flat within the support of the generating distribution. We argue that instead of generalizing linear Euclidean models, it is more fruitful to generalize non-linear Euclidean models. Specifically, we extend the classic Principal Curves from Hastie & Stuetzle to data residing on a complete Riemannian manifold. We show that for elliptical distributions in the tangent of spaces of constant curvature, the standard principal geodesic is a principal curve. The proposed model is simple to compute and avoids many of the pitfalls of traditional geodesic approaches. We empirically demonstrate the effectiveness of the Riemannian principal curves on several manifolds and datasets.
Geometric theory of fundamental interactions. Foundations of unified physics
International Nuclear Information System (INIS)
Pestov, A.B.
2012-01-01
We put forward an idea that regularities of unified physics are in a simple relation: everything in the concept of space and the concept of space in everything. With this hypothesis as a ground, a conceptual structure of a unified geometrical theory of fundamental interactions is created and deductive derivation of its main equations is produced. The formulated theory gives solution of the actual problems, provides opportunity to understand the origin and nature of physical fields, local internal symmetry, time, energy, spin, charge, confinement, dark energy and dark matter, thus conforming the existence of new physics in its unity
The elastic theory of shells using geometric algebra.
Gregory, A L; Lasenby, J; Agarwal, A
2017-03-01
We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.
Type II Superstring Field Theory: Geometric Approach and Operadic Description
Jurco, Branislav
2013-01-01
We outline the construction of type II superstring field theory leading to a geometric and algebraic BV master equation, analogous to Zwiebach's construction for the bosonic string. The construction uses the small Hilbert space. Elementary vertices of the non-polynomial action are described with the help of a properly formulated minimal area problem. They give rise to an infinite tower of superstring field products defining a $\\mathcal{N}=1$ generalization of a loop homotopy Lie algebra, the genus zero part generalizing a homotopy Lie algebra. Finally, we give an operadic interpretation of the construction.
50 years with J. B. Keller's Geometrical Theory of Diffraction in Denmark - Revisiting the Theory
DEFF Research Database (Denmark)
Christiansen, Peter Leth; Albertsen, N. Chr.; Breinbjerg, Olav
2013-01-01
In the introduction, Danish contributions to J. B. Keller's Geometrical Theory of Diffraction are surveyed. The edge diffraction coefficient in the case of scattering by a half-plane with an impedance surface is then analyzed. In short-wavelength scattering theory, the amplitudes of the incident...
International Nuclear Information System (INIS)
Hervik, Sigbjoern; Coley, Alan
2011-01-01
In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the S i - and N-properties, and show that if the curvature tensors of the space possess the N-property, then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null congruence. We also discuss the related Walker metrics.
Geometric control theory for quantum back-action evasion
Energy Technology Data Exchange (ETDEWEB)
Yokotera, Yu; Yamamoto, Naoki [Keio University, Department of Applied Physics and Physico-Informatics, Yokohama (Japan)
2016-12-15
Engineering a sensor system for detecting an extremely tiny signal such as the gravitational-wave force is a very important subject in quantum physics. A major obstacle to this goal is that, in a simple detection setup, the measurement noise is lower bounded by the so-called standard quantum limit (SQL), which is originated from the intrinsic mechanical back-action noise. Hence, the sensor system has to be carefully engineered so that it evades the back-action noise and eventually beats the SQL. In this paper, based on the well-developed geometric control theory for classical disturbance decoupling problem, we provide a general method for designing an auxiliary (coherent feedback or direct interaction) controller for the sensor system to achieve the above-mentioned goal. This general theory is applied to a typical opto-mechanical sensor system. Also, we demonstrate a controller design for a practical situation where several experimental imperfections are present. (orig.)
Geometric control theory for quantum back-action evasion
International Nuclear Information System (INIS)
Yokotera, Yu; Yamamoto, Naoki
2016-01-01
Engineering a sensor system for detecting an extremely tiny signal such as the gravitational-wave force is a very important subject in quantum physics. A major obstacle to this goal is that, in a simple detection setup, the measurement noise is lower bounded by the so-called standard quantum limit (SQL), which is originated from the intrinsic mechanical back-action noise. Hence, the sensor system has to be carefully engineered so that it evades the back-action noise and eventually beats the SQL. In this paper, based on the well-developed geometric control theory for classical disturbance decoupling problem, we provide a general method for designing an auxiliary (coherent feedback or direct interaction) controller for the sensor system to achieve the above-mentioned goal. This general theory is applied to a typical opto-mechanical sensor system. Also, we demonstrate a controller design for a practical situation where several experimental imperfections are present. (orig.)
Geometric invariant theory over the real and complex numbers
Wallach, Nolan R
2017-01-01
Geometric Invariant Theory (GIT) is developed in this text within the context of algebraic geometry over the real and complex numbers. This sophisticated topic is elegantly presented with enough background theory included to make the text accessible to advanced graduate students in mathematics and physics with diverse backgrounds in algebraic and differential geometry. Throughout the book, examples are emphasized. Exercises add to the reader’s understanding of the material; most are enhanced with hints. The exposition is divided into two parts. The first part, ‘Background Theory’, is organized as a reference for the rest of the book. It contains two chapters developing material in complex and real algebraic geometry and algebraic groups that are difficult to find in the literature. Chapter 1 emphasizes the relationship between the Zariski topology and the canonical Hausdorff topology of an algebraic variety over the complex numbers. Chapter 2 develops the interaction between Lie groups and algebraic ...
Group-geometric methods in supergravity and superstring theories
International Nuclear Information System (INIS)
Castellani, L.
1992-01-01
The purpose of this paper is to give a brief and pedagogical account of the group-geometric approach to (super)gravity and superstring theories. The authors summarize the main ideas and apply them to selected examples. Group geometry provides a natural and unified formulation of gravity and gauge theories. The invariance of both are interpreted as diffeomorphisms on a suitable group manifold. This geometrical framework has a fruitful output, in that it provides a systematic algorithm for the gauging of Lie algebras and the construction of (super)gravity or (super)string Lagrangians. The basic idea is to associate fundamental fields to the group generators. This is done by considering first a basis of tangent vectors on the group manifold. These vectors close on the same algebra as the abstract group generators. The dual basis, i.e. the vielbeins (cotangent basis of one-forms) is then identified with the set of fundamental fields. Thus, for example, the vielbein V a and the spin connection ω ab of ordinary Einstein-Cartan gravity are seen as the duals of the tangent vectors corresponding to translations and Lorentz rotations, respectively
Geometrical and topological formulation of local gauge and supergauge theories
International Nuclear Information System (INIS)
Macrae, K.I.
1976-01-01
A geometrical and topological formulation of local gauge and supergauge invariance is presented. Analysis of experiments of the type described by Bohm and Aharanov and in the attempt to understand immersed submanifolds such as the string with internal symmetry, in a geometric setting, are led to the introduction of fiber bundles, superspaces. Many exact classical solutions to the equations of motion were considered for these gauge theories with specific choices of gauge group such as SU 4 . We describe some exact soliton solutions to these theories which have linear Regge trajectories, i.e., their angular momentum is a linear function of their mass squared. Next one discusses the actions and equations of motion for gauge theories whose base manifolds can have arbitrarily dimensioned submanifolds excised from them, manifolds with holes were discussed. These holes can have fractional quark charges when the structure group is, for example, SU 3 or SU 4 . By extending the concept of conservation of energy to include the excised submanifolds, their actions, and their equations of motion were derived showing that they can act as charged particles. Using the fractionality of the quark charges, are led to suggest a topological confinement mechanism for these particles. One also derives the actions and equations of motion for the string from this viewpoint. Some new Lie algebras which have anticommuting elements are introduced. Their gauge theories are described, and the possibility of fermionic actions for the anticommuting pieces is examined. Supersymmetric strings and their supergauge transformations were discussed and an extension was suggested of supersymmetry to immersed minimal submanifolds other than the string. Both quarklike and vectorlike fermions are included. Finally the invariance of both the equations of motion and the gauge conditions under supersymmetry transformations for these submanifolds were described
Natural Connections on Riemannian Product Manifolds
Gribacheva, Dobrinka
2011-01-01
A Riemannian almost product manifold with integrable almost product structure is called a Riemannian product manifold. In the present paper the natural connections on such manifolds are studied, i.e. the linear connections preserving the almost product structure and the Riemannian metric.
Geometrical methods for power network analysis
Energy Technology Data Exchange (ETDEWEB)
Bellucci, Stefano; Tiwari, Bhupendra Nath [Istituto Nazioneale di Fisica Nucleare, Frascati, Rome (Italy). Lab. Nazionali di Frascati; Gupta, Neeraj [Indian Institute of Technology, Kanpur (India). Dept. of Electrical Engineering
2013-02-01
Uses advanced geometrical methods to analyse power networks. Provides a self-contained and tutorial introduction. Includes a fully worked-out example for the IEEE 5 bus system. This book is a short introduction to power system planning and operation using advanced geometrical methods. The approach is based on well-known insights and techniques developed in theoretical physics in the context of Riemannian manifolds. The proof of principle and robustness of this approach is examined in the context of the IEEE 5 bus system. This work addresses applied mathematicians, theoretical physicists and power engineers interested in novel mathematical approaches to power network theory.
Connections and curvatures on complex Riemannian manifolds
International Nuclear Information System (INIS)
Ganchev, G.; Ivanov, S.
1991-05-01
Characteristic connection and characteristic holomorphic sectional curvatures are introduced on a complex Riemannian manifold (not necessarily with holomorphic metric). For the class of complex Riemannian manifolds with holomorphic characteristic connection a classification of the manifolds with (pointwise) constant holomorphic characteristic curvature is given. It is shown that the conformal geometry of complex analytic Riemannian manifolds can be naturally developed on the class of locally conformal holomorphic Riemannian manifolds. Complex Riemannian manifolds locally conformal to the complex Euclidean space are characterized with zero conformal fundamental tensor and zero conformal characteristic tensor. (author). 12 refs
The three-body problem and equivariant Riemannian geometry
Alvarez-Ramírez, M.; García, A.; Meléndez, J.; Reyes-Victoria, J. G.
2017-08-01
We study the planar three-body problem with 1/r2 potential using the Jacobi-Maupertuis metric, making appropriate reductions by Riemannian submersions. We give a different proof of the Gaussian curvature's sign and the completeness of the space reported by Montgomery [Ergodic Theory Dyn. Syst. 25, 921-947 (2005)]. Moreover, we characterize the geodesics contained in great circles.
STUDENTS’ GEOMETRIC THINKING BASED ON VAN HIELE’S THEORY
Directory of Open Access Journals (Sweden)
Harina Fitriyani
2018-02-01
Full Text Available The current study aims to identify the development level of students’ geometric thinking in mathematics education department, Universitas Ahmad Dahlan based on van Hiele’s theory. This is a descriptive qualitative research with the respondents as many as 129 students. In addition to researchers, the instrument used in this study is a test consisting of 25 items multiple choice questions. The data is analyzed by using Milles and Huberman model. The result shows that there were 30,65% of students in pre-visualization level, 21,51% of students in visualizes level, and 29,03% of students in analyze level, 16,67% of students in informal deduction level, 2,15% of students in deduction level, and 0,00% of student in rigor level. Furthermore, findings indicated a transition level among development levels of geometric thinking in pre-analyze, pre-informal deduction, pre-deduction, and pre-rigor that were 20%; 13,44%; 6,45%; 1,08% respectively. The other findings were 40,32% of students were difficult to determine and 4,3% of students cannot be identified.
Optical approximation in the theory of geometric impedance
International Nuclear Information System (INIS)
Stupakov, G.; Bane, K.L.F.; Zagorodnov, I.
2007-02-01
In this paper we introduce an optical approximation into the theory of impedance calculation, one valid in the limit of high frequencies. This approximation neglects diffraction effects in the radiation process, and is conceptually equivalent to the approximation of geometric optics in electromagnetic theory. Using this approximation, we derive equations for the longitudinal impedance for arbitrary offsets, with respect to a reference orbit, of source and test particles. With the help of the Panofsky-Wenzel theorem we also obtain expressions for the transverse impedance (also for arbitrary offsets). We further simplify these expressions for the case of the small offsets that are typical for practical applications. Our final expressions for the impedance, in the general case, involve two dimensional integrals over various cross-sections of the transition. We further demonstrate, for several known axisymmetric examples, how our method is applied to the calculation of impedances. Finally, we discuss the accuracy of the optical approximation and its relation to the diffraction regime in the theory of impedance. (orig.)
A geometrical approach to two-dimensional Conformal Field Theory
Dijkgraaf, Robertus Henricus
1989-09-01
This thesis is organized in the following way. In Chapter 2 we will give a brief introduction to conformal field theory along the lines of standard quantum field theory, without any claims to originality. We introduce the important concepts of the stress-energy tensor, the Virasoro algebra, and primary fields. The general principles are demonstrated by fermionic and bosonic free field theories. This also allows us to discuss some general aspects of moduli spaces of CFT's. In particular, we describe in some detail the space of iiiequivalent toroidal comi)actificalions, giving examples of the quantum equivalences that we already mentioned. In Chapter 3 we will reconsider general quantum field theory from a more geometrical point of view, along the lines of the so-called operator formalism. Crucial to this approach will be the consideration of topology changing amplitudes. After a simple application to 2d topological theories, we proceed to give our second introduction to CFT, stressing the geometry behind it. In Chapter 4 the so-called rational conformal field theories are our object of study. These special CFT's have extended symmetries with only a finite number of representations. If an interpretation as non-linear sigma model exists, this extra symmetry can be seen as a kind of resonance effect due to the commensurability of the size of the string and the target space-time. The structure of rational CFT's is extremely rigid, and one of our results will be that the operator content of these models is—up to some discrete choices—completely determined by the symmetry algebra. The study of rational models is in its rigidity very analogous to finite group theory. In Chapter 5 this analogy is further pursued and substantiated. We will show how one can construct from general grounds rational conformal field theories from finite groups. These models are abstract versions of non-linear o-models describing string propagation on 'orbifoids.' An orbifold is a singular
Energy Technology Data Exchange (ETDEWEB)
Hervik, Sigbjoern [Faculty of Science and Technology, University of Stavanger, N-4036 Stavanger (Norway); Coley, Alan, E-mail: sigbjorn.hervik@uis.no, E-mail: aac@mathstat.dal.ca [Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Scotia B3H 3J5 (Canada)
2011-01-07
In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of the polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the S{sub i}- and N-properties, and show that if the curvature tensors of the space possess the N-property, then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null congruence. We also discuss the related Walker metrics.
STRUCTURE TENSOR IMAGE FILTERING USING RIEMANNIAN L1 AND L∞ CENTER-OF-MASS
Directory of Open Access Journals (Sweden)
Jesus Angulo
2014-06-01
Full Text Available Structure tensor images are obtained by a Gaussian smoothing of the dyadic product of gradient image. These images give at each pixel a n×n symmetric positive definite matrix SPD(n, representing the local orientation and the edge information. Processing such images requires appropriate algorithms working on the Riemannian manifold on the SPD(n matrices. This contribution deals with structure tensor image filtering based on Lp geometric averaging. In particular, L1 center-of-mass (Riemannian median or Fermat-Weber point and L∞ center-of-mass (Riemannian circumcenter can be obtained for structure tensors using recently proposed algorithms. Our contribution in this paper is to study the interest of L1 and L∞ Riemannian estimators for structure tensor image processing. In particular, we compare both for two image analysis tasks: (i structure tensor image denoising; (ii anomaly detection in structure tensor images.
Forward error correction based on algebraic-geometric theory
A Alzubi, Jafar; M Chen, Thomas
2014-01-01
This book covers the design, construction, and implementation of algebraic-geometric codes from Hermitian curves. Matlab simulations of algebraic-geometric codes and Reed-Solomon codes compare their bit error rate using different modulation schemes over additive white Gaussian noise channel model. Simulation results of Algebraic-geometric codes bit error rate performance using quadrature amplitude modulation (16QAM and 64QAM) are presented for the first time and shown to outperform Reed-Solomon codes at various code rates and channel models. The book proposes algebraic-geometric block turbo codes. It also presents simulation results that show an improved bit error rate performance at the cost of high system complexity due to using algebraic-geometric codes and Chase-Pyndiah’s algorithm simultaneously. The book proposes algebraic-geometric irregular block turbo codes (AG-IBTC) to reduce system complexity. Simulation results for AG-IBTCs are presented for the first time.
Classification of non-Riemannian doubled-yet-gauged spacetime
Energy Technology Data Exchange (ETDEWEB)
Morand, Kevin [Universidad Andres Bello, Departamento de Ciencias Fisicas, Santiago de Chile (Chile); Universidad Tecnica Federico Santa Maria, Centro Cientifico-Tecnologico de Valparaiso, Departamento de Fisica, Valparaiso (Chile); Park, Jeong-Hyuck [Sogang University, Department of Physics, Seoul (Korea, Republic of); Institute for Basic Science (IBS), Center for Theoretical Physics of the Universe, Seoul (Korea, Republic of)
2017-10-15
Assuming O(D,D) covariant fields as the 'fundamental' variables, double field theory can accommodate novel geometries where a Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, (n, anti n), 0 ≤ n + anti n ≤ D. Upon these backgrounds, strings become chiral and anti-chiral over n and anti n directions, respectively, while particles and strings are frozen over the n + anti n directions. In particular, we identify (0, 0) as Riemannian manifolds, (1, 0) as non-relativistic spacetime, (1, 1) as Gomis-Ooguri non-relativistic string, (D-1, 0) as ultra-relativistic Carroll geometry, and (D, 0) as Siegel's chiral string. Combined with a covariant Kaluza-Klein ansatz which we further spell, (0, 1) leads to Newton-Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as D = 10, (3, 3) may open a new scheme for the dimensional reduction from ten to four. (orig.)
Transversal Dirac families in Riemannian foliations
International Nuclear Information System (INIS)
Glazebrook, J.F.; Kamber, F.W.
1991-01-01
We describe a family of differential operators parametrized by the transversal vector potentials of a Riemannian foliation relative to the Clifford algebra of the foliation. This family is non-elliptic but in certain ways behaves like a standard Dirac family in the absolute case as a result of its elliptic-like regularity properties. The analytic and topological indices of this family are defined as elements of K-theory in the parameter space. We indicate how the cohomology of the parameter space is described via suitable maps to Fredholm operators. We outline the proof of a theorem of Vafa-Witten type on uniform bounds for the eigenvalues of this family using a spectral flow argument. A determinant operator is also defined with the appropriate zeta function regularization dependent on the codimension of the foliation. With respect to a generalized coupled Dirac-Yang-Mills system, we indicate how chiral anomalies are located relative to the foliation. (orig.)
The geometric β-function in curved space-time under operator regularization
Energy Technology Data Exchange (ETDEWEB)
Agarwala, Susama [Mathematical Institute, Oxford University, Oxford OX2 6GG (United Kingdom)
2015-06-15
In this paper, I compare the generators of the renormalization group flow, or the geometric β-functions, for dimensional regularization and operator regularization. I then extend the analysis to show that the geometric β-function for a scalar field theory on a closed compact Riemannian manifold is defined on the entire manifold. I then extend the analysis to find the generator of the renormalization group flow to conformally coupled scalar-field theories on the same manifolds. The geometric β-function in this case is not defined.
The geometric β-function in curved space-time under operator regularization
International Nuclear Information System (INIS)
Agarwala, Susama
2015-01-01
In this paper, I compare the generators of the renormalization group flow, or the geometric β-functions, for dimensional regularization and operator regularization. I then extend the analysis to show that the geometric β-function for a scalar field theory on a closed compact Riemannian manifold is defined on the entire manifold. I then extend the analysis to find the generator of the renormalization group flow to conformally coupled scalar-field theories on the same manifolds. The geometric β-function in this case is not defined
Dynamic graphs, community detection, and Riemannian geometry
Energy Technology Data Exchange (ETDEWEB)
Bakker, Craig; Halappanavar, Mahantesh; Visweswara Sathanur, Arun
2018-03-29
A community is a subset of a wider network where the members of that subset are more strongly connected to each other than they are to the rest of the network. In this paper, we consider the problem of identifying and tracking communities in graphs that change over time {dynamic community detection} and present a framework based on Riemannian geometry to aid in this task. Our framework currently supports several important operations such as interpolating between and averaging over graph snapshots. We compare these Riemannian methods with entry-wise linear interpolation and that the Riemannian methods are generally better suited to dynamic community detection. Next steps with the Riemannian framework include developing higher-order interpolation methods (e.g. the analogues of polynomial and spline interpolation) and a Riemannian least-squares regression method for working with noisy data.
Liu, Hong; Zhu, Jingping; Wang, Kai
2015-08-24
The geometrical attenuation model given by Blinn was widely used in the geometrical optics bidirectional reflectance distribution function (BRDF) models. Blinn's geometrical attenuation model based on symmetrical V-groove assumption and ray scalar theory causes obvious inaccuracies in BRDF curves and negatives the effects of polarization. Aiming at these questions, a modified polarized geometrical attenuation model based on random surface microfacet theory is presented by combining of masking and shadowing effects and polarized effect. The p-polarized, s-polarized and unpolarized geometrical attenuation functions are given in their separate expressions and are validated with experimental data of two samples. It shows that the modified polarized geometrical attenuation function reaches better physical rationality, improves the precision of BRDF model, and widens the applications for different polarization.
Geometrical theory of nonlinear phase distortion of intense laser beams
International Nuclear Information System (INIS)
Glaze, J.A.; Hunt, J.T.; Speck, D.R.
1975-01-01
Phase distortion arising from whole beam self-focusing of intense laser pulses with arbitrary spatial profiles is treated in the limit of geometrical optics. The constant shape approximation is used to obtain the phase and angular distribution of the geometrical rays in the near field. Conditions for the validity of this approximation are discussed. Geometrical focusing of the aberrated beam is treated for the special case of a beam with axial symmetry. Equations are derived that show both the shift of the focus and the distortion of the intensity distribution that are caused by the nonlinear index of refraction of the optical medium. An illustrative example treats the case of beam distortion in a Nd:Glass amplifier
Geometric representation of the generator of duality in massless and massive p-form field theories
International Nuclear Information System (INIS)
Contreras, Ernesto; Martinez, Yisely; Leal, Lorenzo
2010-01-01
We study the invariance under duality transformations in massless and massive p-form field theories and obtain the Noether generators of the infinitesimal transformations that correspond to this symmetry. These generators can be realized in geometrical representations that generalize the loop representation of the Maxwell field, allowing for a geometrical interpretation which is studied.
The algebraic versus geometric approach to quantum field theory
International Nuclear Information System (INIS)
Schroer, B.
1990-06-01
Some recent developments in algebraic QFT are reviewed and confronted with results obtained by geometric methods. In particular a critical evaluation of the present status of the quantum symmetry discussion is given and the possible relation of the (Gepner-Witten) modularity in conformal QFT 2 and the Tomita modularity (existence of quantum reflections) of the algebraic approach is commented on. (author) 34 refs
Aubrun, Guillaume
2017-01-01
The quest to build a quantum computer is arguably one of the major scientific and technological challenges of the twenty-first century, and quantum information theory (QIT) provides the mathematical framework for that quest. Over the last dozen or so years, it has become clear that quantum information theory is closely linked to geometric functional analysis (Banach space theory, operator spaces, high-dimensional probability), a field also known as asymptotic geometric analysis (AGA). In a nutshell, asymptotic geometric analysis investigates quantitative properties of convex sets, or other geometric structures, and their approximate symmetries as the dimension becomes large. This makes it especially relevant to quantum theory, where systems consisting of just a few particles naturally lead to models whose dimension is in the thousands, or even in the billions. Alice and Bob Meet Banach is aimed at multiple audiences connected through their interest in the interface of QIT and AGA: at quantum information resea...
The invariant charges of the Nambu-Goto theory: Their geometric origin and their completeness
International Nuclear Information System (INIS)
Pohlmeyer, K.; Rehren, K.H.
1988-01-01
We give an alternative construction of the reparametrization invariant 'non-local' conserved charges of the Nambu-Goto theory which elucidates their geometric nature and their completeness property. (orig.)
Existence of localizing solutions in plasticity via the geometric singular perturbation theory
Lee, Min-Gi; Tzavaras, Athanasios
2017-01-01
system has fast and slow time scales, forming a singularly perturbed problem. Geometric singular perturbation theory is applied to this problem to achieve an invariant surface. The flow on the invariant surface is analyzed via the Poincaré
Riemannian geometry of thermodynamics and systems with repulsive power-law interactions.
Ruppeiner, George
2005-07-01
A Riemannian geometric theory of thermodynamics based on the postulate that the curvature scalar R is proportional to the inverse free energy density is used to investigate three-dimensional fluid systems of identical classical point particles interacting with each other via a power-law potential energy gamma r(-alpha) . Such systems are useful in modeling melting transitions. The limit alpha-->infinity corresponds to the hard sphere gas. A thermodynamic limit exists only for short-range (alpha>3) and repulsive (gamma>0) interactions. The geometric theory solutions for given alpha>3 , gamma>0 , and any constant temperature T have the following properties: (1) the thermodynamics follows from a single function b (rho T(-3/alpha) ) , where rho is the density; (2) all solutions are equivalent up to a single scaling constant for rho T(-3/alpha) , related to gamma via the virial theorem; (3) at low density, solutions correspond to the ideal gas; (4) at high density there are solutions with pressure and energy depending on density as expected from solid state physics, though not with a Dulong-Petit heat capacity limit; (5) for 33.7913 a phase transition is required to go between these regimes; (7) for any alpha>3 we may include a first-order phase transition, which is expected from computer simulations; and (8) if alpha-->infinity, the density approaches a finite value as the pressure increases to infinity, with the pressure diverging logarithmically in the density difference.
Geometric symmetries and topological terms in F-theory and field theory
Energy Technology Data Exchange (ETDEWEB)
Kapfer, Andreas
2016-08-25
suggest a new geometric group structure on resolved elliptic fibrations. In the same way we also propose a novel group operation for multi-sections in genus-one fibrations without a proper section. We stress that these arithmetic structures ensure the cancelation of all gauge anomalies in F-theory compactifications on Calabi-Yau manifolds.
New Riemannian Priors on the Univariate Normal Model
Directory of Open Access Journals (Sweden)
Salem Said
2014-07-01
Full Text Available The current paper introduces new prior distributions on the univariate normal model, with the aim of applying them to the classification of univariate normal populations. These new prior distributions are entirely based on the Riemannian geometry of the univariate normal model, so that they can be thought of as “Riemannian priors”. Precisely, if {pθ ; θ ∈ Θ} is any parametrization of the univariate normal model, the paper considers prior distributions G( θ - , γ with hyperparameters θ - ∈ Θ and γ > 0, whose density with respect to Riemannian volume is proportional to exp(−d2(θ, θ - /2γ2, where d2(θ, θ - is the square of Rao’s Riemannian distance. The distributions G( θ - , γ are termed Gaussian distributions on the univariate normal model. The motivation for considering a distribution G( θ - , γ is that this distribution gives a geometric representation of a class or cluster of univariate normal populations. Indeed, G( θ - , γ has a unique mode θ - (precisely, θ - is the unique Riemannian center of mass of G( θ - , γ, as shown in the paper, and its dispersion away from θ - is given by γ. Therefore, one thinks of members of the class represented by G( θ - , γ as being centered around θ - and lying within a typical distance determined by γ. The paper defines rigorously the Gaussian distributions G( θ - , γ and describes an algorithm for computing maximum likelihood estimates of their hyperparameters. Based on this algorithm and on the Laplace approximation, it describes how the distributions G( θ - , γ can be used as prior distributions for Bayesian classification of large univariate normal populations. In a concrete application to texture image classification, it is shown that this leads to an improvement in performance over the use of conjugate priors.
Geometric theory on the elasticity of bio-membranes
Tu, Z. C.; Ou-Yang, Z. C.
2004-01-01
The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status in theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed lipid bilayers, the shape equation and boundary conditions of open lipid bilayers and two-component membranes, the shape equation and in-plane strain equations of cell membrane...
CMC Hypersurfaces on Riemannian and Semi-Riemannian Manifolds
International Nuclear Information System (INIS)
Perdomo, Oscar M.
2012-01-01
In this paper we generalize the explicit formulas for constant mean curvature (CMC) immersion of hypersurfaces of Euclidean spaces, spheres and hyperbolic spaces given in Perdomo (Asian J Math 14(1):73–108, 2010; Rev Colomb Mat 45(1):81–96, 2011) to provide explicit examples of several families of immersions with constant mean curvature and non constant principal curvatures, in semi-Riemannian manifolds with constant sectional curvature. In particular, we prove that every h is an element of [-1,-(2√n-1/n can be realized as the constant curvature of a complete immersion of S 1 n-1 x R in the (n + 1)-dimensional de Sitter space S 1 n+1 . We provide 3 types of immersions with CMC in the Minkowski space, 5 types of immersion with CMC in the de Sitter space and 5 types of immersion with CMC in the anti de Sitter space. At the end of the paper we analyze the families of examples that can be extended to closed hypersurfaces.
Computational Contact Mechanics Geometrically Exact Theory for Arbitrary Shaped Bodies
Konyukhov, Alexander
2013-01-01
This book contains a systematical analysis of geometrical situations leading to contact pairs -- point-to-surface, surface-to-surface, point-to-curve, curve-to-curve and curve-to-surface. Each contact pair is inherited with a special coordinate system based on its geometrical properties such as a Gaussian surface coordinate system or a Serret-Frenet curve coordinate system. The formulation in a covariant form allows in a straightforward fashion to consider various constitutive relations for a certain pair such as anisotropy for both frictional and structural parts. Then standard methods well known in computational contact mechanics such as penalty, Lagrange multiplier methods, combination of both and others are formulated in these coordinate systems. Such formulations require then the powerful apparatus of differential geometry of surfaces and curves as well as of convex analysis. The final goals of such transformations are then ready-for-implementation numerical algorithms within the finite e...
A geometric theory on the elasticity of bio-membranes
International Nuclear Information System (INIS)
Tu, Z C; Ou-Yang, Z C
2004-01-01
The purpose of this paper is to study the shapes and stabilities of bio-membranes within the framework of exterior differential forms. After a brief review of the current status of theoretical and experimental studies on the shapes of bio-membranes, a geometric scheme is proposed to discuss the shape equation of closed lipid bilayers, the shape equation and boundary conditions of open lipid bilayers and two-component membranes, the shape equation and in-plane strain equations of cell membranes with cross-linking structures, and the stabilities of closed lipid bilayers and cell membranes. The key point of this scheme is to deal with the variational problems on surfaces embedded in three-dimensional Euclidean space by using exterior differential forms
Pitfalls of using the geometric-mean combining rule in the density gradient theory
DEFF Research Database (Denmark)
Liang, Xiaodong; Michelsen, Michael Locht; Kontogeorgis, Georgios
2016-01-01
It is popular and attractive to model the interfacial tension using the density gradient theory with the geometric-mean combining rule, in which the same equation of state is used for the interface and bulk phases. The computational efficiency is the most important advantage of this theory. In th...
Chen, Chenglong; Ni, Jiangqun; Shen, Zhaoyi; Shi, Yun Qing
2017-06-01
Geometric transformations, such as resizing and rotation, are almost always needed when two or more images are spliced together to create convincing image forgeries. In recent years, researchers have developed many digital forensic techniques to identify these operations. Most previous works in this area focus on the analysis of images that have undergone single geometric transformations, e.g., resizing or rotation. In several recent works, researchers have addressed yet another practical and realistic situation: successive geometric transformations, e.g., repeated resizing, resizing-rotation, rotation-resizing, and repeated rotation. We will also concentrate on this topic in this paper. Specifically, we present an in-depth analysis in the frequency domain of the second-order statistics of the geometrically transformed images. We give an exact formulation of how the parameters of the first and second geometric transformations influence the appearance of periodic artifacts. The expected positions of characteristic resampling peaks are analytically derived. The theory developed here helps to address the gap left by previous works on this topic and is useful for image security and authentication, in particular, the forensics of geometric transformations in digital images. As an application of the developed theory, we present an effective method that allows one to distinguish between the aforementioned four different processing chains. The proposed method can further estimate all the geometric transformation parameters. This may provide useful clues for image forgery detection.
A geometric view on topologically massive gauge theories
International Nuclear Information System (INIS)
Horvathy, P.A.; Nash, C.
1985-01-01
The topologically massive gauge theory of Deser, Jackiw and Templeton is understood from Souriau's Principle of General Covariance. The non-gauge invariant mass term corresponds to a non-trivial class in the first cohomology group of configuration space, generated by the Chern-Simons secondary characteristic class. Quantization requires this class to be integral
On unified field theories, dynamical torsion and geometrical models: II
International Nuclear Information System (INIS)
Cirilo-Lombardo, D.J.
2011-01-01
We analyze in this letter the same space-time structure as that presented in our previous reference (Part. Nucl, Lett. 2010. V.7, No.5. P.299-307), but relaxing now the condition a priori of the existence of a potential for the torsion. We show through exact cosmological solutions from this model, where the geometry is Euclidean RxO 3 ∼ RxSU(2), the relation between the space-time geometry and the structure of the gauge group. Precisely this relation is directly connected with the relation of the spin and torsion fields. The solution of this model is explicitly compared with our previous ones and we find that: i) the torsion is not identified directly with the Yang-Mills type strength field, ii) there exists a compatibility condition connected with the identification of the gauge group with the geometric structure of the space-time: this fact leads to the identification between derivatives of the scale factor a with the components of the torsion in order to allow the Hosoya-Ogura ansatz (namely, the alignment of the isospin with the frame geometry of the space-time), and iii) of two possible structures of the torsion the 'tratorial' form (the only one studied here) forbid wormhole configurations, leading only to cosmological instanton space-time in eternal expansion
Methods of geometric function theory in classical and modern problems for polynomials
International Nuclear Information System (INIS)
Dubinin, Vladimir N
2012-01-01
This paper gives a survey of classical and modern theorems on polynomials, proved using methods of geometric function theory. Most of the paper is devoted to results of the author and his students, established by applying majorization principles for holomorphic functions, the theory of univalent functions, the theory of capacities, and symmetrization. Auxiliary results and the proofs of some of the theorems are presented. Bibliography: 124 titles.
An introduction to geometric theory of fully nonlinear parabolic equations
International Nuclear Information System (INIS)
Lunardi, A.
1991-01-01
We study a class of nonlinear evolution equations in general Banach space being an abstract version of fully nonlinear parabolic equations. In addition to results of existence, uniqueness and continuous dependence on the data, we give some qualitative results about stability of the stationary solutions, existence and stability of the periodic orbits. We apply such results to some parabolic problems arising from combustion theory. (author). 24 refs
On the motion of matter in the geometrical gauge field theory
International Nuclear Information System (INIS)
Konopleva, N.P.
2005-01-01
In the geometrical gauge field theory, the motion equations of matter (elementary particles) are connected with the field equations. The problems arising from this connection are discussed. For the first time, such problems arose in Einstein's General Relativity. Einstein hoped that solution of these problems will allow explanation of elementary particles nature without making use of quantum mechanics. But, as it turned out, the situation is more difficult. Here the corresponding problems are formulated for the connection of equations of particle motion and field equations in the geometrical gauge field theory. It is shown that appearance of the problems under discussion is an inevitable effect of passage to relativism and local symmetries
On the physical origin for the geometric theory of continuum mechanics
International Nuclear Information System (INIS)
Guenther, H.
1984-01-01
It is explained, that the basic notion for a geometric picture of the continuum mechanics is a four dimensional material manifold. The four dimensional mechanical affinity is then the unified field for any defect distribution in the general time dependent case. The minimal number of geometric relations being valid for any continuum is formulated as a set of pure affine relations. The state variables of the theory are additional tensor fields as e.g. deformation defining a metric. A material with a well defined deformation has a Newton-Cartan structure. Only if defects are included into the dynamical determination by additional equilibrium conditions, the theory has a pseudo relativistic structure. (author)
The algebra of space-time as basis of a quantum field theory of all fermions and interactions
International Nuclear Information System (INIS)
Wolf, A.K.
2005-01-01
In this thesis a construction of a grand unified theory on the base of algebras of vector fields on a Riemannian space-time is described. Hereby from the vector and covector fields a Clifford-geometrical algebra is generated. (HSI)
International Nuclear Information System (INIS)
Teschner, J.
2010-05-01
It was in particular recently argued that the gauge theory in the presence of a certain one-parameter deformation can at low energies effectively be described in terms the quantization of an algebraically integrable system, which is canonically associated to this theory. It seems, however, that the deeper reasons for this relationship between a two- and a fourdimensional theory remain to be understood. A clue in this direction may be seen in the fact that the instanton partition functions which represent the building blocks of the partition functions are obtained by specializing a two-parameter family Z(a,ε 1 ,ε 2 ;q) of instanton partition functions. These functions were identified with the conformal blocks of Liouville theory. This indicates that the relationship between certain gauge theories and Liouville theory involves in particular a two-parametric deformation of the algebraically integrable model associated to the gauge theories on R 4 which ultimately produces Liouville theory as a result. One of my intentions in this paper is to clarify in which sense this point of view is correct. Another piece of motivation comes from relations between fourdimensional gauge theories and the geometric Langlands correspondence. The author feels that the mentioned relations between gauge theory and conformal field theory offer new clues in this regard. It is therefore my second main aim to clarify the relations between the quantization of the Hitchin system, the geometric Langlands correspondence and the Liouville conformal field theory. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Teschner, J. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany). Gruppe Theorie
2010-05-15
It was in particular recently argued that the gauge theory in the presence of a certain one-parameter deformation can at low energies effectively be described in terms the quantization of an algebraically integrable system, which is canonically associated to this theory. It seems, however, that the deeper reasons for this relationship between a two- and a fourdimensional theory remain to be understood. A clue in this direction may be seen in the fact that the instanton partition functions which represent the building blocks of the partition functions are obtained by specializing a two-parameter family Z(a,{epsilon}{sub 1},{epsilon}{sub 2};q) of instanton partition functions. These functions were identified with the conformal blocks of Liouville theory. This indicates that the relationship between certain gauge theories and Liouville theory involves in particular a two-parametric deformation of the algebraically integrable model associated to the gauge theories on R{sup 4} which ultimately produces Liouville theory as a result. One of my intentions in this paper is to clarify in which sense this point of view is correct. Another piece of motivation comes from relations between fourdimensional gauge theories and the geometric Langlands correspondence. The author feels that the mentioned relations between gauge theory and conformal field theory offer new clues in this regard. It is therefore my second main aim to clarify the relations between the quantization of the Hitchin system, the geometric Langlands correspondence and the Liouville conformal field theory. (orig.)
Geometrical interpretation of the topological recursion, and integrable string theories
Eynard, Bertrand
2009-01-01
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative geometry like maps, partitions, Hurwitz numbers, intersection numbers, Gromov-Witten invariants... The problem is thus to understand what they count, or in other words, given a spectral curve, construct an enumerative geometry problem. This is what we do in a semi-heuristic approach in this article. Starting from a spectral curve, i.e. an integrable system, we use its flat connection and flat coordinates, to define a family of worldsheets, whose enumeration is indeed solved by the topological recursion and symplectic invariants. In other words, for any spectral curve, we construct a corresponding string theory, whose target space is a submanifold of the Jacobian.
Geometrical Lagrangian for a Supersymmetric Yang-Mills Theory on the Group Manifold
International Nuclear Information System (INIS)
Borges, M. F.
