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.)

Metrics for Probabilistic Geometries

DEFF Research Database (Denmark)

Tosi, Alessandra; Hauberg, Søren; Vellido, Alfredo

2014-01-01

the distribution over mappings is given by a Gaussian process. We treat the corresponding latent variable model as a Riemannian manifold and we use the expectation of the metric under the Gaussian process prior to define interpolating paths and measure distance between latent points. We show how distances...

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 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

Control of nonholonomic systems from sub-Riemannian geometry to motion planning

CERN Document Server

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.

Riemannian metric optimization on surfaces (RMOS) for intrinsic brain mapping in the Laplace-Beltrami embedding space.

Science.gov (United States)

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.

Geometry and Combinatorics

DEFF Research Database (Denmark)

Kokkendorff, Simon Lyngby

2002-01-01

The subject of this Ph.D.-thesis is somewhere in between continuous and discrete geometry. Chapter 2 treats the geometry of finite point sets in semi-Riemannian hyperquadrics,using a matrix whose entries are a trigonometric function of relative distances in a given point set. The distance...... to the geometry of a simplex in a semi-Riemannian hyperquadric. In chapter 3 we study which finite metric spaces that are realizable in a hyperbolic space in the limit where curvature goes to -∞. We show that such spaces are the so called leaf spaces, the set of degree 1 vertices of weighted trees. We also...... establish results on the limiting geometry of such an isometrically realized leaf space simplex in hyperbolic space, when curvature goes to -∞. Chapter 4 discusses negative type of metric spaces. We give a measure theoretic treatment of this concept and related invariants. The theory developed...

Natural Connections on Riemannian Product Manifolds

OpenAIRE

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.

Applications of Affine and Weyl geometry

CERN Document Server

García-Río, Eduardo; Nikcevic, Stana

2013-01-01

Pseudo-Riemannian geometry is, to a large extent, the study of the Levi-Civita connection, which is the unique torsion-free connection compatible with the metric structure. There are, however, other affine connections which arise in different contexts, such as conformal geometry, contact structures, Weyl structures, and almost Hermitian geometry. In this book, we reverse this point of view and instead associate an auxiliary pseudo-Riemannian structure of neutral signature to certain affine connections and use this correspondence to study both geometries. We examine Walker structures, Riemannia

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.

Pseudo-Riemannian VSI spaces

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.

Pseudo-Riemannian VSI spaces

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.

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)

A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric

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....

A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric

OpenAIRE

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.

Convex functions and optimization methods on Riemannian manifolds

CERN Document Server

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 family of metrics on the moduli space of CP2 instantons

International Nuclear Information System (INIS)

Habermann, L.

1992-01-01

A family of Riemannian metrics on the moduli space of irreducible self-dual connections of instanton number k=1 over CP 2 is considered. We find explicit formulas for these metrics and deduce conclusions concerning the geometry of the instant space. (orig.)

General Geometry and Geometry of Electromagnetism

OpenAIRE

Shahverdiyev, Shervgi S.

2002-01-01

It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is geometry underlying Electromagnetism. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. And it is shown that equation of motion for a particle interacting with electromagnetic...

Differential geometry and topology with a view to dynamical systems

CERN Document Server

Burns, Keith

2005-01-01

MANIFOLDSIntroductionReview of topological conceptsSmooth manifoldsSmooth mapsTangent vectors and the tangent bundleTangent vectors as derivationsThe derivative of a smooth mapOrientationImmersions, embeddings and submersionsRegular and critical points and valuesManifolds with boundarySard's theoremTransversalityStabilityExercisesVECTOR FIELDS AND DYNAMICAL SYSTEMSIntroductionVector fieldsSmooth dynamical systemsLie derivative, Lie bracketDiscrete dynamical systemsHyperbolic fixed points and periodic orbitsExercisesRIEMANNIAN METRICSIntroductionRiemannian metricsStandard geometries on surfacesExercisesRIEMANNIAN CONNECTIONS AND GEODESICSIntroductionAffine connectionsRiemannian connectionsGeodesicsThe exponential mapMinimizing properties of geodesicsThe Riemannian distanceExercisesCURVATUREIntroductionThe curvature tensorThe second fundamental formSectional and Ricci curvaturesJacobi fieldsManifolds of constant curvatureConjugate pointsHorizontal and vertical sub-bundlesThe geodesic flowExercisesTENSORS AND DI...

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.

Methods of information geometry

CERN Document Server

Amari, Shun-Ichi

2000-01-01

Information geometry provides the mathematical sciences with a new framework of analysis. It has emerged from the investigation of the natural differential geometric structure on manifolds of probability distributions, which consists of a Riemannian metric defined by the Fisher information and a one-parameter family of affine connections called the \\alpha-connections. The duality between the \\alpha-connection and the (-\\alpha)-connection together with the metric play an essential role in this geometry. This kind of duality, having emerged from manifolds of probability distributions, is ubiquitous, appearing in a variety of problems which might have no explicit relation to probability theory. Through the duality, it is possible to analyze various fundamental problems in a unified perspective. The first half of this book is devoted to a comprehensive introduction to the mathematical foundation of information geometry, including preliminaries from differential geometry, the geometry of manifolds or probability d...

Statistics on Lie groups: A need to go beyond the pseudo-Riemannian framework

Science.gov (United States)

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.

Clustering in Hilbert simplex geometry

KAUST Repository

Nielsen, Frank

2017-04-03

Clustering categorical distributions in the probability simplex is a fundamental primitive often met in applications dealing with histograms or mixtures of multinomials. Traditionally, the differential-geometric structure of the probability simplex has been used either by (i) setting the Riemannian metric tensor to the Fisher information matrix of the categorical distributions, or (ii) defining the information-geometric structure induced by a smooth dissimilarity measure, called a divergence. In this paper, we introduce a novel computationally-friendly non-Riemannian framework for modeling the probability simplex: Hilbert simplex geometry. We discuss the pros and cons of those three statistical modelings, and compare them experimentally for clustering tasks.

Non-Riemannian geometry

CERN Document Server

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.

A special form of SPD covariance matrix for interpretation and visualization of data manipulated with Riemannian geometry

Science.gov (United States)

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

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

Bosonization in a two-dimensional Riemann Cartan geometry

International Nuclear Information System (INIS)

Denardo, G.; Spallucci, E.

1987-01-01

We study the vacuum functional for a Dirac field in a two dimensional Riemann-Cartan geometry. Torsion is treated as a quantum variable while the metric is considered as a classical background field. Decoupling spinors from the non-Riemannian part of the geometry introduces a chiral Jacobian into the vacuum generating functional. We compute this functional Jacobian determinant by means of the Alvarez method. Finally, we show that the effective action for the background geometry is of the Liouville type and does not preserve any memory of the initial torsion field. (author)

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...

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.)

Information geometry

CERN Document Server

Ay, Nihat; Lê, Hông Vân; Schwachhöfer, Lorenz

2017-01-01

The book provides a comprehensive introduction and a novel mathematical foundation of the field of information geometry with complete proofs and detailed background material on measure theory, Riemannian geometry and Banach space theory. Parametrised measure models are defined as fundamental geometric objects, which can be both finite or infinite dimensional. Based on these models, canonical tensor fields are introduced and further studied, including the Fisher metric and the Amari-Chentsov tensor, and embeddings of statistical manifolds are investigated. This novel foundation then leads to application highlights, such as generalizations and extensions of the classical uniqueness result of Chentsov or the Cramér-Rao inequality. Additionally, several new application fields of information geometry are highlighted, for instance hierarchical and graphical models, complexity theory, population genetics, or Markov Chain Monte Carlo. The book will be of interest to mathematicians who are interested in geometry, inf...

Goedel-type metrics in various dimensions

International Nuclear Information System (INIS)

Guerses, Metin; Karasu, Atalay; Sarioglu, Oezguer

2005-01-01

Goedel-type metrics are introduced and used in producing charged dust solutions in various dimensions. The key ingredient is a (D - 1)-dimensional Riemannian geometry which is then employed in constructing solutions to the Einstein-Maxwell field equations with a dust distribution in D dimensions. The only essential field equation in the procedure turns out to be the source-free Maxwell's equation in the relevant background. Similarly the geodesics of this type of metric are described by the Lorentz force equation for a charged particle in the lower dimensional geometry. It is explicitly shown with several examples that Goedel-type metrics can be used in obtaining exact solutions to various supergravity theories and in constructing spacetimes that contain both closed timelike and closed null curves and that contain neither of these. Among the solutions that can be established using non-flat backgrounds, such as the Tangherlini metrics in (D - 1)-dimensions, there exists a class which can be interpreted as describing black-hole-type objects in a Goedel-like universe

The Finsler spacetime framework. Backgrounds for physics beyond metric geometry

International Nuclear Information System (INIS)

Pfeifer, Christian

2013-11-01

The fundamental structure on which physics is described is the geometric spacetime background provided by a four dimensional manifold equipped with a Lorentzian metric. Most importantly the spacetime manifold does not only provide the stage for physical field theories but its geometry encodes causality, observers and their measurements and gravity simultaneously. This threefold role of the Lorentzian metric geometry of spacetime is one of the key insides of general relativity. During this thesis we extend the background geometry for physics from the metric framework of general relativity to our Finsler spacetime framework and ensure that the threefold role of the geometry of spacetime in physics is not changed. The geometry of Finsler spacetimes is determined by a function on the tangent bundle and includes metric geometry. In contrast to the standard formulation of Finsler geometry our Finsler spacetime framework overcomes the differentiability and existence problems of the geometric objects in earlier attempts to use Finsler geometry as an extension of Lorentzian metric geometry. The development of our nonmetric geometric framework which encodes causality is one central achievement of this thesis. On the basis of our well-defined Finsler spacetime geometry we are able to derive dynamics for the non-metric Finslerian geometry of spacetime from an action principle, obtained from the Einstein-Hilbert action, for the first time. We can complete the dynamics to a non-metric description of gravity by coupling matter fields, also formulated via an action principle, to the geometry of our Finsler spacetimes. We prove that the combined dynamics of the fields and the geometry are consistent with general relativity. Furthermore we demonstrate how to define observers and their measurements solely through the non-metric spacetime geometry. Physical consequence derived on the basis of our Finsler spacetime are: a possible solution to the fly-by anomaly in the solar system; the

The Finsler spacetime framework. Backgrounds for physics beyond metric geometry

Energy Technology Data Exchange (ETDEWEB)

Pfeifer, Christian

2013-11-15

The fundamental structure on which physics is described is the geometric spacetime background provided by a four dimensional manifold equipped with a Lorentzian metric. Most importantly the spacetime manifold does not only provide the stage for physical field theories but its geometry encodes causality, observers and their measurements and gravity simultaneously. This threefold role of the Lorentzian metric geometry of spacetime is one of the key insides of general relativity. During this thesis we extend the background geometry for physics from the metric framework of general relativity to our Finsler spacetime framework and ensure that the threefold role of the geometry of spacetime in physics is not changed. The geometry of Finsler spacetimes is determined by a function on the tangent bundle and includes metric geometry. In contrast to the standard formulation of Finsler geometry our Finsler spacetime framework overcomes the differentiability and existence problems of the geometric objects in earlier attempts to use Finsler geometry as an extension of Lorentzian metric geometry. The development of our nonmetric geometric framework which encodes causality is one central achievement of this thesis. On the basis of our well-defined Finsler spacetime geometry we are able to derive dynamics for the non-metric Finslerian geometry of spacetime from an action principle, obtained from the Einstein-Hilbert action, for the first time. We can complete the dynamics to a non-metric description of gravity by coupling matter fields, also formulated via an action principle, to the geometry of our Finsler spacetimes. We prove that the combined dynamics of the fields and the geometry are consistent with general relativity. Furthermore we demonstrate how to define observers and their measurements solely through the non-metric spacetime geometry. Physical consequence derived on the basis of our Finsler spacetime are: a possible solution to the fly-by anomaly in the solar system; the

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...