2002-01-01
Perhaps one of the main features of Einstein's General Theory of Relativity is that spacetime is not flat itself but curved. Nowadays, however, many of the unifying theories like superstrings on even alternative gravity theories such as teleparalell geometric theories assume flat spacetime for their calculations. This article, an extended account of an earlier author's contribution, it is assumed a curved group manifold as a geometrical background from which a Lagrangian for a supersymmetric N=2, d=5 Yang-Mills - SYM, N=2, d=5 - is built up. The spacetime is a hypersurface embedded in this geometrical scenario, and the geometrical action here obtained can be readily coupled to the five-dimensional supergravity action. The essential idea that underlies this work has its roots in the Einstein-Cartan formulation of gravity and in the 'group manifold approach to gravity and supergravity theories'. The group SYM, N=2, d=5, turns out to be the direct product of supergravity and a general gauge group G:G=GxSU(2,2/1)-bar
Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics
Energy Technology Data Exchange (ETDEWEB)
Novello, Mario [Centro Brasileiro de Pesquisas Fisicas (ICRA/CBPF), Rio de Janeiro, RJ (Brazil). Instituto de Cosmologia Relatividade e Astrofisica; Bittencourt, Eduardo, E-mail: eduardo.bittencourt@icranet.org [Physics Department, La Sapienza University of Rome (Italy)
2015-12-15
We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric ĝ μ υ which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associatedĝ geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces - beyond the uses of General Relativity that is limited only to gravitational processes - is nothing but the relativistic version of the d'Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries-one for each different body in the case of non-gravitational forces-to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical Bridge formalism
Riemannian geometry in an orthogonal frame
Cartan, Elie Joseph
2001-01-01
Foreword by S S Chern. In 1926-27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan's lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. This book has many innovations, such as the n
Harmonic Riemannian Maps on Locally Conformal Kaehler Manifolds
Indian Academy of Sciences (India)
We study harmonic Riemannian maps on locally conformal Kaehler manifolds ( l c K manifolds). We show that if a Riemannian holomorphic map between l c K manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we ...
Geometrical phases from global gauge invariance of nonlinear classical field theories
International Nuclear Information System (INIS)
Garrison, J.C.; Chiao, R.Y.
1988-01-01
We show that the geometrical phases recently discovered in quantum mechanics also occur naturally in the theory of any classical complex multicomponent field satisfying nonlinear equations derived from a Lagrangean with is invariant under gauge transformations of the first kind. Some examples are the paraxial wave equation for nonlinear optics, and Ginzburg-Landau equations for complex order parameters in condensed-matter physics
On N = 1 gauge models from geometric engineering in M-theory
International Nuclear Information System (INIS)
Belhaj, A; Drissi, L B; Rasmussen, J
2003-01-01
We study geometric engineering of four-dimensional N = 1 gauge models from M-theory on a seven-dimensional manifold with G 2 holonomy. The manifold is constructed as a K3 fibration over a three-dimensional base space with ADE geometry. The resulting gauge theory is discussed in the realm of (p, q) webs. We discuss how the anomaly cancellation condition translates into a condition on the associated affine ADE Lie algebras
Spherical-type hypersurfaces in a Riemannian manifold
International Nuclear Information System (INIS)
Ezin, J.P.; Rigoli, M.
1988-06-01
Let M be a compact hypersurface immersed in R n and let K and L be its mean curvature function and scalar curvature respectively. A classical global problem concerning these two geometrical quantities is to find out if assuming that either K or L is constant and under some additional assumptions M is a sphere. It was demonstrated that assuming the immersion to be an embedding, the consistency of K implies M to be spherical. It was also demonstrated that the sphere is the only compact hypersurface with constant scalar curvature embedded in Euclidean space. In this paper we give a generalization of these results when the ambient space is an appropriate Riemannian manifold (N, h). 17 refs
Conformal, Riemannian and Lagrangian geometry the 2000 Barrett lectures
Chang, Sun-Yung A; Grove, Karsten; Yang, Paul C; Freire, Alexandre
2002-01-01
Recent developments in topology and analysis have led to the creation of new lines of investigation in differential geometry. The 2000 Barrett Lectures present the background, context and main techniques of three such lines by means of surveys by leading researchers. The first chapter (by Alice Chang and Paul Yang) introduces new classes of conformal geometric invariants, and then applies powerful techniques in nonlinear differential equations to derive results on compactifications of manifolds and on Yamabe-type variational problems for these invariants. This is followed by Karsten Grove's lectures, which focus on the use of isometric group actions and metric geometry techniques to understand new examples and classification results in Riemannian geometry, especially in connection with positive curvature. The chapter written by Jon Wolfson introduces the emerging field of Lagrangian variational problems, which blends in novel ways the structures of symplectic geometry and the techniques of the modern calculus...
Quantum Riemannian geometry of phase space and nonassociativity
Directory of Open Access Journals (Sweden)
Beggs Edwin J.
2017-04-01
Full Text Available Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.
A sketch to the geometrical N=2-d=5 Yang-Mills theory over a supersymmetric group-manifold - I
International Nuclear Information System (INIS)
Borges, M.; Turin Univ.; Pio, G.
1983-03-01
This work concerns the search and the construction of a geometrical structure for a supersymmetric N=2-d=5 Yang-Mills theory on the group manifold. From criteria established throughout this paper, we build up an ansatz for the curvatures of our theory and then solve the Bianchi identities, whose solution is fundamental for the construction of the geometrical action. (author)
Geometrical theory of ghost and Higgs fields and SU(2/1)
International Nuclear Information System (INIS)
Ne'eman, Y.; Thierry-Mieg, J.
1979-10-01
That a Principal Fiber Bundle provides a precise geometrical representation of Yang-Mills gauge theories has been known since 1963 and used since 1975. This work presents an entirely new domain of applications. The Feynman-DeWitt-Fadeev-Popov ghost-fields required in the renormalization procedure are identified with geometrical objects in the Principal Bundle. This procedure directly yields the BRS equations guaranteeing unitarity and Slavnov-Taylor invariance of the quantum effective Lagrangian. Except for one ghost field and its variation, this entire symmetry thus corresponds to classical notions, in that it is geometrical, and completely independent of the gauge-fixing procedure, which determines the quantized Lagrangian. These results may be used to fix the signs associated with the various ghost loops of quantum supergravity. The result is based upon the identification of a geometrical Z(2) x Z(2) double-gradation of the generalized fields in supergravity: [physical/ghost] fields and [integer/half integer] spins. Then the case of a supergroup as an internal symmetry gauge is considered. Ghosts geometrically associated to odd generators may be identified with the Goldstone-Nambu Higgs-Kibble scalar fields of conventional models with spontaneous symmetry breakdown. As an example, the chiral SU(3)/sub L/ x SU(3)/sub R/ flavor symmetry is realized by gauging the supergroup Q(3).Lastly, the main results concerning asthenodynamics (Weak-EM Unification) as given by the ghost-gauge SU(2/1) supergroup are recalled. 1 table
Mao, Shasha; Xiong, Lin; Jiao, Licheng; Feng, Tian; Yeung, Sai-Kit
2017-05-01
Riemannian optimization has been widely used to deal with the fixed low-rank matrix completion problem, and Riemannian metric is a crucial factor of obtaining the search direction in Riemannian optimization. This paper proposes a new Riemannian metric via simultaneously considering the Riemannian geometry structure and the scaling information, which is smoothly varying and invariant along the equivalence class. The proposed metric can make a tradeoff between the Riemannian geometry structure and the scaling information effectively. Essentially, it can be viewed as a generalization of some existing metrics. Based on the proposed Riemanian metric, we also design a Riemannian nonlinear conjugate gradient algorithm, which can efficiently solve the fixed low-rank matrix completion problem. By experimenting on the fixed low-rank matrix completion, collaborative filtering, and image and video recovery, it illustrates that the proposed method is superior to the state-of-the-art methods on the convergence efficiency and the numerical performance.
Physical principles, geometrical aspects, and locality properties of gauge field theories
International Nuclear Information System (INIS)
Mack, G.; Hamburg Univ.
1981-01-01
Gauge field theories, particularly Yang - Mills theories, are discussed at a classical level from a geometrical point of view. The introductory chapters are concentrated on physical principles and mathematical tools. The main part is devoted to locality problems in gauge field theories. Examples show that locality problems originate from two sources in pure Yang - Mills theories (without matter fields). One is topological and the other is related to the existence of degenerated field configurations of the infinitesimal holonomy groups on some extended region of space or space-time. Nondegenerate field configurations in theories with semisimple gauge groups can be analysed with the help of the concept of a local gauge. Such gauges play a central role in the discussion. (author)
Geometric Methods in the Algebraic Theory of Quadratic Forms : Summer School
2004-01-01
The geometric approach to the algebraic theory of quadratic forms is the study of projective quadrics over arbitrary fields. Function fields of quadrics have been central to the proofs of fundamental results since the renewal of the theory by Pfister in the 1960's. Recently, more refined geometric tools have been brought to bear on this topic, such as Chow groups and motives, and have produced remarkable advances on a number of outstanding problems. Several aspects of these new methods are addressed in this volume, which includes - an introduction to motives of quadrics by Alexander Vishik, with various applications, notably to the splitting patterns of quadratic forms under base field extensions; - papers by Oleg Izhboldin and Nikita Karpenko on Chow groups of quadrics and their stable birational equivalence, with application to the construction of fields which carry anisotropic quadratic forms of dimension 9, but none of higher dimension; - a contribution in French by Bruno Kahn which lays out a general fra...
On the Motion of Matter in the Geometrical Gauge Field Theory
Konopleva, N P
2005-01-01
In the geometrical gauge field theory, the motion equations of matter (elementary particles) are connected with the field equations. In the talk, the problems arising from this connection are discussed. For the first time, such problems arose in Einstein's General Relativity. Einstein hoped that solution of these problems will allow explanation of elementary particles nature without making use of quantum mechanics. But, as it turned out, the situation is more difficult. Here the corresponding problems are formulated for the connection of equations of particle motion and field equations in the geometrical gauge field theory. It is shown that appearance of the problems under discussion is an inevitable effect of passage to relativism and local symmetries.
DEFF Research Database (Denmark)
Sørensen, O.; Rusch, W.
1975-01-01
The geometrical theory of diffraction (GTD) as formulated by R. G. Kouyoumjian has been applied to predict the radiation characteristics of hyperboloidal subreflectors with laterally defocused feeds. In caustic or multicaustic directions the scattered fields are determined using an equivalent ring...... current placed along the edge of the subreflector. The theoretical results are compared to measured amplitude and phase data. In order to improve the agreement, the blocking effects of the feed horn have been accounted for using the geometrical theory of diffraction. The calculated subreflector fields...... have been used to illuminate a paraboloid from which the scattered field is determined by physical optics. The results are compared to those obtained using a laterally defocused equivalent paraboloid....
International Nuclear Information System (INIS)
Kaku, M.
1988-01-01
We present an entirely new approach to closed-string field theory, called Igeometric string field theory R, which avoids the complications found in Becchi-Rouet-Stora-Tyutin string field theory (e.g., ghost counting, infinite overcounting of diagrams, midpoints, lack of modular invariance). Following the analogy with general relativity and Yang-Mills theory, we define a new infinite-dimensional local gauge group, called the unified string group, which uniquely specifies the connection fields, the curvature tensor, the measure and tensor calculus, and finally the action itself. Geometric field theory, when gauge fixed, yields an entirely new class of gauges called the interpolating gauge which allows us to smoothly interpolate between the midpoint gauge and the end-point gauge (''covariantized light-cone gauge''). We can show that geometric string field theory reproduces one copy of the Shapiro-Virasoro model. Surprisingly, after the gauge is broken, a new Iclosed four-string interactionR emerges as the counterpart of the instantaneous four-fermion Coulomb term in QED. This term restores modular invariance and precisely fills the missing region of the complex plane
Deriving the four-string and open-closed string interactions from geometric string field theory
International Nuclear Information System (INIS)
Kaku, M.
1990-01-01
One of the questions concerning the covariant open string field theory is why there are two distinct BRST theories and why the four-string interaction appears in one version but not the other. The authors solve this mystery by showing that both theories are gauge-fixed versions of a higher gauge theory, called the geometric string field theory, with a new field, a string verbein e μσ νρ , which allows us to gauge the string length and σ parametrization. By fixing the gauge, the authors can derive the endpoint gauge (the covariantized light cone gauge), the midpoint gauge of Witten, or the interpolating gauge with arbitrary string length. The authors show explicitly that the four-string interaction is a gauge artifact of the geometric theory (the counterpart of the four-fermion instantaneous Coulomb term of QED). By choosing the interpolating gauge, they produce a new class of four-string interactions which smoothly interpolate between the endpoint gauge and the midpoint gauge (where it vanishes). Similarly, they can extract the closed string as a bound state of the open string, which appears in the endpoint gauge but vanishes in the midpoint gauge. Thus, the four-string and open-closed string interactions do not have to be added to the action as long as the string vierbein is included
Exact solutions for isometric embeddings of pseudo-Riemannian manifolds
International Nuclear Information System (INIS)
Amery, G; Moodley, J
2014-01-01
Embeddings into higher dimensions are of direct importance in the study of higher dimensional theories of our Universe, in high energy physics and in classical general relativity. Theorems have been established that guarantee the existence of local and global codimension-1 embeddings between pseudo-Riemannian manifolds, particularly for Einstein embedding spaces. A technique has been provided to determine solutions to such embeddings. However, general solutions have not yet been found and most known explicit solutions are for embedded spaces with relatively simple Ricci curvature. Motivated by this, we have considered isometric embeddings of 4-dimensional pseudo-Riemannian spacetimes into 5-dimensional Einstein manifolds. We have applied the technique to treat specific 4-dimensional cases of interest in astrophysics and cosmology (including the global monopole exterior and Vaidya-de Sitter-class solutions), and provided novel physical insights into, for example, Einstein-Gauss-Bonnet gravity. Since difficulties arise in solving the 5-dimensional equations for given 4-dimensional spaces, we have also investigated embedded spaces, which admit bulks with a particular metric form. These analyses help to provide insight to the general embedding problem
International Nuclear Information System (INIS)
Ayala, Orlando; Rosa, Bogdan; Wang Lianping
2008-01-01
The effect of air turbulence on the geometric collision kernel of cloud droplets can be predicted if the effects of air turbulence on two kinematic pair statistics can be modeled. The first is the average radial relative velocity and the second is the radial distribution function (RDF). A survey of the literature shows that no theory is available for predicting the radial relative velocity of finite-inertia sedimenting droplets in a turbulent flow. In this paper, a theory for the radial relative velocity is developed, using a statistical approach assuming that gravitational sedimentation dominates the relative motion of droplets before collision. In the weak-inertia limit, the theory reveals a new term making a positive contribution to the radial relative velocity resulting from a coupling between sedimentation and air turbulence on the motion of finite-inertia droplets. The theory is compared to the direct numerical simulations (DNS) results in part 1, showing a reasonable agreement with the DNS data for bidisperse cloud droplets. For droplets larger than 30 μm in radius, a nonlinear drag (NLD) can also be included in the theory in terms of an effective inertial response time and an effective terminal velocity. In addition, an empirical model is developed to quantify the RDF. This, together with the theory for radial relative velocity, provides a parameterization for the turbulent geometric collision kernel. Using this integrated model, we find that turbulence could triple the geometric collision kernel, relative to the stagnant air case, for a droplet pair of 10 and 20 μm sedimenting through a cumulus cloud at R λ =2x10 4 and ε=600 cm 2 s -3 . For the self-collisions of 20 μm droplets, the collision kernel depends sensitively on the flow dissipation rate
Sub-Riemannian geometry and optimal transport
Rifford, Ludovic
2014-01-01
The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.
A geometric theory of swimming: Purcell's swimmer and its symmetrized cousin
International Nuclear Information System (INIS)
Avron, J E; Raz, O
2008-01-01
We develop a qualitative geometric approach to swimming at low Reynolds numbers which avoids solving differential equations and uses instead landscape figures describing the swimming and dissipation. This approach gives complete information about swimmers that swim on a line without rotations and gives the main qualitative features of general swimmers that can also rotate. We illustrate this approach for a symmetric version of Purcell's swimmer, which we solve by elementary analytical means within slender body theory. We then apply the theory to derive the basic qualitative properties of Purcell's swimmer
Zabavnikova, T. A.; Kadashevich, Yu. I.; Pomytkin, S. P.
2018-05-01
A geometric non-linear endochronic theory of inelasticity in tensor parametric form is considered. In the framework of this theory, the creep strains are modelled. The effect of various schemes of applying stresses and changing of material properties on the development of creep strains is studied. The constitutive equations of the model are represented by non-linear systems of ordinary differential equations which are solved in MATLAB environment by implicit difference method. Presented results demonstrate a good qualitative agreement of theoretical data and experimental observations including the description of the tertiary creep and pre-fracture of materials.
Djakou, Audrey Kamta; Darmon, Michel; Fradkin, Larissa; Potel, Catherine
2015-11-01
Diffraction phenomena studied in electromagnetism, acoustics, and elastodynamics are often modeled using integrals, such as the well-known Sommerfeld integral. The far field asymptotic evaluation of such integrals obtained using the method of steepest descent leads to the classical Geometrical Theory of Diffraction (GTD). It is well known that the method of steepest descent is inapplicable when the integrand's stationary phase point coalesces with its pole, explaining why GTD fails in zones where edge diffracted waves interfere with incident or reflected waves. To overcome this drawback, the Uniform geometrical Theory of Diffraction (UTD) has been developed previously in electromagnetism, based on a ray theory, which is particularly easy to implement. In this paper, UTD is developed for the canonical elastodynamic problem of the scattering of a plane wave by a half-plane. UTD is then compared to another uniform extension of GTD, the Uniform Asymptotic Theory (UAT) of diffraction, based on a more cumbersome ray theory. A good agreement between the two methods is obtained in the far field.
Spin foam model for pure gauge theory coupled to quantum gravity
International Nuclear Information System (INIS)
Oriti, Daniele; Pfeiffer, Hendryk
2002-01-01
We propose a spin foam model for pure gauge fields coupled to Riemannian quantum gravity in four dimensions. The model is formulated for the triangulation of a four-manifold which is given merely combinatorially. The Riemannian Barrett-Crane model provides the gravity sector of our model and dynamically assigns geometric data to the given combinatorial triangulation. The gauge theory sector is a lattice gauge theory living on the same triangulation and obtains from the gravity sector the geometric information which is required to calculate the Yang-Mills action. The model is designed so that one obtains a continuum approximation of the gauge theory sector at an effective level, similarly to the continuum limit of lattice gauge theory, when the typical length scale of gravity is much smaller than the Yang-Mills scale
Directory of Open Access Journals (Sweden)
Niangang Jiao
2018-05-01
Full Text Available With the increasing demand for high-resolution remote sensing images for mapping and monitoring the Earth’s environment, geometric positioning accuracy improvement plays a significant role in the image preprocessing step. Based on the statistical learning theory, we propose a new method to improve the geometric positioning accuracy without ground control points (GCPs. Multi-temporal images from the ZY-3 satellite are tested and the bias-compensated rational function model (RFM is applied as the block adjustment model in our experiment. An easy and stable weight strategy and the fast iterative shrinkage-thresholding (FIST algorithm which is widely used in the field of compressive sensing are improved and utilized to define the normal equation matrix and solve it. Then, the residual errors after traditional block adjustment are acquired and tested with the newly proposed inherent error compensation model based on statistical learning theory. The final results indicate that the geometric positioning accuracy of ZY-3 satellite imagery can be improved greatly with our proposed method.
Khlyupin, Aleksey; Aslyamov, Timur
2017-06-01
Realistic fluid-solid interaction potentials are essential in description of confined fluids especially in the case of geometric heterogeneous surfaces. Correlated random field is considered as a model of random surface with high geometric roughness. We provide the general theory of effective coarse-grained fluid-solid potential by proper averaging of the free energy of fluid molecules which interact with the solid media. This procedure is largely based on the theory of random processes. We apply first passage time probability problem and assume the local Markov properties of random surfaces. General expression of effective fluid-solid potential is obtained. In the case of small surface irregularities analytical approximation for effective potential is proposed. Both amorphous materials with large surface roughness and crystalline solids with several types of fcc lattices are considered. It is shown that the wider the lattice spacing in terms of molecular diameter of the fluid, the more obtained potentials differ from classical ones. A comparison with published Monte-Carlo simulations was discussed. The work provides a promising approach to explore how the random geometric heterogeneity affects on thermodynamic properties of the fluids.
Geometric Lagrangian approach to the physical degree of freedom count in field theory
Díaz, Bogar; Montesinos, Merced
2018-05-01
To circumvent some technical difficulties faced by the geometric Lagrangian approach to the physical degree of freedom count presented in the work of Díaz, Higuita, and Montesinos [J. Math. Phys. 55, 122901 (2014)] that prevent its direct implementation to field theory, in this paper, we slightly modify the geometric Lagrangian approach in such a way that its resulting version works perfectly for field theory (and for particle systems, of course). As in previous work, the current approach also allows us to directly get the Lagrangian constraints, a new Lagrangian formula for the counting of the number of physical degrees of freedom, the gauge transformations, and the number of first- and second-class constraints for any action principle based on a Lagrangian depending on the fields and their first derivatives without performing any Dirac's canonical analysis. An advantage of this approach over the previous work is that it also allows us to handle the reducibility of the constraints and to get the off-shell gauge transformations. The theoretical framework is illustrated in 3-dimensional generalized general relativity (Palatini and Witten's exotic actions), Chern-Simons theory, 4-dimensional BF theory, and 4-dimensional general relativity given by Palatini's action with a cosmological constant.
Finsler metrics—a global approach with applications to geometric function theory
Abate, Marco
1994-01-01
Complex Finsler metrics appear naturally in complex analysis. To develop new tools in this area, the book provides a graduate-level introduction to differential geometry of complex Finsler metrics. After reviewing real Finsler geometry stressing global results, complex Finsler geometry is presented introducing connections, Kählerianity, geodesics, curvature. Finally global geometry and complex Monge-Ampère equations are discussed for Finsler manifolds with constant holomorphic curvature, which are important in geometric function theory. Following E. Cartan, S.S. Chern and S. Kobayashi, the global approach carries the full strength of hermitian geometry of vector bundles avoiding cumbersome computations, and thus fosters applications in other fields.
Introduction to global analysis minimal surfaces in Riemannian manifolds
Moore, John Douglas
2017-01-01
During the last century, global analysis was one of the main sources of interaction between geometry and topology. One might argue that the core of this subject is Morse theory, according to which the critical points of a generic smooth proper function on a manifold M determine the homology of the manifold. Morse envisioned applying this idea to the calculus of variations, including the theory of periodic motion in classical mechanics, by approximating the space of loops on M by a finite-dimensional manifold of high dimension. Palais and Smale reformulated Morse's calculus of variations in terms of infinite-dimensional manifolds, and these infinite-dimensional manifolds were found useful for studying a wide variety of nonlinear PDEs. This book applies infinite-dimensional manifold theory to the Morse theory of closed geodesics in a Riemannian manifold. It then describes the problems encountered when extending this theory to maps from surfaces instead of curves. It treats critical point theory for closed param...
Pseudo harmonic morphisms on Riemannian polyhedra
International Nuclear Information System (INIS)
Aprodu, M.A.; Bouziane, T.
2004-10-01
The aim of this paper is to extend the notion of pseudo harmonic morphism (introduced by Loubeau) to the case when the source manifold is an admissible Riemannian polyhedron. We define these maps to be harmonic in the sense of Eells-Fuglede and pseudo-horizontally weakly conformal in our sense. We characterize them by means of germs of harmonic functions on the source polyhedron, in the sense of Korevaar-Schoen, and germs of holomorphic functions on the Kaehler target manifold. (author)
DEFF Research Database (Denmark)
Hirst, Andrew G.; Glazier, Douglas S.; Atkinson, David
2014-01-01
Metabolism fuels all of life’s activities, from biochemical reactions to ecological interactions. According to two intensely debated theories, body size affects metabolism via geometrical influences on the transport of resources and wastes. However, these theories differ crucially in whether...... the size dependence of metabolism is derived from material transport across external surfaces, or through internal resource-transport networks. We show that when body shape changes during growth, these models make opposing predictions. These models are tested using pelagic invertebrates, because...... these animals exhibit highly variable intraspecific scaling relationships for metabolic rate and body shape. Metabolic scaling slopes of diverse integument-breathing species were significantly positively correlated with degree of body flattening or elongation during ontogeny, as expected from surface area...
Higher-order Jordan Osserman pseudo-Riemannian manifolds
International Nuclear Information System (INIS)
Gilkey, Peter B; Ivanova, Raina; Zhang Tan
2002-01-01
We study the higher-order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r, s) for certain values of (r, s). These pseudo-Riemannian manifolds are new and non-trivial examples of higher-order Osserman manifolds
Higher-order Jordan Osserman pseudo-Riemannian manifolds
Energy Technology Data Exchange (ETDEWEB)
Gilkey, Peter B [Mathematics Department, University of Oregon, Eugene, OR 97403 (United States); Ivanova, Raina [Mathematics Department, University of Hawaii - Hilo, 200 W Kawili St, Hilo, HI 96720 (United States); Zhang Tan [Department of Mathematics and Statistics, Murray State University, Murray, KY 42071 (United States)
2002-09-07
We study the higher-order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r, s) for certain values of (r, s). These pseudo-Riemannian manifolds are new and non-trivial examples of higher-order Osserman manifolds.
Miyazaki, Hideki T; Miyazaki, Hiroshi; Miyano, Kenjiro
2003-09-01
We have recently identified the resonant scattering from dielectric bispheres in the specular direction, which has long been known as the specular resonance, to be a type of rainbow (a caustic) and a general phenomenon for bispheres. We discuss the details of the specular resonance on the basis of systematic calculations. In addition to the rigorous theory, which precisely describes the scattering even in the resonance regime, the ray-tracing method, which gives the scattering in the geometrical-optics limit, is used. Specular resonance is explicitly defined as strong scattering in the direction of the specular reflection from the symmetrical axis of the bisphere whose intensity exceeds that of the scattering from noninteracting bispheres. Then the range of parameters for computing a particular specular resonance is specified. This resonance becomes prominent in a wide range of refractive indices (from 1.2 to 2.2) in a wide range of size parameters (from five to infinity) and for an arbitrarily polarized light incident within an angle of 40 degrees to the symmetrical axis. This particular scattering can stay evident even when the spheres are not in contact or the sizes of the spheres are different. Thus specular resonance is a common and robust phenomenon in dielectric bispheres. Furthermore, we demonstrate that various characteristic features in the scattering from bispheres can be explained successfully by using intuitive and simple representations. Most of the significant scatterings other than the specular resonance are also understandable as caustics in geometrical-optics theory. The specular resonance becomes striking at the smallest size parameter among these caustics because its optical trajectory is composed of only the refractions at the surfaces and has an exceptionally large intensity. However, some characteristics are not accounted for by geometrical optics. In particular, the oscillatory behaviors of their scattering intensity are well described by
Strong coupling in F-theory and geometrically non-Higgsable seven-branes
Directory of Open Access Journals (Sweden)
James Halverson
2017-06-01
Full Text Available Geometrically non-Higgsable seven-branes carry gauge sectors that cannot be broken by complex structure deformation, and there is growing evidence that such configurations are typical in F-theory. We study strongly coupled physics associated with these branes. Axiodilaton profiles are computed using Ramanujan's theories of elliptic functions to alternative bases, showing explicitly that the string coupling is O(1 in the vicinity of the brane; that it sources nilpotent SL(2,Z monodromy and therefore the associated brane charges are modular; and that essentially all F-theory compactifications have regions with order one string coupling. It is shown that non-perturbative SU(3 and SU(2 seven-branes are related to weakly coupled counterparts with D7-branes via deformation-induced Hanany–Witten moves on (p,q string junctions that turn them into fundamental open strings; only the former may exist for generic complex structure. D3-brane near these and the Kodaira type II seven-branes probe Argyres–Douglas theories. The BPS states of slightly deformed theories are shown to be dyonic string junctions.
Jupri, A.
2018-05-01
The lack of ability of primary school teachers in deductive thinking, such as doing geometrical proof, is an indispensable issue to be dealt with. In this paper, we report on results of a three-step of the field document study. The study was part of a pilot study for improving deductive thinking ability of primary school teachers. First, we designed geometrical proof problems adapted from literature. Second, we administered an individual written test involving nine master students of primary education program, in which they are having experiences as primary school mathematics teachers. Finally, we analyzed the written work from the view of the Van Hiele theory. The results revealed that even if about the half of the teachers show ability in doing formal proof, still the rest provides inappropriate proving. For further investigation, we wonder whether primary school teachers would show better deductive thinking if the teaching of geometry is designed in a systematic and appropriate manner according to the Van Hiele theory.
G{sub 2}-structures and quantization of non-geometric M-theory backgrounds
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Kupriyanov, Vladislav G. [Centro de Matemática, Computação e Cognição, Universidade de Federal do ABC,Santo André, SP (Brazil); Tomsk State University,Tomsk (Russian Federation); Szabo, Richard J. [Department of Mathematics, Heriot-Watt University,Colin Maclaurin Building, Riccarton, Edinburgh EH14 4AS (United Kingdom); Maxwell Institute for Mathematical Sciences,Edinburgh (United Kingdom); The Higgs Centre for Theoretical Physics,Edinburgh (United Kingdom)
2017-02-20
We describe the quantization of a four-dimensional locally non-geometric M-theory background dual to a twisted three-torus by deriving a phase space star product for deformation quantization of quasi-Poisson brackets related to the nonassociative algebra of octonions. The construction is based on a choice of G{sub 2}-structure which defines a nonassociative deformation of the addition law on the seven-dimensional vector space of Fourier momenta. We demonstrate explicitly that this star product reduces to that of the three-dimensional parabolic constant R-flux model in the contraction of M-theory to string theory, and use it to derive quantum phase space uncertainty relations as well as triproducts for the nonassociative geometry of the four-dimensional configuration space. By extending the G{sub 2}-structure to a Spin(7)-structure, we propose a 3-algebra structure on the full eight-dimensional M2-brane phase space which reduces to the quasi-Poisson algebra after imposing a particular gauge constraint, and whose deformation quantisation simultaneously encompasses both the phase space star products and the configuration space triproducts. We demonstrate how these structures naturally fit in with previous occurences of 3-algebras in M-theory.
International Nuclear Information System (INIS)
Borovikov, V.A.; Kinber, B.E.
1988-01-01
The heuristic criteria (HC) of validity of geometric optics (GO) and of the geometric theory of diffraction (GTD), suggested in [2-7, 13, 14] and based on identifying the physical volume occupied by the ray with the Fresnel volume (FV) introduced in these papers (i.e., the envelope of the first Fresnel zone), are analyzed. Numerous examples of HC invalidity are given, as well as the reasons. In particular, HC provide an incorrect answer for all GO problems with caustics, since in these problems there always exists a ray, whose FV is nonlocal and covers the FV of other rays. The HC are shown to be unsuitable for multiple ray GTD problems, as well as for the simplest problems of diffraction of a cylindrical wave by a half-plane and of a plane wave by a curved half-plane
Contour Propagation With Riemannian Elasticity Regularization
DEFF Research Database (Denmark)
Bjerre, Troels; Hansen, Mads Fogtmann; Sapru, W.
2011-01-01
Purpose/Objective(s): Adaptive techniques allow for correction of spatial changes during the time course of the fractionated radiotherapy. Spatial changes include tumor shrinkage and weight loss, causing tissue deformation and residual positional errors even after translational and rotational image...... the planning CT onto the rescans and correcting to reflect actual anatomical changes. For deformable registration, a free-form, multi-level, B-spline deformation model with Riemannian elasticity, penalizing non-rigid local deformations, and volumetric changes, was used. Regularization parameters was defined...... on the original delineation and tissue deformation in the time course between scans form a better starting point than rigid propagation. There was no significant difference of locally and globally defined regularization. The method used in the present study suggests that deformed contours need to be reviewed...
International Nuclear Information System (INIS)
Wang, Lin; Liu, Xiongwei; Renevier, Nathalie; Stables, Matthew; Hall, George M.
2014-01-01
Due to the increasing size and flexibility of large wind turbine blades, accurate and reliable aeroelastic modelling is playing an important role for the design of large wind turbines. Most existing aeroelastic models are linear models based on assumption of small blade deflections. This assumption is not valid anymore for very flexible blade design because such blades often experience large deflections. In this paper, a novel nonlinear aeroelastic model for large wind turbine blades has been developed by combining BEM (blade element momentum) theory and mixed-form formulation of GEBT (geometrically exact beam theory). The nonlinear aeroelastic model takes account of large blade deflections and thus greatly improves the accuracy of aeroelastic analysis of wind turbine blades. The nonlinear aeroelastic model is implemented in COMSOL Multiphysics and validated with a series of benchmark calculation tests. The results show that good agreement is achieved when compared with experimental data, and its capability of handling large deflections is demonstrated. Finally the nonlinear aeroelastic model is applied to aeroelastic modelling of the parked WindPACT 1.5 MW baseline wind turbine, and reduced flapwise deflection from the nonlinear aeroelastic model is observed compared to the linear aeroelastic code FAST (Fatigue, Aerodynamics, Structures, and Turbulence). - Highlights: • A novel nonlinear aeroelastic model for wind turbine blades is developed. • The model takes account of large blade deflections and geometric nonlinearities. • The model is reliable and efficient for aeroelastic modelling of wind turbine blades. • The accuracy of the model is verified by a series of benchmark calculation tests. • The model provides more realistic aeroelastic modelling than FAST (Fatigue, Aerodynamics, Structures, and Turbulence)
Lorentz Invariance Violation effects on UHECR propagation: A geometrized approach
Torri, Marco Danilo Claudio; Bertini, Stefano; Giammarchi, Marco; Miramonti, Lino
2018-06-01
We explore the possibility to geometrize the interaction of massive fermions with the quantum structure of space-time, trying to create a theoretical background, in order to explain what some recent experimental results seem to implicate on the propagation of Ultra High Energy Cosmic Rays (UHECR). We will investigate part of the phenomenological implications of this approach on the predicted effect of the UHECR suppression, in fact recent evidences seem to involve the modification of the GZK cut-off phenomenon. The search for an effective theory, which can explain this physical effect, is based on Lorentz Invariance Violation (LIV), which is introduced via Modified Dispersion Relations (MDRs). Furthermore we illustrate that this perspective implies a more general geometry of space-time than the usual Riemannian one, indicating, for example, the opportunity to resort to Finsler theory.
Energy Technology Data Exchange (ETDEWEB)
Heller, Marc Andre [Particle Theory and Cosmology Group, Department of Physics,Graduate School of Science, Tohoku University,Aoba-ku, Sendai 980-8578 (Japan); Ikeda, Noriaki [Department of Mathematical Sciences, Ritsumeikan University,Kusatsu, Shiga 525-8577 (Japan); Watamura, Satoshi [Particle Theory and Cosmology Group, Department of Physics,Graduate School of Science, Tohoku University,Aoba-ku, Sendai 980-8578 (Japan)
2017-02-15
We give a systematic derivation of the local expressions of the NS H-flux, geometric F- as well as non-geometric Q- and R-fluxes in terms of bivector β- and two-form B-potentials including vielbeins. They are obtained using a supergeometric method on QP-manifolds by twist of the standard Courant algebroid on the generalized tangent space without flux. Bianchi identities of the fluxes are easily deduced. We extend the discussion to the case of the double space and present a formulation of T-duality in terms of canonical transformations between graded symplectic manifolds. Thus, we find a unified description of geometric as well as non-geometric fluxes and T-duality transformations in double field theory. Finally, the construction is compared to the formerly introduced Poisson Courant algebroid, a Courant algebroid on a Poisson manifold, as a model for R-flux.
Constantinides, E. D.; Marhefka, R. J.
1994-01-01
A uniform geometrical optics (UGO) and an extended uniform geometrical theory of diffraction (EUTD) are developed for evaluating high frequency electromagnetic (EM) fields within transition regions associated with a two and three dimensional smooth caustic of reflected rays and a composite shadow boundary formed by the caustic termination or the confluence of the caustic with the reflection shadow boundary (RSB). The UGO is a uniform version of the classic geometrical optics (GO). It retains the simple ray optical expressions of classic GO and employs a new set of uniform reflection coefficients. The UGO also includes a uniform version of the complex GO ray field that exists on the dark side of the smooth caustic. The EUTD is an extension of the classic uniform geometrical theory of diffraction (UTD) and accounts for the non-ray optical behavior of the UGO reflected field near caustics by using a two-variable transition function in the expressions for the edge diffraction coefficients. It also uniformly recovers the classic UTD behavior of the edge diffracted field outside the composite shadow boundary transition region. The approach employed for constructing the UGO/EUTD solution is based on a spatial domain physical optics (PO) radiation integral representation for the fields which is then reduced using uniform asymptotic procedures. The UGO/EUTD analysis is also employed to investigate the far-zone RCS problem of plane wave scattering from two and three dimensional polynomial defined surfaces, and uniform reflection, zero-curvature, and edge diffraction coefficients are derived. Numerical results for the scattering and diffraction from cubic and fourth order polynomial strips are also shown and the UGO/EUTD solution is validated by comparison to an independent moment method (MM) solution. The UGO/EUTD solution is also compared with the classic GO/UTD solution. The failure of the classic techniques near caustics and composite shadow boundaries is clearly
Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry
International Nuclear Information System (INIS)
Pan, Yiwen
2014-01-01
In this note we generalize the methods of http://dx.doi.org/10.1007/JHEP08(2012)141, http://dx.doi.org/10.1007/JHEP01(2013)072 and http://dx.doi.org/10.1007/JHEP05(2013)017 to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solutions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to M=S 1 ×M 4 , which leads to M being foliated by submanifolds with special properties, such as Quaternion-Kähler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S 3 or T 3 -fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N=1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation
Point interactions in two- and three-dimensional Riemannian manifolds
International Nuclear Information System (INIS)
Erman, Fatih; Turgut, O Teoman
2010-01-01
We present a non-perturbative renormalization of the bound state problem of n bosons interacting with finitely many Dirac-delta interactions on two- and three-dimensional Riemannian manifolds using the heat kernel. We formulate the problem in terms of a new operator called the principal or characteristic operator Φ(E). In order to investigate the problem in more detail, we then restrict the problem to one particle sector. The lower bound of the ground state energy is found for a general class of manifolds, e.g. for compact and Cartan-Hadamard manifolds. The estimate of the bound state energies in the tunneling regime is calculated by perturbation theory. Non-degeneracy and uniqueness of the ground state is proven by the Perron-Frobenius theorem. Moreover, the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics. Renormalization procedure does not lead to any radical change in these cases. Finally, renormalization group equations are derived and the β function is exactly calculated. This work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions, developed by Rajeev.
Gregory, A L; Agarwal, A; Lasenby, J
2017-11-01
We present a novel application of rotors in geometric algebra to represent the change of curvature tensor that is used in shell theory as part of the constitutive law. We introduce a new decomposition of the change of curvature tensor, which has explicit terms for changes of curvature due to initial curvature combined with strain, and changes in rotation over the surface. We use this decomposition to perform a scaling analysis of the relative importance of bending and stretching in flexible tubes undergoing self-excited oscillations. These oscillations have relevance to the lung, in which it is believed that they are responsible for wheezing. The new analysis is necessitated by the fact that the working fluid is air, compared to water in most previous work. We use stereographic imaging to empirically measure the relative importance of bending and stretching energy in observed self-excited oscillations. This enables us to validate our scaling analysis. We show that bending energy is dominated by stretching energy, and the scaling analysis makes clear that this will remain true for tubes in the airways of the lung.
Existence of localizing solutions in plasticity via the geometric singular perturbation theory
Lee, Min-Gi
2017-01-31
Shear bands are narrow zones of intense shear observed during plastic deformations of metals at high strain rates. Because they often precede rupture, their study attracted attention as a mechanism of material failure. Here, we aim to reveal the onset of localization into shear bands using a simple model from viscoplasticity. We exploit the properties of scale invariance of the model to construct a family of self-similar focusing solutions that capture the nonlinear mechanism of shear band formation. The key step is to desingularize a reduced system of singular ordinary differential equations and reduce the problem into the construction of a heteroclinic orbit for an autonomous system of three first-order equations. The associated dynamical system has fast and slow time scales, forming a singularly perturbed problem. Geometric singular perturbation theory is applied to this problem to achieve an invariant surface. The flow on the invariant surface is analyzed via the Poincaré--Bendixson theorem to construct a heteroclinic orbit.
Combining the multilevel fast multipole method with the uniform geometrical theory of diffraction
Directory of Open Access Journals (Sweden)
A. Tzoulis
2005-01-01
Full Text Available The presence of arbitrarily shaped and electrically large objects in the same environment leads to hybridization of the Method of Moments (MoM with the Uniform Geometrical Theory of Diffraction (UTD. The computation and memory complexity of the MoM solution is improved with the Multilevel Fast Multipole Method (MLFMM. By expanding the k-space integrals in spherical harmonics, further considerable amount of memory can be saved without compromising accuracy and numerical speed. However, until now MoM-UTD hybrid methods are restricted to conventional MoM formulations only with Electric Field Integral Equation (EFIE. In this contribution, a MLFMM-UTD hybridization for Combined Field Integral Equation (CFIE is proposed and applied within a hybrid Finite Element - Boundary Integral (FEBI technique. The MLFMM-UTD hybridization is performed at the translation procedure on the various levels of the MLFMM, using a far-field approximation of the corresponding translation operator. The formulation of this new hybrid technique is presented, as well as numerical results.
Geometrical setting of solid mechanics
International Nuclear Information System (INIS)
Fiala, Zdenek
2011-01-01
Highlights: → Solid mechanics within the Riemannian symmetric manifold GL (3, R)/O (3, R). → Generalized logarithmic strain. → Consistent linearization. → Incremental principle of virtual power. → Time-discrete approximation. - Abstract: The starting point in the geometrical setting of solid mechanics is to represent deformation process of a solid body as a trajectory in a convenient space with Riemannian geometry, and then to use the corresponding tools for its analysis. Based on virtual power of internal stresses, we show that such a configuration space is the (globally) symmetric space of symmetric positive-definite real matrices. From this unifying point of view, we shall analyse the logarithmic strain, the stress rate, as well as linearization and intrinsic integration of corresponding evolution equation.
Riemannian multi-manifold modeling and clustering in brain networks
Slavakis, Konstantinos; Salsabilian, Shiva; Wack, David S.; Muldoon, Sarah F.; Baidoo-Williams, Henry E.; Vettel, Jean M.; Cieslak, Matthew; Grafton, Scott T.
2017-08-01
This paper introduces Riemannian multi-manifold modeling in the context of brain-network analytics: Brainnetwork time-series yield features which are modeled as points lying in or close to a union of a finite number of submanifolds within a known Riemannian manifold. Distinguishing disparate time series amounts thus to clustering multiple Riemannian submanifolds. To this end, two feature-generation schemes for brain-network time series are put forth. The first one is motivated by Granger-causality arguments and uses an auto-regressive moving average model to map low-rank linear vector subspaces, spanned by column vectors of appropriately defined observability matrices, to points into the Grassmann manifold. The second one utilizes (non-linear) dependencies among network nodes by introducing kernel-based partial correlations to generate points in the manifold of positivedefinite matrices. Based on recently developed research on clustering Riemannian submanifolds, an algorithm is provided for distinguishing time series based on their Riemannian-geometry properties. Numerical tests on time series, synthetically generated from real brain-network structural connectivity matrices, reveal that the proposed scheme outperforms classical and state-of-the-art techniques in clustering brain-network states/structures.
Palatini approach to Born-Infeld-Einstein theory and a geometric description of electrodynamics
International Nuclear Information System (INIS)
Vollick, Dan N.
2004-01-01
The field equations associated with the Born-Infeld-Einstein action are derived using the Palatini variational technique. In this approach the metric and connection are varied independently and the Ricci tensor is generally not symmetric. For sufficiently small curvatures the resulting field equations can be divided into two sets. One set, involving the antisymmetric part of the Ricci tensor R or μν , consists of the field equation for a massive vector field. The other set consists of the Einstein field equations with an energy momentum tensor for the vector field plus additional corrections. In a vacuum with R or μν =0 the field equations are shown to be the usual Einstein vacuum equations. This extends the universality of the vacuum Einstein equations, discussed by Ferraris et al., to the Born-Infeld-Einstein action. In the simplest version of the theory there is a single coupling constant and by requiring that the Einstein field equations hold to a good approximation in neutron stars it is shown that mass of the vector field exceeds the lower bound on the mass of the photon. Thus, in this case the vector field cannot represent the electromagnetic field and would describe a new geometrical field. In a more general version in which the symmetric and antisymmetric parts of the Ricci tensor have different coupling constants it is possible to satisfy all of the observational constraints if the antisymmetric coupling is much larger than the symmetric coupling. In this case the antisymmetric part of the Ricci tensor can describe the electromagnetic field
Einstein-Dirac theory in spin maximum I
International Nuclear Information System (INIS)
Crumeyrolle, A.
1975-01-01
An unitary Einstein-Dirac theory, first in spin maximum 1, is constructed. An original feature of this article is that it is written without any tetrapod technics; basic notions and existence conditions for spinor structures on pseudo-Riemannian fibre bundles are only used. A coupling gravitation-electromagnetic field is pointed out, in the geometric setting of the tangent bundle over space-time. Generalized Maxwell equations for inductive media in presence of gravitational field are obtained. Enlarged Einstein-Schroedinger theory, gives a particular case of this E.D. theory. E. S. theory is a truncated E.D. theory in spin maximum 1. A close relation between torsion-vector and Schroedinger's potential exists and nullity of torsion-vector has a spinor meaning. Finally the Petiau-Duffin-Kemmer theory is incorporated in this geometric setting [fr
Congedo, Marco; Barachant, Alexandre
2015-01-01
Currently the Riemannian geometry of symmetric positive definite (SPD) matrices is gaining momentum as a powerful tool in a wide range of engineering applications such as image, radar and biomedical data signal processing. If the data is not natively represented in the form of SPD matrices, typically we may summarize them in such form by estimating covariance matrices of the data. However once we manipulate such covariance matrices on the Riemannian manifold we lose the representation in the original data space. For instance, we can evaluate the geometric mean of a set of covariance matrices, but not the geometric mean of the data generating the covariance matrices, the space of interest in which the geometric mean can be interpreted. As a consequence, Riemannian information geometry is often perceived by non-experts as a "black-box" tool and this perception prevents a wider adoption in the scientific community. Hereby we show that we can overcome this limitation by constructing a special form of SPD matrix embedding both the covariance structure of the data and the data itself. Incidentally, whenever the original data can be represented in the form of a generic data matrix (not even square), this special SPD matrix enables an exhaustive and unique description of the data up to second-order statistics. This is achieved embedding the covariance structure of both the rows and columns of the data matrix, allowing naturally a wide range of possible applications and bringing us over and above just an interpretability issue. We demonstrate the method by manipulating satellite images (pansharpening) and event-related potentials (ERPs) of an electroencephalography brain-computer interface (BCI) study. The first example illustrates the effect of moving along geodesics in the original data space and the second provides a novel estimation of ERP average (geometric mean), showing that, in contrast to the usual arithmetic mean, this estimation is robust to outliers. In
Robust Covariance Estimators Based on Information Divergences and Riemannian Manifold
Directory of Open Access Journals (Sweden)
Xiaoqiang Hua
2018-03-01
Full Text Available This paper proposes a class of covariance estimators based on information divergences in heterogeneous environments. In particular, the problem of covariance estimation is reformulated on the Riemannian manifold of Hermitian positive-definite (HPD matrices. The means associated with information divergences are derived and used as the estimators. Without resorting to the complete knowledge of the probability distribution of the sample data, the geometry of the Riemannian manifold of HPD matrices is considered in mean estimators. Moreover, the robustness of mean estimators is analyzed using the influence function. Simulation results indicate the robustness and superiority of an adaptive normalized matched filter with our proposed estimators compared with the existing alternatives.
Roughly isometric minimal immersions into Riemannian manifolds
DEFF Research Database (Denmark)
Markvorsen, Steen
of the intrinsic combinatorial discrete Laplacian, and we will show that they share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in $N$. The intrinsic properties thus obtained may hence serve as roughly invariant descriptors for the original metric space $X$....
1998-01-01
This introduction to the recent exciting developments in the applications of model theory to algebraic geometry, illustrated by E. Hrushovski's model-theoretic proof of the geometric Mordell-Lang Conjecture starts from very basic background and works up to the detailed exposition of Hrushovski's proof, explaining the necessary tools and results from stability theory on the way. The first chapter is an informal introduction to model theory itself, making the book accessible (with a little effort) to readers with no previous knowledge of model theory. The authors have collaborated closely to achieve a coherent and self- contained presentation, whereby the completeness of exposition of the chapters varies according to the existence of other good references, but comments and examples are always provided to give the reader some intuitive understanding of the subject.
DEFF Research Database (Denmark)
Sommer, Stefan Horst; Svane, Anne Marie
2017-01-01
distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hormander condition is satisfied such that the Brownian motions have smooth transition......We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed...... in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal...
Differential calculus on the space of Steiner minimal trees in Riemannian manifolds
International Nuclear Information System (INIS)
Ivanov, A O; Tuzhilin, A A
2001-01-01
It is proved that the length of a minimal spanning tree, the length of a Steiner minimal tree, and the Steiner ratio regarded as functions of finite subsets of a connected complete Riemannian manifold have directional derivatives in all directions. The derivatives of these functions are calculated and some properties of their critical points are found. In particular, a geometric criterion for a finite set to be critical for the Steiner ratio is found. This criterion imposes essential restrictions on the geometry of the sets for which the Steiner ratio attains its minimum, that is, the sets on which the Steiner ratio of the boundary set is equal to the Steiner ratio of the ambient space
On integrability of certain rank 2 sub-Riemannian structures
Czech Academy of Sciences Publication Activity Database
Kruglikov, B.S.; Vollmer, A.; Lukes-Gerakopoulos, Georgios
2017-01-01
Roč. 22, č. 5 (2017), s. 502-519 ISSN 1560-3547 R&D Projects: GA ČR(CZ) GJ17-06962Y Institutional support: RVO:67985815 Keywords : sub-Riemannian geodesic flow * Killing tensor * integral Subject RIV: BN - Astronomy, Celestial Mechanics, Astrophysics OBOR OECD: Astronomy (including astrophysics,space science) Impact factor: 1.562, year: 2016
A Random Riemannian Metric for Probabilistic Shortest-Path Tractography
DEFF Research Database (Denmark)
Hauberg, Søren; Schober, Michael; Liptrot, Matthew George
2015-01-01
of the diffusion tensor as a “random Riemannian metric”, where a geodesic is a distribution over tracts. We approximate this distribution with a Gaussian process and present a probabilistic numerics algorithm for computing the geodesic distribution. We demonstrate SPT improvements on data from the Human Connectome...
A Riemannian scalar measure for diffusion tensor images
Astola, L.J.; Fuster, A.; Florack, L.M.J.
2010-01-01
We study a well-known scalar quantity in Riemannian geometry, the Ricci scalar, in the context of Diffusion Tensor Imaging (DTI), which is an emerging non-invasive medical imaging modality. We derive a physical interpretation for the Ricci scalar and explore experimentally its significance in DTI.
On determining the isometry group of a Riemannian space
International Nuclear Information System (INIS)
Karlhede, A.; Maccallum, M.A.H.
1982-01-01
An extension of the recently discussed algorithm for deciding the equivalence problem for Riemannian metrics is presented. The extension determines the structure constants of the isometry group and enables us to obtain some information about its orbits, including the form of the Killing vectors in canonical coordinates. (author)
Classical field theory on electrodynamics, non-Abelian gauge theories and gravitation
Scheck, Florian
2012-01-01
The book describes Maxwell's equations first in their integral, directly testable form, then moves on to their local formulation. The first two chapters cover all essential properties of Maxwell's equations, including their symmetries and their covariance in a modern notation. Chapter 3 is devoted to Maxwell theory as a classical field theory and to solutions of the wave equation. Chapter 4 deals with important applications of Maxwell theory. It includes topical subjects such as metamaterials with negative refraction index and solutions of Helmholtz' equation in paraxial approximation relevant for the description of laser beams. Chapter 5 describes non-Abelian gauge theories from a classical, geometric point of view, in analogy to Maxwell theory as a prototype, and culminates in an application to the U(2) theory relevant for electroweak interactions. The last chapter 6 gives a concise summary of semi-Riemannian geometry as the framework for the classical field theory of gravitation. The chapter concludes wit...
International Nuclear Information System (INIS)
Thierry-Mieg, J.
1985-01-01
This paper discusses the reinterpretation of the BRS equations of Quantum Field Theory as the Maurer Cartan equation of a classical principal fiber bundle leads to a simple gauge invariant classification of the anomalies in Yang Mills theory and gravity
International Nuclear Information System (INIS)
Thierry-Mieg, J.
1985-01-01
The reinterpretation of the BRS equations of Quantum Field Theory as the Maurer Cartan equation of a classical principal fiber bundle leads to a simple gauge invariant classification of the anomalies in Yang Mills theory and gravity
An extended geometric criterion for chaos in the Dicke model
International Nuclear Information System (INIS)
Li Jiangdan; Zhang Suying
2010-01-01
We extend HBLSL's (Horwitz, Ben Zion, Lewkowicz, Schiffer and Levitan) new Riemannian geometric criterion for chaotic motion to Hamiltonian systems of weak coupling of potential and momenta by defining the 'mean unstable ratio'. We discuss the Dicke model of an unstable Hamiltonian system in detail and show that our results are in good agreement with that of the computation of Lyapunov characteristic exponents.
Geometrical structure of shock waves in general relativity
Energy Technology Data Exchange (ETDEWEB)
Modugno, M [Istituto di Matematica, Universita di Lecce (Italia); Stefani, Gianna [Florence Univ. (Italy)
1979-01-01
A systematic and geometrical analysis of shock structures in a Riemannian manifold is developed. The jump, the infinitesimal jump and the covariant derivative jump of a tensor are defined globally. By means of derivation laws induced on the shock hypersurface, physically significant operators are defined. As physical applications, the charged fluid electromagnetic and gravitational interacting fields are considered.
A geometric theory for semilinear almost-periodic parabolic partial differential equations on RN
International Nuclear Information System (INIS)
Vuillermot, P.A.
1991-01-01
In this short expository article we review various applications of some geometric methods which have been recently devised to investigate the long time behaviour of classical solutions to certain semilinear almost-periodic reaction-diffusion equations on R N . As a consequence, we also show how to construct almost-periodic attractors for such equations and how to investigate their stability properties. The class of problems which we analyse here contains in particular well known equations of population genetics. (author). 17 refs
Application of semiclassical and geometrical optics theories to resonant modes of a coated sphere.
Bambino, Túlio M; Breitschaft, Ana Maria S; Barbosa, Valmar C; Guimarães, Luiz G
2003-03-01
This work deals with some aspects of the resonant scattering of electromagnetic waves by a metallic sphere covered by a dielectric layer, in the weak-absorption approximation. We carry out a geometrical optics treatment of the scattering and develop semiclassical formulas to determine the positions and widths of the system resonances. In addition, we show that the mean lifetime of broad resonances is strongly dependent on the polarization of the incident light.
von Cramon-Taubadel, Noreen; Frazier, Brenda C; Lahr, Marta Mirazón
2007-09-01
Geometric morphometric methods rely on the accurate identification and quantification of landmarks on biological specimens. As in any empirical analysis, the assessment of inter- and intra-observer error is desirable. A review of methods currently being employed to assess measurement error in geometric morphometrics was conducted and three general approaches to the problem were identified. One such approach employs Generalized Procrustes Analysis to superimpose repeatedly digitized landmark configurations, thereby establishing whether repeat measures fall within an acceptable range of variation. The potential problem of this error assessment method (the "Pinocchio effect") is demonstrated and its effect on error studies discussed. An alternative approach involves employing Euclidean distances between the configuration centroid and repeat measures of a landmark to assess the relative repeatability of individual landmarks. This method is also potentially problematic as the inherent geometric properties of the specimen can result in misleading estimates of measurement error. A third approach involved the repeated digitization of landmarks with the specimen held in a constant orientation to assess individual landmark precision. This latter approach is an ideal method for assessing individual landmark precision, but is restrictive in that it does not allow for the incorporation of instrumentally defined or Type III landmarks. Hence, a revised method for assessing landmark error is proposed and described with the aid of worked empirical examples. (c) 2007 Wiley-Liss, Inc.
Srivastava, Vandana; Jarzembski, Maurice A.
1991-01-01
This paper uses Mie theory to treat electromagnetic scattering and to evaluate field enhancement in the forward direction of a small droplet irradiated by a high-energy beam and compares the results of calculations with the field-enhancement evaluation obtained via geometrical optics treatment. Results of this comparison suggest that the field enhancement located at the critical ring region encircling the axis in the forward direction of the droplet can support laser-induced Raman scattering. The results are supported by experimental observations of the interaction of a 120-micron-diam water droplet with a high-energy Nd:YAG laser beam.
Geometrical theory of the relativistic string in t=tau gauge
International Nuclear Information System (INIS)
Barbashov, B.M.; Nesterenko, V.V.
1982-01-01
Using the co-moving frame method and the exterior differential forms in the surface theory the classical theory of the relativistic string in the gauge is constructed. The moving frame on the string world-sheet is chosen in a special form. As a result, the theory of the free relativistic string in the four-dimensional space-time is reduced to the D'Alembert equation for one scalar function
Gay-Balmaz, François; Putkaradze, Vakhtang
2018-01-01
We present a theory for the three-dimensional evolution of tubes with expandable walls conveying fluid. Our theory can accommodate arbitrary deformations of the tube, arbitrary elasticity of the walls, and both compressible and incompressible flows inside the tube. We also present the theory of propagation of shock waves in such tubes and derive the conservation laws and Rankine-Hugoniot conditions in arbitrary spatial configuration of the tubes, and compute several examples of particular sol...
Segmentation of High Angular Resolution Diffusion MRI using Sparse Riemannian Manifold Clustering
Wright, Margaret J.; Thompson, Paul M.; Vidal, René
2015-01-01
We address the problem of segmenting high angular resolution diffusion imaging (HARDI) data into multiple regions (or fiber tracts) with distinct diffusion properties. We use the orientation distribution function (ODF) to represent HARDI data and cast the problem as a clustering problem in the space of ODFs. Our approach integrates tools from sparse representation theory and Riemannian geometry into a graph theoretic segmentation framework. By exploiting the Riemannian properties of the space of ODFs, we learn a sparse representation for each ODF and infer the segmentation by applying spectral clustering to a similarity matrix built from these representations. In cases where regions with similar (resp. distinct) diffusion properties belong to different (resp. same) fiber tracts, we obtain the segmentation by incorporating spatial and user-specified pairwise relationships into the formulation. Experiments on synthetic data evaluate the sensitivity of our method to image noise and the presence of complex fiber configurations, and show its superior performance compared to alternative segmentation methods. Experiments on phantom and real data demonstrate the accuracy of the proposed method in segmenting simulated fibers, as well as white matter fiber tracts of clinical importance in the human brain. PMID:24108748
International Nuclear Information System (INIS)
Bogolubov, Nikolai N. Jr.; Prykarpatsky, Anatoliy K.
2006-12-01
The differential-geometric aspects of generalized de Rham-Hodge complexes naturally related with integrable multi-dimensional differential systems of M. Gromov type, as well as the geometric structure of Chern characteristic classes are studied. Special differential invariants of the Chern type are constructed, their importance for the integrability of multi-dimensional nonlinear differential systems on Riemannian manifolds is discussed. An example of the three-dimensional Davey-Stewartson type nonlinear strongly integrable differential system is considered, its Cartan type connection mapping and related Chern type differential invariants are analyzed. (author)
Riemannian geometry during the second half of the twentieth century
Berger, Marcel
1999-01-01
In the last fifty years of the twentieth century Riemannian geometry has exploded with activity. Berger marks the start of this period with Rauch's pioneering paper of 1951, which contains the first real pinching theorem and an amazing leap in the depth of the connection between geometry and topology. Since then, the field has become so rich that it is almost impossible for the uninitiated to find their way through it. Textbooks on the subject invariably must choose a particular approach, thus narrowing the path. In this book, Berger provides a truly remarkable survey of the main developments in Riemannian geometry in the last fifty years, focusing his main attention on the following five areas: Curvature and topology; the construction of and the classification of space forms; distinguished metrics, especially Einstein metrics; eigenvalues and eigenfunctions of the Laplacian; the study of periodic geodesics and the geodesic flow. Other topics are treated in less detail in a separate section. Berger's survey p...
On the concircular curvature tensor of Riemannian manifolds
International Nuclear Information System (INIS)
Rahman, M.S.; Lal, S.
1990-06-01
Definition of the concircular curvature tensor, Z hijk , along with Z-tensor, Z ij , is given and some properties of Z hijk are described. Tensors identical with Z hijk are shown. A necessary and sufficient condition that a Riemannian V n has zero Z-tensor is found. A number of theorems on concircular symmetric space, concircular recurrent space (Z n -space) and Z n -space with zero Z-tensor are deduced. (author). 6 refs
On Riemannian manifolds (Mn, g) of quasi-constant curvature
International Nuclear Information System (INIS)
Rahman, M.S.
1995-07-01
A Riemannian manifold (M n , g) of quasi-constant curvature is defined. It is shown that an (M n , g) in association with other class of manifolds gives rise, under certain conditions, to a manifold of quasi-constant curvature. Some observations on how a manifold of quasi-constant curvature accounts for a pseudo Ricci-symmetric manifold and quasi-umbilical hypersurface are made. (author). 10 refs
Isometric C1-immersions for pairs of Riemannian metrics
International Nuclear Information System (INIS)
D'Ambra, Giuseppina; Datta, Mahuya
2001-08-01
Let h 1 , h 2 be two Euclidean metrics on R q , and let V be a C ∞ -manifold endowed with two Riemannian metrics g 1 and g 2 . We study the existence of C 1 -immersions f:(V,g 1 ,g 2 )→(R q ,h 1 ,h 2 ) such that f*(h i )=g i for i=1,2. (author)
Chern-Simons as a geometrical set up for three dimensional gauge theories
International Nuclear Information System (INIS)
Lemes, V.E.R; Jesus, C. Linhares de; Sorella, S.P.; Villar, L.C.Q.; Ventura, O.S.
1997-12-01
Three dimensional Yang-Mills gauge theories in the presence of the Chern-Simons action are seen as being generated by the pure topological Chern-Simons term through nonlinear covariant redefinitions of the gauge field. (author)
Finsler geometry, relativity and gauge theories
International Nuclear Information System (INIS)
Asanov, G.S.
1985-01-01
This book provides a self-contained account of the Finslerian techniques which aim to synthesize the ideas of Finslerian metrical generalization of Riemannian geometry to merge with the primary physical concepts of general relativity and gauge field theories. The geometrization of internal symmetries in terms of Finslerian geometry, as well as the formulation of Finslerian generalization of gravitational field equations and equations of motion of matter, are two key points used to expound the techniques. The Clebsch representation of the canonical momentum field is used to formulate the Hamilton-Jacobi theory for homogeneous Lagrangians of classical mechanics. As an auxillary mathematical apparatus, the author uses invariance identities which systematically reflect the covariant properties of geometrical objects. The results of recent studies of special Finsler spaces are also applied. The book adds substantially to the mathematical monographs by Rund (1959) and Rund and Bear (1972), all basic results of the latter being reflected. It is the author's hope that thorough exploration of the materrial presented will tempt the reader to revise the habitual physical concepts supported conventionally by Riemannian geometry. (Auth.)
How the geometric calculus resolves the ordering ambiguity of quantum theory in curved space
International Nuclear Information System (INIS)
Pavsic, Matej
2003-01-01
The long standing problem of the ordering ambiguity in the definition of the Hamilton operator for a point particle in curved space is naturally resolved by using the powerful geometric calculus based on Clifford algebra. The momentum operator is defined to be the vector derivative (the gradient) multiplied by -i; it can be expanded in terms of basis vectors γ μ as p = -iγ μ ∂ μ . The product of two such operators is unambiguous, and such is the Hamiltonian which is just the d'Alembert operator in curved space; the curvature scalar term is not present in the Hamiltonian if we confine our consideration to scalar wavefunctions only. It is also shown that p is Hermitian and a self-adjoint operator: the presence of the basis vectors γ μ compensates the presence of √|g| in the matrix elements and in the scalar product. The expectation value of such an operator follows the classical geodetic line
A method of geometrical factors in the theory and interpretation of formation density logging
International Nuclear Information System (INIS)
Kozhevnikov, D.A.; Khathmullin, I.Ph.
1990-01-01
An interpretational model based on the ''radial geometrical factors concept'' is developed to describe the count-rate of a formation density logging (FDL) multi-spaced tool. The model includes two metrological parameters for each detector-source pair of a multi-spaced probe. These are: sensitivity to formation density, S, and radial sensitivity a. Apart from its universal application, the algorithm also allows some diagnoses of the intermediate zone to be made; that is, to reveal zones of consolidation and fracturing. It is shown that empirical algorithms realizing different forms of ''spine and ribs'' charts may be derived from the general algorithm. There is a practical possibility of resolving problems associated with the vicinity of the borehole wall by means of a triple-spaced FDL tool. It is given a corresponding algorithm and a metrological optimization procedure. The validity of the relations established is substantiated by physical measurements and by Monte-Carlo modelling. (author)
Geometrical theory to predict eccentric photorefraction intensity profiles in the human eye
Roorda, Austin; Campbell, Melanie C. W.; Bobier, W. R.
1995-08-01
In eccentric photorefraction, light returning from the retina of the eye is photographed by a camera focused on the eye's pupil. We use a geometrical model of eccentric photorefraction to generate intensity profiles across the pupil image. The intensity profiles for three different monochromatic aberration functions induced in a single eye are predicted and show good agreement with the measured eccentric photorefraction intensity profiles. A directional reflection from the retina is incorporated into the calculation. Intensity profiles for symmetric and asymmetric aberrations are generated and measured. The latter profile shows a dependency on the source position and the meridian. The magnitude of the effect of thresholding on measured pattern extents is predicted. Monochromatic aberrations in human eyes will cause deviations in the eccentric photorefraction measurements from traditional crescents caused by defocus and may cause misdiagnoses of ametropia or anisometropia. Our results suggest that measuring refraction along the vertical meridian is preferred for screening studies with the eccentric photorefractor.
Boundary Equations and Regularity Theory for Geometric Variational Systems with Neumann Data
Schikorra, Armin
2018-02-01
We study boundary regularity of maps from two-dimensional domains into manifolds which are critical with respect to a generic conformally invariant variational functional and which, at the boundary, intersect perpendicularly with a support manifold. For example, harmonic maps, or H-surfaces, with a partially free boundary condition. In the interior it is known, by the celebrated work of Rivière, that these maps satisfy a system with an antisymmetric potential, from which one can derive the interior regularity of the solution. Avoiding a reflection argument, we show that these maps satisfy along the boundary a system of equations which also exhibits a (nonlocal) antisymmetric potential that combines information from the interior potential and the geometric Neumann boundary condition. We then proceed to show boundary regularity for solutions to such systems.
Lie-Hamilton systems on curved spaces: a geometrical approach
Herranz, Francisco J.; de Lucas, Javier; Tobolski, Mariusz
2017-12-01
A Lie-Hamilton system is a nonautonomous system of first-order ordinary differential equations describing the integral curves of a t-dependent vector field taking values in a finite-dimensional Lie algebra, a Vessiot-Guldberg Lie algebra, of Hamiltonian vector fields relative to a Poisson structure. Its general solution can be written as an autonomous function, the superposition rule, of a generic finite family of particular solutions and a set of constants. We pioneer the study of Lie-Hamilton systems on Riemannian spaces (sphere, Euclidean and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes) as well as on semi-Riemannian spaces (Newtonian spacetimes). Their corresponding constants of motion and superposition rules are obtained explicitly in a geometric way. This work extends the (graded) contraction of Lie algebras to a contraction procedure for Lie algebras of vector fields, Hamiltonian functions, and related symplectic structures, invariants, and superposition rules.
Skorobogatiy, Maksim; Jacobs, Steven; Johnson, Steven; Fink, Yoel
2002-10-21
Perturbation theory formulation of Maxwell's equations gives a theoretically elegant and computationally efficient way of describing small imperfections and weak interactions in electro-magnetic systems. It is generally appreciated that due to the discontinuous field boundary conditions in the systems employing high dielectric contrast profiles standard perturbation formulations fail when applied to the problem of shifted material boundaries. In this paper we developed a novel coupled mode and perturbation theory formulations for treating generic non-uniform (varying along the direction of propagation) perturbations of a waveguide cross-section based on Hamiltonian formulation of Maxwell equations in curvilinear coordinates. We show that our formulation is accurate and rapidly converges to an exact result when used in a coupled mode theory framework even for the high index-contrast discontinuous dielectric profiles. Among others, our formulation allows for an efficient numerical evaluation of induced PMD due to a generic distortion of a waveguide profile, analysis of mode filters, mode converters and other optical elements such as strong Bragg gratings, tapers, bends etc., and arbitrary combinations of thereof. To our knowledge, this is the first time perturbation and coupled mode theories are developed to deal with arbitrary non-uniform profile variations in high index-contrast waveguides.
Directory of Open Access Journals (Sweden)
María Jesús Algar
2013-01-01
Full Text Available An efficient approach for the analysis of surface conformed reflector antennas fed arbitrarily is presented. The near field in a large number of sampling points in the aperture of the reflector is obtained applying the Geometrical Theory of Diffraction (GTD. A new technique named Master Points has been developed to reduce the complexity of the ray-tracing computations. The combination of both GTD and Master Points reduces the time requirements of this kind of analysis. To validate the new approach, several reflectors and the effects on the radiation pattern caused by shifting the feed and introducing different obstacles have been considered concerning both simple and complex geometries. The results of these analyses have been compared with the Method of Moments (MoM results.
Rahmat-Samii, Yahya
1986-01-01
Both offset and symmetric Cassegrain reflector antennas are used in satellite and ground communication systems. It is known that the subreflector diffraction can degrade the performance of these reflectors. A geometrical theory of diffraction/physical optics analysis technique is used to investigate the effects of the extended subreflector, beyond its optical rim, on the reflector efficiency and far-field patterns. Representative numerical results are shown for an offset Cassegrain reflector antenna with different feed illumination tapers and subreflector extensions. It is observed that for subreflector extensions as small as one wavelength, noticeable improvements in the overall efficiencies can be expected. Useful design data are generated for the efficiency curves and far-field patterns.
Geometric approach to the (BRS-) differential algebras of supersymmetric YM-theories
International Nuclear Information System (INIS)
Gieres, F.
1987-01-01
The (BRS-) differential algebra of susy YM-theories is defined in terms of superfields and forms on rigid U(N)-superspace. For d = 4 and N = 1.2 we show that it projects to the ''BRS-component field algebra in the WZ-gauge'' without any supergauge fixing. In this process the supergeometry is destroyed with the result that the final algebra becomes a prototype for a differential algebra which cannot be associated with an ordinary Lie algebra
Geometric methods in the elastic theory of membranes in liquid crystal phases
Ji Xing Liu; Yu Zhang Xie
1999-01-01
This book contains a comprehensive description of the mechanical equilibrium and deformation of membranes as a surface problem in differential geometry. Following the pioneering work by W Helfrich, the fluid membrane is seen as a nematic or smectic - A liquid crystal film and its elastic energy form is deduced exactly from the curvature elastic theory of the liquid crystals. With surface variation the minimization of the energy at fixed osmotical pressure and surface tension gives a completely new surface equation in geometry that involves potential interest in mathematics. The investigations
Do extended bodies move alon.o the geodesics of the Riemannian space-time
International Nuclear Information System (INIS)
Denisov, V.I.; Logunov, A.A.; Mestvirishvili, M.A.
1980-01-01
Motion of a massive self-gravitating body in the gravitational field of a distant massive source has been considered in the post-Newtonian approximation of the arbitrary metric gravitational theory. The comparison of the massive body center of mass acceleration with that of a point one, moving in Riemannian space-time, whose metrics formally is equivalent to the metrics of two moving massive bodies, makes it clear that in any metric gravitation theory, possessing energy-momentum conservation lows for matter and gravitational field, taken together, massive body does not move generally speaking along the geodesics of Riemannian space-time. Application of the obtained general formulae to the system Earth-Sun and using of the experimental results from lunar-laser-ranging has shown that the Earth during its motion along the orbit, oscillates with respect to the reference geodesic of the geometry with the period of 1 hour and the amplitude not less than 10 -2 cm, which is a post-Newtonian quantity. Therefore the deviation of the Earth motion from the geodesic may be observed in a relevant experiment, which will have a post-Newtonian accuracy. The difference in accelerations of the Earth c.m. and a prob body makes up 10 -7 in the post-Newtonian approximation from the value of the Earth acceleration. The ratio of the passive gravitational mass (defined according to Will) to the inertial mass for the Earth is not equal to unity, and differs from it by the value of approximately 10 -8
Geometric optics theory and design of astronomical optical systems using Mathematica
Romano, Antonio
2016-01-01
This text, now in its second edition, presents the mathematical background needed to design many optical combinations that are used in astronomical telescopes and cameras. It uses a novel approach to third-order aberration theory based on Fermat’s principle and the use of particular optical paths (called stigmatic paths) instead of rays, allowing for easier derivation of third-order formulae. Each optical combination analyzed is accompanied by a downloadable Mathematica® notebook that automates its third-order design, eliminating the need for lengthy calculations. The essential aspects of an optical system with an axis of rotational symmetry are introduced first, along with a development of Gaussian optics from Fermat’s principal. A simpler approach to third-order monochromatic aberrations based on both Fermat’s principle and stigmatic paths is then described, followed by a new chapter on fifth-order aberrations and their classification. Several specific optical devices are discussed and analyzed, incl...
Energy Technology Data Exchange (ETDEWEB)
Günaydin, Murat [Institute for Gravitation and the Cosmos, Physics Department,Pennsylvania State University, University Park, PA 16802 (United States); Lüst, Dieter [Arnold Sommerfeld Center for Theoretical Physics, Department für Physik, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München (Germany); Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, Föhringer Ring 6, 80805 München (Germany); Malek, Emanuel [Arnold Sommerfeld Center for Theoretical Physics, Department für Physik, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München (Germany)
2016-11-07
We propose a non-associative phase space algebra for M-theory backgrounds with locally non-geometric fluxes based on the non-associative algebra of octonions. Our proposal is based on the observation that the non-associative algebra of the non-geometric R-flux background in string theory can be obtained by a proper contraction of the simple Malcev algebra generated by imaginary octonions. Furthermore, by studying a toy model of a four-dimensional locally non-geometric M-theory background which is dual to a twisted torus, we show that the non-geometric background is “missing” a momentum mode. The resulting seven-dimensional phase space can thus be naturally identified with the imaginary octonions. This allows us to interpret the full uncontracted algebra of imaginary octonions as the uplift of the string theory R-flux algebra to M-theory, with the contraction parameter playing the role of the string coupling constant g{sub s}.
Boyde, Lars; Ekpenyong, Andrew; Whyte, Graeme; Guck, Jochen
2012-11-20
We present two electromagnetic frameworks to compare the surface stresses on spheroidal particles in the optical stretcher (a dual-beam laser trap that can be used to capture and deform biological cells). The first model is based on geometrical optics (GO) and limited in its applicability to particles that are much greater than the incident wavelength. The second framework is more sophisticated and hinges on the generalized Lorenz-Mie theory (GLMT). Despite the difference in complexity between both theories, the stress profiles computed with GO and GLMT are in good agreement with each other (relative errors are on the order of 1-10%). Both models predict a diminishing of the stresses for larger wavelengths and a strong increase of the stresses for shorter laser-cell distances. Results indicate that surface stresses on a spheroid with an aspect ratio of 1.2 hardly differ from the stresses on a sphere of similar size. Knowledge of the surface stresses and whether or not they redistribute during the stretching process is of crucial importance in real-time applications of the stretcher that aim to discern the viscoelastic properties of cells for purposes of cell characterization, sorting, and medical diagnostics.
International Nuclear Information System (INIS)
Rasolofoson, N.G.
2014-01-01
The properties of a physical system may vary significantly due to the presence of matter or energy. This change can be defined by the deformation of the space which is described as the variation of its curvature. In order to describe this law of physics, we have used differential geometry and studied especially a Schroedinger equation which describes a system evolving with time on a Riemannian manifold of constant curvature. Therefore, we have established and solved the Schroedinger equation using appropriate mathematics tools. As perspective, the study of string theory may be considered. [fr
On the de Rham–Wu decomposition for Riemannian and Lorentzian manifolds
International Nuclear Information System (INIS)
Galaev, Anton S
2014-01-01
It is explained how to find the de Rham decomposition of a Riemannian manifold and the Wu decomposition of a Lorentzian manifold. For that it is enough to find parallel symmetric bilinear forms on the manifold, and do some linear algebra. This result will allow to compute the connected holonomy group of an arbitrary Riemannian or Lorentzian manifold. (paper)
L2-Harmonic Forms on Incomplete Riemannian Manifolds with Positive Ricci Curvature
Directory of Open Access Journals (Sweden)
Junya Takahashi
2018-05-01
Full Text Available We construct an incomplete Riemannian manifold with positive Ricci curvature that has non-trivial L 2 -harmonic forms and on which the L 2 -Stokes theorem does not hold. Therefore, a Bochner-type vanishing theorem does not hold for incomplete Riemannian manifolds.
Energy Technology Data Exchange (ETDEWEB)
Kauder, K. (ed.)
2001-07-01
A method of automatic optimisation of screw rotors was investigated with the intention of obtaining a generally applicable method also for other types of energy conversion systems. As in turbomachinery engineering, compressors and engines are developing into separate fields. For example, while the filling process is a secondary consideration in compressors, it is a key element in screw-type engines. Some of the contribution discuss the new field of power supply in automotive fuel cells. [German] Das vorgelegte Heft 9 der Zeitschrift Schraubenmaschinen setzt mit neuen und fortgeschriebenen Beitraegen die 1993 begonnene Schriftenreihe mit dem Themenschwerpunkt Theorie fort. Erstmalig wird der Versuch unternommen, mit Hilfe unterschiedlicher Optimierungsverfahren in Zusammenarbeit mit der Informatik eine Methode zu entwickeln, mit deren Hilfe im ersten Schritt Schraubenrotoren automatisch optimiert werden koennen. Ziel ist es, dieses Verfahren auch auf andere Energiewandler anzuwenden. Wie im Turbomaschinenbau gehen Kompressoren und Motoren zunehmend getrennte Wege. Waehrend zum Beispiel der Fuellvorgang in Kompressoren eher eine untergeordnete Rolle spielt, wird er beim Schraubenmotor zu einem signifikanten Problem fuer die Gestaltung und Energiewandlung der gesamten Expansion. Kenngroessen beschreiben das vorhandene Optimierungspotential. Zu den neuen Anwendungen gehoert der zukuenftige Einsatz der Schraubenmaschine (Kompressor und Expander) fuer die Energieversorgung in automotiven Brennstoffzellen. Erste Ueberlegungen fuehren in dieses Thema vergleichend ein. (orig.)