A Numerical Framework for Sobolev Metrics on the Space of Curves

DEFF Research Database (Denmark)

Bauer, Martin; Bruveris, Martins; Harms, Philipp

2017-01-01

Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics...

On the (1,1)-tensor bundle with Cheeger–Gromoll type metric

Indian Academy of Sciences (India)

The main purpose of the present paper is to construct Riemannian almost product structures on the (1, 1)-tensor bundle equipped with Cheeger–Gromoll type metric over a Riemannian manifold and present some results concerning these structures. Keywords. Almost product structure; Cheeger–Gromoll type metric; metric ...

Riemannian multi-manifold modeling and clustering in brain networks

Science.gov (United States)

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.

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.

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.)

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

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)

A Riemannian scalar measure for diffusion tensor images

NARCIS (Netherlands)

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.

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...

The Jacobi metric for timelike geodesics in static spacetimes

Science.gov (United States)

Gibbons, G. W.

2016-01-01

It is shown that the free motion of massive particles moving in static spacetimes is given by the geodesics of an energy-dependent Riemannian metric on the spatial sections analogous to Jacobi's metric in classical dynamics. In the massless limit Jacobi's metric coincides with the energy independent Fermat or optical metric. For stationary metrics, it is known that the motion of massless particles is given by the geodesics of an energy independent Finslerian metric of Randers type. The motion of massive particles is governed by neither a Riemannian nor a Finslerian metric. The properies of the Jacobi metric for massive particles moving outside the horizon of a Schwarschild black hole are described. By constrast with the massless case, the Gaussian curvature of the equatorial sections is not always negative.

Single and multiple object tracking using log-euclidean Riemannian subspace and block-division appearance model.

Science.gov (United States)

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.

Characterization of the Unit Tangent Sphere Bundle with $ g $-Natural Metric and Almost Contact B-metric Structure

Directory of Open Access Journals (Sweden)

Farshad Firuzi

2017-06-01

Full Text Available We consider unit tangent sphere bundle of a Riemannian manifold $ (M,g $ as a $ (2n+1 $-dimensional manifold and we equip it with pseudo-Riemannian $ g $-natural almost contact B-metric structure. Then, by computing coefficients of the structure tensor $ F$, we completely characterize the unit tangent sphere bundle equipped to this structure, with respect to the relevant classification of almost contact B-metric structures, and determine a class such that the unit tangent sphere bundle with mentioned structure belongs to it. Also, we find some curvature conditions such that the mentioned structure satisfies each of eleven basic classes.

Geometry of Hamiltonian chaos

DEFF Research Database (Denmark)

Horwitz, Lawrence; Zion, Yossi Ben; Lewkowicz, Meir

2007-01-01

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce ...

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.

Multi-Frequency Polarimetric SAR Classification Based on Riemannian Manifold and Simultaneous Sparse Representation

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.

Complex Monge–Ampère equations and geodesics in the space of Kähler metrics

CERN Document Server

2012-01-01

The purpose of these lecture notes is to provide an introduction to the theory of complex Monge–Ampère operators (definition, regularity issues, geometric properties of solutions, approximation) on compact Kähler manifolds (with or without boundary). These operators are of central use in several fundamental problems of complex differential geometry (Kähler–Einstein equation, uniqueness of constant scalar curvature metrics), complex analysis and dynamics. The topics covered include, the Dirichlet problem (after Bedford–Taylor), Monge–Ampère foliations and laminated currents, polynomial hulls and Perron envelopes with no analytic structure, a self-contained presentation of Krylov regularity results, a modernized proof of the Calabi–Yau theorem (after Yau and Kolodziej), an introduction to infinite dimensional riemannian geometry, geometric structures on spaces of Kähler metrics (after Mabuchi, Semmes and Donaldson), generalizations of the regularity theory of Caffarelli–Kohn–Nirenberg–Spruc...

Metric interpretation of gauge fields in noncommutative geometry

International Nuclear Information System (INIS)

Martinetti, P.

2007-01-01

We shall give an overview of the metric interpretation of gauge fields in noncommutative geometry, via Connes distance formula. Especially we shall focus on the Higgs fields in the standard model, and gauge fields in various models of fiber bundle. (author)

Principal Curves on Riemannian Manifolds.

Science.gov (United States)

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.

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.

Perturbative stability of the approximate Killing field eigenvalue problem

International Nuclear Information System (INIS)

Beetle, Christopher; Wilder, Shawn

2014-01-01

An approximate Killing field may be defined on a compact, Riemannian geometry by solving an eigenvalue problem for a certain elliptic operator. This paper studies the effect of small perturbations in the Riemannian metric on the resulting vector field. It shows that small metric perturbations, as measured using a Sobolev-type supremum norm on the space of Riemannian geometries on a fixed manifold, yield small perturbations in the approximate Killing field, as measured using a Hilbert-type square integral norm. It also discusses applications to the problem of computing the spin of a generic black hole in general relativity. (paper)

Aspects of differential geometry II

CERN Document Server

Gilkey, Peter

2015-01-01

Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology of compact Lie groups an...

Stationary metrics and optical Zermelo-Randers-Finsler geometry

International Nuclear Information System (INIS)

Gibbons, G. W.; Warnick, C. M.; Herdeiro, C. A. R.; Werner, M. C.

2009-01-01

We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullstrand form of the spacetime metric, whereas the data of the Randers problem are encoded in a stationary generalization of the usual optical metric. We discuss how the spacetime viewpoint gives a simple and physical perspective on various issues, including how Finsler geometries with constant flag curvature always map to conformally flat spacetimes and that the Finsler condition maps to either a causality condition or it breaks down at an ergo surface in the spacetime picture. The gauge equivalence in this network of relations is considered as well as the connection to analogue models and the viewpoint of magnetic flows. We provide a variety of examples.

Dark energy and dark matter from hidden symmetry of gravity model with a non-Riemannian volume form

Energy Technology Data Exchange (ETDEWEB)

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.)

Observable traces of non-metricity: New constraints on metric-affine gravity

Science.gov (United States)

Delhom-Latorre, Adrià; Olmo, Gonzalo J.; Ronco, Michele

2018-05-01

Relaxing the Riemannian condition to incorporate geometric quantities such as torsion and non-metricity may allow to explore new physics associated with defects in a hypothetical space-time microstructure. Here we show that non-metricity produces observable effects in quantum fields in the form of 4-fermion contact interactions, thereby allowing us to constrain the scale of non-metricity to be greater than 1 TeV by using results on Bahbah scattering. Our analysis is carried out in the framework of a wide class of theories of gravity in the metric-affine approach. The bound obtained represents an improvement of several orders of magnitude to previous experimental constraints.

The decomposition of global conformal invariants

CERN Document Server

Alexakis, Spyros

2012-01-01

This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question concerns conformally invariant functionals on the space of Riemannian metrics over a given manifold. These functionals act on a metric by first constructing a Riemannian scalar out of it, and then integrating this scalar over the manifold. Suppose this integral remains invariant under conformal re-scalings of the underlying metric. What information can one then deduce about the Riemannian scalar? Dese

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

Global Differential Geometry and Global Analysis

CERN Document Server

Pinkall, Ulrich; Simon, Udo; Wegner, Berd

1991-01-01

All papers appearing in this volume are original research articles and have not been published elsewhere. They meet the requirements that are necessary for publication in a good quality primary journal. E.Belchev, S.Hineva: On the minimal hypersurfaces of a locally symmetric manifold. -N.Blasic, N.Bokan, P.Gilkey: The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary. -J.Bolton, W.M.Oxbury, L.Vrancken, L.M. Woodward: Minimal immersions of RP2 into CPn. -W.Cieslak, A. Miernowski, W.Mozgawa: Isoptics of a strictly convex curve. -F.Dillen, L.Vrancken: Generalized Cayley surfaces. -A.Ferrandez, O.J.Garay, P.Lucas: On a certain class of conformally flat Euclidean hypersurfaces. -P.Gauduchon: Self-dual manifolds with non-negative Ricci operator. -B.Hajduk: On the obstruction group toexistence of Riemannian metrics of positive scalar curvature. -U.Hammenstaedt: Compact manifolds with 1/4-pinched negative curvature. -J.Jost, Xiaowei Peng: The geometry of moduli spaces of stabl...

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...

Clustering in Hilbert simplex geometry

KAUST Repository

Nielsen, Frank; Sun, Ke

2017-01-01

has been used either by (i) setting the Riemannian metric tensor to the Fisher information matrix of the categorical distributions, or (ii) defining the information-geometric structure induced by a smooth dissimilarity measure, called a divergence

Covariant electrodynamics in linear media: Optical metric

Science.gov (United States)

Thompson, Robert T.

2018-03-01

While the postulate of covariance of Maxwell's equations for all inertial observers led Einstein to special relativity, it was the further demand of general covariance—form invariance under general coordinate transformations, including between accelerating frames—that led to general relativity. Several lines of inquiry over the past two decades, notably the development of metamaterial-based transformation optics, has spurred a greater interest in the role of geometry and space-time covariance for electrodynamics in ponderable media. I develop a generally covariant, coordinate-free framework for electrodynamics in general dielectric media residing in curved background space-times. In particular, I derive a relation for the spatial medium parameters measured by an arbitrary timelike observer. In terms of those medium parameters I derive an explicit expression for the pseudo-Finslerian optical metric of birefringent media and show how it reduces to a pseudo-Riemannian optical metric for nonbirefringent media. This formulation provides a basis for a unified approach to ray and congruence tracing through media in curved space-times that may smoothly vary among positively refracting, negatively refracting, and vacuum.

Real tunneling geometries and the large-scale topology of the universe

International Nuclear Information System (INIS)

Gibbons, G.W.; Hartle, J.B.

1990-01-01

If the topology and geometry of spacetime are quantum-mechanically variable, then the particular classical large-scale topology and geometry observed in our universe must be statistical predictions of its initial condition. This paper examines the predictions of the ''no boundary'' initial condition for the present large-scale topology and geometry. Finite-action real tunneling solutions of Einstein's equation are important for such predictions. These consist of compact Riemannian (Euclidean) geometries joined to a Lorentzian cosmological geometry across a spacelike surface of vanishing extrinsic curvature. The classification of such solutions is discussed and general constraints on their topology derived. For example, it is shown that, if the Euclidean Ricci tensor is positive, then a real tunneling solution can nucleate only a single connected Lorentzian spacetime (the unique conception theorem). Explicit examples of real tunneling solutions driven by a cosmological constant are exhibited and their implications for cosmic baldness described. It is argued that the most probable large-scale spacetime predicted by the real tunneling solutions of the ''no-boundary'' initial condition has the topology RxS 3 with the de Sitter metric

Differential geometry bundles, connections, metrics and curvature

CERN Document Server

Taubes, Clifford Henry

2011-01-01

Bundles, connections, metrics and curvature are the 'lingua franca' of modern differential geometry and theoretical physics. This book will supply a graduate student in mathematics or theoretical physics with the fundamentals of these objects. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the

Riemannian geometry of Hamiltonian chaos: hints for a general theory.

Science.gov (United States)

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.

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 ...

Intrinsic Losses Based on Information Geometry and Their Applications

Directory of Open Access Journals (Sweden)

Yao Rong

2017-08-01

Full Text Available One main interest of information geometry is to study the properties of statistical models that do not depend on the coordinate systems or model parametrization; thus, it may serve as an analytic tool for intrinsic inference in statistics. In this paper, under the framework of Riemannian geometry and dual geometry, we revisit two commonly-used intrinsic losses which are respectively given by the squared Rao distance and the symmetrized Kullback–Leibler divergence (or Jeffreys divergence. For an exponential family endowed with the Fisher metric and α -connections, the two loss functions are uniformly described as the energy difference along an α -geodesic path, for some α ∈ { − 1 , 0 , 1 } . Subsequently, the two intrinsic losses are utilized to develop Bayesian analyses of covariance matrix estimation and range-spread target detection. We provide an intrinsically unbiased covariance estimator, which is verified to be asymptotically efficient in terms of the intrinsic mean square error. The decision rules deduced by the intrinsic Bayesian criterion provide a geometrical justification for the constant false alarm rate detector based on generalized likelihood ratio principle.

An introduction to differential geometry

CERN Document Server

Willmore, T J

2012-01-01

This text employs vector methods to explore the classical theory of curves and surfaces. Topics include basic theory of tensor algebra, tensor calculus, calculus of differential forms, and elements of Riemannian geometry. 1959 edition.

Hadamard States for the Klein-Gordon Equation on Lorentzian Manifolds of Bounded Geometry

Science.gov (United States)

Gérard, Christian; Oulghazi, Omar; Wrochna, Michał

2017-06-01

We consider the Klein-Gordon equation on a class of Lorentzian manifolds with Cauchy surface of bounded geometry, which is shown to include examples such as exterior Kerr, Kerr-de Sitter spacetime and the maximal globally hyperbolic extension of the Kerr outer region. In this setup, we give an approximate diagonalization and a microlocal decomposition of the Cauchy evolution using a time-dependent version of the pseudodifferential calculus on Riemannian manifolds of bounded geometry. We apply this result to construct all pure regular Hadamard states (and associated Feynman inverses), where regular refers to the state's two-point function having Cauchy data given by pseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both the choice of (pure, regular) Hadamard state and the underlying spacetime metric.

Thermodynamic metrics and optimal paths.

Science.gov (United States)

Sivak, David A; Crooks, Gavin E

2012-05-11

A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. To this end, we derive a friction tensor that induces a Riemannian manifold on the space of thermodynamic states. Within the linear-response regime, this metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols with many useful properties. We discuss the connection to the existing thermodynamic length formalism, and demonstrate the utility of this metric by solving for optimal control parameter protocols in a simple nonequilibrium model.

Riemannian computing in computer vision

CERN Document Server

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...

Radar orthogonality and radar length in Finsler and metric spacetime geometry

Science.gov (United States)

Pfeifer, Christian

2014-09-01

The radar experiment connects the geometry of spacetime with an observers measurement of spatial length. We investigate the radar experiment on Finsler spacetimes which leads to a general definition of radar orthogonality and radar length. The directions radar orthogonal to an observer form the spatial equal time surface an observer experiences and the radar length is the physical length the observer associates to spatial objects. We demonstrate these concepts on a forth order polynomial Finsler spacetime geometry which may emerge from area metric or premetric linear electrodynamics or in quantum gravity phenomenology. In an explicit generalization of Minkowski spacetime geometry we derive the deviation from the Euclidean spatial length measure in an observers rest frame explicitly.

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.)

Space–time and spatial geodesic orbits in Schwarzschild geometry

Science.gov (United States)

Resca, Lorenzo

2018-05-01

Geodesic orbit equations in the Schwarzschild geometry of general relativity reduce to ordinary conic sections of Newtonian mechanics and gravity for material particles in the non-relativistic limit. On the contrary, geodesic orbit equations for a proper spatial submanifold of Schwarzschild metric at any given coordinate-time correspond to an unphysical gravitational repulsion in the non-relativistic limit. This demonstrates at a basic level the centrality and critical role of relativistic time and its intimate pseudo-Riemannian connection with space. Correspondingly, a commonly popularised depiction of geodesic orbits of planets as resulting from the curvature of space produced by the Sun, represented as a rubber sheet dipped in the middle by the weighing of that massive body, is mistaken and misleading for the essence of relativity, even in the non-relativistic limit.

Riemann-Cartan geometry of nonlinear disclination mechanics

KAUST Repository

Yavari, A.

2012-03-23

In the continuous theory of defects in nonlinear elastic solids, it is known that a distribution of disclinations leads, in general, to a non-trivial residual stress field. To study this problem, we consider the particular case of determining the residual stress field of a cylindrically symmetric distribution of parallel wedge disclinations. We first use the tools of differential geometry to construct a Riemannian material manifold in which the body is stress-free. This manifold is metric compatible, has zero torsion, but has non-vanishing curvature. The problem then reduces to embedding this manifold in Euclidean 3-space following the procedure of a classical nonlinear elastic problem. We show that this embedding can be elegantly accomplished by using Cartan\\'s method of moving frames and compute explicitly the residual stress field for various distributions in the case of a neo-Hookean material. © 2012 The Author(s).

Nonlinear Methods in Riemannian and Kählerian Geometry

CERN Document Server

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 ...

Comparison of exit time moment spectra for extrinsic metric balls

DEFF Research Database (Denmark)

Hurtado, Ana; Markvorsen, Steen; Palmer, Vicente

2012-01-01

We prove explicit upper and lower bounds for the $L^1$-moment spectra for the Brownian motion exit time from extrinsic metric balls of submanifolds $P^m$ in ambient Riemannian spaces $N^n$. We assume that $P$ and $N$ both have controlled radial curvatures (mean curvature and sectional curvature...... obtain new intrinsic comparison results for the exit time spectra for metric balls in the ambient manifolds $N^n$ themselves....

Natural metrics and least-committed priors for articulated tracking

DEFF Research Database (Denmark)

Hauberg, Søren; Sommer, Stefan Horst; Pedersen, Kim Steenstrup

2012-01-01

of joint positions, which is embedded in a high dimensional Euclidean space. This Riemannian manifold inherits the metric from the embedding space, such that distances are measured as the combined physical length that joints travel during movements. We then develop a least-committed Brownian motion model...

Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry

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...

Wave fields in Weyl spaces and conditions for the existence of a preferred pseudo-Riemannian structure

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.)

A View on Optimal Transport from Noncommutative Geometry

Directory of Open Access Journals (Sweden)

Francesco D'Andrea

2010-07-01

Full Text Available We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space R^n, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

Unification of Electromagnetism and Gravitation in the Framework of General Geometry

OpenAIRE

Shahverdiyev, Shervgi

2005-01-01

A new geometry, called General geometry, is constructed. It is proven that its the most simplest special case is geometry underlying Electromagnetism. Another special case is Riemannian geometry. Action for electromagnetic field and Maxwell equations are derived from curvature function of geometry underlying Electromagnetism. It is shown that equation of motion for a particle interacting with electromagnetic field coincides exactly with equation for geodesics of geometry underlying Electromag...

Geometry of quantum computation with qutrits.

Science.gov (United States)

Li, Bin; Yu, Zu-Huan; Fei, Shao-Ming

2013-01-01

Determining the quantum circuit complexity of a unitary operation is an important problem in quantum computation. By using the mathematical techniques of Riemannian geometry, we investigate the efficient quantum circuits in quantum computation with n qutrits. We show that the optimal quantum circuits are essentially equivalent to the shortest path between two points in a certain curved geometry of SU(3(n)). As an example, three-qutrit systems are investigated in detail.

Piecewise linear manifolds: Einstein metrics and Ricci flows

International Nuclear Information System (INIS)

Schrader, Robert

2016-01-01

This article provides an attempt to extend concepts from the theory of Riemannian manifolds to piecewise linear (p.l.) spaces. In particular we propose an analogue of the Ricci tensor, which we give the name of an Einstein vector field . On a given set of p.l. spaces we define and discuss (normalized) Einstein flows. p.l. Einstein metrics are defined and examples are provided. Criteria for flows to approach Einstein metrics are formulated. Second variations of the total scalar curvature at a specific Einstein space are calculated. (paper)

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.

Segmentation of High Angular Resolution Diffusion MRI using Sparse Riemannian Manifold Clustering

Science.gov (United States)

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

Geodesics in (Rn, d1

Directory of Open Access Journals (Sweden)

Mehmet KILIÇ

2016-09-01

Full Text Available The notion of geodesic, which may be regarded as an extension of the line segment in Euclidean geometry to the space we study in, has an important place in many branches of geometry, such as Riemannian geometry, Metric geometry, to name but a few. In this article, the concept of geodesic in a metric space will be introduced, then geodesics in the space (Rn, d1 will be characterized. Furthermore, some examples will be presented to demonstrate the effectiveness of the main result.

A Short Description of Electromagnetism Using the Finsler Geometry

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Otilia Lungu

2011-12-01

Full Text Available Abstract. It is well known that a Randers metric is a deformation of a Riemannian metric alfa(x,y=sqrt(a_ij(xy^iy^j using a 1-form beta(x,y=beta_i(xy^i. In this paper we are replacing beta(x,y with beta_2(x,y=sqrt(beta_ij(xy^iy^j. We obtain a new space and we are going to study some of its properties.Key words: electromagnetism, Finsler space, Randers spaces.

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.

The geometry of classical Regge calculus

International Nuclear Information System (INIS)

Barrett, J.W.

1987-01-01

Standard notions of Riemannian geometry are applied to the case of piecewise-flat manifolds. Particular care is taken to explain how one may define some particular vectors and tensors in an invariant way at points of a conical singularity. The geometry surrounding the equations of motion and the energy-momentum of the piecewise-flat manifold is developed in detail. The resolution theorem is presented, which states that on certain resolution hypersurfaces there is a clear connection between the energy-momentum of the piecewise-flat manifold and the Regge equations of motion. (author)

Curvature properties of four-dimensional Walker metrics

International Nuclear Information System (INIS)

Chaichi, M; Garcia-Rio, E; Matsushita, Y

2005-01-01

A Walker n-manifold is a semi-Riemannian manifold, which admits a field of parallel null r-planes, r ≤ n/2. In the present paper we study curvature properties of a Walker 4-manifold (M, g) which admits a field of parallel null 2-planes. The metric g is necessarily of neutral signature (+ + - -). Such a Walker 4-manifold is the lowest dimensional example not of Lorentz type. There are three functions of coordinates which define a Walker metric. Some recent work shows that a Walker 4-manifold of restricted type whose metric is characterized by two functions exhibits a large variety of symplectic structures, Hermitian structures, Kaehler structures, etc. For such a restricted Walker 4-manifold, we shall study mainly curvature properties, e.g., conditions for a Walker metric to be Einstein, Osserman, or locally conformally flat, etc. One of our main results is the exact solutions to the Einstein equations for a restricted Walker 4-manifold

Complex differential geometry

CERN Document Server

Zheng, Fangyang

2002-01-01

The theory of complex manifolds overlaps with several branches of mathematics, including differential geometry, algebraic geometry, several complex variables, global analysis, topology, algebraic number theory, and mathematical physics. Complex manifolds provide a rich class of geometric objects, for example the (common) zero locus of any generic set of complex polynomials is always a complex manifold. Yet complex manifolds behave differently than generic smooth manifolds; they are more coherent and fragile. The rich yet restrictive character of complex manifolds makes them a special and interesting object of study. This book is a self-contained graduate textbook that discusses the differential geometric aspects of complex manifolds. The first part contains standard materials from general topology, differentiable manifolds, and basic Riemannian geometry. The second part discusses complex manifolds and analytic varieties, sheaves and holomorphic vector bundles, and gives a brief account of the surface classifi...

Covariant Schrödinger semigroups on Riemannian manifolds

CERN Document Server

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...

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

Deriving principal channel metrics from bank and long-profile geometry with the R package cmgo

Science.gov (United States)

Golly, Antonius; Turowski, Jens M.

2017-09-01

Landscape patterns result from landscape forming processes. This link can be exploited in geomorphological research by reversely analyzing the geometrical content of landscapes to develop or confirm theories of the underlying processes. Since rivers represent a dominant control on landscape formation, there is a particular interest in examining channel metrics in a quantitative and objective manner. For example, river cross-section geometry is required to model local flow hydraulics, which in turn determine erosion and thus channel dynamics. Similarly, channel geometry is crucial for engineering purposes, water resource management, and ecological restoration efforts. These applications require a framework to capture and derive the data. In this paper we present an open-source software tool that performs the calculation of several channel metrics (length, slope, width, bank retreat, knickpoints, etc.) in an objective and reproducible way based on principal bank geometry that can be measured in the field or in a GIS. Furthermore, the software provides a framework to integrate spatial features, for example the abundance of species or the occurrence of knickpoints. The program is available at https://github.com/AntoniusGolly/cmgo and is free to use, modify, and redistribute under the terms of the GNU General Public License version 3 as published by the Free Software Foundation.