DEFF Research Database (Denmark)
Andersen, Jørgen Ellegaard; Borot, Gaëtan; Orantin, Nicolas
We propose a general theory whose main component are functorial assignments ∑→Ω∑ ∈ E (∑), for a large class of functors E from a certain category of bordered surfaces (∑'s) to a suitable a target category of topological vector spaces. The construction is done by summing appropriate compositions...... as Poisson structures on the moduli space of flat connections. The theory has a wider scope than that and one expects that many functorial objects in low-dimensional geometry and topology should have a GR construction. The geometric recursion has various projections to topological recursion (TR) and we...... in particular show it retrieves all previous variants and applications of TR. We also show that, for any initial data for topological recursion, one can construct initial data for GR with values in Frobenius algebra-valued continuous functions on Teichmueller space, such that the ωg,n of TR are obtained...
Symmetry reduction for nonlinear wave equations in Riemannian and pseudo-Riemannian spaces
International Nuclear Information System (INIS)
Grundland, A.M.; Harnad, J.; Winternitz, P.
1984-01-01
The authors show how group theory can be systematically employed to reduce nonlinear partial differential equations in n independent variables to partial differential equations in fewer variables and in particular, to ordinary differential equations. (Auth.)
Relativistic theory of gravitation
International Nuclear Information System (INIS)
Logunov, A.A.; Mestvirishvili, M.A.
1986-01-01
In the present paper a relativistic theory of gravitation (RTG) is unambiguously constructed on the basis of the special relativity and geometrization principle. In this a gravitational field is treated as the Faraday--Maxwell spin-2 and spin-0 physical field possessing energy and momentum. The source of a gravitational field is the total conserved energy-momentum tensor of matter and of a gravitational field in Minkowski space. In the RTG the conservation laws are strictly fulfilled for the energy-moment and for the angular momentum of matter and a gravitational field. The theory explains the whole available set of experiments on gravity. By virtue of the geometrization principle, the Riemannian space in our theory is of field origin, since it appears as an effective force space due to the action of a gravitational field on matter. The RTG leads to an exceptionally strong prediction: The universe is not closed but just ''flat.'' This suggests that in the universe a ''missing mass'' should exist in a form of matter
Relativistic theory of gravitation
International Nuclear Information System (INIS)
Logunov, A.A.; Mestvirishvilli, M.A.
1985-01-01
In the present paper a relativistic theory of gravitation (RTG) is constructed in a unique way on the basis of the special relativity and geometrization principle. In this, a gravitational field is treated as the Faraday-Maxwell spin-2 and spin-0 physical field possessing energy and momentum. The source of a gravitational field is the total conserved energy-momentum tensor of matter and of a gravitational field in Minkowski space. In the RTG, the conservation laws are strictly fulfilled for the energy-momentum and for the angular momentum of matter and a gravitational field. The theory explains the whole available set of experiments on gravitation. In virtue of the geometrization principle, the Riemannian space in our theory is of field origin, since it appears as an effective force space due to the action of a gravitational field on matter. The RTg leads to an exceptionally strong prediction: The Universe is not closed but just ''flat''. This suggests that in the Universe a ''hidden mass'' should exist in some form of matter
Directory of Open Access Journals (Sweden)
Zoran Gligoric
2014-01-01
Full Text Available Underground mine projects are often associated with diverse sources of uncertainties. Having the ability to plan for these uncertainties plays a key role in the process of project evaluation and is increasingly recognized as critical to mining project success. To make the best decision, based on the information available, it is necessary to develop an adequate model incorporating the uncertainty of the input parameters. The model is developed on the basis of full discounted cash flow analysis of an underground zinc mine project. The relationships between input variables and economic outcomes are complex and often nonlinear. Fuzzy-interval grey system theory is used to forecast zinc metal prices while geometric Brownian motion is used to forecast operating costs over the time frame of the project. To quantify the uncertainty in the parameters within a project, such as capital investment, ore grade, mill recovery, metal content of concentrate, and discount rate, we have applied the concept of interval numbers. The final decision related to project acceptance is based on the net present value of the cash flows generated by the simulation over the time project horizon.
Color Texture Image Retrieval Based on Local Extrema Features and Riemannian Distance
Directory of Open Access Journals (Sweden)
Minh-Tan Pham
2017-10-01
Full Text Available A novel efficient method for content-based image retrieval (CBIR is developed in this paper using both texture and color features. Our motivation is to represent and characterize an input image by a set of local descriptors extracted from characteristic points (i.e., keypoints within the image. Then, dissimilarity measure between images is calculated based on the geometric distance between the topological feature spaces (i.e., manifolds formed by the sets of local descriptors generated from each image of the database. In this work, we propose to extract and use the local extrema pixels as our feature points. Then, the so-called local extrema-based descriptor (LED is generated for each keypoint by integrating all color, spatial as well as gradient information captured by its nearest local extrema. Hence, each image is encoded by an LED feature point cloud and Riemannian distances between these point clouds enable us to tackle CBIR. Experiments performed on several color texture databases including Vistex, STex, color Brodazt, USPtex and Outex TC-00013 using the proposed approach provide very efficient and competitive results compared to the state-of-the-art methods.
The Hodge theory of projective manifolds
de Cataldo, Mark Andrea
2007-01-01
This book is a written-up and expanded version of eight lectures on the Hodge theory of projective manifolds. It assumes very little background and aims at describing how the theory becomes progressively richer and more beautiful as one specializes from Riemannian, to Kähler, to complex projective manifolds. Though the proof of the Hodge Theorem is omitted, its consequences - topological, geometrical and algebraic - are discussed at some length. The special properties of complex projective manifolds constitute an important body of knowledge and readers are guided through it with the help of selected exercises. Despite starting with very few prerequisites, the concluding chapter works out, in the meaningful special case of surfaces, the proof of a special property of maps between complex projective manifolds, which was discovered only quite recently.
Quantum mechanics on Riemannian manifold in Schwinger's quantization approach II
International Nuclear Information System (INIS)
Chepilko, N.M.; Romanenko, A.V.
2001-01-01
The extended Schwinger quantization procedure is used for constructing quantum mechanics on a manifold with a group structure. The considered manifold M is a homogeneous Riemannian space with the given action of an isometry transformation group. Using the identification of M with the quotient space G/H, where H is the isotropy group of an arbitrary fixed point of M, we show that quantum mechanics on G/H possesses a gauge structure, described by a gauge potential that is the connection 1-form of the principal fiber bundle G(G/H, H). The coordinate representation of quantum mechanics and the procedure for selecting the physical sector of the states are developed. (orig.)
Spinorial Characterizations of Surfaces into 3-dimensional Pseudo-Riemannian Space Forms
International Nuclear Information System (INIS)
Lawn, Marie-Amélie; Roth, Julien
2011-01-01
We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. This generalizes a recent work of the first author for spacelike immersed Lorentzian surfaces in ℝ 2,1 to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well as for spacelike and timelike immersions of surfaces of signature (0, 2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.
Brandenburg, Jan Gerit; Alessio, Maristella; Civalleri, Bartolomeo; Peintinger, Michael F; Bredow, Thomas; Grimme, Stefan
2013-09-26
We extend the previously developed geometrical correction for the inter- and intramolecular basis set superposition error (gCP) to periodic density functional theory (DFT) calculations. We report gCP results compared to those from the standard Boys-Bernardi counterpoise correction scheme and large basis set calculations. The applicability of the method to molecular crystals as the main target is tested for the benchmark set X23. It consists of 23 noncovalently bound crystals as introduced by Johnson et al. (J. Chem. Phys. 2012, 137, 054103) and refined by Tkatchenko et al. (J. Chem. Phys. 2013, 139, 024705). In order to accurately describe long-range electron correlation effects, we use the standard atom-pairwise dispersion correction scheme DFT-D3. We show that a combination of DFT energies with small atom-centered basis sets, the D3 dispersion correction, and the gCP correction can accurately describe van der Waals and hydrogen-bonded crystals. Mean absolute deviations of the X23 sublimation energies can be reduced by more than 70% and 80% for the standard functionals PBE and B3LYP, respectively, to small residual mean absolute deviations of about 2 kcal/mol (corresponding to 13% of the average sublimation energy). As a further test, we compute the interlayer interaction of graphite for varying distances and obtain a good equilibrium distance and interaction energy of 6.75 Å and -43.0 meV/atom at the PBE-D3-gCP/SVP level. We fit the gCP scheme for a recently developed pob-TZVP solid-state basis set and obtain reasonable results for the X23 benchmark set and the potential energy curve for water adsorption on a nickel (110) surface.
International Nuclear Information System (INIS)
Khaneja, Navin; Brockett, Roger; Glaser, Steffen J.
2002-01-01
Radio-frequency pulses are used in nuclear-magnetic-resonance spectroscopy to produce unitary transfer of states. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation, and to optimize the sensitivity of the experiments. Many coherence-transfer experiments in NMR, involving a network of coupled spins, use temporary spin decoupling to produce desired effective Hamiltonians. In this paper, we demonstrate that significant time can be saved in producing an effective Hamiltonian if spin decoupling is avoided. We provide time-optimal pulse sequences for producing an important class of effective Hamiltonians in three-spin networks. These effective Hamiltonians are useful for coherence-transfer experiments in three-spin systems and implementation of indirect swap and Λ 2 (U) gates in the context of NMR quantum computing. It is shown that computing these time-optimal pulses can be reduced to geometric problems that involve computing sub-Riemannian geodesics. Using these geometric ideas, explicit expressions for the minimum time required for producing these effective Hamiltonians, transfer of coherence, and implementation of indirect swap gates, in a three-spin network are derived (Theorems 1 and 2). It is demonstrated that geometric control techniques provide a systematic way of finding time-optimal pulse sequences for transferring coherence and synthesizing unitary transformations in quantum networks, with considerable time savings (e.g., 42.3% for constructing indirect swap gates)
Relativistic gravitation theory
International Nuclear Information System (INIS)
Logunov, A.A.; Mestvirishvili, M.A.
1984-01-01
On the basis of the special relativity and geometrization principle a relativistic gravitation theory (RGT) is unambiguously constructed with the help of a notion of a gravitational field as a physical field in Faraday-Maxwell spirit, which posesses energy momentum and spins 2 and 0. The source of gravitation field is a total conserved energy-momentum tensor for matter and for gravitation field in Minkowski space. In the RGT conservation laws for the energy momentum and angular momentum of matter and gravitational field hold rigorously. The theory explains the whole set of gravitation experiments. Here, due to the geometrization principle the Riemannian space is of a field origin since this space arises effectively as a result of the gravitation field origin since this space arises effectively as a result of the gravitation field action on the matter. The RGT astonishing prediction is that the Universe is not closed but ''flat''. It means that in the Universe there should exist a ''missing'' mass in some form of matter
On discrete geometrodynamical theories in physics
International Nuclear Information System (INIS)
Towe, J.P.
1988-01-01
In this dissertation the author considers two topological-geometrical models (based upon a single suggestive formalism) in which a geometrodynamics is both feasible and pedagogically advantageous. Specifically he considers the topology which is constituted by the real domains of the two broad classes of rotation groups: those characterized by the commutator and anti-commutator algebras. He then adopts a Riemannian geometric structure and shows that the monistically geometric interpretation of this formalism restricts displacements on the proposed manifold to integral multiples of universal constant. Secondly, he demonstrates that in the context under consideration, this constraint affects a very interesting ontological reduction: the unification of quantum mechanics with a discrete, multidimensional extension of general relativity. A particularly interesting features of this unification is that is includes and requires the choice of an SL (2,R) direct-product SU (3)-symmetric realization of the proposed, generic formalism which is a lattice of spins ℎ and ℎ/2. If the vertices of this lattice are associated with the fundamental particles, then the resulting theory predicts and precludes the same interactions as the standard supersymmetry theory. In addition to the ontological reduction which is provided, and the restriction to supersymmetry, the proposed theory may also represent a scientifically useful extension of conventional theory in that it suggests a means of understanding the apparently large energy productions of the quasars and relates Planck's constant to the size of the universe
Directory of Open Access Journals (Sweden)
Feng Qi
2014-10-01
Full Text Available The authors find the absolute monotonicity and complete monotonicity of some functions involving trigonometric functions and related to estimates the lower bounds of the first eigenvalue of Laplace operator on Riemannian manifolds.
On some hypersurfaces with time like normal bundle in pseudo Riemannian space forms
International Nuclear Information System (INIS)
Kashani, S.M.B.
1995-12-01
In this work we classify immersed hypersurfaces with constant sectional curvature in pseudo Riemannian space forms if the normal bundle is time like and the mean curvature is constant. (author). 9 refs
Spinorial characterizations of surfaces into 3-dimensional psuedo-Riemannian space forms
Lawn , Marie-Amélie; Roth , Julien
2011-01-01
9 pages; We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in $\\mathbb{R}^{2,1}$ to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well ...
Steiner minimal trees in small neighbourhoods of points in Riemannian manifolds
Chikin, V. M.
2017-07-01
In contrast to the Euclidean case, almost no Steiner minimal trees with concrete boundaries on Riemannian manifolds are known. A result describing the types of Steiner minimal trees on a Riemannian manifold for arbitrary small boundaries is obtained. As a consequence, it is shown that for sufficiently small regular n-gons with n≥ 7 their boundaries without a longest side are Steiner minimal trees. Bibliography: 22 titles.
Razeto-Barry, Pablo; Díaz, Javier; Vásquez, Rodrigo A
2012-06-01
The general theories of molecular evolution depend on relatively arbitrary assumptions about the relative distribution and rate of advantageous, deleterious, neutral, and nearly neutral mutations. The Fisher geometrical model (FGM) has been used to make distributions of mutations biologically interpretable. We explored an FGM-based molecular model to represent molecular evolutionary processes typically studied by nearly neutral and selection models, but in which distributions and relative rates of mutations with different selection coefficients are a consequence of biologically interpretable parameters, such as the average size of the phenotypic effect of mutations and the number of traits (complexity) of organisms. A variant of the FGM-based model that we called the static regime (SR) represents evolution as a nearly neutral process in which substitution rates are determined by a dynamic substitution process in which the population's phenotype remains around a suboptimum equilibrium fitness produced by a balance between slightly deleterious and slightly advantageous compensatory substitutions. As in previous nearly neutral models, the SR predicts a negative relationship between molecular evolutionary rate and population size; however, SR does not have the unrealistic properties of previous nearly neutral models such as the narrow window of selection strengths in which they work. In addition, the SR suggests that compensatory mutations cannot explain the high rate of fixations driven by positive selection currently found in DNA sequences, contrary to what has been previously suggested. We also developed a generalization of SR in which the optimum phenotype can change stochastically due to environmental or physiological shifts, which we called the variable regime (VR). VR models evolution as an interplay between adaptive processes and nearly neutral steady-state processes. When strong environmental fluctuations are incorporated, the process becomes a selection model
Ruffino, Fabio Ferrari
2013-01-01
Given a cohomology theory, there is a well-known abstract way to define the dual homology theory using the theory of spectra. In [4] the author provides a more geometric construction of the homology theory, using a generalization of the bordism groups. Such a generalization involves in its definition the vector bundle modification, which is a particular case of the Gysin map. In this paper we provide a more natural variant of that construction, which replaces the vector bundle modification wi...
Li, Jing; Hong, Wenxue
2014-12-01
The feature extraction and feature selection are the important issues in pattern recognition. Based on the geometric algebra representation of vector, a new feature extraction method using blade coefficient of geometric algebra was proposed in this study. At the same time, an improved differential evolution (DE) feature selection method was proposed to solve the elevated high dimension issue. The simple linear discriminant analysis was used as the classifier. The result of the 10-fold cross-validation (10 CV) classification of public breast cancer biomedical dataset was more than 96% and proved superior to that of the original features and traditional feature extraction method.
Covariant Schrödinger semigroups on Riemannian manifolds
Güneysu, Batu
2017-01-01
This monograph discusses covariant Schrödinger operators and their heat semigroups on noncompact Riemannian manifolds and aims to fill a gap in the literature, given the fact that the existing literature on Schrödinger operators has mainly focused on scalar Schrödinger operators on Euclidean spaces so far. In particular, the book studies operators that act on sections of vector bundles. In addition, these operators are allowed to have unbounded potential terms, possibly with strong local singularities. The results presented here provide the first systematic study of such operators that is sufficiently general to simultaneously treat the natural operators from quantum mechanics, such as magnetic Schrödinger operators with singular electric potentials, and those from geometry, such as squares of Dirac operators that have smooth but endomorphism-valued and possibly unbounded potentials. The book is largely self-contained, making it accessible for graduate and postgraduate students alike. Since it also inc...
Haddout, Soufiane
2018-01-01
The equations of motion of a bicycle are highly nonlinear and rolling of wheels without slipping can only be expressed by nonholonomic constraint equations. A geometrical theory of general nonholonomic constrained systems on fibered manifolds and their jet prolongations, based on so-called Chetaev-type constraint forces, was proposed and developed in the last decade by O. Krupková (Rossi) in 1990's. Her approach is suitable for study of all kinds of mechanical systems-without restricting to Lagrangian, time-independent, or regular ones, and is applicable to arbitrary constraints (holonomic, semiholonomic, linear, nonlinear or general nonholonomic). The goal of this paper is to apply Krupková's geometric theory of nonholonomic mechanical systems to study a concrete problem in nonlinear nonholonomic dynamics, i.e., autonomous bicycle. The dynamical model is preserved in simulations in its original nonlinear form without any simplifying. The results of numerical solutions of constrained equations of motion, derived within the theory, are in good agreement with measurements and thus they open the possibility of direct application of the theory to practical situations.
Divergence theorem for symmetric (0,2)-tensor fields on a semi-Riemannian manifold with boundary
International Nuclear Information System (INIS)
Ezin, J.P.; Mouhamadou Hassirou; Tossa, J.
2005-08-01
We prove in this paper a divergence theorem for symmetric (0,2)-tensors on a semi-Riemannian manifold with boundary. As a consequence we establish the complete divergence theorem on a semi-Riemannian manifold with any kinds of smooth boundaries. This result contains the previous attempts to write this theorem on a semi-Riemannian manifold as Unal results. A vanishing theorem for gradient timelike Killing vector fields on Einstein semi-Riemannian manifolds is obtained. As a tool, an induced volume form is defined for a degenerate boundary by using a star like operator that we define on degenerate submanifolds. (author)
On Certain Conceptual Anomalies in Einstein's Theory of Relativity
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Crothers S. J.
2008-01-01
Full Text Available There are a number of conceptual anomalies occurring in the Standard exposition of Einstein's Theory of Relativity. These anomalies relate to issues in both mathematics and in physics and penetrate to the very heart of Einstein's theory. This paper reveals and amplifies a few such anomalies, including the fact that Einstein's field equations for the so-called static vacuum configuration, $R_{mu u} = 0$, violates his Principle of Equivalence, and is therefore erroneous. This has a direct bearing on the usual concept of conservation of energy for the gravitational field and the conventional formulation for localisation of energy using Einstein's pseudo-tensor. Misconceptions as to the relationship between Minkowski spacetime and Special Relativity are also discussed, along with their relationships to the pseudo-Riemannian metric manifold of Einstein's gravitational field, and their fundamental geometric structures pertaining to spherical symmetry.
On Certain Conceptual Anomalies in Einstein's Theory of Relativity
Directory of Open Access Journals (Sweden)
Crothers S. J.
2008-01-01
Full Text Available There are a number of conceptual anomalies occurring in the Standard exposition of Einstein’s Theory of Relativity. These anomalies relate to issues in both mathematics and in physics and penetrate to the very heart of Einstein’s theory. This paper reveals and amplifies a few such anomalies, including the fact that Einstein’s field equations for the so-called static vacuum configuration, R = 0 , violates his Principle of Equiv- alence, and is therefore erroneous. This has a direct bearing on the usual concept of conservation of energy for the gravitational field and the conventional formulation for localisation of energy using Einstein’s pseudo-tensor. Misconceptions as to the relationship between Minkowski spacetime and Special Relativity are also discussed, along with their relationships to the pseudo-Riemannian metric manifold of Einstein’s gravitational field, and their fundamental geometric structures pertaining to spherical symmetry.
Bilinear Regularized Locality Preserving Learning on Riemannian Graph for Motor Imagery BCI.
Xie, Xiaofeng; Yu, Zhu Liang; Gu, Zhenghui; Zhang, Jun; Cen, Ling; Li, Yuanqing
2018-03-01
In off-line training of motor imagery-based brain-computer interfaces (BCIs), to enhance the generalization performance of the learned classifier, the local information contained in test data could be used to improve the performance of motor imagery as well. Further considering that the covariance matrices of electroencephalogram (EEG) signal lie on Riemannian manifold, in this paper, we construct a Riemannian graph to incorporate the information of training and test data into processing. The adjacency and weight in Riemannian graph are determined by the geodesic distance of Riemannian manifold. Then, a new graph embedding algorithm, called bilinear regularized locality preserving (BRLP), is derived upon the Riemannian graph for addressing the problems of high dimensionality frequently arising in BCIs. With a proposed regularization term encoding prior information of EEG channels, the BRLP could obtain more robust performance. Finally, an efficient classification algorithm based on extreme learning machine is proposed to perform on the tangent space of learned embedding. Experimental evaluations on the BCI competition and in-house data sets reveal that the proposed algorithms could obtain significantly higher performance than many competition algorithms after using same filter process.
Directory of Open Access Journals (Sweden)
Frédéric Barbaresco
2016-11-01
Full Text Available We introduce the symplectic structure of information geometry based on Souriau’s Lie group thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its moment space, defining physical observables like energy, heat, and moment as pure geometrical objects. Using geometric Planck temperature of Souriau model and symplectic cocycle notion, the Fisher metric is identified as a Souriau geometric heat capacity. The Souriau model is based on affine representation of Lie group and Lie algebra that we compare with Koszul works on G/K homogeneous space and bijective correspondence between the set of G-invariant flat connections on G/K and the set of affine representations of the Lie algebra of G. In the framework of Lie group thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables, and a new variational principal for thermodynamics is built through an invariant Poincaré-Cartan-Souriau integral. The Souriau-Fisher metric is linked to KKS (Kostant–Kirillov–Souriau 2-form that associates a canonical homogeneous symplectic manifold to the co-adjoint orbits. We apply this model in the framework of information geometry for the action of an affine group for exponential families, and provide some illustrations of use cases for multivariate gaussian densities. Information geometry is presented in the context of the seminal work of Fréchet and his Clairaut-Legendre equation. The Souriau model of statistical physics is validated as compatible with the Balian gauge model of thermodynamics. We recall the precursor work of Casalis on affine group invariance for natural exponential families.
Energy Technology Data Exchange (ETDEWEB)
Ebata, T [Tohoku Univ., Sendai (Japan). Coll. of General Education
1976-06-01
The geometrical distribution inferred from the inelastic cross section is assumed to be proportional to the partial waves. The precocious scaling and the Q/sup 2/-dependence of various quantities are treated from the geometrical point of view. It is shown that the approximate conservation of the orbital angular momentum may be a very practical rule to understand the helicity structure of various hadronic and electromagnetic reactions. The rule can be applied to inclusive reactions as well. The model is also applied to large angle processes. Through the discussion, it is suggested that many peculiar properties of the quark-parton can be ascribed to the geometrical effects.
Directory of Open Access Journals (Sweden)
Uwe Schneider
Full Text Available When fractionation schemes for hypofractionation and stereotactic body radiotherapy are considered, a reliable cell survival model at high dose is needed for calculating doses of similar biological effectiveness. An alternative to the LQ-model is the track-event theory which is based on the probabilities for one- and two two-track events. A one-track-event (OTE is always represented by at least two simultaneous double strand breaks. A two-track-event (TTE results in one double strand break. Therefore at least two two-track-events on the same or different chromosomes are necessary to produce an event which leads to cell sterilization. It is obvious that the probabilities of OTEs and TTEs must somehow depend on the geometrical structure of the chromatin. In terms of the track-event theory the ratio ε of the probabilities of OTEs and TTEs includes the geometrical dependence and is obtained in this work by simple Monte Carlo simulations.For this work it was assumed that the anchors of loop forming chromatin are most sensitive to radiation induced cell deaths. Therefore two adjacent tetranucleosomes representing the loop anchors were digitized. The probability ratio ε of OTEs and TTEs was factorized into a radiation quality dependent part and a geometrical part: ε = εion ∙ εgeo. εgeo was obtained for two situations, by applying Monte Carlo simulation for DNA on the tetranucleosomes itself and for linker DNA. Low energy electrons were represented by randomly distributed ionizations and high energy electrons by ionizations which were simulated on rays. εion was determined for electrons by using results from nanodosimetric measurements. The calculated ε was compared to the ε obtained from fits of the track event model to 42 sets of experimental human cell survival data.When the two tetranucleosomes are in direct contact and the hits are randomly distributed εgeo and ε are 0.12 and 0.85, respectively. When the hits are simulated on rays
Castro, C
2004-01-01
We construct the Extended Relativity Theory in Born-Clifford-Phase spaces with an upper and lower length scales (infrared/ultraviolet cutoff). The invariance symmetry leads naturally to the real Clifford algebra Cl (2, 6, R ) and complexified Clifford Cl_C ( 4 ) algebra related to Twistors. We proceed with an extensive review of Smith's 8D model based on the Clifford algebra Cl ( 1 ,7) that reproduces at low energies the physics of the Standard Model and Gravity; including the derivation of all the coupling constants, particle masses, mixing angles, ....with high precision. Further results by Smith are discussed pertaining the interplay among Clifford, Jordan, Division and Exceptional Lie algebras within the hierarchy of dimensions D = 26, 27, 28 related to bosonic string, M, F theory. Two Geometric actions are presented like the Clifford-Space extension of Maxwell's Electrodynamics, Brandt's action related the 8D spacetime tangent-bundle involving coordinates and velocities (Finsler geometries) followed by a...
The Green-Schwarz mechanism and geometric anomaly relations in 2d (0,2) F-theory vacua
Weigand, Timo; Xu, Fengjun
2018-04-01
We study the structure of gauge and gravitational anomalies in 2d N = (0 , 2) theories obtained by compactification of F-theory on elliptically fibered Calabi-Yau 5-folds. Abelian gauge anomalies, induced at 1-loop in perturbation theory, are cancelled by a generalized Green-Schwarz mechanism operating at the level of chiral scalar fields in the 2d supergravity theory. We derive closed expressions for the gravitational and the non-abelian and abelian gauge anomalies including the Green-Schwarz counterterms. These expressions involve topological invariants of the underlying elliptic fibration and the gauge background thereon. Cancellation of anomalies in the effective theory predicts intricate topological identities which must hold on every elliptically fibered Calabi-Yau 5-fold. We verify these relations in a non-trivial example, but their proof from a purely mathematical perspective remains as an interesting open problem. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of anomalies in 6d N = (1 , 0) and 4d N = 1 theories obtained from F-theory.
Gauging of 1D-space translations for nonrelativistic matter - Geometric bags
International Nuclear Information System (INIS)
Stichel, P.C.
2000-01-01
We develop in a systematic fashion the idea of gauging 1D-space translations with fixed Newtonian time for nonrelativistic matter (particles and fields). By starting with a nonrelativistic free theory we obtain its minimal gauge invariant extension by introducing two gauge fields with a Maxwellian self interaction. We fix the gauge so that the residual symmetry group is the Galilei group and construct a representation of the extended Galilei algebra. The reduced N-particle Lagrangian describes geodesic motion in a (N-1)-dimensional (Pseudo-) Riemannian space. The singularity of the metric for negative gauge coupling leads in classical dynamics to the formation of geometric bags in the case of two or three particles. The ordering problem within the quantization scheme for N-particles is solved by canonical quantization of a pseudoclassical Schroedinger theory obtained by adding to the continuum generalization of the point-particle Lagrangian an appropriate quantum correction. We solve the two-particle bound state problem for both signs of the gauge coupling. At the end we speculate on the possible physical relevance of the new interaction induced by the gauge fields
Nonlinear Methods in Riemannian and Kählerian Geometry
Jost, Jürgen
1991-01-01
In this book, I present an expanded version of the contents of my lectures at a Seminar of the DMV (Deutsche Mathematiker Vereinigung) in Düsseldorf, June, 1986. The title "Nonlinear methods in complex geometry" already indicates a combination of techniques from nonlinear partial differential equations and geometric concepts. In older geometric investigations, usually the local aspects attracted more attention than the global ones as differential geometry in its foundations provides approximations of local phenomena through infinitesimal or differential constructions. Here, all equations are linear. If one wants to consider global aspects, however, usually the presence of curvature Ieads to a nonlinearity in the equations. The simplest case is the one of geodesics which are described by a system of second ordernonlinear ODE; their linearizations are the Jacobi fields. More recently, nonlinear PDE played a more and more pro~inent röle in geometry. Let us Iist some of the most important ones: - harmonic maps ...
Niethammer, Marc; Hart, Gabriel L; Pace, Danielle F; Vespa, Paul M; Irimia, Andrei; Van Horn, John D; Aylward, Stephen R
2011-01-01
Standard image registration methods do not account for changes in image appearance. Hence, metamorphosis approaches have been developed which jointly estimate a space deformation and a change in image appearance to construct a spatio-temporal trajectory smoothly transforming a source to a target image. For standard metamorphosis, geometric changes are not explicitly modeled. We propose a geometric metamorphosis formulation, which explains changes in image appearance by a global deformation, a deformation of a geometric model, and an image composition model. This work is motivated by the clinical challenge of predicting the long-term effects of traumatic brain injuries based on time-series images. This work is also applicable to the quantification of tumor progression (e.g., estimating its infiltrating and displacing components) and predicting chronic blood perfusion changes after stroke. We demonstrate the utility of the method using simulated data as well as scans from a clinical traumatic brain injury patient.
The implicit function theorem history, theory, and applications
Krantz, Steven G
2003-01-01
The implicit function theorem is part of the bedrock of mathematics analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis. There are many different forms of the implicit function theorem, including (i) the classical formulation for Ck functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth function, (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash-Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present volume. The history of the implicit function theorem is a lively and complex store, and intimately bound up with the development of fundamental ideas in a...
Construction of harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces
International Nuclear Information System (INIS)
Konderak, J.
1988-09-01
Defined here is an orthogonal multiplication for vector spaces with indefinite nondegenerate scalar product. This is then used, via the Hopf construction, to obtain harmonic maps between pseudo-Riemannian spheres and hyperbolic spaces. Examples of harmonic maps are constructed using Clifford algebras. (author). 6 refs
Frè, Pietro Giuseppe
2013-01-01
‘Gravity, a Geometrical Course’ presents general relativity (GR) in a systematic and exhaustive way, covering three aspects that are homogenized into a single texture: i) the mathematical, geometrical foundations, exposed in a self consistent contemporary formalism, ii) the main physical, astrophysical and cosmological applications, updated to the issues of contemporary research and observations, with glimpses on supergravity and superstring theory, iii) the historical development of scientific ideas underlying both the birth of general relativity and its subsequent evolution. The book is divided in two volumes. Volume One is dedicated to the development of the theory and basic physical applications. It guides the reader from the foundation of special relativity to Einstein field equations, illustrating some basic applications in astrophysics. A detailed account of the historical and conceptual development of the theory is combined with the presentation of its mathematical foundations. Differe...
Statistics on Lie groups: A need to go beyond the pseudo-Riemannian framework
Miolane, Nina; Pennec, Xavier
2015-01-01
Lie groups appear in many fields from Medical Imaging to Robotics. In Medical Imaging and particularly in Computational Anatomy, an organ's shape is often modeled as the deformation of a reference shape, in other words: as an element of a Lie group. In this framework, if one wants to model the variability of the human anatomy, e.g. in order to help diagnosis of diseases, one needs to perform statistics on Lie groups. A Lie group G is a manifold that carries an additional group structure. Statistics on Riemannian manifolds have been well studied with the pioneer work of Fréchet, Karcher and Kendall [1, 2, 3, 4] followed by others [5, 6, 7, 8, 9]. In order to use such a Riemannian structure for statistics on Lie groups, one needs to define a Riemannian metric that is compatible with the group structure, i.e a bi-invariant metric. However, it is well known that general Lie groups which cannot be decomposed into the direct product of compact and abelian groups do not admit a bi-invariant metric. One may wonder if removing the positivity of the metric, thus asking only for a bi-invariant pseudo-Riemannian metric, would be sufficient for most of the groups used in Computational Anatomy. In this paper, we provide an algorithmic procedure that constructs bi-invariant pseudo-metrics on a given Lie group G. The procedure relies on a classification theorem of Medina and Revoy. However in doing so, we prove that most Lie groups do not admit any bi-invariant (pseudo-) metric. We conclude that the (pseudo-) Riemannian setting is not the richest setting if one wants to perform statistics on Lie groups. One may have to rely on another framework, such as affine connection space.
Fluid mechanics a geometrical point of view
Rajeev, S G
2018-01-01
Fluid Mechanics: A Geometrical Point of View emphasizes general principles of physics illustrated by simple examples in fluid mechanics. Advanced mathematics (e.g., Riemannian geometry and Lie groups) commonly used in other parts of theoretical physics (e.g. General Relativity or High Energy Physics) are explained and applied to fluid mechanics. This follows on from the author's book Advanced Mechanics (Oxford University Press, 2013). After introducing the fundamental equations (Euler and Navier-Stokes), the book provides particular cases: ideal and viscous flows, shocks, boundary layers, instabilities, and transients. A restrained look at integrable systems (KdV) leads into a formulation of an ideal fluid as a hamiltonian system. Arnold's deep idea, that the instability of a fluid can be understood using the curvature of the diffeomorphism group, will be explained. Leray's work on regularity of Navier-Stokes solutions, and the modern developments arising from it, will be explained in language for physicists...
Information geometric methods for complexity
Felice, Domenico; Cafaro, Carlo; Mancini, Stefano
2018-03-01
Research on the use of information geometry (IG) in modern physics has witnessed significant advances recently. In this review article, we report on the utilization of IG methods to define measures of complexity in both classical and, whenever available, quantum physical settings. A paradigmatic example of a dramatic change in complexity is given by phase transitions (PTs). Hence, we review both global and local aspects of PTs described in terms of the scalar curvature of the parameter manifold and the components of the metric tensor, respectively. We also report on the behavior of geodesic paths on the parameter manifold used to gain insight into the dynamics of PTs. Going further, we survey measures of complexity arising in the geometric framework. In particular, we quantify complexity of networks in terms of the Riemannian volume of the parameter space of a statistical manifold associated with a given network. We are also concerned with complexity measures that account for the interactions of a given number of parts of a system that cannot be described in terms of a smaller number of parts of the system. Finally, we investigate complexity measures of entropic motion on curved statistical manifolds that arise from a probabilistic description of physical systems in the presence of limited information. The Kullback-Leibler divergence, the distance to an exponential family and volumes of curved parameter manifolds, are examples of essential IG notions exploited in our discussion of complexity. We conclude by discussing strengths, limits, and possible future applications of IG methods to the physics of complexity.
Geometrical aspects of quantum spaces
International Nuclear Information System (INIS)
Ho, P.M.
1996-01-01
Various geometrical aspects of quantum spaces are presented showing the possibility of building physics on quantum spaces. In the first chapter the authors give the motivations for studying noncommutative geometry and also review the definition of a Hopf algebra and some general features of the differential geometry on quantum groups and quantum planes. In Chapter 2 and Chapter 3 the noncommutative version of differential calculus, integration and complex structure are established for the quantum sphere S 1 2 and the quantum complex projective space CP q (N), on which there are quantum group symmetries that are represented nonlinearly, and are respected by all the aforementioned structures. The braiding of S q 2 and CP q (N) is also described. In Chapter 4 the quantum projective geometry over the quantum projective space CP q (N) is developed. Collinearity conditions, coplanarity conditions, intersections and anharmonic ratios is described. In Chapter 5 an algebraic formulation of Reimannian geometry on quantum spaces is presented where Riemannian metric, distance, Laplacian, connection, and curvature have their quantum counterparts. This attempt is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective space and the two-sheeted space. The quantum group of general coordinate transformations on some quantum spaces is also given
Geometric approximation algorithms
Har-Peled, Sariel
2011-01-01
Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.
Geometrical optical illusionists.
Wade, Nicholas J
2014-01-01
Geometrical optical illusions were given this title by Oppel in 1855. Variants on such small distortions of visual space were illustrated thereafter, many of which bear the names of those who first described them. Some original forms of the geometrical optical illusions are shown together with 'perceptual portraits' of those who described them. These include: Roget, Chevreul, Fick, Zöllner, Poggendorff, Hering, Kundt, Delboeuf Mach, Helmholtz, Hermann, von Bezold, Müller-Lyer, Lipps, Thiéry, Wundt, Münsterberg, Ebbinghaus, Titchener, Ponzo, Luckiesh, Sander, Ehrenstein, Gregory, Heard, White, Shepard, and. Lingelbach. The illusions are grouped under the headings of orientation, size, the combination of size and orientation, and contrast. Early theories of illusions, before geometrical optical illusions were so named, are mentioned briefly.
Directory of Open Access Journals (Sweden)
Hamid M. Sedighi
Full Text Available This paper investigates the dynamic pull-in instability of vibrating micro-beams undergoing large deflection under electrosatically actuation. The governing equation of motion is derived based on the modified couple stress theory. Homotopy Perturbation Method is employed to produce the high accuracy approximate solution as well as the second-order frequency- amplitude relationship. The nonlinear governing equation of micro beam vibrations predeformed by an electric field includes both even and odd nonlinearities. The influences of basic non-dimensional parameters on the pull-in instability as well as the natural frequency are studied. It is demonstrated that two terms in series expansions are sufficient to produce high accuracy solution of the micro-structure. The accuracy of proposed asymptotic approach is validated via numerical results. The phase portrait of the system exhibits periodic and homoclinic orbits.
Problem of energy-momentum and theory of gravitation
International Nuclear Information System (INIS)
Logunov, A.A.; Folomeshkin, V.N.