Smarandache Spaces as a New Extension of the Basic Space-Time of General Relativity

Directory of Open Access Journals (Sweden)

Rabounski D.

2010-04-01

Full Text Available This short letter manifests how Smarandache geometries can be employed in order to extend the “classical” basis of the General Theory of Relativity (Riemannian geometry through joining the properties of two or more (different geometries in the same single space. Perspectives in this way seem much profitable: the basic space-time of General Relativity can be extended to not only metric geometries, but even to non-metric ones (where no distances can be measured, or to spaces of the mixed kind which possess the properties of both metric and non-metric spaces (the latter should be referred to as “semi-metric spaces”. If both metric and non-metric properties possessed at the same (at least one point of a space, it is one of Smarandache geometries, and should be re- ferred to as “Smarandache semi-metric space”. Such spaces can be introduced accord- ing to the mathematical apparatus of physically observable quantities (chronometric invariants, if we consider a breaking of the observable space metric in the continuous background of the fundamental metric tensor.

Some Early Ideas on the Metric Geometry of Thermodynamics

Science.gov (United States)

Ruppeiner, George

2016-11-01

It is a pleasure to write for this 90th anniversary volume of Journal of Low Temperature Physics dedicated to Horst Meyer at Duke University. I was a PhD student with Horst in the period 1975-1980, working in experimental low temperature physics. While in Horst's group, I also did a theoretical physics project on the side. This project in the metric geometry of thermodynamics was motivated by my work in Horst's lab, and helped me to understand the theory of critical phenomena, very much in play in Horst's lab. In this paper, I explain the essence of my theory project and give a few accounts of its future development, focussing on topics where I interacted with Horst. I pay particular attention to the pure fluid critical point.

Bilinear Regularized Locality Preserving Learning on Riemannian Graph for Motor Imagery BCI.

Science.gov (United States)

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.

Riemannian geometry of thermodynamics and systems with repulsive power-law interactions.

Science.gov (United States)

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.

Nonparametric Information Geometry: From Divergence Function to Referential-Representational Biduality on Statistical Manifolds

Directory of Open Access Journals (Sweden)

Jun Zhang

2013-12-01

Full Text Available Divergence functions are the non-symmetric “distance” on the manifold, Μθ, of parametric probability density functions over a measure space, (Χ,μ. Classical information geometry prescribes, on Μθ: (i a Riemannian metric given by the Fisher information; (ii a pair of dual connections (giving rise to the family of α-connections that preserve the metric under parallel transport by their joint actions; and (iii a family of divergence functions ( α-divergence defined on Μθ x Μθ, which induce the metric and the dual connections. Here, we construct an extension of this differential geometric structure from Μθ (that of parametric probability density functions to the manifold, Μ, of non-parametric functions on X, removing the positivity and normalization constraints. The generalized Fisher information and α-connections on M are induced by an α-parameterized family of divergence functions, reflecting the fundamental convex inequality associated with any smooth and strictly convex function. The infinite-dimensional manifold, M, has zero curvature for all these α-connections; hence, the generally non-zero curvature of M can be interpreted as arising from an embedding of Μθ into Μ. Furthermore, when a parametric model (after a monotonic scaling forms an affine submanifold, its natural and expectation parameters form biorthogonal coordinates, and such a submanifold is dually flat for α = ± 1, generalizing the results of Amari’s α-embedding. The present analysis illuminates two different types of duality in information geometry, one concerning the referential status of a point (measurable function expressed in the divergence function (“referential duality” and the other concerning its representation under an arbitrary monotone scaling (“representational duality”.

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.

Geometry as an aspect of dynamics

International Nuclear Information System (INIS)

Videira, A.L.L.; Barros, A.L.R.; Fernandes, N.C.

1982-07-01

Contrary to the predominant way of doing physics, it is shown that the geometric structure of a general differentiable space-time manifold can be determined by means of the introduction in that manifold of a minimal set of fundamental dynamical quantities associated to a free particle endowed with the fundamental property of momentum. Thus, general relativistic physics implies a general pseudo-Riemannian geometry, whereas the physics of the special theory of relativity is tied up with Minkowski space-time, and Newtonian dynamics is bound to Newtonian space-time. While in the relativistic instance, the Riemannian character of the manifold is basically fixed by means only of the Hamiltonian state function of the free particle (its kynetic energy), in the latter case, it has to resort, perhaps not unexpectedly, to the two dynamical entities mass and energy, separately. (Author) [pt

Geometry as an aspect of dynamics

International Nuclear Information System (INIS)

Videira, A.L.L.; Barros, A.L.R.; Fernandes, N.C.

1983-12-01

Contrary to the predominant way of doing physics, it is shown that the geometric structure of a general differentiable space-time manifold can be determined by means of the introduction in that manifold of a minimal set of fundamental dynamical quantities associated to a particle endowed with the fundamental property of covariant momentum. Thus, general relativistic physics implies a general pseudo-Riemannian geometry, whereas the physics of the special theory of relativity is tied up with Minkowski space-time, and Newtonian dynamics is bound to Newtonian space-time. While in the relativistic instance, the Riemannian character of the manifold is basically fixed by means only of the Hamiltonian state function of the particle (its energy), in the latter case, one have to resort, perhaps not unexpectedly, to the two dynamical entities mass energy, separately. (Author) [pt

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)

Canonical differential geometry of string backgrounds

International Nuclear Information System (INIS)

Schuller, Frederic P.; Wohlfarth, Mattias N.R.

2006-01-01

String backgrounds and D-branes do not possess the structure of Lorentzian manifolds, but that of manifolds with area metric. Area metric geometry is a true generalization of metric geometry, which in particular may accommodate a B-field. While an area metric does not determine a connection, we identify the appropriate differential geometric structure which is of relevance for the minimal surface equation in such a generalized geometry. In particular the notion of a derivative action of areas on areas emerges naturally. Area metric geometry provides new tools in differential geometry, which promise to play a role in the description of gravitational dynamics on D-branes

Tensors and their applications

CERN Document Server

Islam, Nazrul

2006-01-01

About the Book: The book is written is in easy-to-read style with corresponding examples. The main aim of this book is to precisely explain the fundamentals of Tensors and their applications to Mechanics, Elasticity, Theory of Relativity, Electromagnetic, Riemannian Geometry and many other disciplines of science and engineering, in a lucid manner. The text has been explained section wise, every concept has been narrated in the form of definition, examples and questions related to the concept taught. The overall package of the book is highly useful and interesting for the people associated with the field. Contents: Preliminaries Tensor Algebra Metric Tensor and Riemannian Metric Christoffel`s Symbols and Covariant Differentiation Riemann-Christoffel Tensor The e-Systems and the Generalized Krönecker Deltas Geometry Analytical Mechanics Curvature of a Curve, Geodesic Parallelism of Vectors Ricci`s Coefficients of Rotation and Congruence Hyper Surfaces

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)

Information geometry of density matrices and state estimation

International Nuclear Information System (INIS)

Brody, Dorje C

2011-01-01

Given a pure state vector |x) and a density matrix ρ-hat, the function p(x|ρ-hat)= defines a probability density on the space of pure states parameterised by density matrices. The associated Fisher-Rao information measure is used to define a unitary invariant Riemannian metric on the space of density matrices. An alternative derivation of the metric, based on square-root density matrices and trace norms, is provided. This is applied to the problem of quantum-state estimation. In the simplest case of unitary parameter estimation, new higher-order corrections to the uncertainty relations, applicable to general mixed states, are derived. (fast track communication)

Introduction to global analysis minimal surfaces in Riemannian manifolds

CERN Document Server

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...

Classical boundary-value problem in Riemannian quantum gravity and self-dual Taub-NUT-(anti)de Sitter geometries

International Nuclear Information System (INIS)

Akbar, M.M.; D'Eath, P.D.

2003-01-01

The classical boundary-value problem of the Einstein field equations is studied with an arbitrary cosmological constant, in the case of a compact (S 3 ) boundary given a biaxial Bianchi-IX positive-definite three-metric, specified by two radii (a,b). For the simplest, four-ball, topology of the manifold with this boundary, the regular classical solutions are found within the family of Taub-NUT-(anti)de Sitter metrics with self-dual Weyl curvature. For arbitrary choice of positive radii (a,b), we find that there are three solutions for the infilling geometry of this type. We obtain exact solutions for them and for their Euclidean actions. The case of negative cosmological constant is investigated further. For reasonable squashing of the three-sphere, all three infilling solutions have real-valued actions which possess a 'cusp catastrophe' structure with a non-self-intersecting 'catastrophe manifold' implying that the dominant contribution comes from the unique real positive-definite solution on the ball. The positive-definite solution exists even for larger deformations of the three-sphere, as long as a certain inequality between a and b holds. The action of this solution is proportional to -a 3 for large a (∼b) and hence larger radii are favoured. The same boundary-value problem with more complicated interior topology containing a 'bolt' is investigated in a forthcoming paper

Contribution to the establishment and resolution of the Schroedinger equation in a Riemannian manifold with constant curvature

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

Steiner minimal trees in small neighbourhoods of points in Riemannian manifolds

Science.gov (United States)

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.

Topics in modern differential geometry

CERN Document Server

Verstraelen, Leopold

2017-01-01

A variety of introductory articles is provided on a wide range of topics, including variational problems on curves and surfaces with anisotropic curvature. Experts in the fields of Riemannian, Lorentzian and contact geometry present state-of-the-art reviews of their topics. The contributions are written on a graduate level and contain extended bibliographies. The ten chapters are the result of various doctoral courses which were held in 2009 and 2010 at universities in Leuven, Serbia, Romania and Spain.

Topology, ergodic theory, real algebraic geometry Rokhlin's memorial

CERN Document Server

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

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

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.

VIII International Meeting on Lorentzian Geometry

CERN Document Server

Flores, José; Palomo, Francisco; GeLoMa 2016; Lorentzian geometry and related topics

2017-01-01

This volume contains a collection of research papers and useful surveys by experts in the field which provide a representative picture of the current status of this fascinating area. Based on contributions from the VIII International Meeting on Lorentzian Geometry, held at the University of Málaga, Spain, this volume covers topics such as distinguished (maximal, trapped, null, spacelike, constant mean curvature, umbilical...) submanifolds, causal completion of spacetimes, stationary regions and horizons in spacetimes, solitons in semi-Riemannian manifolds, relation between Lorentzian and Finslerian geometries and the oscillator spacetime. In the last decades Lorentzian geometry has experienced a significant impulse, which has transformed it from just a mathematical tool for general relativity to a consolidated branch of differential geometry, interesting in and of itself. Nowadays, this field provides a framework where many different mathematical techniques arise with applications to multiple parts of mathem...

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)

Spinorial characterizations of surfaces into 3-dimensional psuedo-Riemannian space forms

OpenAIRE

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 ...

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

On Finsler Geometry and Applications in Mechanics: Review and New Perspectives

Directory of Open Access Journals (Sweden)

J. D. Clayton

2015-01-01

direction as well as position, and a number of connections emerge associated with various covariant derivatives involving affine and nonlinear coefficients. Finsler geometry encompasses Riemannian, Euclidean, and Minkowskian geometries as special cases, and thus it affords great generality for describing a number of phenomena in physics. Here, descriptions of finite deformation of continuous media are of primary focus. After a review of necessary mathematical definitions and derivations, prior work involving application of Finsler geometry in continuum mechanics of solids is reviewed. A new theoretical description of continua with microstructure is then outlined, merging concepts from Finsler geometry and phase field theories of materials science.