1977-01-01
General properties of geometrised theories of gravitation are considered. Covariant formulation of conservation laws in arbitrary riemannian space-time is given. In the Einstein theory the symmetric as well as canonical energy-momentum tensor of the system ''matter plus gravitational field'' and in particular, the energy-momentum of free gravitational waves, turns out to be equal to zero. To understand the origin of the problems and difficulties concerning the energy-momentum in the Einstein theory, the gravitational filed is considered in the usual framework of the Lorentz invariant field theory, just like any other physical field. Combination of the approach proposed with the Einstein's idea of geometrization makes it possible to formulate the geometrised gravitation theory, in which there are no inner contradictions, the energy-momentum of gravitational field is defined precisely and all the known experimental facts are described successfully. For strong gravitational fields the predictions of the quasilinear geometrised theory under consideration are different from those of the gravitational theory in the Einstein formulation. Black holes are absent in the theory. Evaluation of the energy-flux of gravitational waves leads to unambiguous results and shows that the gravitational waves transfer the positive-definite energy
Nonlinear closure relations theory for transport processes in nonequilibrium systems
International Nuclear Information System (INIS)
Sonnino, Giorgio
2009-01-01
A decade ago, a macroscopic theory for closure relations has been proposed for systems out of Onsager's region. This theory is referred to as the thermodynamic field theory (TFT). The aim of this work was to determine the nonlinear flux-force relations that respect the thermodynamic theorems for systems far from equilibrium. We propose a formulation of the TFT where one of the basic restrictions, namely, the closed-form solution for the skew-symmetric piece of the transport coefficients, has been removed. In addition, the general covariance principle is replaced by the De Donder-Prigogine thermodynamic covariance principle (TCP). The introduction of TCP requires the application of an appropriate mathematical formalism, which is referred to as the entropy-covariant formalism. By geometrical arguments, we prove the validity of the Glansdorff-Prigogine universal criterion of evolution. A new set of closure equations determining the nonlinear corrections to the linear ('Onsager') transport coefficients is also derived. The geometry of the thermodynamic space is non-Riemannian. However, it tends to be Riemannian for high values of the entropy production. In this limit, we recover the transport equations found by the old theory. Applications of our approach to transport in magnetically confined plasmas, materials submitted to temperature, and electric potential gradients or to unimolecular triangular chemical reactions can be found at references cited herein. Transport processes in tokamak plasmas are of particular interest. In this case, even in the absence of turbulence, the state of the plasma remains close to (but, it is not in) a state of local equilibrium. This prevents the transport relations from being linear.
Discrete Curvature Theories and Applications
Sun, Xiang
2016-08-25
Discrete Di erential Geometry (DDG) concerns discrete counterparts of notions and methods in di erential geometry. This thesis deals with a core subject in DDG, discrete curvature theories on various types of polyhedral surfaces that are practically important for free-form architecture, sunlight-redirecting shading systems, and face recognition. Modeled as polyhedral surfaces, the shapes of free-form structures may have to satisfy di erent geometric or physical constraints. We study a combination of geometry and physics { the discrete surfaces that can stand on their own, as well as having proper shapes for the manufacture. These proper shapes, known as circular and conical meshes, are closely related to discrete principal curvatures. We study curvature theories that make such surfaces possible. Shading systems of freeform building skins are new types of energy-saving structures that can re-direct the sunlight. From these systems, discrete line congruences across polyhedral surfaces can be abstracted. We develop a new curvature theory for polyhedral surfaces equipped with normal congruences { a particular type of congruences de ned by linear interpolation of vertex normals. The main results are a discussion of various de nitions of normality, a detailed study of the geometry of such congruences, and a concept of curvatures and shape operators associated with the faces of a triangle mesh. These curvatures are compatible with both normal congruences and the Steiner formula. In addition to architecture, we consider the role of discrete curvatures in face recognition. We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold, which is an extension of the classical notion of asymptotic directions. We get a simple expression of these cones for polyhedral surfaces, as well as convergence and approximation theorems. We use the asymptotic cones as facial descriptors and demonstrate the
Johnstone, PT
2014-01-01
Focusing on topos theory's integration of geometric and logical ideas into the foundations of mathematics and theoretical computer science, this volume explores internal category theory, topologies and sheaves, geometric morphisms, other subjects. 1977 edition.
Existence of parallel spinors on non-simply-connected Riemannian manifolds
International Nuclear Information System (INIS)
McInnes, B.
1997-04-01
It is well known, and important for applications, that Ricci-flat Riemannian manifolds of non-generic holonomy always admit a parallel [covariant constant] spinor if they are simply connected. The non-simply-connected case is much more subtle, however. We show that a parallel spinor can still be found in this case provided that the [real] dimension is not a multiple of four, and provided that the spin structure is carefully chosen. (author). 10 refs
International Nuclear Information System (INIS)
Saveliev, M.V.
1983-01-01
In the framework of the algebraic approach a construction of exactly integrable two-dimensional Riemannian manifolds embedded into enveloping Euclidean (pseudo-Euclidean) space Rsub(N) of an arbitrary dimension is presented. The construction is based on a reformulation of the Gauss, Peterson-Codazzi and Ricci equations in the form of a Lax-type representation in two-dimensional space. Here the Lax pair operators take the values in algebra SO(N)
Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group
Ardentov, Andrei A.; Sachkov, Yuri L.
2017-12-01
We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations R+ and a discrete group of reflections Z2 × Z2 × Z2. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i. e., the set of minimizers for each terminal point in the Engel group.
From lattice BF gauge theory to area-angle Regge calculus
International Nuclear Information System (INIS)
Bonzom, Valentin
2009-01-01
We consider Riemannian 4D BF lattice gauge theory, on a triangulation of spacetime. Introducing the simplicity constraints which turn BF theory into simplicial gravity, some geometric quantities of Regge calculus, areas, and 3D and 4D dihedral angles, are identified. The parallel transport conditions are taken care of to ensure a consistent gluing of simplices. We show that these gluing relations, together with the simplicity constraints, contain the constraints of area-angle Regge calculus in a simple way, via the group structure of the underlying BF gauge theory. This provides a precise road from constrained BF theory to area-angle Regge calculus. Doing so, a framework combining variables of lattice BF theory and Regge calculus is built. The action takes a form a la Regge and includes the contribution of the Immirzi parameter. In the absence of simplicity constraints, the standard spin foam model for BF theory is recovered. Insertions of local observables are investigated, leading to Casimir insertions for areas and reproducing for 3D angles known results obtained through angle operators on spin networks. The present formulation is argued to be suitable for deriving spin foam models from discrete path integrals and to unravel their geometric content.
Directory of Open Access Journals (Sweden)
Fan Yang
2015-07-01
Full Text Available Normally, polarimetric SAR classification is a high-dimensional nonlinear mapping problem. In the realm of pattern recognition, sparse representation is a very efficacious and powerful approach. As classical descriptors of polarimetric SAR, covariance and coherency matrices are Hermitian semidefinite and form a Riemannian manifold. Conventional Euclidean metrics are not suitable for a Riemannian manifold, and hence, normal sparse representation classification cannot be applied to polarimetric SAR directly. This paper proposes a new land cover classification approach for polarimetric SAR. There are two principal novelties in this paper. First, a Stein kernel on a Riemannian manifold instead of Euclidean metrics, combined with sparse representation, is employed for polarimetric SAR land cover classification. This approach is named Stein-sparse representation-based classification (SRC. Second, using simultaneous sparse representation and reasonable assumptions of the correlation of representation among different frequency bands, Stein-SRC is generalized to simultaneous Stein-SRC for multi-frequency polarimetric SAR classification. These classifiers are assessed using polarimetric SAR images from the Airborne Synthetic Aperture Radar (AIRSAR sensor of the Jet Propulsion Laboratory (JPL and the Electromagnetics Institute Synthetic Aperture Radar (EMISAR sensor of the Technical University of Denmark (DTU. Experiments on single-band and multi-band data both show that these approaches acquire more accurate classification results in comparison to many conventional and advanced classifiers.
Tran, Van Tan; Nguyen, Minh Thao; Tran, Quoc Tri
2017-10-12
Density functional theory and the multiconfigurational CASSCF/CASPT2 method have been employed to study the low-lying states of VGe n -/0 (n = 1-4) clusters. For VGe -/0 and VGe 2 -/0 clusters, the relative energies and geometrical structures of the low-lying states are reported at the CASSCF/CASPT2 level. For the VGe 3 -/0 and VGe 4 -/0 clusters, the computational results show that due to the large contribution of the Hartree-Fock exact exchange, the hybrid B3LYP, B3PW91, and PBE0 functionals overestimate the energies of the high-spin states as compared to the pure GGA BP86 and PBE functionals and the CASPT2 method. On the basis of the pure GGA BP86 and PBE functionals and the CASSCF/CASPT2 results, the ground states of anionic and neutral clusters are defined, the relative energies of the excited states are computed, and the electron detachment energies of the anionic clusters are evaluated. The computational results are employed to give new assignments for all features in the photoelectron spectra of VGe 3 - and VGe 4 - clusters.
Ingram, Kieran I M; Häller, L Jonas L; Kaltsoyannis, Nikolas
2006-05-28
Gradient corrected density functional theory has been used to calculate the geometric and electronic structures of the family of molecules [UO2(H2O)m(OH)n](2 - n) (n + m = 5). Comparisons are made with previous experimental and theoretical structural and spectroscopic data. r(U-O(yl)) is found to lengthen as water molecules are replaced by hydroxides in the equatorial plane, and the nu(sym) and nu(asym) uranyl vibrational wavenumbers decrease correspondingly. GGA functionals (BP86, PW91 and PBE) are generally found to perform better for the cationic complexes than for the anions. The inclusion of solvent effects using continuum models leads to spurious low frequency imaginary vibrational modes and overall poorer agreement with experimental data for nu(sym) and nu(asym). Analysis of the molecular orbital structure is performed in order to trace the origin of the lengthening and weakening of the U-O(yl) bond as waters are replaced by hydroxides. No evidence is found to support previous suggestions of a competition for U 6d atomic orbitals in U-O(yl) and U-O(hydroxide)pi bonding. Rather, the lengthening and weakening of U-O(yl) is attributed to reduced ionic bonding generated in part by the sigma-donating ability of the hydroxide ligands.
Yoshida, Tatsusada; Hayashi, Takahisa; Mashima, Akira; Chuman, Hiroshi
2015-10-01
One of the most challenging problems in computer-aided drug discovery is the accurate prediction of the binding energy between a ligand and a protein. For accurate estimation of net binding energy ΔEbind in the framework of the Hartree-Fock (HF) theory, it is necessary to estimate two additional energy terms; the dispersion interaction energy (Edisp) and the basis set superposition error (BSSE). We previously reported a simple and efficient dispersion correction, Edisp, to the Hartree-Fock theory (HF-Dtq). In the present study, an approximation procedure for estimating BSSE proposed by Kruse and Grimme, a geometrical counterpoise correction (gCP), was incorporated into HF-Dtq (HF-Dtq-gCP). The relative weights of the Edisp (Dtq) and BSSE (gCP) terms were determined to reproduce ΔEbind calculated with CCSD(T)/CBS or /aug-cc-pVTZ (HF-Dtq-gCP (scaled)). The performance of HF-Dtq-gCP (scaled) was compared with that of B3LYP-D3(BJ)-bCP (dispersion corrected B3LYP with the Boys and Bernadi counterpoise correction (bCP)), by taking ΔEbind (CCSD(T)-bCP) of small non-covalent complexes as 'a golden standard'. As a critical test, HF-Dtq-gCP (scaled)/6-31G(d) and B3LYP-D3(BJ)-bCP/6-31G(d) were applied to the complex model for HIV-1 protease and its potent inhibitor, KNI-10033. The present results demonstrate that HF-Dtq-gCP (scaled) is a useful and powerful remedy for accurately and promptly predicting ΔEbind between a ligand and a protein, albeit it is a simple correction procedure. Copyright © 2015 Elsevier Ltd. All rights reserved.
Covariant Theory of Gravitation in the Spacetime with Finsler Structure
Huang, Xin-Bing
2007-01-01
The theory of gravitation in the spacetime with Finsler structure is constructed. It is shown that the theory keeps general covariance. Such theory reduces to Einstein's general relativity when the Finsler structure is Riemannian. Therefore, this covariant theory of gravitation is an elegant realization of Einstein's thoughts on gravitation in the spacetime with Finsler structure.
Lectures in geometric combinatorics
Thomas, Rekha R
2006-01-01
This book presents a course in the geometry of convex polytopes in arbitrary dimension, suitable for an advanced undergraduate or beginning graduate student. The book starts with the basics of polytope theory. Schlegel and Gale diagrams are introduced as geometric tools to visualize polytopes in high dimension and to unearth bizarre phenomena in polytopes. The heart of the book is a treatment of the secondary polytope of a point configuration and its connections to the state polytope of the toric ideal defined by the configuration. These polytopes are relatively recent constructs with numerous connections to discrete geometry, classical algebraic geometry, symplectic geometry, and combinatorics. The connections rely on Gr�bner bases of toric ideals and other methods from commutative algebra. The book is self-contained and does not require any background beyond basic linear algebra. With numerous figures and exercises, it can be used as a textbook for courses on geometric, combinatorial, and computational as...
International Nuclear Information System (INIS)
Catoni, Francesco; Cannata, Roberto; Zampetti, Paolo
2005-08-01
The Riemann and Lorentz constant curvature surfaces are investigated from an Euclidean point of view. The four surfaces (constant positive and constant negative curvatures with definite and non-definite fine elements) are represented as surfaces in a Riemannian or in a particular semi-Riemannian flat space and it is shown that the complex and the hyperbolic numbers allow to obtain the same equations for the corresponding Riemann and Lorentz surfaces, respectively. Moreover it is shown that the geodesics on the Lorentz surfaces states, from a physical point of view, a link between curvature and fields. This result is obtained just as a consequence of the space-time geometrical symmetry, without invoking the famous Einstein general relativity postulate [it
Semiclassical quantum gravity: statistics of combinatorial Riemannian geometries
International Nuclear Information System (INIS)
Bombelli, L.; Corichi, A.; Winkler, O.
2005-01-01
This paper is a contribution to the development of a framework, to be used in the context of semiclassical canonical quantum gravity, in which to frame questions about the correspondence between discrete spacetime structures at ''quantum scales'' and continuum, classical geometries at large scales. Such a correspondence can be meaningfully established when one has a ''semiclassical'' state in the underlying quantum gravity theory, and the uncertainties in the correspondence arise both from quantum fluctuations in this state and from the kinematical procedure of matching a smooth geometry to a discrete one. We focus on the latter type of uncertainty, and suggest the use of statistical geometry as a way to quantify it. With a cell complex as an example of discrete structure, we discuss how to construct quantities that define a smooth geometry, and how to estimate the associated uncertainties. We also comment briefly on how to combine our results with uncertainties in the underlying quantum state, and on their use when considering phenomenological aspects of quantum gravity. (Abstract Copyright [2005], Wiley Periodicals, Inc.)
Geometrical spin symmetry and spin
International Nuclear Information System (INIS)
Pestov, I. B.
2011-01-01
Unification of General Theory of Relativity and Quantum Mechanics leads to General Quantum Mechanics which includes into itself spindynamics as a theory of spin phenomena. The key concepts of spindynamics are geometrical spin symmetry and the spin field (space of defining representation of spin symmetry). The essence of spin is the bipolar structure of geometrical spin symmetry induced by the gravitational potential. The bipolar structure provides a natural derivation of the equations of spindynamics. Spindynamics involves all phenomena connected with spin and provides new understanding of the strong interaction.
Seeley-Gilkey coefficients for the fourth-order operators on a Riemannian manifold
International Nuclear Information System (INIS)
Gusynin, V.P.
1989-01-01
A new covariant method for computing the coefficients in the heat kernel expansion is suggested. It allows one to calculate Seeley-Gilkey coefficients for both minimal and nonminimal differential operators acting on a vector bundle over a Riemannian manifold. The coefficients for the fourth-order minimal operators in arbitrary dimension of the space are calculated. In contrast to the second-order operators the coefficients for the fourth-order (and higher) operators turn out to be essentially dependent on the space dimension. The algorithmic character of the method suggested allows one to calculate coefficients by computer using the analytical calculation system. 19 refs.; 1 fig
Control of nonholonomic systems from sub-Riemannian geometry to motion planning
Jean, Frédéric
2014-01-01
Nonholonomic systems are control systems which depend linearly on the control. Their underlying geometry is the sub-Riemannian geometry, which plays for these systems the same role as Euclidean geometry does for linear systems. In particular the usual notions of approximations at the first order, that are essential for control purposes, have to be defined in terms of this geometry. The aim of these notes is to present these notions of approximation and their application to the motion planning problem for nonholonomic systems.
Duality on Geodesics of Cartan Distributions and Sub-Riemannian Pseudo-Product Structures
Directory of Open Access Journals (Sweden)
Ishikawa Goo
2015-06-01
Full Text Available Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in (15, that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.
Towards a theory of macroscopic gravity
International Nuclear Information System (INIS)
Zalaletdinov, R.M.
1993-01-01
By averaging out Cartan's structure equations for a four-dimensional Riemannian space over space regions, the structure equations for the averaged space have been derived with the procedure being valid on an arbitrary Riemannian space. The averaged space is characterized by a metric, Riemannian and non-Riemannian curvature 2-forms, and correlation 2-, 3- and 4-forms, an affine deformation 1-form being due to the non-metricity of one of two connection 1-forms. Using the procedure for the space-time averaging of the Einstein equations produces the averaged ones with the terms of geometric correction by the correlation tensors. The equations of motion for averaged energy momentum, obtained by averaging out the coritracted Bianchi identifies, also include such terms. Considering the gravitational induction tensor to be the Riemannian curvature tensor (the non-Riemannian one is then the field tensor), a theorem is proved which relates the algebraic structure of the averaged microscopic metric to that of the induction tensor. It is shown that the averaged Einstein equations can be put in the form of the Einstein equations with the conserved macroscopic energy-momentum tensor of a definite structure including the correlation functions. By using the high-frequency approximation of Isaacson with second-order correction to the microscopic metric, the self-consistency and compatibility of the equations and relations obtained are shown. Macrovacuum turns out to be Ricci non-flat, the macrovacuum source being defined in terms of the correlation functions. In the high-frequency limit the equations are shown to become Isaacson's ones with the macrovacuum source becoming Isaacson's stress tensor for gravitational waves. 17 refs
Dynamos driven by poloidal flows in untwisted, curved and flat Riemannian diffusive flux tubes
International Nuclear Information System (INIS)
De Andrade, L.C.G.
2010-01-01
Recently Vishik anti-fast dynamo theorem has been tested against non-stretching flux tubes (Phys. Plasmas, 15 (2008)). In this paper, another anti dynamo theorem, called Cowling's theorem, which states that axisymmetric magnetic fields cannot support dynamo action, is carefully tested against thick tubular and curved Riemannian untwisted flows, as well as thin flux tubes in diffusive and diffusion less media. In the non-diffusive media Cowling's theorem is not violated in thin Riemann-flat untwisted flux tubes, where the Frenet curvature is negative. Nevertheless the diffusion action in the thin flux tube leads to a dynamo action driven by poloidal flows as shown by Love and Gubbins (Geophysical Res., 23 (1996) 857) in the context of geo dynamos. Actually it is shown that a slow dynamo action is obtained. In this case the Frenet and Riemann curvature still vanishes. In the case of magnetic filaments in diffusive media dynamo action is obtained when the Frenet scalar curvature is negative. Since the Riemann curvature tensor can be expressed in terms of the Frenet curvature of the magnetic flux tube axis, this result can be analogous to a recent result obtained by Chicone, Latushkin and Smith, which states that geodesic curvature in compact Riemannian manifolds can drive dynamo action in the manifold. It is also shown that in the absence of diffusion, magnetic energy does not grow but magnetic toroidal magnetic field can be generated by the poloidal field, what is called a plasma dynamo.
Geometrical interpretation of extended supergravity
International Nuclear Information System (INIS)
Townsend, P.K.; Nieuwenhuizen, P.van
1977-01-01
SO 2 extended supergravity is shown to be a geometrical theory, whose underlying gauge group is OSp(4,2). The couplings which gauge the SO 2 symmetry as well as the accompanying cosmological and masslike terms are directly obtained, and the usual SO 2 model is obtained after a Wigner-Inoenue group contraction. (Auth.)
Cartan's geometrical structure of supergravity
International Nuclear Information System (INIS)
Baaklini, N.S.
1977-06-01
The geometrical partnership of the vierbein and the spin-3/2 field in the structure of the supergravity Lagrangian is emphasized. Both fields are introduced as component of the same matrix differential form. The only local symmetry of the theory is SL(2,C)
Classical field theory. On electrodynamics, non-Abelian gauge theories and gravitation. 2. ed.
Energy Technology Data Exchange (ETDEWEB)
Scheck, Florian
2018-04-01
Scheck's successful textbook presents a comprehensive treatment, ideally suited for a one-semester course. The textbook describes Maxwell's equations first in their integral, directly testable form, then moves on to their local formulation. The first two chapters cover all essential properties of Maxwell's equations, including their symmetries and their covariance in a modern notation. Chapter 3 is devoted to Maxwell's theory as a classical field theory and to solutions of the wave equation. Chapter 4 deals with important applications of Maxwell's theory. It includes topical subjects such as metamaterials with negative refraction index and solutions of Helmholtz' equation in paraxial approximation relevant for the description of laser beams. Chapter 5 describes non-Abelian gauge theories from a classical, geometric point of view, in analogy to Maxwell's theory as a prototype, and culminates in an application to the U(2) theory relevant for electroweak interactions. The last chapter 6 gives a concise summary of semi-Riemannian geometry as the framework for the classical field theory of gravitation. The chapter concludes with a discussion of the Schwarzschild solution of Einstein's equations and the classical tests of general relativity. The new concept of this edition presents the content divided into two tracks: the fast track for master's students, providing the essentials, and the intensive track for all wanting to get in depth knowledge of the field. Cleary labeled material and sections guide students through the preferred level of treatment. Numerous problems and worked examples will provide successful access to Classical Field Theory.
Theoretical frameworks for the learning of geometrical reasoning
Jones, Keith
1998-01-01
With the growth in interest in geometrical ideas it is important to be clear about the nature of geometrical reasoning and how it develops. This paper provides an overview of three theoretical frameworks for the learning of geometrical reasoning: the van Hiele model of thinking in geometry, Fischbein’s theory of figural concepts, and Duval’s cognitive model of geometrical reasoning. Each of these frameworks provides theoretical resources to support research into the development of geometrical...
On bivariate geometric distribution
Directory of Open Access Journals (Sweden)
K. Jayakumar
2013-05-01
Full Text Available Characterizations of bivariate geometric distribution using univariate and bivariate geometric compounding are obtained. Autoregressive models with marginals as bivariate geometric distribution are developed. Various bivariate geometric distributions analogous to important bivariate exponential distributions like, Marshall-Olkin’s bivariate exponential, Downton’s bivariate exponential and Hawkes’ bivariate exponential are presented.
Visualizing the Geometric Series.
Bennett, Albert B., Jr.
1989-01-01
Mathematical proofs often leave students unconvinced or without understanding of what has been proved, because they provide no visual-geometric representation. Presented are geometric models for the finite geometric series when r is a whole number, and the infinite geometric series when r is the reciprocal of a whole number. (MNS)
Dark energy and dark matter from hidden symmetry of gravity model with a non-Riemannian volume form
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Guendelman, Eduardo [Ben-Gurion University of the Negev, Department of Physics, Beersheba (Israel); Nissimov, Emil; Pacheva, Svetlana [Bulgarian Academy of Sciences, Institute for Nuclear Research and Nuclear Energy, Sofia (Bulgaria)
2015-10-15
We show that dark energy and dark matter can be described simultaneously by ordinary Einstein gravity interacting with a single scalar field provided the scalar field Lagrangian couples in a symmetric fashion to two different spacetime volume forms (covariant integration measure densities) on the spacetime manifold - one standard Riemannian given by √(-g) (square root of the determinant of the pertinent Riemannian metric) and another non-Riemannian volume form independent of the Riemannian metric, defined in terms of an auxiliary antisymmetric tensor gauge field of maximal rank. Integration of the equations of motion of the latter auxiliary gauge field produce an a priori arbitrary integration constant that plays the role of a dynamically generated cosmological constant or dark energy. Moreover, the above modified scalar field action turns out to possess a hidden Noether symmetry whose associated conserved current describes a pressureless ''dust'' fluid which we can identify with the dark matter completely decoupled from the dark energy. The form of both the dark energy and dark matter that results from the above class of models is insensitive to the specific form of the scalar field Lagrangian. By adding an appropriate perturbation, which breaks the above hidden symmetry and along with this couples dark matter and dark energy, we also suggest a way to obtain growing dark energy in the present universe's epoch without evolution pathologies. (orig.)
A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds
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Qiang Ru
2013-01-01
Full Text Available We study the asymptotic behavior of the parabolic Monge-Ampère equation in , in , where is a compact complete Riemannian manifold, λ is a positive real parameter, and is a smooth function. We show a meaningful asymptotic result which is more general than those in Huisken, 1997.
Zimmermann, Ralf
2016-01-01
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm converges locally and exhibits a linear rate of convergence.
DEFF Research Database (Denmark)
Zimmermann, Ralf
2017-01-01
We derive a numerical algorithm for evaluating the Riemannian logarithm on the Stiefel manifold with respect to the canonical metric. In contrast to the optimization-based approach known from the literature, we work from a purely matrix-algebraic perspective. Moreover, we prove that the algorithm...... converges locally and exhibits a linear rate of convergence....
Asymptotic geometric analysis, part I
Artstein-Avidan, Shiri
2015-01-01
The authors present the theory of asymptotic geometric analysis, a field which lies on the border between geometry and functional analysis. In this field, isometric problems that are typical for geometry in low dimensions are substituted by an "isomorphic" point of view, and an asymptotic approach (as dimension tends to infinity) is introduced. Geometry and analysis meet here in a non-trivial way. Basic examples of geometric inequalities in isomorphic form which are encountered in the book are the "isomorphic isoperimetric inequalities" which led to the discovery of the "concentration phenomen
Kruse, Holger; Grimme, Stefan
2012-04-21
A semi-empirical counterpoise-type correction for basis set superposition error (BSSE) in molecular systems is presented. An atom pair-wise potential corrects for the inter- and intra-molecular BSSE in supermolecular Hartree-Fock (HF) or density functional theory (DFT) calculations. This geometrical counterpoise (gCP) denoted scheme depends only on the molecular geometry, i.e., no input from the electronic wave-function is required and hence is applicable to molecules with ten thousands of atoms. The four necessary parameters have been determined by a fit to standard Boys and Bernadi counterpoise corrections for Hobza's S66×8 set of non-covalently bound complexes (528 data points). The method's target are small basis sets (e.g., minimal, split-valence, 6-31G*), but reliable results are also obtained for larger triple-ζ sets. The intermolecular BSSE is calculated by gCP within a typical error of 10%-30% that proves sufficient in many practical applications. The approach is suggested as a quantitative correction in production work and can also be routinely applied to estimate the magnitude of the BSSE beforehand. The applicability for biomolecules as the primary target is tested for the crambin protein, where gCP removes intramolecular BSSE effectively and yields conformational energies comparable to def2-TZVP basis results. Good mutual agreement is also found with Jensen's ACP(4) scheme, estimating the intramolecular BSSE in the phenylalanine-glycine-phenylalanine tripeptide, for which also a relaxed rotational energy profile is presented. A variety of minimal and double-ζ basis sets combined with gCP and the dispersion corrections DFT-D3 and DFT-NL are successfully benchmarked on the S22 and S66 sets of non-covalent interactions. Outstanding performance with a mean absolute deviation (MAD) of 0.51 kcal/mol (0.38 kcal/mol after D3-refit) is obtained at the gCP-corrected HF-D3/(minimal basis) level for the S66 benchmark. The gCP-corrected B3LYP-D3/6-31G* model
Kruse, Holger; Grimme, Stefan
2012-04-01
A semi-empirical counterpoise-type correction for basis set superposition error (BSSE) in molecular systems is presented. An atom pair-wise potential corrects for the inter- and intra-molecular BSSE in supermolecular Hartree-Fock (HF) or density functional theory (DFT) calculations. This geometrical counterpoise (gCP) denoted scheme depends only on the molecular geometry, i.e., no input from the electronic wave-function is required and hence is applicable to molecules with ten thousands of atoms. The four necessary parameters have been determined by a fit to standard Boys and Bernadi counterpoise corrections for Hobza's S66×8 set of non-covalently bound complexes (528 data points). The method's target are small basis sets (e.g., minimal, split-valence, 6-31G*), but reliable results are also obtained for larger triple-ζ sets. The intermolecular BSSE is calculated by gCP within a typical error of 10%-30% that proves sufficient in many practical applications. The approach is suggested as a quantitative correction in production work and can also be routinely applied to estimate the magnitude of the BSSE beforehand. The applicability for biomolecules as the primary target is tested for the crambin protein, where gCP removes intramolecular BSSE effectively and yields conformational energies comparable to def2-TZVP basis results. Good mutual agreement is also found with Jensen's ACP(4) scheme, estimating the intramolecular BSSE in the phenylalanine-glycine-phenylalanine tripeptide, for which also a relaxed rotational energy profile is presented. A variety of minimal and double-ζ basis sets combined with gCP and the dispersion corrections DFT-D3 and DFT-NL are successfully benchmarked on the S22 and S66 sets of non-covalent interactions. Outstanding performance with a mean absolute deviation (MAD) of 0.51 kcal/mol (0.38 kcal/mol after D3-refit) is obtained at the gCP-corrected HF-D3/(minimal basis) level for the S66 benchmark. The gCP-corrected B3LYP-D3/6-31G* model
Geodesic B-Preinvex Functions and Multiobjective Optimization Problems on Riemannian Manifolds
Directory of Open Access Journals (Sweden)
Sheng-lan Chen
2014-01-01
Full Text Available We introduce a class of functions called geodesic B-preinvex and geodesic B-invex functions on Riemannian manifolds and generalize the notions to the so-called geodesic quasi/pseudo B-preinvex and geodesic quasi/pseudo B-invex functions. We discuss the links among these functions under appropriate conditions and obtain results concerning extremum points of a nonsmooth geodesic B-preinvex function by using the proximal subdifferential. Moreover, we study a differentiable multiobjective optimization problem involving new classes of generalized geodesic B-invex functions and derive Kuhn-Tucker-type sufficient conditions for a feasible point to be an efficient or properly efficient solution. Finally, a Mond-Weir type duality is formulated and some duality results are given for the pair of primal and dual programming.
Seeley-Gilkey coefficients for fourth-order operators on Riemannian manifold
International Nuclear Information System (INIS)
Gusynin, V.P.
1990-01-01
The covariant pseudodifferential-operator method of Widom is developed for computing the coefficients in the heat kernel expansion. It allows one to calculate Seeley-Gilkey coefficients for both minimal and nonminimal differential operators acting on a vector bundle over a riemannian manifold. The coefficients for the fourth-order minimal operators in arbitrary dimensions of space are calculated. In contrast to the second-order operators the coefficients for the fourth-order (and higher) operators turn out to be essentially dependent on the space dimension. The algorithmic character of the method allows one to calculate the coefficients by computer using an analytical calculation system. The method also permits a simple generalization to manifolds with torsion and supermanifolds. (orig.)
Baust, Maximilian; Weinmann, Andreas; Wieczorek, Matthias; Lasser, Tobias; Storath, Martin; Navab, Nassir
2016-08-01
In this paper, we consider combined TV denoising and diffusion tensor fitting in DTI using the affine-invariant Riemannian metric on the space of diffusion tensors. Instead of first fitting the diffusion tensors, and then denoising them, we define a suitable TV type energy functional which incorporates the measured DWIs (using an inverse problem setup) and which measures the nearness of neighboring tensors in the manifold. To approach this functional, we propose generalized forward- backward splitting algorithms which combine an explicit and several implicit steps performed on a decomposition of the functional. We validate the performance of the derived algorithms on synthetic and real DTI data. In particular, we work on real 3D data. To our knowledge, the present paper describes the first approach to TV regularization in a combined manifold and inverse problem setup.
A Divergence Median-based Geometric Detector with A Weighted Averaging Filter
Hua, Xiaoqiang; Cheng, Yongqiang; Li, Yubo; Wang, Hongqiang; Qin, Yuliang
2018-01-01
To overcome the performance degradation of the classical fast Fourier transform (FFT)-based constant false alarm rate detector with the limited sample data, a divergence median-based geometric detector on the Riemannian manifold of Heimitian positive definite matrices is proposed in this paper. In particular, an autocorrelation matrix is used to model the correlation of sample data. This method of the modeling can avoid the poor Doppler resolution as well as the energy spread of the Doppler filter banks result from the FFT. Moreover, a weighted averaging filter, conceived from the philosophy of the bilateral filtering in image denoising, is proposed and combined within the geometric detection framework. As the weighted averaging filter acts as the clutter suppression, the performance of the geometric detector is improved. Numerical experiments are given to validate the effectiveness of our proposed method.
Harmonic and geometric analysis
Citti, Giovanna; Pérez, Carlos; Sarti, Alessandro; Zhong, Xiao
2015-01-01
This book presents an expanded version of four series of lectures delivered by the authors at the CRM. Harmonic analysis, understood in a broad sense, has a very wide interplay with partial differential equations and in particular with the theory of quasiconformal mappings and its applications. Some areas in which real analysis has been extremely influential are PDE's and geometric analysis. Their foundations and subsequent developments made extensive use of the Calderón–Zygmund theory, especially the Lp inequalities for Calderón–Zygmund operators (Beurling transform and Riesz transform, among others) and the theory of Muckenhoupt weights. The first chapter is an application of harmonic analysis and the Heisenberg group to understanding human vision, while the second and third chapters cover some of the main topics on linear and multilinear harmonic analysis. The last serves as a comprehensive introduction to a deep result from De Giorgi, Moser and Nash on the regularity of elliptic partial differen...
Geometrical formulation of the conformal Ward identity
International Nuclear Information System (INIS)
Kachkachi, M.
2002-08-01
In this paper we use deep ideas in complex geometry that proved to be very powerful in unveiling the Polyakov measure on the moduli space of Riemann surfaces and lead to obtain the partition function of perturbative string theory for 2, 3, 4 loops. Indeed, a geometrical interpretation of the conformal Ward identity in two dimensional conformal field theory is proposed: the conformal anomaly is interpreted as a deformation of the complex structure of the basic Riemann surface. This point of view is in line with the modern trend of geometric quantizations that are based on deformations of classical structures. Then, we solve the conformal Ward identity by using this geometrical formalism. (author)
Refined geometric transition and qq-characters
Kimura, Taro; Mori, Hironori; Sugimoto, Yuji
2018-01-01
We show the refinement of the prescription for the geometric transition in the refined topological string theory and, as its application, discuss a possibility to describe qq-characters from the string theory point of view. Though the suggested way to operate the refined geometric transition has passed through several checks, it is additionally found in this paper that the presence of the preferred direction brings a nontrivial effect. We provide the modified formula involving this point. We then apply our prescription of the refined geometric transition to proposing the stringy description of doubly quantized Seiberg-Witten curves called qq-characters in certain cases.
Ghost properties of generalized theories of gravitation
International Nuclear Information System (INIS)
Mann, R.B.; Moffat, J.W.
1982-01-01
We investigate theories of gravitation, in which spacetime is non-Riemannian and the metric g/sub munu/ is nonsymmetric, for ghosts and tachyons, using a spin-projection operator formalism. Ghosts are removed not by gauge invariance but by a Lagrange multiplier W/sub μ/, which occurs due to the breaking of projective invariance in the theory. Unified theories based on a Lagrangian containing a term lambdag/sup munu/g/sub / are proved to contain ghosts or tachyons
Federal Laboratory Consortium — Purpose: The mission of the Geometric Design Laboratory (GDL) is to support the Office of Safety Research and Development in research related to the geometric design...
Torsion in a gravity theory with SO(k) x SO(d-k) as tangent group
International Nuclear Information System (INIS)
Viswanathan, K.S.; Wong, B.; Simon Fraser Univ., Burnaby, British Columbia
1985-01-01
We consider a d-dimensional theory of gravity where the tangent group is SO(k) x SO(d-k) rather than SO(d) as in riemannian theories. This theory has nonvanishing torsion (which is required if the theory is to yield gauge fields). The torsion is determined consistently in terms of vielbein derivatives. (orig.)
Gahm, Jin Kyu; Shi, Yonggang
2018-05-01
Surface mapping methods play an important role in various brain imaging studies from tracking the maturation of adolescent brains to mapping gray matter atrophy patterns in Alzheimer's disease. Popular surface mapping approaches based on spherical registration, however, have inherent numerical limitations when severe metric distortions are present during the spherical parameterization step. In this paper, we propose a novel computational framework for intrinsic surface mapping in the Laplace-Beltrami (LB) embedding space based on Riemannian metric optimization on surfaces (RMOS). Given a diffeomorphism between two surfaces, an isometry can be defined using the pullback metric, which in turn results in identical LB embeddings from the two surfaces. The proposed RMOS approach builds upon this mathematical foundation and achieves general feature-driven surface mapping in the LB embedding space by iteratively optimizing the Riemannian metric defined on the edges of triangular meshes. At the core of our framework is an optimization engine that converts an energy function for surface mapping into a distance measure in the LB embedding space, which can be effectively optimized using gradients of the LB eigen-system with respect to the Riemannian metrics. In the experimental results, we compare the RMOS algorithm with spherical registration using large-scale brain imaging data, and show that RMOS achieves superior performance in the prediction of hippocampal subfields and cortical gyral labels, and the holistic mapping of striatal surfaces for the construction of a striatal connectivity atlas from substantia nigra. Copyright © 2018 Elsevier B.V. All rights reserved.
Hu, Weiming; Li, Xi; Luo, Wenhan; Zhang, Xiaoqin; Maybank, Stephen; Zhang, Zhongfei
2012-12-01
Object appearance modeling is crucial for tracking objects, especially in videos captured by nonstationary cameras and for reasoning about occlusions between multiple moving objects. Based on the log-euclidean Riemannian metric on symmetric positive definite matrices, we propose an incremental log-euclidean Riemannian subspace learning algorithm in which covariance matrices of image features are mapped into a vector space with the log-euclidean Riemannian metric. Based on the subspace learning algorithm, we develop a log-euclidean block-division appearance model which captures both the global and local spatial layout information about object appearances. Single object tracking and multi-object tracking with occlusion reasoning are then achieved by particle filtering-based Bayesian state inference. During tracking, incremental updating of the log-euclidean block-division appearance model captures changes in object appearance. For multi-object tracking, the appearance models of the objects can be updated even in the presence of occlusions. Experimental results demonstrate that the proposed tracking algorithm obtains more accurate results than six state-of-the-art tracking algorithms.
Geometric quantization and general relativity
International Nuclear Information System (INIS)
Souriau, J.-M.
1977-01-01
The purpose of geometric quantization is to give a rigorous mathematical content to the 'correspondence principle' between classical and quantum mechanics. The main tools are borrowed on one hand from differential geometry and topology (differential manifolds, differential forms, fiber bundles, homology and cohomology, homotopy), on the other hand from analysis (functions of positive type, infinite dimensional group representations, pseudo-differential operators). Some satisfactory results have been obtained in the study of dynamical systems, but some fundamental questions are still waiting for an answer. The 'geometric quantization of fields', where some further well known difficulties arise, is still in a preliminary stage. In particular, the geometric quantization on the gravitational field is still a mere project. The situation is even more uncertain due to the fact that there is no experimental evidence of any quantum gravitational effect which could give us a hint towards what we are supposed to look for. The first level of both Quantum Theory, and General Relativity describes passive matter: influence by the field without being a source of it (first quantization and equivalence principle respectively). In both cases this is only an approximation (matter is always a source). But this approximation turns out to be the least uncertain part of the description, because on one hand the first quantization avoids the problems of renormalization and on the other hand the equivalence principle does not imply any choice of field equations (it is known that one can modify Einstein equations at short distances without changing their geometrical properties). (Auth.)
Inferring imagined speech using EEG signals: a new approach using Riemannian manifold features
Nguyen, Chuong H.; Karavas, George K.; Artemiadis, Panagiotis
2018-02-01
Objective. In this paper, we investigate the suitability of imagined speech for brain-computer interface (BCI) applications. Approach. A novel method based on covariance matrix descriptors, which lie in Riemannian manifold, and the relevance vector machines classifier is proposed. The method is applied on electroencephalographic (EEG) signals and tested in multiple subjects. Main results. The method is shown to outperform other approaches in the field with respect to accuracy and robustness. The algorithm is validated on various categories of speech, such as imagined pronunciation of vowels, short words and long words. The classification accuracy of our methodology is in all cases significantly above chance level, reaching a maximum of 70% for cases where we classify three words and 95% for cases of two words. Significance. The results reveal certain aspects that may affect the success of speech imagery classification from EEG signals, such as sound, meaning and word complexity. This can potentially extend the capability of utilizing speech imagery in future BCI applications. The dataset of speech imagery collected from total 15 subjects is also published.
a Super Voxel-Based Riemannian Graph for Multi Scale Segmentation of LIDAR Point Clouds
Li, Minglei
2018-04-01
Automatically segmenting LiDAR points into respective independent partitions has become a topic of great importance in photogrammetry, remote sensing and computer vision. In this paper, we cast the problem of point cloud segmentation as a graph optimization problem by constructing a Riemannian graph. The scale space of the observed scene is explored by an octree-based over-segmentation with different depths. The over-segmentation produces many super voxels which restrict the structure of the scene and will be used as nodes of the graph. The Kruskal coordinates are used to compute edge weights that are proportional to the geodesic distance between nodes. Then we compute the edge-weight matrix in which the elements reflect the sectional curvatures associated with the geodesic paths between super voxel nodes on the scene surface. The final segmentation results are generated by clustering similar super voxels and cutting off the weak edges in the graph. The performance of this method was evaluated on LiDAR point clouds for both indoor and outdoor scenes. Additionally, extensive comparisons to state of the art techniques show that our algorithm outperforms on many metrics.
Geometric U-folds in four dimensions
Lazaroiu, C. I.; Shahbazi, C. S.
2018-01-01
We describe a general construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued together. We show that smooth non-trivial U-folds of this type can exist only in theories where both the scalar and space-time manifolds have non-trivial fundamental group and in addition the scalar map of the solution is homotopically non-trivial. Consistency with string theory requires smooth geometric U-folds to be glued using subgroups of the effective discrete U-duality group, implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter. We construct simple examples of geometric U-folds in a generalization of the axion-dilaton model of \
The geometric semantics of algebraic quantum mechanics.
Cruz Morales, John Alexander; Zilber, Boris
2015-08-06
In this paper, we will present an ongoing project that aims to use model theory as a suitable mathematical setting for studying the formalism of quantum mechanics. We argue that this approach provides a geometric semantics for such a formalism by means of establishing a (non-commutative) duality between certain algebraic and geometric objects. © 2015 The Author(s) Published by the Royal Society. All rights reserved.
International Nuclear Information System (INIS)
Cariglia, Marco; Alves, Filipe Kelmer
2015-01-01
This work originates from part of a final year undergraduate research project on the Eisenhart lift for Hamiltonian systems. The Eisenhart lift is a procedure to describe trajectories of a classical natural Hamiltonian system as geodesics in an enlarged space. We point out that it can be easily obtained from basic principles of Hamiltonian dynamics, and as such it represents a useful didactical way to introduce graduate students to several modern concepts of geometry applied to physics: curved spaces, both Riemannian and Lorentzian, conformal transformations, geometrization of interactions and extra dimensions, and geometrization of dynamical symmetries. For all these concepts the Eisenhart lift can be used as a theoretical tool that provides easily achievable examples, with the added benefit of also being a topic of current research with several applications, among which are included the study of dynamical systems and non-relativistic holography. (paper)
Wang, Dong; Wang, Haifeng; Hu, P
2015-01-21
Using density functional theory calculations with HSE 06 functional, we obtained the structures of spin-polarized radicals on rutile TiO2(110), which is crucial to understand the photooxidation at the atomic level, and further calculate the thermodynamic stabilities of these radicals. By analyzing the results, we identify the structural features for hole trapping in the system, and reveal the mutual effects among the geometric structures, the energy levels of trapped hole states and their hole trapping capacities. Furthermore, the results from HSE 06 functional are compared to those from DFT + U and the stability trend of radicals against the number of slabs is tested. The effect of trapped holes on two important steps of the oxygen evolution reaction, i.e. water dissociation and the oxygen removal, is investigated and discussed.
Geometric Transformations in Engineering Geometry
Directory of Open Access Journals (Sweden)
I. F. Borovikov
2015-01-01
Full Text Available Recently, for business purposes, in view of current trends and world experience in training engineers, research and faculty staff there has been a need to transform traditional courses of descriptive geometry into the course of engineering geometry in which the geometrical transformations have to become its main section. On the basis of critical analysis the paper gives suggestions to improve a presentation technique of this section both in the classroom and in academic literature, extend an application scope of geometrical transformations to solve the position and metric tasks and simulation of surfaces, as well as to design complex engineering configurations, which meet a number of pre-specified conditions.The article offers to make a number of considerable amendments to the terms and definitions used in the existing courses of descriptive geometry. It draws some conclusions and makes the appropriate proposals on feasibility of coordination in teaching the movement transformation in the courses of analytical and descriptive geometry. This will provide interdisciplinary team teaching and allow students to be convinced that a combination of analytical and graphic ways to solve geometric tasks is useful and reasonable.The traditional sections of learning courses need to be added with a theory of projective and bi-rational transformations. In terms of application simplicity and convenience it is enough to consider the central transformations when solving the applied tasks. These transformations contain a beam of sub-invariant (low-invariant straight lines on which the invariant curve induces non-involution and involution projectivities. The expediency of nonlinear transformations application is shown in the article by a specific example of geometric modeling of the interfacing surface "spar-blade".Implementation of these suggestions will contribute to a real transformation of a traditional course of descriptive geometry to the engineering geometry
Matter coupled to quantum gravity in group field theory
International Nuclear Information System (INIS)
Ryan, James
2006-01-01
We present an account of a new model incorporating 3d Riemannian quantum gravity and matter at the group field theory level. We outline how the Feynman diagram amplitudes of this model are spin foam amplitudes for gravity coupled to matter fields and discuss some features of the model. To conclude, we describe some related future work
Darling, Jeremy
A new field of study, "real-time cosmology," is now possible. This involves observing a dynamic universe that can be seen to change over human timescales. Most cosmological observations are geometrical, using standard candles or rulers to measure the expansion history and curvature as light propagates through the universe. Real-time cosmological measurements are dynamical, revealing the changing geometry of the universe - thus often providing geometrical distances independent of the canonical cosmological distance ladder - and are typically orthogonal to customary cosmological tests. This field of inquiry is no longer far-fetched, and this proposal demonstrates using extant data that many types of measurement are now within a factor of a few of being detectable, but the theory will very soon lag the observational capabilities. The Gaia mission will provide astrometry and proper motions of roughly 100 microarcseconds per year for half a million quasars by the end of its 5-year mission, but the theory for how to employ these data for cosmological tests has not been established. This project will develop the theory, models, and methods needed to make optimal use of the Gaia extragalactic proper motion measurements and to make significant new cosmological tests, distance measurements, and mass measurements. Gaia data can provide rich cosmological tests that are nearly model-independent. This work will build the theoretical framework enabling Gaia to measure or constrain: (1) The real-time growth and recession of structures, providing mass and distance measurements, (2) Extragalactic parallax for a statistical sample and individual galaxies, thus providing geometric distances, (3) The primordial stochastic long-period gravitational wave background, which deflects quasar light in a quadrupolar proper motion pattern, and (4) Cosmic shear, rotation, bulk motion, and local voids that may manifest as an apparent acceleration attributed to dark energy. One can also test the
A Comment on the geometry of some scalar-tensor theories
Energy Technology Data Exchange (ETDEWEB)
Lindstrom, U
1986-08-01
We show that the scalar field in scalar-tensor theories such as the Jordan-Brans-Dicke theory has an interpretation as a potential for the torsion in a Riemannian manifold. The relation is similar to that of the metric to the connection.
On the geometry of Riemannian manifolds with a Lie structure at infinity
Directory of Open Access Journals (Sweden)
Bernd Ammann
2004-01-01
Full Text Available We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.
Geometric flows and (some of) their physical applications
Bakas, Ioannis
2005-01-01
The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis of non-linear sigma models and in general relativity. They are divided into classes of intrinsic and extrinsic curvature flows. Here, we review the main aspects of intrinsic geometric flows driven by the Ricci curvature, in various forms, and explain the intimate relation between Ricci and Calabi flows on Kahler manifolds using the notion of super-evolution. The integration of these flows on two-dimensional surfaces relies on the introduction of a novel class of infinite dimensional algebras with infinite growth. It is also explained in this context how Kac's K_2 simple Lie algebra can be used to construct metrics on S^2 with prescribed scalar curvature equal to the sum of any holomorphic function and its complex conjugate; applications of this special problem to general re...
Najafi, M. N.
2018-04-01
The coupling of the c = ‑2, c=\\frac{1}{2} and c = 0 conformal field theories are numerically considered in this paper. As the prototypes of the couplings, (c_1=-2)\\oplus (c_2=0) and (c_1=-2)\\oplus (c_2=\\frac{1}{2}) , we consider the Bak–Tang–Weisenfeld (BTW) model on the 2D square critical site-percolation and the BTW model on Ising-correlated percolation lattices respectively. Some geometrical techniques are used to characterize the presumable conformal symmetry of the resultant systems. Based on the numerical analysis of the diffusivity parameter (κ) in the Schramm–Loewner evolution (SLE) theory we propose that the algebra of the central charges of the coupled models is closed. This result is based on the analysis of the conformal loop ensemble (CLE) analysis. The diffusivity parameter in each case is obtained by calculating the fractal dimension of loops (and the corresponding exponent of mean-square root distance), the direct SLE mapping method, the left passage probability and the winding angle analysis. More precisely we numerically show that the coupling (c_1=-2)\\oplus (c_2=\\frac{1}{2}) results to 2D self-avoiding walk (SAW) fixed point corresponding to c = 0 conformal field theory, whereas the coupling (c_1=-2)\\oplus (c_2=0) results to the 2D critical Ising fixed point corresponding to the c=\\frac{1}{2} conformal field theory.
The perception of geometrical structure from congruence
Lappin, Joseph S.; Wason, Thomas D.
1989-01-01
The principle function of vision is to measure the environment. As demonstrated by the coordination of motor actions with the positions and trajectories of moving objects in cluttered environments and by rapid recognition of solid objects in varying contexts from changing perspectives, vision provides real-time information about the geometrical structure and location of environmental objects and events. The geometric information provided by 2-D spatial displays is examined. It is proposed that the geometry of this information is best understood not within the traditional framework of perspective trigonometry, but in terms of the structure of qualitative relations defined by congruences among intrinsic geometric relations in images of surfaces. The basic concepts of this geometrical theory are outlined.
Geometric and engineering drawing
Morling, K
2010-01-01
The new edition of this successful text describes all the geometric instructions and engineering drawing information that are likely to be needed by anyone preparing or interpreting drawings or designs with plenty of exercises to practice these principles.
Differential geometric structures
Poor, Walter A
2007-01-01
This introductory text defines geometric structure by specifying parallel transport in an appropriate fiber bundle and focusing on simplest cases of linear parallel transport in a vector bundle. 1981 edition.
Geometric ghosts and unitarity
International Nuclear Information System (INIS)
Ne'eman, Y.
1980-09-01
A review is given of the geometrical identification of the renormalization ghosts and the resulting derivation of Unitarity equations (BRST) for various gauges: Yang-Mills, Kalb-Ramond, and Soft-Group-Manifold
Asymptotic and geometrical quantization
International Nuclear Information System (INIS)
Karasev, M.V.; Maslov, V.P.
1984-01-01
The main ideas of geometric-, deformation- and asymptotic quantizations are compared. It is shown that, on the one hand, the asymptotic approach is a direct generalization of exact geometric quantization, on the other hand, it generates deformation in multiplication of symbols and Poisson brackets. Besides investigating the general quantization diagram, its applications to the calculation of asymptotics of a series of eigenvalues of operators possessing symmetry groups are considered
EARLY HISTORY OF GEOMETRIC PROBABILITY AND STEREOLOGY
Directory of Open Access Journals (Sweden)
Magdalena Hykšová
2012-03-01
Full Text Available The paper provides an account of the history of geometric probability and stereology from the time of Newton to the early 20th century. It depicts the development of two parallel ways: on one hand, the theory of geometric probability was formed with minor attention paid to other applications than those concerning spatial chance games. On the other hand, practical rules of the estimation of area or volume fraction and other characteristics, easily deducible from geometric probability theory, were proposed without the knowledge of this branch. A special attention is paid to the paper of J.-É. Barbier published in 1860, which contained the fundamental stereological formulas, but remained almost unnoticed both by mathematicians and practicians.
Two particle entanglement and its geometric duals
Energy Technology Data Exchange (ETDEWEB)
Wasay, Muhammad Abdul [University of Agriculture, Department of Physics, Faisalabad (Pakistan); Quaid-i-Azam University Campus, National Centre for Physics, Islamabad (Pakistan); Bashir, Asma [University of Agriculture, Department of Physics, Faisalabad (Pakistan)
2017-12-15
We show that for a system of two entangled particles, there is a dual description to the particle equations in terms of classical theory of conformally stretched spacetime. We also connect these entangled particle equations with Finsler geometry. We show that this duality translates strongly coupled quantum equations in the pilot-wave limit to weakly coupled geometric equations. (orig.)
Two particle entanglement and its geometric duals
International Nuclear Information System (INIS)
Wasay, Muhammad Abdul; Bashir, Asma
2017-01-01
We show that for a system of two entangled particles, there is a dual description to the particle equations in terms of classical theory of conformally stretched spacetime. We also connect these entangled particle equations with Finsler geometry. We show that this duality translates strongly coupled quantum equations in the pilot-wave limit to weakly coupled geometric equations. (orig.)
Geometrical method of decoupling
Directory of Open Access Journals (Sweden)
C. Baumgarten
2012-12-01
Full Text Available The computation of tunes and matched beam distributions are essential steps in the analysis of circular accelerators. If certain symmetries—like midplane symmetry—are present, then it is possible to treat the betatron motion in the horizontal, the vertical plane, and (under certain circumstances the longitudinal motion separately using the well-known Courant-Snyder theory, or to apply transformations that have been described previously as, for instance, the method of Teng and Edwards. In a preceding paper, it has been shown that this method requires a modification for the treatment of isochronous cyclotrons with non-negligible space charge forces. Unfortunately, the modification was numerically not as stable as desired and it was still unclear, if the extension would work for all conceivable cases. Hence, a systematic derivation of a more general treatment seemed advisable. In a second paper, the author suggested the use of real Dirac matrices as basic tools for coupled linear optics and gave a straightforward recipe to decouple positive definite Hamiltonians with imaginary eigenvalues. In this article this method is generalized and simplified in order to formulate a straightforward method to decouple Hamiltonian matrices with eigenvalues on the real and the imaginary axis. The decoupling of symplectic matrices which are exponentials of such Hamiltonian matrices can be deduced from this in a few steps. It is shown that this algebraic decoupling is closely related to a geometric “decoupling” by the orthogonalization of the vectors E[over →], B[over →], and P[over →], which were introduced with the so-called “electromechanical equivalence.” A mathematical analysis of the problem can be traced down to the task of finding a structure-preserving block diagonalization of symplectic or Hamiltonian matrices. Structure preservation means in this context that the (sequence of transformations must be symplectic and hence canonical. When
Sasakian manifolds and M-theory
International Nuclear Information System (INIS)
Figueroa-O’Farrill, José; Santi, Andrea
2016-01-01
We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of η-Einstein Sasakian manifolds, both in Riemannian and Lorentzian settings, characterizing them in terms of generalized Killing spinors. We propose a definition of supersymmetric M-theory backgrounds on such a geometry and find a new class of such backgrounds, extending previous work of Haupt, Lukas and Stelle. (paper)
Foliated control theory and its applications
Energy Technology Data Exchange (ETDEWEB)
Jones, L E [Department of Mathematics, State University of New York at Stony Brook, Stony Brook (United States)
2002-08-15
The control theorems and fibered control theorems due to Chapman, Ferry and Quinn, concerning controlled h-cobordisms and controlled homotopy equivalences, are reviewed. Some foliated control theorems, due to Farrell and Jones, are formulated and deduced from the fibered control theorems. The role that foliated control theory plays in proving the Borel conjecture for closed Riemannian manifolds having non-positive sectional curvat and in calculating Whitehead groups for the fundamental group of such manifolds, is described. (author)
General relativity: An introduction to the theory of the gravitational field
International Nuclear Information System (INIS)
Stephani, H.
1985-01-01
The entire treatment presented here is framed by questions which led to and now lead out of the general theory of relativity: can an absolute acceleration be defined meaningfully? Do gravitational effects propagate with infinite velocity as Newton required? Can the general theory correctly reflect the dynamics of the whole universe while consistently describing stellar evolution? Can a theory which presupposes measurement of properties of space through the interaction of matter be made compatible with a theory in which dimensions of the objects measured are so small that location loses meaning? The book gives the mathematics necessary to understand the theory and begins in Riemannian geometry. Contents, abridged: Foundations of Riemannian geometry. Foundations of Einstein's theory of gravitation. Linearised theory of gravitation, far fields and gravitational waves. Invariant characterisation of exact solutions. Gravitational collapse and black holes. Cosmology. Non-Einsteinian theories of gravitation. Index
Geometric convergence of some two-point Pade approximations
International Nuclear Information System (INIS)
Nemeth, G.
1983-01-01
The geometric convergences of some two-point Pade approximations are investigated on the real positive axis and on certain infinite sets of the complex plane. Some theorems concerning the geometric convergence of Pade approximations are proved, and bounds on geometric convergence rates are given. The results may be interesting considering the applications both in numerical computations and in approximation theory. As a specific case, the numerical calculations connected with the plasma dispersion function may be performed. (D.Gy.)
A fast method for linear waves based on geometrical optics
Stolk, C.C.
2009-01-01
We develop a fast method for solving the one-dimensional wave equation based on geometrical optics. From geometrical optics (e.g., Fourier integral operator theory or WKB approximation) it is known that high-frequency waves split into forward and backward propagating parts, each propagating with the
A note on the geometric unification of gravity and electromagnetism
International Nuclear Information System (INIS)
Coley, A.
1984-01-01
In recent years there have been many authors that have sought a geometrically unified theory of gravity and electromagnetism. It will be argued that the motivation behind the search for such a unified theory on geometric grounds alone is both erroneous and misleading. It is felt that any new unified theory of gravity and electromagnetism must include an explanation of why the existing theory is inadequate, and should provide clear physical reasons for introducing new fields (or field equations) that appear in the theory. (author)
Aspects of the geometrical approach to supermanifolds
International Nuclear Information System (INIS)
Rogers, A.
1984-01-01
Various topics in the theory and application of the geometrical approach to supermanifolds are discussed. The construction of the superspace used in supergravity over an arbitrary spacetime manifold is described. Super Lie groups and their relation to graded Lie algebras (and more general structures referred to as 'graded Lie modules') are discussed, with examples. Certain supermanifolds, allowed in the geometric approach (using the fine topology), but having no analogue in the algebraic approach, are discussed. Finally lattice supersymmetry, and its relation to the differential geometry of supermanifolds, is discussed. (orig.)
On pseudoparticle solutions in Yang's theory of gravity
International Nuclear Information System (INIS)
Mielke, E.W.
1980-03-01
Within the framework of differential geometry, Yang's parallel-displacement gauge theory is considered with respect to ''pure'' gravitational fields. In a four-dimensional Riemannian manifold it is shown that the double self-dual solutions obey Einstein's vacuum equations with cosmological term, whereas the double anti-self-dual configurations satisfy the Rainich conditions of Wheeler's geometrodynamics. Conformal methods reveal that the gravitational analogue of the ''instanton'' or pseudoparticle solution of Yang-Mills theory was already known to Riemann. (author)
The Geometric Theory of Roof Reflector Resonators
1976-12-01
reflector, if properly oriented, (The terms "roof-top prism ," "right-angle prism ," and - incorrectly - " Porro prism " are encountered in .the literature...Q-switch prisms ) in laser resonators have been infrequent compared to the attention given spherical mirrors. This chapter summarizes the relevant...designator (Refs 42 and 43). In one experiment, a 900 roof prism was tested in a resonator with a 70% reflecting filat mirror. Thus, in Fig. 2, the right roof
Dynamics of inequalities in geometric function theory
Directory of Open Access Journals (Sweden)
Reich Simeon
2001-01-01
Full Text Available A domain in the complex plane which is star-like with respect to a boundary point can be approximated by domains which are star-like with respect to interior points. This approximation process can be viewed dynamically as an evolution of the null points of the underlying holomorphic functions from the interior of the open unit disk towards a boundary point. We trace these dynamics analytically in terms of the Alexander–Nevanlinna and Robertson inequalities by using the framework of complex dynamical systems and hyperbolic monotonicity.
A Geometrical Application of Number Theory
Srinivasan, V. K.
2013-01-01
Any quadruple of natural numbers {a, b, c, d} is called a "Pythagorean quadruple" if it satisfies the relationship "a[superscript 2] + b[superscript 2] + c[superscript 2]". This "Pythagorean quadruple" can always be identified with a rectangular box of dimensions "a greater than 0," "b greater than…
Geometric invariant theory for polarized curves
Bini, Gilberto; Melo, Margarida; Viviani, Filippo
2014-01-01
We investigate GIT quotients of polarized curves. More specifically, we study the GIT problem for the Hilbert and Chow schemes of curves of degree d and genus g in a projective space of dimension d-g, as d decreases with respect to g. We prove that the first three values of d at which the GIT quotients change are given by d=a(2g-2) where a=2, 3.5, 4. We show that, for a>4, L. Caporaso's results hold true for both Hilbert and Chow semistability. If 3.5
International Nuclear Information System (INIS)
La, H.
1992-01-01
A new geometric formulation of Liouville gravity based on the area preserving diffeo-morphism is given and a possible alternative to reinterpret Liouville gravity is suggested, namely, a scalar field coupled to two-dimensional gravity with a curvature constraint
A Geometric Dissection Problem
Indian Academy of Sciences (India)
Home; Journals; Resonance – Journal of Science Education; Volume 7; Issue 7. A Geometric Dissection Problem. M N Deshpande. Think It Over Volume 7 Issue 7 July 2002 pp 91-91. Fulltext. Click here to view fulltext PDF. Permanent link: https://www.ias.ac.in/article/fulltext/reso/007/07/0091-0091. Author Affiliations.
Geometric statistical inference
International Nuclear Information System (INIS)
Periwal, Vipul
1999-01-01
A reparametrization-covariant formulation of the inverse problem of probability is explicitly solved for finite sample sizes. The inferred distribution is explicitly continuous for finite sample size. A geometric solution of the statistical inference problem in higher dimensions is outlined
Geometric Series via Probability
Tesman, Barry
2012-01-01
Infinite series is a challenging topic in the undergraduate mathematics curriculum for many students. In fact, there is a vast literature in mathematics education research on convergence issues. One of the most important types of infinite series is the geometric series. Their beauty lies in the fact that they can be evaluated explicitly and that…
Pragmatic geometric model evaluation
Pamer, Robert
2015-04-01
Quantification of subsurface model reliability is mathematically and technically demanding as there are many different sources of uncertainty and some of the factors can be assessed merely in a subjective way. For many practical applications in industry or risk assessment (e. g. geothermal drilling) a quantitative estimation of possible geometric variations in depth unit is preferred over relative numbers because of cost calculations for different scenarios. The talk gives an overview of several factors that affect the geometry of structural subsurface models that are based upon typical geological survey organization (GSO) data like geological maps, borehole data and conceptually driven construction of subsurface elements (e. g. fault network). Within the context of the trans-European project "GeoMol" uncertainty analysis has to be very pragmatic also because of different data rights, data policies and modelling software between the project partners. In a case study a two-step evaluation methodology for geometric subsurface model uncertainty is being developed. In a first step several models of the same volume of interest have been calculated by omitting successively more and more input data types (seismic constraints, fault network, outcrop data). The positions of the various horizon surfaces are then compared. The procedure is equivalent to comparing data of various levels of detail and therefore structural complexity. This gives a measure of the structural significance of each data set in space and as a consequence areas of geometric complexity are identified. These areas are usually very data sensitive hence geometric variability in between individual data points in these areas is higher than in areas of low structural complexity. Instead of calculating a multitude of different models by varying some input data or parameters as it is done by Monte-Carlo-simulations, the aim of the second step of the evaluation procedure (which is part of the ongoing work) is to
On the inverse problem of the calculus of variations in field theory
International Nuclear Information System (INIS)
Henneaux, M.
1984-01-01
The inverse problem of the calculus of variations is investigated in the case of field theory. Uniqueness of the action principle is demonstrated for the vector Laplace equation in a non-decomposable Riemannian space, as well as for the harmonic map equation. (author)
A geometric construction of traveling waves in a bioremediation model
Beck, M.A.; Doelman, A.; Kaper, T.J.
2006-01-01
Bioremediation is a promising technique for cleaning contaminated soil. We study an idealized bioremediation model involving a substrate (contaminant to be removed), electron acceptor (added nutrient), and microorganisms in a one-dimensional soil column. Using geometric singular perturbation theory,
A new approach toward geometrical concept of black hole thermodynamics
Energy Technology Data Exchange (ETDEWEB)
Hendi, Seyed Hossein [Shiraz University, Physics Department and Biruni Observatory, College of Sciences, Shiraz (Iran, Islamic Republic of); Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha (Iran, Islamic Republic of); Panahiyan, Shahram; Panah, Behzad Eslam; Momennia, Mehrab [Shiraz University, Physics Department and Biruni Observatory, College of Sciences, Shiraz (Iran, Islamic Republic of)
2015-10-15
Motivated by the energy representation of Riemannian metric, in this paper we study different approaches toward the geometrical concept of black hole thermodynamics. We investigate thermodynamical Ricci scalar of Weinhold, Ruppeiner and Quevedo metrics and show that their number and location of divergences do not coincide with phase transition points arisen from heat capacity. Next, we introduce a new metric to solve these problems. We show that the denominator of the Ricci scalar of the new metric contains terms which coincide with different types of phase transitions. We elaborate the effectiveness of the new metric and shortcomings of the previous metrics with some examples. Furthermore, we find a characteristic behavior of the new thermodynamical Ricci scalar which enables one to distinguish two types of phase transitions. In addition, we generalize the new metric for the cases of more than two extensive parameters and show that in these cases the divergencies of thermodynamical Ricci scalar coincide with phase transition points of the heat capacity. (orig.)
A new approach toward geometrical concept of black hole thermodynamics
International Nuclear Information System (INIS)
Hendi, Seyed Hossein; Panahiyan, Shahram; Panah, Behzad Eslam; Momennia, Mehrab
2015-01-01
Motivated by the energy representation of Riemannian metric, in this paper we study different approaches toward the geometrical concept of black hole thermodynamics. We investigate thermodynamical Ricci scalar of Weinhold, Ruppeiner and Quevedo metrics and show that their number and location of divergences do not coincide with phase transition points arisen from heat capacity. Next, we introduce a new metric to solve these problems. We show that the denominator of the Ricci scalar of the new metric contains terms which coincide with different types of phase transitions. We elaborate the effectiveness of the new metric and shortcomings of the previous metrics with some examples. Furthermore, we find a characteristic behavior of the new thermodynamical Ricci scalar which enables one to distinguish two types of phase transitions. In addition, we generalize the new metric for the cases of more than two extensive parameters and show that in these cases the divergencies of thermodynamical Ricci scalar coincide with phase transition points of the heat capacity. (orig.)
Haisch, B. M.
1976-01-01
A tensor formulation of the equation of radiative transfer is derived in a seven-dimensional Riemannian space such that the resulting equation constitutes a divergence in any coordinate system. After being transformed to a spherically symmetric comoving coordinate system, the transfer equation contains partial derivatives in angle and frequency, as well as optical depth due to the effects of aberration and the Doppler shift. However, by virtue of the divergence form of this equation, the divergence theorem may be applied to yield a numerical differencing scheme which is expected to be stable and to conserve luminosity. It is shown that the equation of transfer derived by this method in a Lagrangian coordinate system may be reduced to that given by Castor (1972), although it is, of course, desirable to leave the equation in divergence form.
International Nuclear Information System (INIS)
Audretsch, J.; Gaehler, F.; Straumann, N.
1984-01-01
Previous axiomatic approaches to general relativity which led to a Weylian structure of space-time are supplemented by a physical condition which implies the existence of a preferred pseudo-Riemannian structure. It is stipulated that the trajectories of the short wave limit of classical massive fields agree with the geodesics of the Weyl connection and it is shown that this is equivalent to the vanishing of the covariant derivative of a ''mass function'' of nontrivial Weyl type.This in turn is proven to be equivalent to the existence of a preferred metric of the conformal structure such that the Weyl connection is reducible to a connection of the bundle of orthonormal frames belonging to this distinguished metric. (orig.)
Geometric structures on loop and path spaces
Indian Academy of Sciences (India)
Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 4, September 2010, pp. 417–428. .... that a vector field X in a Riemannian manifold (M,g) is locally gradient-like .... Note that if we apply this map to γ (t) itself, we get a curve x(t) ∈ Tγ(0)M. Now we.
Dynamics in geometrical confinement
Kremer, Friedrich
2014-01-01
This book describes the dynamics of low molecular weight and polymeric molecules when they are constrained under conditions of geometrical confinement. It covers geometrical confinement in different dimensionalities: (i) in nanometer thin layers or self supporting films (1-dimensional confinement) (ii) in pores or tubes with nanometric diameters (2-dimensional confinement) (iii) as micelles embedded in matrices (3-dimensional) or as nanodroplets.The dynamics under such conditions have been a much discussed and central topic in the focus of intense worldwide research activities within the last two decades. The present book discusses how the resulting molecular mobility is influenced by the subtle counterbalance between surface effects (typically slowing down molecular dynamics through attractive guest/host interactions) and confinement effects (typically increasing the mobility). It also explains how these influences can be modified and tuned, e.g. through appropriate surface coatings, film thicknesses or pore...
Geometric information provider platform
Directory of Open Access Journals (Sweden)
Meisam Yousefzadeh
2015-07-01
Full Text Available Renovation of existing buildings is known as an essential stage in reduction of the energy loss. Considerable part of renovation process depends on geometric reconstruction of building based on semantic parameters. Following many research projects which were focused on parameterizing the energy usage, various energy modelling methods were developed during the last decade. On the other hand, by developing accurate measuring tools such as laser scanners, the interests of having accurate 3D building models are rapidly growing. But the automation of 3D building generation from laser point cloud or detection of specific objects in that is still a challenge. The goal is designing a platform through which required geometric information can be efficiently produced to support energy simulation software. Developing a reliable procedure which extracts required information from measured data and delivers them to a standard energy modelling system is the main purpose of the project.
Geometric transitions on non-Kaehler manifolds
International Nuclear Information System (INIS)
Knauf, A.
2007-01-01
We study geometric transitions on the supergravity level using the basic idea of an earlier paper (M. Becker et al., 2004), where a pair of non-Kaehler backgrounds was constructed, which are related by a geometric transition. Here we embed this idea into an orientifold setup. The non-Kaehler backgrounds we obtain in type IIA are non-trivially fibered due to their construction from IIB via T-duality with Neveu-Schwarz flux. We demonstrate that these non-Kaehler manifolds are not half-flat and show that a symplectic structure exists on them at least locally. We also review the construction of new non-Kaehler backgrounds in type I and heterotic theory. They are found by a series of T- and S-duality and can be argued to be related by geometric transitions as well. A local toy model is provided that fulfills the flux equations of motion in IIB and the torsional relation in heterotic theory, and that is consistent with the U-duality relating both theories. For the heterotic theory we also propose a global solution that fulfills the torsional relation because it is similar to the Maldacena-Nunez background. (Abstract Copyright [2007], Wiley Periodicals, Inc.)
Plasma geometric optics analysis and computation
International Nuclear Information System (INIS)
Smith, T.M.
1983-01-01
Important practical applications in the generation, manipulation, and diagnosis of laboratory thermonuclear plasmas have created a need for elaborate computational capabilities in the study of high frequency wave propagation in plasmas. A reduced description of such waves suitable for digital computation is provided by the theory of plasma geometric optics. The existing theory is beset by a variety of special cases in which the straightforward analytical approach fails, and has been formulated with little attention to problems of numerical implementation of that analysis. The standard field equations are derived for the first time from kinetic theory. A discussion of certain terms previously, and erroneously, omitted from the expansion of the plasma constitutive relation is given. A powerful but little known computational prescription for determining the geometric optics field in the neighborhood of caustic singularities is rigorously developed, and a boundary layer analysis for the asymptotic matching of the plasma geometric optics field across caustic singularities is performed for the first time with considerable generality. A proper treatment of birefringence is detailed, wherein a breakdown of the fundamental perturbation theory is identified and circumvented. A general ray tracing computer code suitable for applications to radiation heating and diagnostic problems is presented and described
Developing geometrical reasoning
Brown, Margaret; Jones, Keith; Taylor, Ron; Hirst, Ann
2004-01-01
This paper summarises a report (Brown, Jones & Taylor, 2003) to the UK Qualifications and Curriculum Authority of the work of one geometry group. The group was charged with developing and reporting on teaching ideas that focus on the development of geometrical reasoning at the secondary school level. The group was encouraged to explore what is possible both within and beyond the current requirements of the UK National Curriculum and the Key Stage 3 strategy, and to consider the whole atta...
Geometrically Consistent Mesh Modification
Bonito, A.
2010-01-01
A new paradigm of adaptivity is to execute refinement, coarsening, and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy. We refer to this collectively as geometrically consistent (GC) mesh modification. We discuss the concept of discrete GC, show the failure of naive approaches, and propose and analyze a simple algorithm that is GC and accuracy preserving. © 2010 Society for Industrial and Applied Mathematics.
Geometric transitions, flops and non-Kahler manifolds: I
International Nuclear Information System (INIS)
Becker, Melanie; Dasgupta, Keshav; Knauf, Anke; Tatar, Radu
2004-01-01
We construct a duality cycle which provides a complete supergravity description of geometric transitions in type II theories via a flop in M-theory. This cycle connects the different supergravity descriptions before and after the geometric transitions. Our construction reproduces many of the known phenomena studied earlier in the literature and allows us to describe some new and interesting aspects in a simple and elegant fashion. A precise supergravity description of new torsional manifolds that appear on the type IIA side with branes and fluxes and the corresponding geometric transition are obtained. A local description of new G2 manifolds that are circle fibrations over non-Kahler manifolds is presented
Geometric leaf placement strategies
International Nuclear Information System (INIS)
Fenwick, J D; Temple, S W P; Clements, R W; Lawrence, G P; Mayles, H M O; Mayles, W P M
2004-01-01
Geometric leaf placement strategies for multileaf collimators (MLCs) typically involve the expansion of the beam's-eye-view contour of a target by a uniform MLC margin, followed by movement of the leaves until some point on each leaf end touches the expanded contour. Film-based dose-distribution measurements have been made to determine appropriate MLC margins-characterized through an index d 90 -for multileaves set using one particular strategy to straight lines lying at various angles to the direction of leaf travel. Simple trigonometric relationships exist between different geometric leaf placement strategies and are used to generalize the results of the film work into d 90 values for several different strategies. Measured d 90 values vary both with angle and leaf placement strategy. A model has been derived that explains and describes quite well the observed variations of d 90 with angle. The d 90 angular variations of the strategies studied differ substantially, and geometric and dosimetric reasoning suggests that the best strategy is the one with the least angular variation. Using this criterion, the best straightforwardly implementable strategy studied is a 'touch circle' approach for which semicircles are imagined to be inscribed within leaf ends, the leaves being moved until the semicircles just touch the expanded target outline
Studies in geometric quantization
International Nuclear Information System (INIS)
Tuynman, G.M.
1988-01-01
This thesis contains five chapters, of which the first, entitled 'What is prequantization, and what is geometric quantization?', is meant as an introduction to geometric quantization for the non-specialist. The second chapter, entitled 'Central extensions and physics' deals with the notion of central extensions of manifolds and elaborates and proves the statements made in the first chapter. Central extensions of manifolds occur in physics as the freedom of a phase factor in the quantum mechanical state vector, as the phase factor in the prequantization process of classical mechanics and it appears in mathematics when studying central extension of Lie groups. In this chapter the connection between these central extensions is investigated and a remarkable similarity between classical and quantum mechanics is shown. In chapter three a classical model is given for the hydrogen atom including spin-orbit and spin-spin interaction. The method of geometric quantization is applied to this model and the results are discussed. In the final chapters (4 and 5) an explicit method to calculate the operators corresponding to classical observables is given when the phase space is a Kaehler manifold. The obtained formula are then used to quantise symplectic manifolds which are irreducible hermitian symmetric spaces and the results are compared with other quantization procedures applied to these manifolds (in particular to Berezin's quantization). 91 refs.; 3 tabs
Can EPR non-locality be geometrical?
International Nuclear Information System (INIS)
Ne'eman, Y.
1995-01-01
The presence in Quantum Mechanics of non-local correlations is one of the two fundamentally non-intuitive features of that theory. The non-local correlations themselves fall into two classes: EPR and Geometrical. The non-local characteristics of the geometrical type are well-understood and are not suspected of possibly generating acausal features, such as faster-than-light propagation of information. This has especially become true since the emergence of a geometrical treatment for the relevant gauge theories, i.e. Fiber Bundle geometry, in which the quantum non-localities are seen to correspond to pure homotopy considerations. This aspect is reviewed in section 2. Contrary-wise, from its very conception, the EPR situation was felt to be paradoxical. It has been suggested that the non-local features of EPR might also derive from geometrical considerations, like all other non-local characteristics of QM. In[7], one of the authors was able to point out several plausibility arguments for this thesis, emphasizing in particular similarities between the non-local correlations provided by any gauge field theory and those required by the preservation of the quantum numbers of the original EPR state-vector, throughout its spatially-extended mode. The derivation was, however, somewhat incomplete, especially because of the apparent difference between, on the one hand, the closed spatial loops arising in the analysis of the geometrical non-localities, from Aharonov-Bohm and Berry phases to magnetic monopoles and instantons, and on the other hand, in the EPR case, the open line drawn by the positions of the two moving decay products of the disintegrating particle. In what follows, the authors endeavor to remove this obstacle and show that as in all other QM non-localities, EPR is somehow related to closed loops, almost involving homotopy considerations. They develop this view in section 3
Geometrical model of multiple production
International Nuclear Information System (INIS)
Chikovani, Z.E.; Jenkovszky, L.L.; Kvaratshelia, T.M.; Struminskij, B.V.
1988-01-01
The relation between geometrical and KNO-scaling and their violation is studied in a geometrical model of multiple production of hadrons. Predictions concerning the behaviour of correlation coefficients at future accelerators are given
Geometric Computing for Freeform Architecture
Wallner, J.; Pottmann, Helmut
2011-01-01
Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and construction-aware design processes of freeform architecture. We report on our work in this area
Gauge theory and gravitation: an approach to a fiber bundle formalism
International Nuclear Information System (INIS)
Mello, L.A. de.
1986-01-01
The thesis is composed of two different parts. A formal complete and rigorous mathematical part-of topics of differential manilfolds, exterior calculus, riemannian geometry, principal fiber bundle (p.f.) with connections and linear connections and a second part of application of this mathematical formalism concerning physical theories, particularly the Maxwell eletromagnetism (EM), gauge theory of Yang-Mills (Y-M), the GRT, and the gravitation theory of Einstein-Cartan. (E.C.) [pt
Scale-covariant theory of gravitation and astrophysical applications
International Nuclear Information System (INIS)
Canuto, V.; Adams, P.J.; Hsieh, S.; Tsiang, E.