Geometry, algebra and applications from mechanics to cryptography

CERN Document Server

Encinas, Luis; Gadea, Pedro; María, Mª

2016-01-01

This volume collects contributions written by different experts in honor of Prof. Jaime Muñoz Masqué. It covers a wide variety of research topics, from differential geometry to algebra, but particularly focuses on the geometric formulation of variational calculus; geometric mechanics and field theories; symmetries and conservation laws of differential equations, and pseudo-Riemannian geometry of homogeneous spaces. It also discusses algebraic applications to cryptography and number theory. It offers state-of-the-art contributions in the context of current research trends. The final result is a challenging panoramic view of connecting problems that initially appear distant.

Information geometry of Gaussian channels

International Nuclear Information System (INIS)

Monras, Alex; Illuminati, Fabrizio

2010-01-01

We define a local Riemannian metric tensor in the manifold of Gaussian channels and the distance that it induces. We adopt an information-geometric approach and define a metric derived from the Bures-Fisher metric for quantum states. The resulting metric inherits several desirable properties from the Bures-Fisher metric and is operationally motivated by distinguishability considerations: It serves as an upper bound to the attainable quantum Fisher information for the channel parameters using Gaussian states, under generic constraints on the physically available resources. Our approach naturally includes the use of entangled Gaussian probe states. We prove that the metric enjoys some desirable properties like stability and covariance. As a by-product, we also obtain some general results in Gaussian channel estimation that are the continuous-variable analogs of previously known results in finite dimensions. We prove that optimal probe states are always pure and bounded in the number of ancillary modes, even in the presence of constraints on the reduced state input in the channel. This has experimental and computational implications. It limits the complexity of optimal experimental setups for channel estimation and reduces the computational requirements for the evaluation of the metric: Indeed, we construct a converging algorithm for its computation. We provide explicit formulas for computing the multiparametric quantum Fisher information for dissipative channels probed with arbitrary Gaussian states and provide the optimal observables for the estimation of the channel parameters (e.g., bath couplings, squeezing, and temperature).

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...

Analysis of correlation between full-waveform metrics, scan geometry and land-cover: an application over forests

Directory of Open Access Journals (Sweden)

F. Pirotti

2013-10-01

Full Text Available For a correct use of metrics derived from processing of the full-waveform return signal from airborne laser scanner sensors any correlation which is not related to properties of the reflecting target must be known and, if possible, removed. In the following article we report on an analysis of correlation between several metrics extracted from the full-waveform return signal and scan characteristics (mainly range and type of land-cover (urban, grasslands, forests. The metrics taken in consideration are the amplitude, normalized amplitude, width (full width at half maximum, asymmetry indicators, left and right energy content, and the cross-section calculated from width and normalized amplitude considering the range effect. The results show that scan geometry in this case does not have a significant impact scans over forest cover, except for range affecting amplitude and width distribution. Over complex targets such as vegetation canopy, other factors such as incidence angle have little meaning, therefore corrections of range effect are the most meaningful. A strong correlation with the type of land-cover is also shown by the distribution of the values of the metrics in the different areas taken in consideration.

10th China-Japan Geometry Conference

CERN Document Server

Miyaoka, Reiko; Tang, Zizhou; Zhang, Weiping

2016-01-01

Since the year 2000, we have witnessed several outstanding results in geometry that have solved long-standing problems such as the Poincaré conjecture, the Yau–Tian–Donaldson conjecture, and the Willmore conjecture. There are still many important and challenging unsolved problems including, among others, the Strominger–Yau–Zaslow conjecture on mirror symmetry, the relative Yau–Tian–Donaldson conjecture in Kähler geometry, the Hopf conjecture, and the Yau conjecture on the first eigenvalue of an embedded minimal hypersurface of the sphere. For the younger generation to approach such problems and obtain the required techniques, it is of the utmost importance to provide them with up-to-date information from leading specialists. The geometry conference for the friendship of China and Japan has achieved this purpose during the past 10 years. Their talks deal with problems at the highest level, often accompanied with solutions and ideas, which extend across various fields in Riemannian geometry, sympl...

Metrics of quantum states

International Nuclear Information System (INIS)

Ma Zhihao; Chen Jingling

2011-01-01

In this work we study metrics of quantum states, which are natural generalizations of the usual trace metric and Bures metric. Some useful properties of the metrics are proved, such as the joint convexity and contractivity under quantum operations. Our result has a potential application in studying the geometry of quantum states as well as the entanglement detection.

Arcmancer: Geodesics and polarized radiative transfer library

Science.gov (United States)

Pihajoki, Pauli; Mannerkoski, Matias; Nättilä, Joonas; Johansson, Peter H.

2018-05-01

Arcmancer computes geodesics and performs polarized radiative transfer in user-specified spacetimes. The library supports Riemannian and semi-Riemannian spaces of any dimension and metric; it also supports multiple simultaneous coordinate charts, embedded geometric shapes, local coordinate systems, and automatic parallel propagation. Arcmancer can be used to solve various problems in numerical geometry, such as solving the curve equation of motion using adaptive integration with configurable tolerances and differential equations along precomputed curves. It also provides support for curves with an arbitrary acceleration term and generic tools for generating ray initial conditions and performing parallel computation over the image, among other tools.

Partner Symmetries, Group Foliation and ASD Ricci-Flat Metrics without Killing Vectors

Directory of Open Access Journals (Sweden)

Andrei A. Malykh

2013-11-01

Full Text Available We demonstrate how a combination of our recently developed methods of partner symmetries, symmetry reduction in group parameters and a new version of the group foliation method can produce noninvariant solutions of complex Monge-Ampère equation (CMA and provide a lift from invariant solutions of CMA satisfying Boyer-Finley equation to non-invariant ones. Applying these methods, we obtain a new noninvariant solution of CMA and the corresponding Ricci-flat anti-self-dual Einstein-Kähler metric with Euclidean signature without Killing vectors, together with Riemannian curvature two-forms. There are no singularities of the metric and curvature in a bounded domain if we avoid very special choices of arbitrary functions of a single variable in our solution. This metric does not describe gravitational instantons because the curvature is not concentrated in a bounded domain.

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.

The positive action conjecture and asymptotically euclidean metrics in quantum gravity

International Nuclear Information System (INIS)

Gibbons, G.W.; Pope, C.N.

1979-01-01

The positive action conjecture requires that the action of any asymptotically Euclidean 4-dimensional Riemannian metric be positive, vanishing if and only if the space is flat. Because any Ricci flat, asymptotically Euclidean metric has zero action and is local extremum of the action which is a local minimum at flat space, the conjecture requires that there are no Ricci flat asymptotically Euclidean metrics other than flat space, which would establish that flat space is the only local minimum. We prove this for metrics on R 4 and a large class of more complicated topologies and for self-dual metrics. We show that if Rsupμsubμ >= 0 there are no bound states of the Dirac equation and discuss the relevance to possible baryon non-conserving processes mediated by gravitational instantons. We conclude that these are forbidden in the lowest stationary phase approximation. We give a detailed discussion of instantons invariant under an SU(2) or SO(3) isometry group. We find all regular solutions, none of which is asymptotically Euclidean and all of which possess a further Killing vector. In an appendix we construct an approximate self-dual metric on K3 - the only simply connected compact manifold which admits a self-dual metric. (orig.) [de

Geometry and dynamics in Gromov hyperbolic metric spaces with an emphasis on non-proper settings

CERN Document Server

Das, Tushar; Urbański, Mariusz

2016-01-01

This book presents the foundations of the theory of groups and semigroups acting isometrically on Gromov hyperbolic metric spaces. Particular emphasis is paid to the geometry of their limit sets and on behavior not found in the proper setting. The authors provide a number of examples of groups which exhibit a wide range of phenomena not to be found in the finite-dimensional theory. The book contains both introductory material to help beginners as well as new research results, and closes with a list of attractive unsolved problems.

Combined Tensor Fitting and TV Regularization in Diffusion Tensor Imaging Based on a Riemannian Manifold Approach.

Science.gov (United States)

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.

Lectures on coarse geometry

CERN Document Server

Roe, John

2003-01-01

Coarse geometry is the study of spaces (particularly metric spaces) from a 'large scale' point of view, so that two spaces that look the same from a great distance are actually equivalent. This point of view is effective because it is often true that the relevant geometric properties of metric spaces are determined by their coarse geometry. Two examples of important uses of coarse geometry are Gromov's beautiful notion of a hyperbolic group and Mostow's proof of his famous rigidity theorem. The first few chapters of the book provide a general perspective on coarse structures. Even when only metric coarse structures are in view, the abstract framework brings the same simplification as does the passage from epsilons and deltas to open sets when speaking of continuity. The middle section reviews notions of negative curvature and rigidity. Modern interest in large scale geometry derives in large part from Mostow's rigidity theorem and from Gromov's subsequent 'large scale' rendition of the crucial properties of n...

Conformal changes of metrics and the initial-value problem of general relativity

International Nuclear Information System (INIS)

Mielke, E.W.

1977-01-01

Conformal techniques are reviewed with respect to applications to the initial-value problem of general relativity. Invariant transverse traceless decompositions of tensors, one of its main tools, are related to representations of the group of 'conformeomorphisms' acting on the space of all Riemannian metrics on M. Conformal vector fields, a kernel in the decomposition, are analyzed on compact manifolds with constant scalar curvature. The realization of arbitrary functions as scalar curvature of conformally equivalent metrics, a generalization of Yamabe's (Osaka Math. J.; 12:12 (1960)) conjecture, is applied to the Hamiltonian constraint and to the issue of positive energy of gravitational fields. Various approaches to the solution of the initial-value equations produced by altering the scaling behaviour of the second fundamental form are compared. (author)

Electroweak vacuum geometry

International Nuclear Information System (INIS)

Lepora, N.; Kibble, T.

1999-01-01

We analyse symmetry breaking in the Weinberg-Salam model paying particular attention to the underlying geometry of the theory. In this context we find two natural metrics upon the vacuum manifold: an isotropic metric associated with the scalar sector, and a squashed metric associated with the gauge sector. Physically, the interplay between these metrics gives rise to many of the non-perturbative features of Weinberg-Salam theory. (author)

Spinning geometry = Twisted geometry

International Nuclear Information System (INIS)

Freidel, Laurent; Ziprick, Jonathan

2014-01-01

It is well known that the SU(2)-gauge invariant phase space of loop gravity can be represented in terms of twisted geometries. These are piecewise-linear-flat geometries obtained by gluing together polyhedra, but the resulting geometries are not continuous across the faces. Here we show that this phase space can also be represented by continuous, piecewise-flat three-geometries called spinning geometries. These are composed of metric-flat three-cells glued together consistently. The geometry of each cell and the manner in which they are glued is compatible with the choice of fluxes and holonomies. We first remark that the fluxes provide each edge with an angular momentum. By studying the piecewise-flat geometries which minimize edge lengths, we show that these angular momenta can be literally interpreted as the spin of the edges: the geometries of all edges are necessarily helices. We also show that the compatibility of the gluing maps with the holonomy data results in the same conclusion. This shows that a spinning geometry represents a way to glue together the three-cells of a twisted geometry to form a continuous geometry which represents a point in the loop gravity phase space. (paper)

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)

Cartan for beginners differential geometry via moving frames and exterior differential systems

CERN Document Server

Ivey, Thomas A

2016-01-01

Two central aspects of Cartan's approach to differential geometry are the theory of exterior differential systems (EDS) and the method of moving frames. This book presents thorough and modern treatments of both subjects, including their applications to both classic and contemporary problems in geometry. It begins with the classical differential geometry of surfaces and basic Riemannian geometry in the language of moving frames, along with an elementary introduction to exterior differential systems. Key concepts are developed incrementally, with motivating examples leading to definitions, theorems, and proofs. Once the basics of the methods are established, the authors develop applications and advanced topics. One notable application is to complex algebraic geometry, where they expand and update important results from projective differential geometry. As well, the book features an introduction to G-structures and a treatment of the theory of connections. The techniques of EDS are also applied to obtain explici...