1977-01-01
By associating the mathematical operation of scale transformation with the physics of using different dynamical systems to measure space-time distances, we formulate a scale-covariant theory of gravitation. Corresponding to each dynamical system of units is a gauge condition which determines the otherwise arbitrary gauge function. For gravitational units, the gauge condition is chosen so that the standard Einstein equations are recovered. Assuming the atomic units, derivable from atomic dynamics, to be distinct from the gravitational units, a different gauge condition must be imposed. It is suggested that Dirac's large-number hypothesis be used for the determination of this condition so that gravitational phenomena can be described in atomic units. The result allows a natural interpretation of the possible variation of the gravitational constant without compromising the validity of general relativity. A geometrical interpretation of the scale-covariant theory is possible if the covariant tensors in Riemannian space are replaced by cocovariant cotensors in an integrable Weyl space. A scale-invariant action principle is constructed from the metrical potentials of the integrable Weyl space. Application of the dynamical equations in atomic units to cosmology yields a family of homogeneous solutions characterized by R approx. t for large cosmological times. Equations of motion in atomic units are solved for spherically symmetric gravitational fields. Expressions for perihelion shift and light deflection are derived. They do not differ from the predictions of general relativity except for secular variations, having the age of the universe as a time scale. Similar variations of periods and radii for planetary orbits are also derived
Manfredini, Maria; Morbidelli, Daniele; Polidoro, Sergio; Uguzzoni, Francesco
2015-01-01
The analysis of PDEs is a prominent discipline in mathematics research, both in terms of its theoretical aspects and its relevance in applications. In recent years, the geometric properties of linear and nonlinear second order PDEs of elliptic and parabolic type have been extensively studied by many outstanding researchers. This book collects contributions from a selected group of leading experts who took part in the INdAM meeting "Geometric methods in PDEs", on the occasion of the 70th birthday of Ermanno Lanconelli. They describe a number of new achievements and/or the state of the art in their discipline of research, providing readers an overview of recent progress and future research trends in PDEs. In particular, the volume collects significant results for sub-elliptic equations, potential theory and diffusion equations, with an emphasis on comparing different methodologies and on their implications for theory and applications. .
Geometric Constructions with the Computer.
Chuan, Jen-chung
The computer can be used as a tool to represent and communicate geometric knowledge. With the appropriate software, a geometric diagram can be manipulated through a series of animation that offers more than one particular snapshot as shown in a traditional mathematical text. Geometric constructions with the computer enable the learner to see and…
Geometrical Aspects of non-gravitational interactions
Roldan, Omar; Barros Jr, C. C.
2016-01-01
In this work we look for a geometric description of non-gravitational forces. The basic ideas are proposed studying the interaction between a punctual particle and an electromagnetic external field. For this purpose, we introduce the concept of proper space-time, that allow us to describe this interaction in a way analogous to the one that the general relativity theory does for gravitation. The field equations that define this geometry are similar to the Einstein's equations, where in general...
Geometric (Berry) phases in neutron molecular spectroscopy
International Nuclear Information System (INIS)
Lovesey, S.W.
1992-02-01
A theory of neutron scattering by nuclei in a molecule, accompanied by an electronic transition, is formulated with attention to gauge potentials and geometric phases in the Born-Oppenheimer scheme. Non-degenerate and nearly degenerate electronic levels are considered. For nearly degenerate levels it is shown that, the cross-section is free of the singular structure which characterizes the corresponding gauge potential for the phase, and much larger than for well separated electronic states. (author)
Corrochano, Eduardo Bayro
2010-01-01
This book presents contributions from a global selection of experts in the field. This useful text offers new insights and solutions for the development of theorems, algorithms and advanced methods for real-time applications across a range of disciplines. Written in an accessible style, the discussion of all applications is enhanced by the inclusion of numerous examples, figures and experimental analysis. Features: provides a thorough discussion of several tasks for image processing, pattern recognition, computer vision, robotics and computer graphics using the geometric algebra framework; int
Geometric multipartite entanglement measures
International Nuclear Information System (INIS)
Paz-Silva, Gerardo A.; Reina, John H.
2007-01-01
Within the framework of constructions for quantifying entanglement, we build a natural scenario for the assembly of multipartite entanglement measures based on Hopf bundle-like mappings obtained through Clifford algebra representations. Then, given the non-factorizability of an arbitrary two-qubit density matrix, we give an alternate quantity that allows the construction of two types of entanglement measures based on their arithmetical and geometrical averages over all pairs of qubits in a register of size N, and thus fully characterize its degree and type of entanglement. We find that such an arithmetical average is both additive and strongly super additive
Geometric correlations and multifractals
International Nuclear Information System (INIS)
Amritkar, R.E.
1991-07-01
There are many situations where the usual statistical methods are not adequate to characterize correlations in the system. To characterize such situations we introduce mutual correlation dimensions which describe geometric correlations in the system. These dimensions allow us to distinguish between variables which are perfectly correlated with or without a phase lag, variables which are uncorrelated and variables which are partially correlated. We demonstrate the utility of our formalism by considering two examples from dynamical systems. The first example is about the loss of memory in chaotic signals and describes auto-correlations while the second example is about synchronization of chaotic signals and describes cross-correlations. (author). 19 refs, 6 figs
Immagini e Concetti in Geometria=The Figural and the Conceptual Components of Geometrical Concepts.
Mariotti, Maria Alessandra
1992-01-01
Discusses geometrical reasoning in the framework of the theory of Figural Concepts to highlight the interaction between the figural and conceptual components of geometrical concepts. Examples of students' difficulties and errors in geometrical reasoning are interpreted according to the internal tension that appears in figural concepts resulting…
A geometric form of the canonical commutation
International Nuclear Information System (INIS)
Guz, W.
1987-01-01
Some aspects of a geometric approach to quantum theory, in which the quantum-mechanical position and momentum operators are represented by covariant derivatives, are here developed. Here, the previously estabilished formalism of Caianiello and his co-workers is extended to the case of an integrable almost complex Hermitian manifold. The general theory is then applied to the two-dimensional case, where the structure of the 'quantum geometry' induced in the manifold by the quantum-mechanical CCR can be explicitly determined
Geometric Algebra Techniques in Flux Compactifications
International Nuclear Information System (INIS)
Coman, Ioana Alexandra; Lazaroiu, Calin Iuliu; Babalic, Elena Mirela
2016-01-01
We study “constrained generalized Killing (s)pinors,” which characterize supersymmetric flux compactifications of supergravity theories. Using geometric algebra techniques, we give conceptually clear and computationally effective methods for translating supersymmetry conditions into differential and algebraic constraints on collections of differential forms. In particular, we give a synthetic description of Fierz identities, which are an important ingredient of such problems. As an application, we show how our approach can be used to efficiently treat N=1 compactification of M-theory on eight manifolds and prove that we recover results previously obtained in the literature.
On the geometrization of electromagnetism by torsion
International Nuclear Information System (INIS)
Fonseca Neto, J.B. da.
1984-01-01
The possibility of electromagnetism geometrization using an four dimension Cartan geometry is investigated. The Lagrangian density which presents dual invariance for dyons electrodynamics formulated in term of two potentials is constructed. This theory by association of two potentials with track and with torsion pseudo-track and of the field with torsion covariant divergent is described. The minimum coupling of particle gravitational field of scalar and spinorial fields with dyon geometry theory by the minimum coupling of these fields with Cartan geometry was obtained. (author)
Geometric interpretation of optimal iteration strategies
International Nuclear Information System (INIS)
Jones, R.B.
1977-01-01
The relationship between inner and outer iteration errors is extremely complex, and even formal description of total error behavior is difficult. Inner and outer iteration error propagation is analyzed in a variational formalism for a reactor model describing multidimensional, one-group theory. In a generalization the work of Akimov and Sabek, the number of inner iterations performed during each outer serial that minimizes the total computation time is determined. The generalized analysis admits a geometric interpretation of total error behavior. The results can be applied to both transport and diffusion theory computer methods. 1 figure
Moduli stabilization in non-geometric backgrounds
International Nuclear Information System (INIS)
Becker, Katrin; Becker, Melanie; Vafa, Cumrun; Walcher, Johannes
2007-01-01
Type II orientifolds based on Landau-Ginzburg models are used to describe moduli stabilization for flux compactifications of type II theories from the world-sheet CFT point of view. We show that for certain types of type IIB orientifolds which have no Kaehler moduli and are therefore intrinsically non-geometric, all moduli can be explicitly stabilized in terms of fluxes. The resulting four-dimensional theories can describe Minkowski as well as anti-de Sitter vacua. This construction provides the first string vacuum with all moduli frozen and leading to a 4D Minkowski background
In the realm of the geometric transitions
International Nuclear Information System (INIS)
Alexander, Stephon; Becker, Katrin; Becker, Melanie; Dasgupta, Keshav; Knauf, Anke; Tatar, Radu
2005-01-01
We complete the duality cycle by constructing the geometric transition duals in the type IIB, type I and heterotic theories. We show that in the type IIB theory the background on the closed string side is a Kaehler deformed conifold, as expected, even though the mirror type IIA backgrounds are non-Kaehler (both before and after the transition). On the other hand, the type I and heterotic backgrounds are non-Kaehler. Therefore, on the heterotic side these backgrounds give rise to new torsional manifolds that have not been studied before. We show the consistency of these backgrounds by verifying the torsional equation
String theory compactifications
Graña, Mariana
2017-01-01
The lectures in this book provide graduate students and non-specialist researchers with a concise introduction to the concepts and formalism required to reduce the ten-dimensional string theories to the observable four-dimensional space-time - a procedure called string compactification. The text starts with a very brief introduction to string theory, first working out its massless spectrum and showing how the condition on the number of dimensions arises. It then dwells on the different possible internal manifolds, from the simplest to the most relevant phenomenologically, thereby showing that the most elegant description is through an extension of ordinary Riemannian geometry termed generalized geometry, which was first introduced by Hitchin. Last but not least, the authors review open problems in string phenomenology, such as the embedding of the Standard Model and obtaining de Sitter solutions.
Towards spectral geometric methods for Euclidean quantum gravity
Panine, Mikhail; Kempf, Achim
2016-04-01
The unification of general relativity with quantum theory will also require a coming together of the two quite different mathematical languages of general relativity and quantum theory, i.e., of differential geometry and functional analysis, respectively. Of particular interest in this regard is the field of spectral geometry, which studies to which extent the shape of a Riemannian manifold is describable in terms of the spectra of differential operators defined on the manifold. Spectral geometry is hard because it is highly nonlinear, but linearized spectral geometry, i.e., the task to determine small shape changes from small spectral changes, is much more tractable and may be iterated to approximate the full problem. Here, we generalize this approach, allowing, in particular, nonequal finite numbers of shape and spectral degrees of freedom. This allows us to study how well the shape degrees of freedom are encoded in the eigenvalues. We apply this strategy numerically to a class of planar domains and find that the reconstruction of small shape changes from small spectral changes is possible if enough eigenvalues are used. While isospectral nonisometric shapes are known to exist, we find evidence that generically shaped isospectral nonisometric shapes, if existing, are exceedingly rare.
The construction of periodic unfolding operators on some compact Riemannian manifolds
DEFF Research Database (Denmark)
Dobberschütz, Sören; Böhm, Michael
2014-01-01
The notion of periodic unfolding has become a standard tool in the theory of periodic homogenization. However, all the results obtained so far are only applicable to the "flat" Euclidean space R n. In this paper, we present a generalization of the method of periodic unfolding applicable to struct...
Gauge field vacuum structure in geometrical aspect
International Nuclear Information System (INIS)
Konopleva, N.P.
2003-01-01
Vacuum conception is one of the main conceptions of quantum field theory. Its meaning in classical field theory is also very profound. In this case the vacuum conception is closely connected with ideas of the space-time geometry. The global and local geometrical space-time conceptions lead to different vacuum definitions and therefore to different ways of physical theory construction. Some aspects of the gauge field vacuum structure are analyzed. It is shown that in the gauge field theory the vacuum Einstein equation solutions describe the relativistic vacuum as common vacuum of all gauge fields and its sources. Instantons (both usual and hyperbolical) are regarded as nongravitating matter, because they have zero energy-momentum tensors and correspond to vacuum Einstein equations
Material inhomogeneities and their evolution a geometric approach
Epstein, Marcelo
2007-01-01
Presents a unified treatment of the inhomogeneity theory using some of the tools of modern differential geometry. This book deals with the geometrical description of uniform bodies and their homogeneity conditions. It also develops a theory of material evolution and discusses its relevance in various applied contexts.
Unification and geometrization of physics in the cosmological context
International Nuclear Information System (INIS)
Heller, M.; Watykanskie Obserwatorium Astronomiczne, Vatican
1991-01-01
Einstein belived that a good physical theory should posses an ''inner perfection''. Trace the inner perfection of the present gauge theories by contemplating their geometric structures (in terms of fibre bundles). The search for the ultimate symmetry of the unification of physics unavoidably leads to the unification of physics and cosmology. 4 figs., 23 refs. (author)
Geometrical determination of the constant of motion in General Relativity
International Nuclear Information System (INIS)
Catoni, F.; Cannata, R.; Zampetti, P.
2009-01-01
In recent time a theorem, due to E. Beltrami, through which the integration of the geodesic equations of a curved manifold is obtained by means of a merely geometric method, has been revisited. This way of dealing with the problem is well in accordance with the geometric spirit of the Theory of General Relativity. In this paper we show another relevant consequence of this method. Actually, the constants of the motion, introduced in this geometrical way that is completely independent of Newton theory, are related to the conservation laws for test particles in the Einstein theory. These conservation laws may be compared with the conservation laws of Newton. In particular, by the conservation of energy (E) and the L z component of angular momentum, the equivalence of the conservation laws for the Schwarzschild field is verified and the difference between Newton and Einstein theories for the rotating bodies (Kerr metric) is obtained in a straightforward way.
International Nuclear Information System (INIS)
Noga, M.T.
1984-01-01
This thesis addresses a number of important problems that fall within the framework of the new discipline of Computational Geometry. The list of topics covered includes sorting and selection, convex hull algorithms, the L 1 hull, determination of the minimum encasing rectangle of a set of points, the Euclidean and L 1 diameter of a set of points, the metric traveling salesman problem, and finding the superrange of star-shaped and monotype polygons. The main theme of all the work was to develop a set of very fast state-of-the-art algorithms that supersede any rivals in terms of speed and ease of implementation. In some cases existing algorithms were refined; for others new techniques were developed that add to the present database of fast adaptive geometric algorithms. What emerges is a collection of techniques that is successful at merging modern tools developed in analysis of algorithms with those of classical geometry
Geometrization of quantum physics
International Nuclear Information System (INIS)
Ol'khov, O.A.
2009-01-01
It is shown that the Dirac equation for a free particle can be considered as a description of specific distortion of the space Euclidean geometry (space topological defect). This approach is based on the possibility of interpretation of the wave function as vector realizing representation of the fundamental group of the closed topological space-time 4-manifold. Mass and spin appear to be topological invariants. Such a concept explains all so-called 'strange' properties of quantum formalism: probabilities, wave-particle duality, nonlocal instantaneous correlation between noninteracting particles (EPR-paradox) and so on. Acceptance of the suggested geometrical concept means rejection of atomistic concept where all matter is considered as consisting of more and more small elementary particles. There are no any particles a priory, before measurement: the notions of particles appear as a result of classical interpretation of the contact of the region of the curved space with a device
Geometrization of quantum physics
Ol'Khov, O. A.
2009-12-01
It is shown that the Dirac equation for free particle can be considered as a description of specific distortion of the space euclidean geometry (space topological defect). This approach is based on possibility of interpretation of the wave function as vector realizing representation of the fundamental group of the closed topological space-time 4-manifold. Mass and spin appear to be topological invariants. Such concept explains all so called “strange” properties of quantum formalism: probabilities, wave-particle duality, nonlocal instantaneous correlation between noninteracting particles (EPR-paradox) and so on. Acceptance of suggested geometrical concept means rejection of atomistic concept where all matter is considered as consisting of more and more small elementary particles. There is no any particles a priori, before measurement: the notions of particles appear as a result of classical interpretation of the contact of the region of the curved space with a device.
Havelka, Jan
2008-01-01
Tato diplomová práce se zabývá akcelerací geometrických transformací obrazu s využitím GPU a architektury NVIDIA (R) CUDA TM. Časově kritické části kódu jsou přesunuty na GPU a vykonány paralelně. Jedním z výsledků je demonstrační aplikace pro porovnání výkonnosti obou architektur: CPU, a GPU v kombinaci s CPU. Pro referenční implementaci jsou použity vysoce optimalizované algoritmy z knihovny OpenCV, od firmy Intel. This master's thesis deals with acceleration of geometrical image transfo...
Yang Mills instantons, geometrical aspects
International Nuclear Information System (INIS)
Stora, R.
1977-09-01
The word instanton has been coined by analogy with the word soliton. They both refer to solutions of elliptic non linear field equations with boundary conditions at infinity (of euclidean space time in the first case, euclidean space in the second case) lying on the set of classical vacua in such a way that stable topological properties emerge, susceptible to survive quantum effects, if those are small. Under this assumption, instantons are believed to be relevant to the description of tunnelling effects between classical vacua and signal some characteristics of the vacuum at the quantum level, whereas solitons should be associated with particles, i.e. discrete points in the mass spectrum. In one case the euclidean action is finite, in the other case, the energy is finite. From the mathematical point of view, the geometrical phenomena associated with the existence of solitons have forced physicists to learn rudiments of algebraic topology. The study of euclidean classical Yang Mills fields involves naturally mathematical items falling under the headings: differential geometry (fibre bundles, connections); differential topology (characteristic classes, index theory) and more recently algebraic geometry. These notes are divided as follows: a first section is devoted to a description of the physicist's views; a second section is devoted to the mathematician's vie
Geometrical description of fields
International Nuclear Information System (INIS)
Sokolik, H.
1979-01-01
The author suggests a purely algebraic interpretation of interaction. The main idea is to consider interaction as a deformation of an inhomogeneous algebra composed of momentum operators and an arbitrary group admitting the equation of the theory. The only difference between this approach and the conventional one is that the generalized momentum operators do not commute with aech other, due not merely to the introduction of some external interaction field, but to the change of the structure of the algebra from which the theory stems
Geometric mechanics of periodic pleated origami.
Wei, Z Y; Guo, Z V; Dudte, L; Liang, H Y; Mahadevan, L
2013-05-24
Origami structures are mechanical metamaterials with properties that arise almost exclusively from the geometry of the constituent folds and the constraint of piecewise isometric deformations. Here we characterize the geometry and planar and nonplanar effective elastic response of a simple periodically folded Miura-ori structure, which is composed of identical unit cells of mountain and valley folds with four-coordinated ridges, defined completely by two angles and two lengths. We show that the in-plane and out-of-plane Poisson's ratios are equal in magnitude, but opposite in sign, independent of material properties. Furthermore, we show that effective bending stiffness of the unit cell is singular, allowing us to characterize the two-dimensional deformation of a plate in terms of a one-dimensional theory. Finally, we solve the inverse design problem of determining the geometric parameters for the optimal geometric and mechanical response of these extreme structures.
Coated sphere scattering by geometric optics approximation.
Mengran, Zhai; Qieni, Lü; Hongxia, Zhang; Yinxin, Zhang
2014-10-01
A new geometric optics model has been developed for the calculation of light scattering by a coated sphere, and the analytic expression for scattering is presented according to whether rays hit the core or not. The ray of various geometric optics approximation (GOA) terms is parameterized by the number of reflections in the coating/core interface, the coating/medium interface, and the number of chords in the core, with the degeneracy path and repeated path terms considered for the rays striking the core, which simplifies the calculation. For the ray missing the core, the various GOA terms are dealt with by a homogeneous sphere. The scattering intensity of coated particles are calculated and then compared with those of Debye series and Aden-Kerker theory. The consistency of the results proves the validity of the method proposed in this work.
Geometric modeling in probability and statistics
Calin, Ovidiu
2014-01-01
This book covers topics of Informational Geometry, a field which deals with the differential geometric study of the manifold probability density functions. This is a field that is increasingly attracting the interest of researchers from many different areas of science, including mathematics, statistics, geometry, computer science, signal processing, physics and neuroscience. It is the authors’ hope that the present book will be a valuable reference for researchers and graduate students in one of the aforementioned fields. This textbook is a unified presentation of differential geometry and probability theory, and constitutes a text for a course directed at graduate or advanced undergraduate students interested in applications of differential geometry in probability and statistics. The book contains over 100 proposed exercises meant to help students deepen their understanding, and it is accompanied by software that is able to provide numerical computations of several information geometric objects. The reader...
A practical guide to experimental geometrical optics
Garbovskiy, Yuriy A
2017-01-01
A concise, yet deep introduction to experimental, geometrical optics, this book begins with fundamental concepts and then develops the practical skills and research techniques routinely used in modern laboratories. Suitable for students, researchers and optical engineers, this accessible text teaches readers how to build their own optical laboratory and to design and perform optical experiments. It uses a hands-on approach which fills a gap between theory-based textbooks and laboratory manuals, allowing the reader to develop their practical skills in this interdisciplinary field, and also explores the ways in which this knowledge can be applied to the design and production of commercial optical devices. Including supplementary online resources to help readers track and evaluate their experimental results, this text is the ideal companion for anyone with a practical interest in experimental geometrical optics.
Discrete geometric structures for architecture
Pottmann, Helmut
2010-06-13
The emergence of freeform structures in contemporary architecture raises numerous challenging research problems, most of which are related to the actual fabrication and are a rich source of research topics in geometry and geometric computing. The talk will provide an overview of recent progress in this field, with a particular focus on discrete geometric structures. Most of these result from practical requirements on segmenting a freeform shape into planar panels and on the physical realization of supporting beams and nodes. A study of quadrilateral meshes with planar faces reveals beautiful relations to discrete differential geometry. In particular, we discuss meshes which discretize the network of principal curvature lines. Conical meshes are among these meshes; they possess conical offset meshes at a constant face/face distance, which in turn leads to a supporting beam layout with so-called torsion free nodes. This work can be generalized to a variety of multilayer structures and laid the ground for an adapted curvature theory for these meshes. There are also efforts on segmenting surfaces into planar hexagonal panels. Though these are less constrained than planar quadrilateral panels, this problem is still waiting for an elegant solution. Inspired by freeform designs in architecture which involve circles and spheres, we present a new kind of triangle mesh whose faces\\' in-circles form a packing, i.e., the in-circles of two triangles with a common edge have the same contact point on that edge. These "circle packing (CP) meshes" exhibit an aesthetic balance of shape and size of their faces. They are closely tied to sphere packings on surfaces and to various remarkable structures and patterns which are of interest in art, architecture, and design. CP meshes constitute a new link between architectural freeform design and computational conformal geometry. Recently, certain timber structures motivated us to study discrete patterns of geodesics on surfaces. This
International Nuclear Information System (INIS)
Kaku, M.
1987-01-01
In this article, the authors summarize the rapid progress in constructing string field theory actions, such as the development of the covariant BRST theory. They also present the newer geometric formulation of string field theory, from which the BRST theory and the older light cone theory can be derived from first principles. This geometric formulation allows us to derive the complete field theory of strings from two geometric principles, in the same way that general relativity and Yang-Mills theory can be derived from two principles based on global and local symmetry. The geometric formalism therefore reduces string field theory to a problem of finding an invariant under a new local gauge group they call the universal string group (USG). Thus, string field theory is the gauge theory of the universal string group in much the same way that Yang-Mills theory is the gauge theory of SU(N). The geometric formulation places superstring theory on the same rigorous group theoretical level as general relativity and gauge theory
Regular Polygons and Geometric Series.
Jarrett, Joscelyn A.
1982-01-01
Examples of some geometric illustrations of limits are presented. It is believed the limit concept is among the most important topics in mathematics, yet many students do not have good intuitive feelings for the concept, since it is often taught very abstractly. Geometric examples are suggested as meaningful tools. (MP)
Geometric Invariants and Object Recognition.
1992-08-01
University of Chicago Press. Maybank , S.J. [1992], "The Projection of Two Non-coplanar Conics", in Geometric Invariance in Machine Vision, eds. J.L...J.L. Mundy and A. Zisserman, MIT Press, Cambridge, MA. Mundy, J.L., Kapur, .. , Maybank , S.J., and Quan, L. [1992a] "Geometric Inter- pretation of
Transmuted Complementary Weibull Geometric Distribution
Directory of Open Access Journals (Sweden)
Ahmed Z. A fify
2014-12-01
Full Text Available This paper provides a new generalization of the complementary Weibull geometric distribution that introduced by Tojeiro et al. (2014, using the quadratic rank transmutation map studied by Shaw and Buckley (2007. The new distribution is referred to as transmuted complementary Weibull geometric distribution (TCWGD. The TCWG distribution includes as special cases the complementary Weibull geometric distribution (CWGD, complementary exponential geometric distribution(CEGD,Weibull distribution (WD and exponential distribution (ED. Various structural properties of the new distribution including moments, quantiles, moment generating function and RØnyi entropy of the subject distribution are derived. We proposed the method of maximum likelihood for estimating the model parameters and obtain the observed information matrix. A real data set are used to compare the exibility of the transmuted version versus the complementary Weibull geometric distribution.
Geometric singular perturbation analysis of systems with friction
DEFF Research Database (Denmark)
Bossolini, Elena
This thesis is concerned with the application of geometric singular perturbation theory to mechanical systems with friction. The mathematical background on geometric singular perturbation theory, on the blow-up method, on non-smooth dynamical systems and on regularization is presented. Thereafter......, two mechanical problems with two diﬀerent formulations of the friction force are introduced and analysed. The ﬁrst mechanical problem is a one-dimensional spring-block model describing earthquake faulting. The dynamics of earthquakes is naturally a multiple timescale problem: the timescale...... scales. The action of friction is generally explained as the loss and restoration of linkages between the surface asperities at the molecular scale. However, the consequences of friction are noticeable at much larger scales, like hundreds of kilometers. By using geometric singular perturbation theory...
Modeling Geometric-Temporal Context With Directional Pyramid Co-Occurrence for Action Recognition.
Yuan, Chunfeng; Li, Xi; Hu, Weiming; Ling, Haibin; Maybank, Stephen J
2014-02-01
In this paper, we present a new geometric-temporal representation for visual action recognition based on local spatio-temporal features. First, we propose a modified covariance descriptor under the log-Euclidean Riemannian metric to represent the spatio-temporal cuboids detected in the video sequences. Compared with previously proposed covariance descriptors, our descriptor can be measured and clustered in Euclidian space. Second, to capture the geometric-temporal contextual information, we construct a directional pyramid co-occurrence matrix (DPCM) to describe the spatio-temporal distribution of the vector-quantized local feature descriptors extracted from a video. DPCM characterizes the co-occurrence statistics of local features as well as the spatio-temporal positional relationships among the concurrent features. These statistics provide strong descriptive power for action recognition. To use DPCM for action recognition, we propose a directional pyramid co-occurrence matching kernel to measure the similarity of videos. The proposed method achieves the state-of-the-art performance and improves on the recognition performance of the bag-of-visual-words (BOVWs) models by a large margin on six public data sets. For example, on the KTH data set, it achieves 98.78% accuracy while the BOVW approach only achieves 88.06%. On both Weizmann and UCF CIL data sets, the highest possible accuracy of 100% is achieved.
Uniform asymptotic theory of edge diffraction
Lewis, R.M.; Boersma, J.; Oughstun, K.E.
1992-01-01
Geometrical optics fails to account for the phenomenon of diffraction, i.e., the existence of nonzero fields in the geometrical shadow. Keller's geometrical theory of diffraction accounts for this phenomenon by providing correction terms to the geometrical optics field, in the form of a
Uniform asymptotic theory of edge diffraction
Lewis, R.M.; Boersma, J.
1969-01-01
Geometrical optics fails to account for the phenomenon of diffraction, i.e., the existence of nonzero fields in the geometrical shadow. Keller's geometrical theory of diffraction accounts for this phenomenon by providing correction terms to the geometrical optics field, in the form of a
Geometric low-energy effective action in a doubled spacetime
Ma, Chen-Te; Pezzella, Franco
2018-05-01
The ten-dimensional supergravity theory is a geometric low-energy effective theory and the equations of motion for its fields can be obtained from string theory by computing β functions. With d compact dimensions, an O (d , d ; Z) geometric structure can be added to it giving the supergravity theory with T-duality manifest. In this paper, this is constructed through the use of a suitable star product whose role is the one to implement the weak constraint on the fields and the gauge parameters in order to have a closed gauge symmetry algebra. The consistency of the action here proposed is based on the orthogonality of the momenta associated with fields in their triple star products in the cubic terms defined for d ≥ 1. This orthogonality holds also for an arbitrary number of star products of fields for d = 1. Finally, we extend our analysis to the double sigma model, non-commutative geometry and open string theory.
Time as a geometric property of space
Directory of Open Access Journals (Sweden)
James Michael Chappell
2016-11-01
Full Text Available The proper description of time remains a key unsolved problem in science. Newton conceived of time as absolute and universal which it `flows equably without relation to anything external'}. In the nineteenth century, the four-dimensional algebraic structure of the quaternions developed by Hamilton, inspired him to suggest that they could provide a unified representation of space and time. With the publishing of Einstein's theory of special relativity these ideas then lead to the generally accepted Minkowski spacetime formulation in 1908. Minkowski, though, rejected the formalism of quaternions suggested by Hamilton and adopted rather an approach using four-vectors. The Minkowski framework is indeed found to provide a versatile formalism for describing the relationship between space and time in accordance with Einstein's relativistic principles, but nevertheless fails to provide more fundamental insights into the nature of time itself. In order to answer this question we begin by exploring the geometric properties of three-dimensional space that we model using Clifford geometric algebra, which is found to contain sufficient complexity to provide a natural description of spacetime. This description using Clifford algebra is found to provide a natural alternative to the Minkowski formulation as well as providing new insights into the nature of time. Our main result is that time is the scalar component of a Clifford space and can be viewed as an intrinsic geometric property of three-dimensional space without the need for the specific addition of a fourth dimension.
Geometric inequalities for black holes
International Nuclear Information System (INIS)
Dain, Sergio
2013-01-01
Full text: A geometric inequality in General Relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the Kerr-Newman black hole satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities, which are valid in the dynamical and strong field regime, play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this talk I will review recent results in this subject. (author)
Geometric Computing for Freeform Architecture
Wallner, J.
2011-06-03
Geometric computing has recently found a new field of applications, namely the various geometric problems which lie at the heart of rationalization and construction-aware design processes of freeform architecture. We report on our work in this area, dealing with meshes with planar faces and meshes which allow multilayer constructions (which is related to discrete surfaces and their curvatures), triangles meshes with circle-packing properties (which is related to conformal uniformization), and with the paneling problem. We emphasize the combination of numerical optimization and geometric knowledge.
Optical traps with geometric aberrations
International Nuclear Information System (INIS)
Roichman, Yael; Waldron, Alex; Gardel, Emily; Grier, David G.
2006-01-01
We assess the influence of geometric aberrations on the in-plane performance of optical traps by studying the dynamics of trapped colloidal spheres in deliberately distorted holographic optical tweezers. The lateral stiffness of the traps turns out to be insensitive to moderate amounts of coma, astigmatism, and spherical aberration. Moreover holographic aberration correction enables us to compensate inherent shortcomings in the optical train, thereby adaptively improving its performance. We also demonstrate the effects of geometric aberrations on the intensity profiles of optical vortices, whose readily measured deformations suggest a method for rapidly estimating and correcting geometric aberrations in holographic trapping systems
Geometric inequalities for black holes
Energy Technology Data Exchange (ETDEWEB)
Dain, Sergio [Universidad Nacional de Cordoba (Argentina)
2013-07-01
Full text: A geometric inequality in General Relativity relates quantities that have both a physical interpretation and a geometrical definition. It is well known that the parameters that characterize the Kerr-Newman black hole satisfy several important geometric inequalities. Remarkably enough, some of these inequalities also hold for dynamical black holes. This kind of inequalities, which are valid in the dynamical and strong field regime, play an important role in the characterization of the gravitational collapse. They are closed related with the cosmic censorship conjecture. In this talk I will review recent results in this subject. (author)
Invariant Theory (IT) & Standard Monomial Theory (SMT)
Indian Academy of Sciences (India)
2013-07-06
Jul 6, 2013 ... Why invariant theory? (continued). Now imagine algebraic calculations being made, with the two different sets of co-ordinates, about something of geometrical or physical interest concerning the configuration of points, ...
The Geometric Phase in Quantum Systems
International Nuclear Information System (INIS)
Pascazio, S
2003-01-01
The discovery of the geometric phase is one of the most interesting and intriguing findings of the last few decades. It led to a deeper understanding of the concept of phase in quantum mechanics and motivated a surge of interest in fundamental quantum mechanical issues, disclosing unexpected applications in very diverse fields of physics. Although the key ideas underlying the existence of a purely geometrical phase had already been proposed in 1956 by Pancharatnam, it was Michael Berry who revived this issue 30 years later. The clarity of Berry's seminal paper, in 1984, was extraordinary. Research on the topic flourished at such a pace that it became difficult for non-experts to follow the many different theoretical ideas and experimental proposals which ensued. Diverse concepts in independent areas of mathematics, physics and chemistry were being applied, for what was (and can still be considered) a nascent arena for theory, experiments and technology. Although collections of papers by different authors appeared in the literature, sometimes with ample introductions, surprisingly, to the best of my knowledge, no specific and exhaustive book has ever been written on this subject. The Geometric Phase in Quantum Systems is the first thorough book on geometric phases and fills an important gap in the physical literature. Other books on the subject will undoubtedly follow. But it will take a fairly long time before other authors can cover that same variety of concepts in such a comprehensive manner. The book is enjoyable. The choice of topics presented is well balanced and appropriate. The appendices are well written, understandable and exhaustive - three rare qualities. I also find it praiseworthy that the authors decided to explicitly carry out most of the calculations, avoiding, as much as possible, the use of the joke 'after a straightforward calculation, one finds...' This was one of the sentences I used to dislike most during my undergraduate studies. A student is
Topological charge on the lattice: a field theoretical view of the geometrical approach
International Nuclear Information System (INIS)
Rastelli, L.; Rossi, P.; Vicari, E.
1997-01-01
We construct sequences of ''field theoretical'' lattice topological charge density operators which formally approach geometrical definitions in 2D CP N-1 models and 4D SU(N) Yang-Mills theories. The analysis of these sequences of operators suggests a new way of looking at the geometrical method, showing that geometrical charges can be interpreted as limits of sequences of field theoretical (analytical) operators. In perturbation theory, renormalization effects formally tend to vanish along such sequences. But, since the perturbative expansion is asymptotic, this does not necessarily lead to well-behaved geometrical limits. It indeed leaves open the possibility that non-perturbative renormalizations survive. (orig.)
Cox, David A
2012-01-01
Praise for the First Edition ". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!"—Monatshefte fur Mathematik Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields. Covering classic applications of the theory, such as solvability by radicals, geometric constructions, and finite fields, Galois Theory, Second Edition delves into novel topics like Abel’s theory of Abelian equations, casus irreducibili, and the Galo
Discrete geometric structures for architecture
Pottmann, Helmut
2010-01-01
. The talk will provide an overview of recent progress in this field, with a particular focus on discrete geometric structures. Most of these result from practical requirements on segmenting a freeform shape into planar panels and on the physical realization
Geometric Rationalization for Freeform Architecture
Jiang, Caigui
2016-01-01
The emergence of freeform architecture provides interesting geometric challenges with regards to the design and manufacturing of large-scale structures. To design these architectural structures, we have to consider two types of constraints. First
Geometrical optics in general relativity
Loinger, A.
2006-01-01
General relativity includes geometrical optics. This basic fact has relevant consequences that concern the physical meaning of the discontinuity surfaces propagated in the gravitational field - as it was first emphasized by Levi-Civita.
Mobile Watermarking against Geometrical Distortions
Directory of Open Access Journals (Sweden)
Jing Zhang
2015-08-01
Full Text Available Mobile watermarking robust to geometrical distortions is still a great challenge. In mobile watermarking, efficient computation is necessary because mobile devices have very limited resources due to power consumption. In this paper, we propose a low-complexity geometrically resilient watermarking approach based on the optimal tradeoff circular harmonic function (OTCHF correlation filter and the minimum average correlation energy Mellin radial harmonic (MACE-MRH correlation filter. By the rotation, translation and scale tolerance properties of the two kinds of filter, the proposed watermark detector can be robust to geometrical attacks. The embedded watermark is weighted by a perceptual mask which matches very well with the properties of the human visual system. Before correlation, a whitening process is utilized to improve watermark detection reliability. Experimental results demonstrate that the proposed watermarking approach is computationally efficient and robust to geometrical distortions.
On quantum field theory in gravitational background
International Nuclear Information System (INIS)
Haag, R.; Narnhofer, H.; Stein, U.
1984-02-01
We discuss Quantum Fields on Riemannian space-time. A principle of local definitness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non inertial motion are added. (orig.)
Geometric inequalities methods of proving
Sedrakyan, Hayk
2017-01-01
This unique collection of new and classical problems provides full coverage of geometric inequalities. Many of the 1,000 exercises are presented with detailed author-prepared-solutions, developing creativity and an arsenal of new approaches for solving mathematical problems. This book can serve teachers, high-school students, and mathematical competitors. It may also be used as supplemental reading, providing readers with new and classical methods for proving geometric inequalities. .
Geometric methods for discrete dynamical systems
Easton, Robert W
1998-01-01
This book looks at dynamics as an iteration process where the output of a function is fed back as an input to determine the evolution of an initial state over time. The theory examines errors which arise from round-off in numerical simulations, from the inexactness of mathematical models used to describe physical processes, and from the effects of external controls. The author provides an introduction accessible to beginning graduate students and emphasizing geometric aspects of the theory. Conley''s ideas about rough orbits and chain-recurrence play a central role in the treatment. The book will be a useful reference for mathematicians, scientists, and engineers studying this field, and an ideal text for graduate courses in dynamical systems.
A Geometrical Approach to Bell's Theorem
Rubincam, David Parry
2000-01-01
Bell's theorem can be proved through simple geometrical reasoning, without the need for the Psi function, probability distributions, or calculus. The proof is based on N. David Mermin's explication of the Einstein-Podolsky-Rosen-Bohm experiment, which involves Stern-Gerlach detectors which flash red or green lights when detecting spin-up or spin-down. The statistics of local hidden variable theories for this experiment can be arranged in colored strips from which simple inequalities can be deduced. These inequalities lead to a demonstration of Bell's theorem. Moreover, all local hidden variable theories can be graphed in such a way as to enclose their statistics in a pyramid, with the quantum-mechanical result lying a finite distance beneath the base of the pyramid.
A geometrical description of local and global anomalies
International Nuclear Information System (INIS)
Catenacci, R.; Pirola, G.P.
1990-01-01
The general topological framework for testing the possible occurrence of anomalies in gauge theories can be constructed in terms of the theory of group actions on line bundles through the introduction of a suitable group cohomology. In this Letter, we generalize this construction in such a way that it can be applied to a larger class of theories, allowing for a noncontractible configuration space and a nonconnected 'gauge' group. This construction find applications to the problem of the lifts of principal group actions. As a physical application, we compare the mechanisms of the anomalies cancelation in gauge and string theories, through a geometrical splitting of local and global anomalies. (orig.)
Geometric Mixing, Peristalsis, and the Geometric Phase of the Stomach.
Arrieta, Jorge; Cartwright, Julyan H E; Gouillart, Emmanuelle; Piro, Nicolas; Piro, Oreste; Tuval, Idan
2015-01-01
Mixing fluid in a container at low Reynolds number--in an inertialess environment--is not a trivial task. Reciprocating motions merely lead to cycles of mixing and unmixing, so continuous rotation, as used in many technological applications, would appear to be necessary. However, there is another solution: movement of the walls in a cyclical fashion to introduce a geometric phase. We show using journal-bearing flow as a model that such geometric mixing is a general tool for using deformable boundaries that return to the same position to mix fluid at low Reynolds number. We then simulate a biological example: we show that mixing in the stomach functions because of the "belly phase," peristaltic movement of the walls in a cyclical fashion introduces a geometric phase that avoids unmixing.
Geometric Mixing, Peristalsis, and the Geometric Phase of the Stomach.
Directory of Open Access Journals (Sweden)
Jorge Arrieta
Full Text Available Mixing fluid in a container at low Reynolds number--in an inertialess environment--is not a trivial task. Reciprocating motions merely lead to cycles of mixing and unmixing, so continuous rotation, as used in many technological applications, would appear to be necessary. However, there is another solution: movement of the walls in a cyclical fashion to introduce a geometric phase. We show using journal-bearing flow as a model that such geometric mixing is a general tool for using deformable boundaries that return to the same position to mix fluid at low Reynolds number. We then simulate a biological example: we show that mixing in the stomach functions because of the "belly phase," peristaltic movement of the walls in a cyclical fashion introduces a geometric phase that avoids unmixing.
Space-time-matter analytic and geometric structures
Brüning, Jochen
2018-01-01
At the boundary of mathematics and mathematical physics, this monograph explores recent advances in the mathematical foundations of string theory and cosmology. The geometry of matter and the evolution of geometric structures as well as special solutions, singularities and stability properties of the underlying partial differential equations are discussed.
SIAM Conference on Geometric Design and Computing. Final Technical Report
Energy Technology Data Exchange (ETDEWEB)
None
2002-03-11
The SIAM Conference on Geometric Design and Computing attracted 164 domestic and international researchers, from academia, industry, and government. It provided a stimulating forum in which to learn about the latest developments, to discuss exciting new research directions, and to forge stronger ties between theory and applications. Final Report
A geometrical approach to free-field quantization
International Nuclear Information System (INIS)
Tabensky, R.; Valle, J.W.F.
1977-01-01
A geometrical approach to the quantization of free relativistic fields is given. Complex probability amplitudes are assigned to the solutions of the classical evolution equation. It is assumed that the evolution is stricly classical, according to the scalar unitary representation of the Poincare group in a functional space. The theory is equivalent to canonical quantization [pt
Eisenhart, Luther Pfahler
2005-01-01
This concise text by a prominent mathematician deals chiefly with manifolds dominated by the geometry of paths. Topics include asymmetric and symmetric connections, the projective geometry of paths, and the geometry of sub-spaces. 1927 edition.
Geometric procedures for civil engineers
Tonias, Elias C
2016-01-01
This book provides a multitude of geometric constructions usually encountered in civil engineering and surveying practice. A detailed geometric solution is provided to each construction as well as a step-by-step set of programming instructions for incorporation into a computing system. The volume is comprised of 12 chapters and appendices that may be grouped in three major parts: the first is intended for those who love geometry for its own sake and its evolution through the ages, in general, and, more specifically, with the introduction of the computer. The second section addresses geometric features used in the book and provides support procedures used by the constructions presented. The remaining chapters and the appendices contain the various constructions. The volume is ideal for engineering practitioners in civil and construction engineering and allied areas.
Topology, ergodic theory, real algebraic geometry Rokhlin's memorial
Turaev, V
2001-01-01
This book is dedicated to the memory of the outstanding Russian mathematician, V. A. Rokhlin (1919-1984). It is a collection of research papers written by his former students and followers, who are now experts in their fields. The topics in this volume include topology (the Morse-Novikov theory, spin bordisms in dimension 6, and skein modules of links), real algebraic geometry (real algebraic curves, plane algebraic surfaces, algebraic links, and complex orientations), dynamics (ergodicity, amenability, and random bundle transformations), geometry of Riemannian manifolds, theory of Teichmüller
An introduction to geometrical physics
Aldrovandi, R
1995-01-01
This book stresses the unifying power of the geometrical framework in bringing together concepts from the different areas of physics. Common underpinnings of optics, elasticity, gravitation, relativistic fields, particle mechanics and other subjects are underlined. It attempts to extricate the notion of space currently in the physical literature from the metric connotation.The book's goal is to present mathematical ideas associated with geometrical physics in a rather introductory language. Included are many examples from elementary physics and also, for those wishing to reach a higher level o
Geometric scaling as traveling waves
International Nuclear Information System (INIS)
Munier, S.; Peschanski, R.
2003-01-01
We show the relevance of the nonlinear Fisher and Kolmogorov-Petrovsky-Piscounov (KPP) equation to the problem of high energy evolution of the QCD amplitudes. We explain how the traveling wave solutions of this equation are related to geometric scaling, a phenomenon observed in deep-inelastic scattering experiments. Geometric scaling is for the first time shown to result from an exact solution of nonlinear QCD evolution equations. Using general results on the KPP equation, we compute the velocity of the wave front, which gives the full high energy dependence of the saturation scale
Geometric integration for particle accelerators
International Nuclear Information System (INIS)
Forest, Etienne
2006-01-01
This paper is a very personal view of the field of geometric integration in accelerator physics-a field where often work of the highest quality is buried in lost technical notes or even not published; one has only to think of Simon van der Meer Nobel prize work on stochastic cooling-unpublished in any refereed journal. So I reconstructed the relevant history of geometrical integration in accelerator physics as much as I could by talking to collaborators and using my own understanding of the field. The reader should not be too surprised if this account is somewhere between history, science and perhaps even fiction
Geometric integration for particle accelerators
Forest, Étienne
2006-05-01
This paper is a very personal view of the field of geometric integration in accelerator physics—a field where often work of the highest quality is buried in lost technical notes or even not published; one has only to think of Simon van der Meer Nobel prize work on stochastic cooling—unpublished in any refereed journal. So I reconstructed the relevant history of geometrical integration in accelerator physics as much as I could by talking to collaborators and using my own understanding of the field. The reader should not be too surprised if this account is somewhere between history, science and perhaps even fiction.
Lattice degeneracies of geometric fermions
International Nuclear Information System (INIS)
Raszillier, H.
1983-05-01
We give the minimal numbers of degrees of freedom carried by geometric fermions on all lattices of maximal symmetries in d = 2, 3, and 4 dimensions. These numbers are lattice dependent, but in the (free) continuum limit, part of the degrees of freedom have to escape to infinity by a Wilson mechanism built in, and 2sup(d) survive for any lattice. On self-reciprocal lattices we compare the minimal numbers of degrees of freedom of geometric fermions with the minimal numbers of naive fermions on these lattices and argue that these numbers are equal. (orig.)
Gravity, general relativity theory and alternative theories
International Nuclear Information System (INIS)
Zel'dovich, Ya.B.; Grishchuk, L.P.; Moskovskij Gosudarstvennyj Univ.
1986-01-01
The main steps in plotting the current gravitation theory and some prospects of its subsequent development are reviewed. The attention is concentrated on a comparison of the relativistic gravitational field with other physical fields. Two equivalent formulations of the general relativity (GR) - geometrical and field-theoretical - are considered in detail. It is shown that some theories of gravity constructed as the field theories at a flat background space-time are in fact just different formulations of GR and not alternative theories
International Nuclear Information System (INIS)
Kikkawa, Keiji; Nakanishi, Noboru; Nariai, Hidekazu
1983-01-01
These proceedings contain the articles presented at the named symposium. They deal with geometrical aspects of gauge theory and gravitation, special problems in gauge theories, quantum field theory in curved space-time, quantum gravity, supersymmetry including supergravity, and grand unification. See hints under the relevant topics. (HSI)
Classification of mammographic masses using geometric symmetry and fractal analysis
Energy Technology Data Exchange (ETDEWEB)
Guo Qi; Ruiz, V.F. [Cybernetics, School of Systems Engineering, Univ. of Reading (United Kingdom); Shao Jiaqing [Dept. of Electronics, Univ. of Kent (United Kingdom); Guo Falei [WanDe Industrial Engineering Co. (China)
2007-06-15
In this paper, we propose a fuzzy symmetry measure based on geometrical operations to characterise shape irregularity of mammographic mass lesion. Group theory, a powerful tool in the investigation of geometric transformation, is employed in our work to define and describe the underlying mathematical relations. We investigate the usefulness of fuzzy symmetry measure in combination with fractal analysis for classification of masses. Comparative studies show that fuzzy symmetry measure is useful for shape characterisation of mass lesions and is a good complementary feature for benign-versus-malignant classification of masses. (orig.)
Geometric singularities and spectra of Landau-Ginzburg models
International Nuclear Information System (INIS)
Greene, B.R.; Roan, S.S.; Yau, S.T.
1991-01-01
Some mathematical and physical aspects of superconformal string compactification in weighted projective space are discussed. In particular, we recast the path integral argument establishing the connection between Landau-Ginsburg conformal theories and Calabi-Yau string compactification in a geometric framework. We then prove that the naive expression for the vanishing of the first Chern class for a complete intersection (adopted from the smooth case) is sufficient to ensure that the resulting variety, which is generically singular, can be resolved to a smooth Calabi-Yau space. This justifies much analysis which has recently been expended on the study of Landau-Ginzburg models. Furthermore, we derive some simple formulae for the determination of the Witten index in these theories which are complementary to those derived using semiclassical reasoning by Vafa. Finally, we also comment on the possible geometrical significance of unorbifolded Landau-Ginzburg theories. (orig.)
Height and Tilt Geometric Texture
DEFF Research Database (Denmark)
Andersen, Vedrana; Desbrun, Mathieu; Bærentzen, Jakob Andreas
2009-01-01
compromise between functionality and simplicity: it can efficiently handle and process geometric texture too complex to be represented as a height field, without having recourse to full blown mesh editing algorithms. The height-and-tilt representation proposed here is fully intrinsic to the mesh, making...
In Defence of Geometrical Algebra
Blasjo, V.N.E.
The geometrical algebra hypothesis was once the received interpretation of Greek mathematics. In recent decades, however, it has become anathema to many. I give a critical review of all arguments against it and offer a consistent rebuttal case against the modern consensus. Consequently, I find that
Geometric scaling in exclusive processes
International Nuclear Information System (INIS)
Munier, S.; Wallon, S.
2003-01-01
We show that according to the present understanding of the energy evolution of the observables measured in deep-inelastic scattering, the photon-proton scattering amplitude has to exhibit geometric scaling at each impact parameter. We suggest a way to test this experimentally at HERA. A qualitative analysis based on published data is presented and discussed. (orig.)
Geometric origin of central charges
International Nuclear Information System (INIS)
Lukierski, J.; Rytel, L.
1981-05-01
The complete set of N(N-1) central charge generators for D=4 N-extended super Poincare algebra is obtained by suitable contraction of OSp (2N; 4) superalgebra. The superspace realizations of the spinorial generators with central charges are derived. The conjugate set of N(N-1) additional bosonic superspace coordinates is introduced in an unique and geometric way. (author)
Vergence, Vision, and Geometric Optics
Keating, Michael P.
1975-01-01
Provides a definition of vergence in terms of the curvature of the wave fronts, and gives examples to illustrate the advantages of this approach. The vergence treatment of geometrical optics provides both conceptual and algebraic advantages, particularly for the life science student, over the traditional object distance-image distance-focal length…
Geometric phases and quantum computation
International Nuclear Information System (INIS)
Vedral, V.
2005-01-01
Full text: In my lectures I will talk about the notion of the geometric phase and explain its relevance for both fundamental quantum mechanics as well as quantum computation. The phase will be at first introduced via the idea of Pancharatnam which involves interference of three or more light beams. This notion will then be generalized to the evolving quantum systems. I will discuss both pure and mixed states as well as unitary and non-unitary evolutions. I will also show how the concept of the vacuum induced geometric phase arises in quantum optics. A simple measurement scheme involving a Mach Zehnder interferometer will be presented and will be used to illustrate all the concepts in the lecture. Finally, I will expose a simple generalization of the geometric phase to evolving degenerate states. This will be seen to lead to the possibility of universal quantum computation using geometric effects only. Moreover, this contains a promise of intrinsically fault tolerant quantum information processing, whose prospects will be outlined at the end of the lecture. (author)
A geometric viewpoint on generalized hydrodynamics
Directory of Open Access Journals (Sweden)
Benjamin Doyon
2018-01-01
Full Text Available Generalized hydrodynamics (GHD is a large-scale theory for the dynamics of many-body integrable systems. It consists of an infinite set of conservation laws for quasi-particles traveling with effective (“dressed” velocities that depend on the local state. We show that these equations can be recast into a geometric dynamical problem. They are conservation equations with state-independent quasi-particle velocities, in a space equipped with a family of metrics, parametrized by the quasi-particles' type and speed, that depend on the local state. In the classical hard rod or soliton gas picture, these metrics measure the free length of space as perceived by quasi-particles; in the quantum picture, they weigh space with the density of states available to them. Using this geometric construction, we find a general solution to the initial value problem of GHD, in terms of a set of integral equations where time appears explicitly. These integral equations are solvable by iteration and provide an extremely efficient solution algorithm for GHD.
Brain Morphometry on Congenital Hand Deformities based on Teichmüller Space Theory.
Peng, Hao; Wang, Xu; Duan, Ye; Frey, Scott H; Gu, Xianfeng
2015-01-01
Congenital Hand Deformities (CHD) are usually occurred between fourth and eighth week after the embryo is formed. Failure of the transformation from arm bud cells to upper limb can lead to an abnormal appearing/functioning upper extremity which is presented at birth. Some causes are linked to genetics while others are affected by the environment, and the rest have remained unknown. CHD patients develop prehension through the use of their hands, which affect the brain as time passes. In recent years, CHD have gain increasing attention and researches have been conducted on CHD, both surgically and psychologically. However, the impacts of CHD on brain structure are not well-understood so far. Here, we propose a novel approach to apply Teichmüller space theory and conformal welding method to study brain morphometry in CHD patients. Conformal welding signature reflects the geometric relations among different functional areas on the cortex surface, which is intrinsic to the Riemannian metric, invariant under conformal deformation, and encodes complete information of the functional area boundaries. The computational algorithm is based on discrete surface Ricci flow, which has theoretic guarantees for the existence and uniqueness of the solutions. In practice, discrete Ricci flow is equivalent to a convex optimization problem, therefore has high numerically stability. In this paper, we compute the signatures of contours on general 3D surfaces with surface Ricci flow method, which encodes both global and local surface contour information. Then we evaluated the signatures of pre-central and post-central gyrus on healthy control and CHD subjects for analyzing brain cortical morphometry. Preliminary experimental results from 3D MRI data of CHD/control data demonstrate the effectiveness of our method. The statistical comparison between left and right brain gives us a better understanding on brain morphometry of subjects with Congenital Hand Deformities, in particular, missing
The Effect of Bulk Tachyon Field on the Dynamics of Geometrical Tachyon
International Nuclear Information System (INIS)
Papantonopoulos, Eleftherios; Pappa, Ioanna; Zamarias, Vassilios
2007-01-01
We study the dynamics of the geometrical tachyon field on an unstable D3-brane in the background of a bulk tachyon field of a D3-brane solution of Type-0 string theory. We find that the geometrical tachyon potential is modified by a function of the bulk tachyon and inflation occurs at weak string coupling, where the bulk tachyon condenses, near the top of the geometrical tachyon potential. We also find a late accelerating phase when the bulk tachyon asymptotes to zero and the geometrical tachyon field reaches the minimum of the potential
Geometric Properties of Grassmannian Frames for and
Directory of Open Access Journals (Sweden)
Benedetto John J
2006-01-01
Full Text Available Grassmannian frames are frames satisfying a min-max correlation criterion. We translate a geometrically intuitive approach for two- and three-dimensional Euclidean space ( and into a new analytic method which is used to classify many Grassmannian frames in this setting. The method and associated algorithm decrease the maximum frame correlation, and hence give rise to the construction of specific examples of Grassmannian frames. Many of the results are known by other techniques, and even more generally, so that this paper can be viewed as tutorial. However, our analytic method is presented with the goal of developing it to address unresovled problems in -dimensional Hilbert spaces which serve as a setting for spherical codes, erasure channel modeling, and other aspects of communications theory.
International Nuclear Information System (INIS)
Bergmann, P.G.
1980-01-01
A problem of construction of the unitary field theory is discussed. The preconditions of the theory are briefly described. The main attention is paid to the geometrical interpretation of physical fields. The meaning of the conceptions of diversity and exfoliation is elucidated. Two unitary field theories are described: the Weyl conformic geometry and Calitzy five-dimensioned theory. It is proposed to consider supersymmetrical theories as a new approach to the problem of a unitary field theory. It is noted that the supergravitational theories are really unitary theories, since the fields figuring there do not assume invariant expansion
A geometrical interpretation of renormalisation group flow
International Nuclear Information System (INIS)
Dolan, B.P.
1993-05-01
The renormalisation group (RG) equation in D-dimensional Euclidean space, R D , is analysed from a geometrical point of view. A general form of the RG equation is derived which is applicable to composite operators as well tensor operators (on R D ) which may depend on the Euclidean metric. It is argued that physical N-point amplitudes should be interpreted as rank N co-variant tensors on the space of couplings, G, and that the RG equation can be viewed as an equation for Lie transport on G with respect to the vector field generated by the β-functions of the theory. In one sense it is nothing more than the definition of a Lie derivative. The source of the anomalous dimensions can be interpreted as being due to the change of the basis vectors on G under Lie transport. The RG equation acts as a bridge between Euclidean space and coupling constant space in that the effect on amplitudes of a diffeomorphism of R D (that of dilations) is completely equivalent to a diffeomorphism of G generated by the β-functions of the theory. A form of the RG equation for operators is also given. These ideas are developed in detail for the example of massive λΦ 4 theory in 4 dimensions. (orig.)
The differential-geometric aspects of integrable dynamical systems
International Nuclear Information System (INIS)
Prykarpatsky, Y.A.; Samoilenko, A.M.; Prykarpatsky, A.K.; Bogolubov, N.N. Jr.; Blackmore, D.L.
2007-05-01
The canonical reduction method on canonically symplectic manifolds is analyzed in detail, and the relationships with the geometric properties of associated principal fiber bundles endowed with connection structures are described. Some results devoted to studying geometrical properties of nonabelian Yang-Mills type gauge field equations are presented. A symplectic theory approach is developed for partially solving the problem of algebraic-analytical construction of integral submanifold embeddings for integrable (via the abelian and nonabelian Liouville-Arnold theorems) Hamiltonian systems on canonically symplectic phase spaces. The fundamental role of the so-called Picard-Fuchs type equations is revealed, and their differential-geometric and algebraic properties are studied in detail. Some interesting examples of integrable Hamiltonian systems are are studied in detail in order to demonstrate the ease of implementation and effectiveness of the procedure for investigating the integral submanifold embedding mapping. (author)
Auto-focusing accelerating hyper-geometric laser beams
International Nuclear Information System (INIS)
Kovalev, A A; Kotlyar, V V; Porfirev, A P
2016-01-01
We derive a new solution to the paraxial wave equation that defines a two-parameter family of three-dimensional structurally stable vortex annular auto-focusing hyper-geometric (AH) beams, with their complex amplitude expressed via a degenerate hyper-geometric function. The AH beams are found to carry an orbital angular momentum and be auto-focusing, propagating on an accelerating path toward a focus, where the annular intensity pattern is ‘sharply’ reduced in diameter. An explicit expression for the complex amplitude of vortex annular auto-focusing hyper-geometric-Gaussian beams is derived. The experiment has been shown to be in good agreement with theory. (paper)
Stochastic pump effect and geometric phases in dissipative and stochastic systems
Energy Technology Data Exchange (ETDEWEB)
Sinitsyn, Nikolai [Los Alamos National Laboratory
2008-01-01
The success of Berry phases in quantum mechanics stimulated the study of similar phenomena in other areas of physics, including the theory of living cell locomotion and motion of patterns in nonlinear media. More recently, geometric phases have been applied to systems operating in a strongly stochastic environment, such as molecular motors. We discuss such geometric effects in purely classical dissipative stochastic systems and their role in the theory of the stochastic pump effect (SPE).
Geometric Description of Fibre Bundle Surface for Birkhoff System
International Nuclear Information System (INIS)
Li-Mei, Cao; Hua-Fei, Sun; Zhen-Ning, Zhang
2009-01-01
A fibre bundle surface for the Birkhoff system is constructed. The metric and the Riemannian connection of the surface are defined and the representation of the Gaussian curvature of this surface is presented. Finally, three examples for the Birkhoff system are given to illustrate our results. (general)
From the Weyl theory to a theory of locally anisotropic space-time
International Nuclear Information System (INIS)
Bogoslovsky, G.Yu.
1991-01-01
It is shown that Weyl ideas, pertaining to local conformal invariance, find natural embodiment within the framework of a relativistic theory based on a viable Finslerian model of space-time. This is associated with the peculiar property of the conformal invariant Finslerian metric which describes a locally anisotropic space of events. The local conformal transformations of the Riemannian metric tensor leave invariant rest masses as well as all observables and thus appear as local gauge transformations. The corresponding Finslerian theory of gravitation turns out, as a result, to be an Abelian gauge theory. It satisfies the principle of correspondence with Einstein theory and predicts a number of nontrivial physical effects accessible for experimental test under laboratory conditions. 13 refs
On chromatic and geometrical calibration
DEFF Research Database (Denmark)
Folm-Hansen, Jørgen
1999-01-01
The main subject of the present thesis is different methods for the geometrical and chromatic calibration of cameras in various environments. For the monochromatic issues of the calibration we present the acquisition of monochrome images, the classic monochrome aberrations and the various sources...... the correct interpolation method is described. For the chromatic issues of calibration we present the acquisition of colour and multi-spectral images, the chromatic aberrations and the various lens/camera based non-uniformities of the illumination of the image plane. It is described how the monochromatic...... to design calibration targets for both geometrical and chromatic calibration are described. We present some possible systematical errors on the detection of the objects in the calibration targets, if viewed in a non orthogonal angle, if the intensities are uneven or if the image blurring is uneven. Finally...
Geometrical approach to tumor growth.
Escudero, Carlos
2006-08-01
Tumor growth has a number of features in common with a physical process known as molecular beam epitaxy. Both growth processes are characterized by the constraint of growth development to the body border, and surface diffusion of cells and particles at the growing edge. However, tumor growth implies an approximate spherical symmetry that makes necessary a geometrical treatment of the growth equations. The basic model was introduced in a former paper [C. Escudero, Phys. Rev. E 73, 020902(R) (2006)], and in the present work we extend our analysis and try to shed light on the possible geometrical principles that drive tumor growth. We present two-dimensional models that reproduce the experimental observations, and analyze the unexplored three-dimensional case, for which interesting conclusions on tumor growth are derived.
Geometrical interpretation of optical absorption
Energy Technology Data Exchange (ETDEWEB)
Monzon, J. J.; Barriuso, A. G.; Sanchez-Soto, L. L. [Departamento de Optica, Facultad de Fisica, Universidad Complutense, E-28040 Madrid (Spain); Montesinos-Amilibia, J. M. [Departamento de Geometria y Topologia, Facultad de Matematicas, Universidad Complutense, E-28040 Madrid (Spain)
2011-08-15
We reinterpret the transfer matrix for an absorbing system in very simple geometrical terms. In appropriate variables, the system appears as performing a Lorentz transformation in a (1 + 3)-dimensional space. Using homogeneous coordinates, we map that action on the unit sphere, which is at the realm of the Klein model of hyperbolic geometry. The effects of absorption appear then as a loxodromic transformation, that is, a rhumb line crossing all the meridians at the same angle.
Parametric FEM for geometric biomembranes
Bonito, Andrea; Nochetto, Ricardo H.; Sebastian Pauletti, M.
2010-05-01
We consider geometric biomembranes governed by an L2-gradient flow for bending energy subject to area and volume constraints (Helfrich model). We give a concise derivation of a novel vector formulation, based on shape differential calculus, and corresponding discretization via parametric FEM using quadratic isoparametric elements and a semi-implicit Euler method. We document the performance of the new parametric FEM with a number of simulations leading to dumbbell, red blood cell and toroidal equilibrium shapes while exhibiting large deformations.
Geometrical approach to tumor growth
Escudero, Carlos
2006-01-01
Tumor growth has a number of features in common with a physical process known as molecular beam epitaxy. Both growth processes are characterized by the constraint of growth development to the body border, and surface diffusion of cells/particles at the growing edge. However, tumor growth implies an approximate spherical symmetry that makes necessary a geometrical treatment of the growth equations. The basic model was introduced in a former article [C. Escudero, Phys. Rev. E 73, 020902(R) (200...
Geometric mean for subspace selection.
Tao, Dacheng; Li, Xuelong; Wu, Xindong; Maybank, Stephen J
2009-02-01
Subspace selection approaches are powerful tools in pattern classification and data visualization. One of the most important subspace approaches is the linear dimensionality reduction step in the Fisher's linear discriminant analysis (FLDA), which has been successfully employed in many fields such as biometrics, bioinformatics, and multimedia information management. However, the linear dimensionality reduction step in FLDA has a critical drawback: for a classification task with c classes, if the dimension of the projected subspace is strictly lower than c - 1, the projection to a subspace tends to merge those classes, which are close together in the original feature space. If separate classes are sampled from Gaussian distributions, all with identical covariance matrices, then the linear dimensionality reduction step in FLDA maximizes the mean value of the Kullback-Leibler (KL) divergences between different classes. Based on this viewpoint, the geometric mean for subspace selection is studied in this paper. Three criteria are analyzed: 1) maximization of the geometric mean of the KL divergences, 2) maximization of the geometric mean of the normalized KL divergences, and 3) the combination of 1 and 2. Preliminary experimental results based on synthetic data, UCI Machine Learning Repository, and handwriting digits show that the third criterion is a potential discriminative subspace selection method, which significantly reduces the class separation problem in comparing with the linear dimensionality reduction step in FLDA and its several representative extensions.
Control of the spin geometric phase in semiconductor quantum rings.
Nagasawa, Fumiya; Frustaglia, Diego; Saarikoski, Henri; Richter, Klaus; Nitta, Junsaku
2013-01-01
Since the formulation of the geometric phase by Berry, its relevance has been demonstrated in a large variety of physical systems. However, a geometric phase of the most fundamental spin-1/2 system, the electron spin, has not been observed directly and controlled independently from dynamical phases. Here we report experimental evidence on the manipulation of an electron spin through a purely geometric effect in an InGaAs-based quantum ring with Rashba spin-orbit coupling. By applying an in-plane magnetic field, a phase shift of the Aharonov-Casher interference pattern towards the small spin-orbit-coupling regions is observed. A perturbation theory for a one-dimensional Rashba ring under small in-plane fields reveals that the phase shift originates exclusively from the modulation of a pure geometric-phase component of the electron spin beyond the adiabatic limit, independently from dynamical phases. The phase shift is well reproduced by implementing two independent approaches, that is, perturbation theory and non-perturbative transport simulations.
Geometrization and Generalization of the Kowalevski Top
Dragović, Vladimir
2010-08-01
A new view on the Kowalevski top and the Kowalevski integration procedure is presented. For more than a century, the Kowalevski 1889 case, has attracted full attention of a wide community as the highlight of the classical theory of integrable systems. Despite hundreds of papers on the subject, the Kowalevski integration is still understood as a magic recipe, an unbelievable sequence of skillful tricks, unexpected identities and smart changes of variables. The novelty of our present approach is based on our four observations. The first one is that the so-called fundamental Kowalevski equation is an instance of a pencil equation of the theory of conics which leads us to a new geometric interpretation of the Kowalevski variables w, x 1, x 2 as the pencil parameter and the Darboux coordinates, respectively. The second is observation of the key algebraic property of the pencil equation which is followed by introduction and study of a new class of discriminantly separable polynomials. All steps of the Kowalevski integration procedure are now derived as easy and transparent logical consequences of our theory of discriminantly separable polynomials. The third observation connects the Kowalevski integration and the pencil equation with the theory of multi-valued groups. The Kowalevski change of variables is now recognized as an example of a two-valued group operation and its action. The final observation is surprising equivalence of the associativity of the two-valued group operation and its action to the n = 3 case of the Great Poncelet Theorem for pencils of conics.
Geometric constructions for repulsive gravity and quantization
International Nuclear Information System (INIS)
Hohmann, Manuel
2010-11-01
In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N>1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N=2. We further show that this theorem does not hold for N>2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite
Geometric constructions for repulsive gravity and quantization
Energy Technology Data Exchange (ETDEWEB)
Hohmann, Manuel
2010-11-15
In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N>1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N=2. We further show that this theorem does not hold for N>2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite
International Nuclear Information System (INIS)
Mukhi, S.
1986-01-01
A simple recursive algorithm is presented which generates the reparametrization-invariant background-field expansion for non-linear sigma-models on manifolds with an arbitrary riemannian metric. The method is also applicable to Wess-Zumino terms and to counterterms. As an example, the general-metric model is expanded to sixth order and compared with previous results. For locally symmetric spaces, we actually obtain a general formula for the nth order term. The method is shown to facilitate the study of models with Wess-Zumino terms. It is demonstrated that, for chiral models, the Wess-Zumino term is unrenormalized to all orders in perturbation theory even when the model is not conformally invariant. (orig.)
The geometric Langlands twist in five and six dimensions
International Nuclear Information System (INIS)
Bak, Dongsu; Gustavsson, Andreas
2015-01-01
Abelian 6d (2,0) theory has SO(5) R symmetry. We twist this theory by identifying the R symmetry group with the SO(5) subgroup of the SO(1,5) Lorentz group. This twisted theory can be put on any five-manifold M, times R, while preserving one scalar supercharge. We subsequently assume the existence of one unit normalized Killing vector field on M, and we find a corresponding SO(4) twist that preserves two supercharges and is a generalization of the geometric Langlands twist of 4d SYM. We generalize the story to non-Abelian gauge group for the corresponding 5d SYM theories on M. We derive a vanishing theorem for BPS contact instantons by identifying the 6d potential energy and its BPS bound, in the 5d theory. To this end we need to perform a Wick rotation that complexifies the gauge field.
An algebraic geometric approach to separation of variables
Schöbel, Konrad
2015-01-01
Konrad Schöbel aims to lay the foundations for a consequent algebraic geometric treatment of variable separation, which is one of the oldest and most powerful methods to construct exact solutions for the fundamental equations in classical and quantum physics. The present work reveals a surprising algebraic geometric structure behind the famous list of separation coordinates, bringing together a great range of mathematics and mathematical physics, from the late 19th century theory of separation of variables to modern moduli space theory, Stasheff polytopes and operads. "I am particularly impressed by his mastery of a variety of techniques and his ability to show clearly how they interact to produce his results.” (Jim Stasheff) Contents The Foundation: The Algebraic Integrability Conditions The Proof of Concept: A Complete Solution for the 3-Sphere The Generalisation: A Solution for Spheres of Arbitrary Dimension The Perspectives: Applications and Generalisations Target Groups Scientists in the fie...
Exact Solutions for Einstein's Hyperbolic Geometric Flow
International Nuclear Information System (INIS)
He Chunlei
2008-01-01
In this paper we investigate the Einstein's hyperbolic geometric flow and obtain some interesting exact solutions for this kind of flow. Many interesting properties of these exact solutions have also been analyzed and we believe that these properties of Einstein's hyperbolic geometric flow are very helpful to understanding the Einstein equations and the hyperbolic geometric flow
Geometric Structure-Preserving Discretization Schemes for Nonlinear Elasticity
2015-08-13
sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies. The main...conditions. 15. SUBJECT TERMS geometric theory for nonlinear elasticity, discrete exterior calculus 16. SECURITY CLASSIFICATION OF: 17. LIMITATION...complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress
Geometric Semantic Genetic Programming Algorithm and Slump Prediction
Xu, Juncai; Shen, Zhenzhong; Ren, Qingwen; Xie, Xin; Yang, Zhengyu
2017-01-01
Research on the performance of recycled concrete as building material in the current world is an important subject. Given the complex composition of recycled concrete, conventional methods for forecasting slump scarcely obtain satisfactory results. Based on theory of nonlinear prediction method, we propose a recycled concrete slump prediction model based on geometric semantic genetic programming (GSGP) and combined it with recycled concrete features. Tests show that the model can accurately p...
Hydrodynamic Limit with Geometric Correction of Stationary Boltzmann Equation
Wu, Lei
2014-01-01
We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. We prove the solution can be approximated in $L^{\\infty}$ by the sum of interior solution which satisfies steady incompressible Navier-Stokes-Fourier system, and boundary layer with geometric correction. Also, we construct a counterexample to the classical theory which states the behavior of solution near boundary can be described by the Knudsen layer derived from the Milne problem.
Supersymmetric sigma models and composite Yang-Mills theory
International Nuclear Information System (INIS)
Lukierski, J.
1980-04-01
We describe two types of supersymmetric sigma models: with field values in supercoset space and with superfields. The notion of Riemannian symmetric pair (H,G/H) is generalized to supergroups. Using the supercoset approach the superconformal-invariant model of composite U(n) Yang-Mills fields in introduced. In the framework of the superfield approach we present with some details two versions of the composite N=1 supersymmetric Yang-Mills theory in four dimensions with U(n) and U(m) x U(n) local invariance. We argue that especially the superfield sigma models can be used for the description of pre-QCD supersymmetric dynamics. (author)
Geometrically Induced Interactions and Bifurcations
Binder, Bernd
2010-01-01
In order to evaluate the proper boundary conditions in spin dynamics eventually leading to the emergence of natural and artificial solitons providing for strong interactions and potentials with monopole charges, the paper outlines a new concept referring to a curvature-invariant formalism, where superintegrability is given by a special isometric condition. Instead of referring to the spin operators and Casimir/Euler invariants as the generator of rotations, a curvature-invariant description is introduced utilizing a double Gudermann mapping function (generator of sine Gordon solitons and Mercator projection) cross-relating two angular variables, where geometric phases and rotations arise between surfaces of different curvature. Applying this stereographic projection to a superintegrable Hamiltonian can directly map linear oscillators to Kepler/Coulomb potentials and/or monopoles with Pöschl-Teller potentials and vice versa. In this sense a large scale Kepler/Coulomb (gravitational, electro-magnetic) wave dynamics with a hyperbolic metric could be mapped as a geodesic vertex flow to a local oscillator singularity (Dirac monopole) with spherical metrics and vice versa. Attracting fixed points and dynamic constraints are given by special isometries with magic precession angles. The nonlinear angular encoding directly provides for a Shannon mutual information entropy measure of the geodesic phase space flow. The emerging monopole patterns show relations to spiral Fresnel holography and Berry/Aharonov-Bohm geometric phases subject to bifurcation instabilities and singularities from phase ambiguities due to a local (entropy) overload. Neutral solitons and virtual patterns emerging and mediating in the overlap region between charged or twisted holographic patterns are visualized and directly assigned to the Berry geometric phase revealing the role of photons, neutrons, and neutrinos binding repulsive charges in Coulomb, strong and weak interaction.
Moving walls and geometric phases
Energy Technology Data Exchange (ETDEWEB)
Facchi, Paolo, E-mail: paolo.facchi@ba.infn.it [Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari (Italy); INFN, Sezione di Bari, I-70126 Bari (Italy); Garnero, Giancarlo, E-mail: giancarlo.garnero@uniba.it [Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari (Italy); INFN, Sezione di Bari, I-70126 Bari (Italy); Marmo, Giuseppe [Dipartimento di Scienze Fisiche and MECENAS, Università di Napoli “Federico II”, I-80126 Napoli (Italy); INFN, Sezione di Napoli, I-80126 Napoli (Italy); Samuel, Joseph [Raman Research Institute, 560080 Bangalore (India)
2016-09-15
We unveil the existence of a non-trivial Berry phase associated to the dynamics of a quantum particle in a one dimensional box with moving walls. It is shown that a suitable choice of boundary conditions has to be made in order to preserve unitarity. For these boundary conditions we compute explicitly the geometric phase two-form on the parameter space. The unboundedness of the Hamiltonian describing the system leads to a natural prescription of renormalization for divergent contributions arising from the boundary.
Geometric approach to soliton equations
International Nuclear Information System (INIS)
Sasaki, R.
1979-09-01
A class of nonlinear equations that can be solved in terms of nxn scattering problem is investigated. A systematic geometric method of exploiting conservation laws and related equations, the so-called prolongation structure, is worked out. The nxn problem is reduced to nsub(n-1)x(n-1) problems and finally to 2x2 problems, which have been comprehensively investigated recently by the author. A general method of deriving the infinite numbers of polynomial conservation laws for an nxn problem is presented. The cases of 3x3 and 2x2 problems are discussed explicitly. (Auth.)
Geometric Rationalization for Freeform Architecture
Jiang, Caigui
2016-06-20
The emergence of freeform architecture provides interesting geometric challenges with regards to the design and manufacturing of large-scale structures. To design these architectural structures, we have to consider two types of constraints. First, aesthetic constraints are important because the buildings have to be visually impressive. Sec- ond, functional constraints are important for the performance of a building and its e cient construction. This thesis contributes to the area of architectural geometry. Specifically, we are interested in the geometric rationalization of freeform architec- ture with the goal of combining aesthetic and functional constraints and construction requirements. Aesthetic requirements typically come from designers and architects. To obtain visually pleasing structures, they favor smoothness of the building shape, but also smoothness of the visible patterns on the surface. Functional requirements typically come from the engineers involved in the construction process. For exam- ple, covering freeform structures using planar panels is much cheaper than using non-planar ones. Further, constructed buildings have to be stable and should not collapse. In this thesis, we explore the geometric rationalization of freeform archi- tecture using four specific example problems inspired by real life applications. We achieve our results by developing optimization algorithms and a theoretical study of the underlying geometrical structure of the problems. The four example problems are the following: (1) The design of shading and lighting systems which are torsion-free structures with planar beams based on quad meshes. They satisfy the functionality requirements of preventing light from going inside a building as shad- ing systems or reflecting light into a building as lighting systems. (2) The Design of freeform honeycomb structures that are constructed based on hex-dominant meshes with a planar beam mounted along each edge. The beams intersect without
Field guide to geometrical optics
Greivenkamp, John E
2004-01-01
This Field Guide derives from the treatment of geometrical optics that has evolved from both the undergraduate and graduate programs at the Optical Sciences Center at the University of Arizona. The development is both rigorous and complete, and it features a consistent notation and sign convention. This volume covers Gaussian imagery, paraxial optics, first-order optical system design, system examples, illumination, chromatic effects, and an introduction to aberrations. The appendices provide supplemental material on radiometry and photometry, the human eye, and several other topics.
Geometric phase from dielectric matrix
International Nuclear Information System (INIS)
Banerjee, D.
2005-10-01
The dielectric property of the anisotropic optical medium is found by considering the polarized photon as two component spinor of spherical harmonics. The Geometric Phase of a polarized photon has been evaluated in two ways: the phase two-form of the dielectric matrix through a twist and the Pancharatnam phase (GP) by changing the angular momentum of the incident polarized photon over a closed triangular path on the extended Poincare sphere. The helicity in connection with the spin angular momentum of the chiral photon plays the key role in developing these phase holonomies. (author)