Geometry of isotropic convex bodies

CERN Document Server

Brazitikos, Silouanos; Valettas, Petros; Vritsiou, Beatrice-Helen

2014-01-01

The study of high-dimensional convex bodies from a geometric and analytic point of view, with an emphasis on the dependence of various parameters on the dimension stands at the intersection of classical convex geometry and the local theory of Banach spaces. It is also closely linked to many other fields, such as probability theory, partial differential equations, Riemannian geometry, harmonic analysis and combinatorics. It is now understood that the convexity assumption forces most of the volume of a high-dimensional convex body to be concentrated in some canonical way and the main question is whether, under some natural normalization, the answer to many fundamental questions should be independent of the dimension. The aim of this book is to introduce a number of well-known questions regarding the distribution of volume in high-dimensional convex bodies, which are exactly of this nature: among them are the slicing problem, the thin shell conjecture and the Kannan-Lov�sz-Simonovits conjecture. This book prov...

On Averaging Rotations

DEFF Research Database (Denmark)

Gramkow, Claus

1999-01-01

In this article two common approaches to averaging rotations are compared to a more advanced approach based on a Riemannian metric. Very offten the barycenter of the quaternions or matrices that represent the rotations are used as an estimate of the mean. These methods neglect that rotations belo...... approximations to the Riemannian metric, and that the subsequent corrections are inherient in the least squares estimation. Keywords: averaging rotations, Riemannian metric, matrix, quaternion......In this article two common approaches to averaging rotations are compared to a more advanced approach based on a Riemannian metric. Very offten the barycenter of the quaternions or matrices that represent the rotations are used as an estimate of the mean. These methods neglect that rotations belong...

The Ricci flow part IV : long-time solutions and related topics

CERN Document Server

Chow, Bennett; Glickenstein, David; Isenberg, James

2015-01-01

Ricci flow is a powerful technique using a heat-type equation to deform Riemannian metrics on manifolds to better metrics in the search for geometric decompositions. With the fourth part of their volume on techniques and applications of the theory, the authors discuss long-time solutions of the Ricci flow and related topics. In dimension 3, Perelman completed Hamilton's program to prove Thurston's geometrization conjecture. In higher dimensions the Ricci flow has remarkable properties, which indicates its usefulness to understand relations between the geometry and topology of manifolds. This b

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

W-geometry

International Nuclear Information System (INIS)

Hull, C.M.

1993-01-01

The geometric structure of theories with gauge fields of spins two and higher should involve a higher spin generalisation of Riemannian geometry. Such geometries are discussed and the case of W ∝ -gravity is analysed in detail. While the gauge group for gravity in d dimensions is the diffeomorphism group of the space-time, the gauge group for a certain W-gravity theory (which is W ∝ -gravity in the case d=2) is the group of symplectic diffeomorphisms of the cotangent bundle of the space-time. Gauge transformations for W-gravity gauge fields are given by requiring the invariance of a generalised line element. Densities exist and can be constructed from the line element (generalising √detg μν ) only if d=1 or d=2, so that only for d=1,2 can actions be constructed. These two cases and the corresponding W-gravity actions are considered in detail. In d=2, the gauge group is effectively only a subgroup of the symplectic diffeomorphisms group. Some of the constraints that arise for d=2 are similar to equations arising in the study of self-dual four-dimensional geometries and can be analysed using twistor methods, allowing contact to be made with other formulations of W-gravity. While the twistor transform for self-dual spaces with one Killing vector reduces to a Legendre transform, that for two Killing vectors gives a generalisation of the Legendre transform. (orig.)

Conformal pure radiation with parallel rays

International Nuclear Information System (INIS)

Leistner, Thomas; Paweł Nurowski

2012-01-01

We define pure radiation metrics with parallel rays to be n-dimensional pseudo-Riemannian metrics that admit a parallel null line bundle K and whose Ricci tensor vanishes on vectors that are orthogonal to K. We give necessary conditions in terms of the Weyl, Cotton and Bach tensors for a pseudo-Riemannian metric to be conformal to a pure radiation metric with parallel rays. Then, we derive conditions in terms of the tractor calculus that are equivalent to the existence of a pure radiation metric with parallel rays in a conformal class. We also give analogous results for n-dimensional pseudo-Riemannian pp-waves. (paper)

Variational study of spectral shifts. II

International Nuclear Information System (INIS)

Peton, A.

1979-01-01

In a static gravitational field the paths of light are curved. This property can be a priori stated for a V 3 Riemannian manifold: through any two points of V 3 it is possible to draw two families of curves, the straight lines of Euclidean geometry and the photon trajectories z. A fibration of the Galilean space-time can be performed in an original way, by taking the z-trajectories of the photons as the base, the isochronic surfaces as fibres, and 'the equal length time on a z trajectory to reach a given point' as the equivalence relation. The straight lines of Euclidean geometry can then carry the classical mechanics time t, and the z trajectories can carry the optics time (T). These times are related by d(T)=F(x,t)dt. If the Universe is classed as a pseudo-Riemannian manifold of normal hyperbolic type Csup(infinity), the time (T) determined above can be taken as the time coordinate in V 4 . Under these conditions d(S) 2 =F 2 ds 2 , where d(S) 2 is the metric of the Riemannian manifold, conforming to the metric ds 2 and allowing (T) as the cosmic time. The results previously achieved by the author (Peton, 1979) can be used to find 1+zsub(G)=F(Asub(s), tsub(s))/F(Asub(O),tsub(O)) where zsub(G) denotes the shift of the spectral lines due to the metric. In the case of relative motion between O and S, 1+z'=(1+zsub(G))(1+βsub(r))(1-β 2 )sup(-1/2)). The Doppler-Fizeau effect therefore appears as a result of the application of the Fermat principle. (Auth.)

Metric Structures on Fibered Manifolds Through Partitions of Unity

Directory of Open Access Journals (Sweden)

Hulya Kadioglu

2016-05-01

Full Text Available The notion of partitions of unity is extremely useful as it allows one to extend local constructions on Euclidean patches to global ones. It is widely used in many fields in mathematics. Therefore, prolongation of this useful tool to another manifold may help constructing many geometric structures. In this paper, we construct a partition of unity on a fiber bundle by using a given partition of unity on the base manifold. On the other hand we show that the converse is also possible if it is a vector bundle. As an application, we define a Riemannian metric on the fiber bundle by using induced partition of unity on the fiber bundle.

A Metric for Heterotic Moduli

Science.gov (United States)

Candelas, Philip; de la Ossa, Xenia; McOrist, Jock

2017-12-01

Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in {α^{\\backprime}}, in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kähler, as is required by supersymmetry. Checking the metric is Kähler is intricate and the anomaly cancellation equation for the H field plays an essential role. The Kähler potential nevertheless takes a remarkably simple form: it is the Kähler potential of special geometry with the Kähler form replaced by the {α^{\\backprime}}-corrected hermitian form.

The metric and curvature properties of H-space

International Nuclear Information System (INIS)

Hansen, R.O.; Newman, E.T.; Penrose, R.; Tod, K.P.

1978-01-01

The space H of asymptotically (left-) shear-free cuts of the future null infinity (good cuts) of an asymptotically flat space-time M is defined. The connection between this space and the asymptotic projective twistor space of M is discussed, and this relation is used to prove that H is four-complex-dimensional for sufficiently 'calm' gravitational radiation in M. The metric on H-space is defined by a simple contour integral expression and is found to be complex Riemannian. The good cut equation governing H-space is solved to three orders by a Taylor series and the solution is used to demonstrate that the curvature of H-space is always a self dual (left flat) solution of the Einstein vacuum equations. (author)

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

Axioms of spheres in lightlike geometry of submanifolds

Indian Academy of Sciences (India)

Introduction. The notion of axioms of planes for Riemannian manifolds was originally introduced by. Cartan [2]. In [8], Leung and Nomizu generalized the notion of axioms of planes to the axioms of spheres on Riemannian manifolds. In [7], Kumar et al. studied the axioms of spheres and planes for indefinite Riemannian ...

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

Science.gov (United States)

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.

Does Negative Type Characterize the Round Sphere?

DEFF Research Database (Denmark)

Kokkendorff, Simon Lyngby

2007-01-01

We discuss the measure theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio 1 must be a round sphere, was put forward in a previous paper....... We resolve this conjecture in the class of Riemannian symmetric spaces by showing, that a Riemannian manifold with symmetry ratio 1 must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres....

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

Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes.

Science.gov (United States)

Cafaro, Carlo; Alsing, Paul M

2018-04-01

The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.

Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes

Science.gov (United States)

Cafaro, Carlo; Alsing, Paul M.

2018-04-01

The relevance of the concept of Fisher information is increasing in both statistical physics and quantum computing. From a statistical mechanical standpoint, the application of Fisher information in the kinetic theory of gases is characterized by its decrease along the solutions of the Boltzmann equation for Maxwellian molecules in the two-dimensional case. From a quantum mechanical standpoint, the output state in Grover's quantum search algorithm follows a geodesic path obtained from the Fubini-Study metric on the manifold of Hilbert-space rays. Additionally, Grover's algorithm is specified by constant Fisher information. In this paper, we present an information geometric characterization of the oscillatory or monotonic behavior of statistically parametrized squared probability amplitudes originating from special functional forms of the Fisher information function: constant, exponential decay, and power-law decay. Furthermore, for each case, we compute both the computational speed and the availability loss of the corresponding physical processes by exploiting a convenient Riemannian geometrization of useful thermodynamical concepts. Finally, we briefly comment on the possibility of using the proposed methods of information geometry to help identify a suitable trade-off between speed and thermodynamic efficiency in quantum search algorithms.

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

Projective geometry and projective metrics

CERN Document Server

Busemann, Herbert

2005-01-01

The basic results and methods of projective and non-Euclidean geometry are indispensable for the geometer, and this book--different in content, methods, and point of view from traditional texts--attempts to emphasize that fact. Results of special theorems are discussed in detail only when they are needed to develop a feeling for the subject or when they illustrate a general method. On the other hand, an unusual amount of space is devoted to the discussion of the fundamental concepts of distance, motion, area, and perpendicularity.Topics include the projective plane, polarities and conic sectio

The geometry of warped product singularities

Science.gov (United States)

Stoica, Ovidiu Cristinel

In this article, the degenerate warped products of singular semi-Riemannian manifolds are studied. They were used recently by the author to handle singularities occurring in General Relativity, in black holes and at the big-bang. One main result presented here is that a degenerate warped product of semi-regular semi-Riemannian manifolds with the warping function satisfying a certain condition is a semi-regular semi-Riemannian manifold. The connection and the Riemann curvature of the warped product are expressed in terms of those of the factor manifolds. Examples of singular semi-Riemannian manifolds which are semi-regular are constructed as warped products. Applications include cosmological models and black holes solutions with semi-regular singularities. Such singularities are compatible with a certain reformulation of the Einstein equation, which in addition holds at semi-regular singularities too.

Riemannian theory of Hamiltonian chaos and Lyapunov exponents

Science.gov (United States)

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.

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)

Optical geometry

International Nuclear Information System (INIS)

Robinson, I.; Trautman, A.

1988-01-01

The geometry of classical physics is Lorentzian; but weaker geometries are often more appropriate: null geodesics and electromagnetic fields, for example, are well known to be objects of conformal geometry. To deal with a single null congruence, or with the radiative electromagnetic fields associated with it, even less is needed: flag geometry for the first, optical geometry, with which this paper is chiefly concerned, for the second. The authors establish a natural one-to-one correspondence between optical geometries, considered locally, and three-dimensional Cauchy-Riemann structures. A number of Lorentzian geometries are shown to be equivalent from the optical point of view. For example the Goedel universe, the Taub-NUT metric and Hauser's twisting null solution have an optical geometry isomorphic to the one underlying the Robinson congruence in Minkowski space. The authors present general results on the problem of lifting a CR structure to a Lorentz manifold and, in particular, to Minkowski space; and exhibit the relevance of the deviation form to this problem

Metric diffusion along foliations

CERN Document Server

Walczak, Szymon M

2017-01-01

Up-to-date research in metric diffusion along compact foliations is presented in this book. Beginning with fundamentals from the optimal transportation theory and the theory of foliations; this book moves on to cover Wasserstein distance, Kantorovich Duality Theorem, and the metrization of the weak topology by the Wasserstein distance. Metric diffusion is defined, the topology of the metric space is studied and the limits of diffused metrics along compact foliations are discussed. Essentials on foliations, holonomy, heat diffusion, and compact foliations are detailed and vital technical lemmas are proved to aide understanding. Graduate students and researchers in geometry, topology and dynamics of foliations and laminations will find this supplement useful as it presents facts about the metric diffusion along non-compact foliation and provides a full description of the limit for metrics diffused along foliation with at least one compact leaf on the two dimensions.

a Super Voxel-Based Riemannian Graph for Multi Scale Segmentation of LIDAR Point Clouds

Science.gov (United States)

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.

Sub-Riemannian geometry and time optimal control of three spin systems: Quantum gates and coherence transfer

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)

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

Metrical and dynamical aspects in complex analysis

CERN Document Server

2017-01-01

The central theme of this reference book is the metric geometry of complex analysis in several variables. Bridging a gap in the current literature, the text focuses on the fine behavior of the Kobayashi metric of complex manifolds and its relationships to dynamical systems, hyperbolicity in the sense of Gromov and operator theory, all very active areas of research. The modern points of view expressed in these notes, collected here for the first time, will be of interest to academics working in the fields of several complex variables and metric geometry. The different topics are treated coherently and include expository presentations of the relevant tools, techniques and objects, which will be particularly useful for graduate and PhD students specializing in the area.

General very special relativity in Finsler cosmology

International Nuclear Information System (INIS)

Kouretsis, A. P.; Stathakopoulos, M.; Stavrinos, P. C.

2009-01-01

General very special relativity (GVSR) is the curved space-time of very special relativity (VSR) proposed by Cohen and Glashow. The geometry of general very special relativity possesses a line element of Finsler geometry introduced by Bogoslovsky. We calculate the Einstein field equations and derive a modified Friedmann-Robertson-Walker cosmology for an osculating Riemannian space. The Friedmann equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial-spurionic vector field introduced into the metric gives back an estimation of the energy evolution and inflation.

Jacobi-Maupertuis metric and Kepler equation

Science.gov (United States)

Chanda, Sumanto; Gibbons, Gary William; Guha, Partha

This paper studies the application of the Jacobi-Eisenhart lift, Jacobi metric and Maupertuis transformation to the Kepler system. We start by reviewing fundamentals and the Jacobi metric. Then we study various ways to apply the lift to Kepler-related systems: first as conformal description and Bohlin transformation of Hooke’s oscillator, second in contact geometry and third in Houri’s transformation [T. Houri, Liouville integrability of Hamiltonian systems and spacetime symmetry (2016), www.geocities.jp/football_physician/publication.html], coupled with Milnor’s construction [J. Milnor, On the geometry of the Kepler problem, Am. Math. Mon. 90 (1983) 353-365] with eccentric anomaly.

Flow equation, conformal symmetry, and anti-de Sitter geometry

Science.gov (United States)

Aoki, Sinya; Yokoyama, Shuichi

2018-03-01

We argue that the anti-de Sitter (AdS) geometry in d+1 dimensions naturally emerges from an arbitrary conformal field theory in d dimensions using the free flow equation. We first show that an induced metric defined from the flowed field generally corresponds to the quantum information metric, called the Bures or Helstrom metric, if the flowed field is normalized appropriately. We next verify that the induced metric computed explicitly with the free flow equation always becomes the AdS metric when the theory is conformal. We finally prove that the conformal symmetry in d dimensions converts to the AdS isometry in d+1 dimensions after d-dimensional quantum averaging. This guarantees the emergence of AdS geometry without explicit calculation.

Physical meaning of the optical reference geometry

International Nuclear Information System (INIS)

Abramowicz, M.A.

1990-09-01

I show that contrary to a popular misconception the optical reference geometry, introduced a few years ago as a formally possible metric of a 3-space corresponding to a static spacetime, is quite satisfactory also from the physical point of view. The optical reference geometry has a clear physical meaning, as it may be constructed experimentally by measuring light round travel time between static observers. Distances and directions in the optical reference geometry are more strongly connected to experiment than distances and directions in the widely used directly projected metric (discussed e.g. in Landau and Lifshitz textbook. In addition, the optical reference geometry is more natural and convenient than the directly projected one in application to dynamics. In the optical geometry dynamical behaviour of matter is described by concepts and formulae identical to those well known in Newtonian dynamics on a given two dimensional (curved) surface. (author). 22 refs

Towards Video Quality Metrics Based on Colour Fractal Geometry

Directory of Open Access Journals (Sweden)

Richard Noël

2010-01-01

Full Text Available Vision is a complex process that integrates multiple aspects of an image: spatial frequencies, topology and colour. Unfortunately, so far, all these elements were independently took into consideration for the development of image and video quality metrics, therefore we propose an approach that blends together all of them. Our approach allows for the analysis of the complexity of colour images in the RGB colour space, based on the probabilistic algorithm for calculating the fractal dimension and lacunarity. Given that all the existing fractal approaches are defined only for gray-scale images, we extend them to the colour domain. We show how these two colour fractal features capture the multiple aspects that characterize the degradation of the video signal, based on the hypothesis that the quality degradation perceived by the user is directly proportional to the modification of the fractal complexity. We claim that the two colour fractal measures can objectively assess the quality of the video signal and they can be used as metrics for the user-perceived video quality degradation and we validated them through experimental results obtained for an MPEG-4 video streaming application; finally, the results are compared against the ones given by unanimously-accepted metrics and subjective tests.

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...

On Darboux's approach to R-separability of variables. Classification of conformally flat 4-dimensional binary metrics

International Nuclear Information System (INIS)

Szereszewski, A; Sym, A

2015-01-01

The standard method of separation of variables in PDEs called the Stäckel–Robertson–Eisenhart (SRE) approach originated in the papers by Robertson (1928 Math. Ann. 98 749–52) and Eisenhart (1934 Ann. Math. 35 284–305) on separability of variables in the Schrödinger equation defined on a pseudo-Riemannian space equipped with orthogonal coordinates, which in turn were based on the purely classical mechanics results by Paul Stäckel (1891, Habilitation Thesis, Halle). These still fundamental results have been further extended in diverse directions by e.g. Havas (1975 J. Math. Phys. 16 1461–8; J. Math. Phys. 16 2476–89) or Koornwinder (1980 Lecture Notes in Mathematics 810 (Berlin: Springer) pp 240–63). The involved separability is always ordinary (factor R = 1) and regular (maximum number of independent parameters in separation equations). A different approach to separation of variables was initiated by Gaston Darboux (1878 Ann. Sci. E.N.S. 7 275–348) which has been almost completely forgotten in today’s research on the subject. Darboux’s paper was devoted to the so-called R-separability of variables in the standard Laplace equation. At the outset he did not make any specific assumption about the separation equations (this is in sharp contrast to the SRE approach). After impressive calculations Darboux obtained a complete solution of the problem. He found not only eleven cases of ordinary separability Eisenhart (1934 Ann. Math. 35 284–305) but also Darboux–Moutard–cyclidic metrics (Bôcher 1894 Ueber die Reihenentwickelungen der Potentialtheorie (Leipzig: Teubner)) and non-regularly separable Dupin-cyclidic metrics as well. In our previous paper Darboux’s approach was extended to the case of the stationary Schrödinger equation on Riemannian spaces admitting orthogonal coordinates. In particular the class of isothermic metrics was defined (isothermicity of the metric is a necessary condition for its R-separability). An important sub

The geometry of higher-order Lagrange spaces applications to mechanics and physics

CERN Document Server

Miron, Radu

1997-01-01

This monograph is devoted to the problem of the geometrizing of Lagrangians which depend on higher-order accelerations It presents a construction of the geometry of the total space of the bundle of the accelerations of order k>=1 A geometrical study of the notion of the higher-order Lagrange space is conducted, and the old problem of prolongation of Riemannian spaces to k-osculator manifolds is solved Also, the geometrical ground for variational calculus on the integral of actions involving higher-order Lagrangians is dealt with Applications to higher-order analytical mechanics and theoretical physics are included as well Audience This volume will be of interest to scientists whose work involves differential geometry, mechanics of particles and systems, calculus of variation and optimal control, optimization, optics, electromagnetic theory, and biology

On A Nonlinear Generalization of Sparse Coding and Dictionary Learning.

Science.gov (United States)

Xie, Yuchen; Ho, Jeffrey; Vemuri, Baba

2013-01-01

Existing dictionary learning algorithms are based on the assumption that the data are vectors in an Euclidean vector space ℝ d , and the dictionary is learned from the training data using the vector space structure of ℝ d and its Euclidean L 2 -metric. However, in many applications, features and data often originated from a Riemannian manifold that does not support a global linear (vector space) structure. Furthermore, the extrinsic viewpoint of existing dictionary learning algorithms becomes inappropriate for modeling and incorporating the intrinsic geometry of the manifold that is potentially important and critical to the application. This paper proposes a novel framework for sparse coding and dictionary learning for data on a Riemannian manifold, and it shows that the existing sparse coding and dictionary learning methods can be considered as special (Euclidean) cases of the more general framework proposed here. We show that both the dictionary and sparse coding can be effectively computed for several important classes of Riemannian manifolds, and we validate the proposed method using two well-known classification problems in computer vision and medical imaging analysis.

Invariance group of the Finster metric function

International Nuclear Information System (INIS)

Asanov, G.S.

1985-01-01

An invariance group of the Finsler metric function is introduced and studied that directly generalized the respective concept (a group of Euclidean rolations) of the Rieman geometry. A sequential description of the isotopic invariance of physical fields on the base of the Finsler geometry is possible in terms of this group

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

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

On Averaging Rotations

DEFF Research Database (Denmark)

Gramkow, Claus

2001-01-01

In this paper two common approaches to averaging rotations are compared to a more advanced approach based on a Riemannian metric. Very often the barycenter of the quaternions or matrices that represent the rotations are used as an estimate of the mean. These methods neglect that rotations belong ...... approximations to the Riemannian metric, and that the subsequent corrections are inherent in the least squares estimation.......In this paper two common approaches to averaging rotations are compared to a more advanced approach based on a Riemannian metric. Very often the barycenter of the quaternions or matrices that represent the rotations are used as an estimate of the mean. These methods neglect that rotations belong...

A New Look to Massive Neutron Cores

OpenAIRE

Bel, Ll.

2002-01-01

We reconsider the problem of modelling static spherically symmetric perfect fluid configurations with an equation of state from a point of view of that requires the use of the concept of principal transform of a 3-dimensional Riemannian metric. We discuss from this new point of view the meaning of those familiar quantities that we call density, pressure and geometry in a relativistic context. This is not simple semantics. To prove it we apply the new ideas to recalculate the maximum mass that...

Riemannian and Lorentzian flow-cut theorems

Science.gov (United States)

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.

Momentum-space cigar geometry in topological phases

Science.gov (United States)

Palumbo, Giandomenico

2018-01-01

In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of three-dimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.

Torsional heterotic geometries

International Nuclear Information System (INIS)

Becker, Katrin; Sethi, Savdeep

2009-01-01

We construct new examples of torsional heterotic backgrounds using duality with orientifold flux compactifications. We explain how duality provides a perturbative solution to the type I/heterotic string Bianchi identity. The choice of connection used in the Bianchi identity plays an important role in the construction. We propose the existence of a much larger landscape of compact torsional geometries using string duality. Finally, we present some quantum exact metrics that correspond to NS5-branes placed on an elliptic space. These metrics describe how torus isometries are broken by NS flux.

Geometry on the space of geometries

International Nuclear Information System (INIS)

Christodoulakis, T.; Zanelli, J.

1988-06-01

We discuss the geometric structure of the configuration space of pure gravity. This is an infinite dimensional manifold, M, where each point represents one spatial geometry g ij (x). The metric on M is dictated by geometrodynamics, and from it, the Christoffel symbols and Riemann tensor can be found. A ''free geometry'' tracing a geodesic on the manifold describes the time evolution of space in the strong gravity limit. In a regularization previously introduced by the authors, it is found that M does not have the same dimensionality, D, everywhere, and that D is not a scalar, although it is covariantly constant. In this regularization, it is seen that the path integral measure can be absorbed in a renormalization of the cosmological constant. (author). 19 refs

A Lorentzian quantum geometry

Energy Technology Data Exchange (ETDEWEB)

Grotz, Andreas

2011-10-07

In this thesis, a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems is proposed. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to classical differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations in Minkowski space and on a globally hyperbolic Lorentzian manifold. When removing the regularization, the objects of our quantum geometry reduce to the common objects of spin geometry on Lorentzian manifolds, up to higher order curvature corrections.

A Lorentzian quantum geometry

International Nuclear Information System (INIS)

Grotz, Andreas

2011-01-01

In this thesis, a formulation of a Lorentzian quantum geometry based on the framework of causal fermion systems is proposed. After giving the general definition of causal fermion systems, we deduce space-time as a topological space with an underlying causal structure. Restricting attention to systems of spin dimension two, we derive the objects of our quantum geometry: the spin space, the tangent space endowed with a Lorentzian metric, connection and curvature. In order to get the correspondence to classical differential geometry, we construct examples of causal fermion systems by regularizing Dirac sea configurations in Minkowski space and on a globally hyperbolic Lorentzian manifold. When removing the regularization, the objects of our quantum geometry reduce to the common objects of spin geometry on Lorentzian manifolds, up to higher order curvature corrections.

Embeddings for the Schwarzschild metric: classification and new results

International Nuclear Information System (INIS)

Paston, S A; Sheykin, A A

2012-01-01

We suggest a method to search the embeddings of Riemannian spaces with a high enough symmetry in a flat ambient space. It is based on a procedure of construction surfaces with a given symmetry. The method is used to classify the embeddings of the Schwarzschild metric which have the symmetry of this solution, and all such embeddings in a six-dimensional ambient space (i.e. a space with a minimal possible dimension) are constructed. Four of the six possible embeddings are already known, while the two others are new. One of the new embeddings is asymptotically flat, while the other embeddings in a six-dimensional ambient space do not have this property. The asymptotically flat embedding can be of use in the analysis of the many-body problem, as well as for the development of gravity description as a theory of a surface in a flat ambient space. (paper)

Classroom reconstruction of the Schwarzschild metric

OpenAIRE

Kassner, Klaus

2015-01-01

A promising way to introduce general relativity in the classroom is to study the physical implications of certain given metrics, such as the Schwarzschild one. This involves lower mathematical expenditure than an approach focusing on differential geometry in its full glory and permits to emphasize physical aspects before attacking the field equations. Even so, in terms of motivation, lacking justification of the metric employed may pose an obstacle. The paper discusses how to establish the we...

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.)

Integrable systems, geometry, and topology

CERN Document Server

Terng, Chuu-Lian

2006-01-01

The articles in this volume are based on lectures from a program on integrable systems and differential geometry held at Taiwan's National Center for Theoretical Sciences. As is well-known, for many soliton equations, the solutions have interpretations as differential geometric objects, and thereby techniques of soliton equations have been successfully applied to the study of geometric problems. The article by Burstall gives a beautiful exposition on isothermic surfaces and their relations to integrable systems, and the two articles by Guest give an introduction to quantum cohomology, carry out explicit computations of the quantum cohomology of flag manifolds and Hirzebruch surfaces, and give a survey of Givental's quantum differential equations. The article by Heintze, Liu, and Olmos is on the theory of isoparametric submanifolds in an arbitrary Riemannian manifold, which is related to the n-wave equation when the ambient manifold is Euclidean. Mukai-Hidano and Ohnita present a survey on the moduli space of ...

The geometry of entanglement and Grover's algorithm

International Nuclear Information System (INIS)

Iwai, Toshihiro; Hayashi, Naoki; Mizobe, Kimitake

2008-01-01

A measure of entanglement with respect to a bipartite partition of n-qubit has been defined and studied from the viewpoint of Riemannian geometry (Iwai 2007 J. Phys. A: Math. Theor. 40 12161). This paper has two aims. One is to study further the geometry of entanglement, and the other is to investigate Grover's search algorithms, both the original and the fixed-point ones, in reference with entanglement. As the distance between the maximally entangled states and the separable states is known already in the previous paper, this paper determines the set of maximally entangled states nearest to a typical separable state which is used as an initial state in Grover's search algorithms, and to find geodesic segments which realize the above-mentioned distance. As for Grover's algorithms, it is already known that while the initial and the target states are separable, the algorithms generate sequences of entangled states. This fact is confirmed also in the entanglement measure proposed in the previous paper, and then a split Grover algorithm is proposed which generates sequences of separable states only with respect to the bipartite partition

Absolute Monotonicity of Functions Related To Estimates of First Eigenvalue of Laplace Operator on Riemannian Manifolds

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.

Modern approaches to discrete curvature

CERN Document Server

Romon, Pascal

2017-01-01

 This book provides a valuable glimpse into discrete curvature, a rich new field of research which blends discrete mathematics, differential geometry, probability and computer graphics. It includes a vast collection of ideas and tools which will offer something new to all interested readers. Discrete geometry has arisen as much as a theoretical development as in response to unforeseen challenges coming from applications. Discrete and continuous geometries have turned out to be intimately connected. Discrete curvature is the key concept connecting them through many bridges in numerous fields: metric spaces, Riemannian and Euclidean geometries, geometric measure theory, topology, partial differential equations, calculus of variations, gradient flows, asymptotic analysis, probability, harmonic analysis, graph theory, etc. In spite of its crucial importance both in theoretical mathematics and in applications, up to now, almost no books have provided a coherent outlook on this emerging field.

An iterative method to reconstruct the refractive index of a medium from time-of-flight measurements

Science.gov (United States)

Schröder, Udo; Schuster, Thomas

2016-08-01

The article deals with a classical inverse problem: the computation of the refractive index of a medium from ultrasound time-of-flight measurements. This problem is very popular in seismics but also for tomographic problems in inhomogeneous media. For example ultrasound vector field tomography needs a priori knowledge of the sound speed. According to Fermat’s principle ultrasound signals travel along geodesic curves of a Riemannian metric which is associated with the refractive index. The inverse problem thus consists of determining the index of refraction from integrals along geodesics curves associated with the integrand leading to a nonlinear problem. In this article we describe a numerical solver for this problem scheme based on an iterative minimization method for an appropriate Tikhonov functional. The outcome of the method is a stable approximation of the sought index of refraction as well as a corresponding set of geodesic curves. We prove some analytical convergence results for this method and demonstrate its performance by means of several numerical experiments. Another novelty in this article is the explicit representation of the backprojection operator for the ray transform in Riemannian geometry and its numerical realization relying on a corresponding phase function that is determined by the metric. This gives a natural extension of the conventional backprojection from 2D computerized tomography to inhomogeneous geometries. The authors dedicate this article to Prof Todd Quinto on the occasion of his 65th birthday.

Noncommutative geometry inspired black holes in Rastall gravity

Energy Technology Data Exchange (ETDEWEB)

Ma, Meng-Sen [Shanxi Datong University, Institute of Theoretical Physics, Datong (China); Shanxi Datong University, Department of Physics, Datong (China); Zhao, Ren [Shanxi Datong University, Institute of Theoretical Physics, Datong (China)

2017-09-15

Under two different metric ansatzes, the noncommutative geometry inspired black holes (NCBH) in the framework of Rastall gravity are derived and analyzed. We consider the fluid-type matter with the Gaussian-distribution smeared mass density. Taking a Schwarzschild-like metric ansatz, it is shown that the noncommutative geometry inspired Schwarzschild black hole (NCSBH) in Rastall gravity, unlike its counterpart in general relativity (GR), is not a regular black hole. It has at most one event horizon. After showing a finite maximal temperature, the black hole will leave behind a point-like massive remnant at zero temperature. Considering a more general metric ansatz and a special equation of state of the matter, we also find a regular NCBH in Rastall gravity, which has a similar geometric structure and temperature to that of NCSBH in GR. (orig.)

Cosmological solutions and finite time singularities in Finslerian geometry

Science.gov (United States)

Paul, Nupur; de, S. S.; Rahaman, Farook

2018-03-01

We consider a very general scenario of our universe where its geometry is characterized by the Finslerian structure on the underlying spacetime manifold, a generalization of the Riemannian geometry. Now considering a general energy-momentum tensor for matter sector, we derive the gravitational field equations in such spacetime. Further, to depict the cosmological dynamics in such spacetime proposing an interesting equation of state identified by a sole parameter γ which for isotropic limit is simply the barotropic equation of state p = (γ ‑ 1)ρ (γ ∈ ℝ being the barotropic index), we solve the background dynamics. The dynamics offers several possibilities depending on this sole parameter as follows: (i) only an exponential expansion, or (ii) a finite time past singularity (big bang) with late accelerating phase, or (iii) a nonsingular universe exhibiting an accelerating scenario at late time which finally predicts a big rip type singularity. We also discuss several energy conditions and the possibility of cosmic bounce. Finally, we establish the first law of thermodynamics in such spacetime.

Finsler-type modification of the Coulomb law

Science.gov (United States)

Itin, Yakov; Lämmerzahl, Claus; Perlick, Volker

2014-12-01

Finsler geometry is a natural generalization of pseudo-Riemannian geometry. It can be motivated e.g. by a modified version of the Ehlers-Pirani-Schild axiomatic approach to space-time theory. Also, some scenarios of quantum gravity suggest a modified dispersion relation which could be phrased in terms of Finsler geometry. On a Finslerian space-time, the universality of free fall is still satisfied but local Lorentz invariance is violated in a way not covered by standard Lorentz invariance violation schemes. In this paper we consider a Finslerian modification of Maxwell's equations. The corrections to the Coulomb potential and to the hydrogen energy levels are computed. We find that the Finsler metric corrections yield a splitting of the energy levels. Experimental data provide bounds for the Finsler parameters.

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

Proceedings – Mathematical Sciences | Indian Academy of Sciences

Indian Academy of Sciences (India)

The main purpose of the present paper is to construct Riemannian almost product structures on the (1, 1)-tensor bundle equipped with Cheeger–Gromoll type metric over a Riemannian manifold and present some results concerning these structures.

Geometry of Theory Space and RG Flows

Science.gov (United States)

Kar, Sayan

The space of couplings of a given theory is the arena of interest in this article. Equipped with a metric ansatz akin to the Fisher information matrix in the space of parameters in statistics (similar metrics in physics are the Zamolodchikov metric or the O'Connor-Stephens metric) we investigate the geometry of theory space through a study of specific examples. We then look into renormalisation group flows in theory space and make an attempt to characterise such flows via its isotropic expansion, rotation and shear. Consequences arising from the evolution equation for the isotropic expansion are discussed. We conclude by pointing out generalisations and pose some open questions.

Relativistic sonic geometry for isothermal accretion in the Kerr metric

Science.gov (United States)

Arif Shaikh, Md

2018-03-01

We linearly perturb advective isothermal transonic accretion onto rotating astrophysical black holes to study the emergence of the relativistic acoustic spacetime and to investigate how the salient features of this spacetime is influenced by the spin angular momentum of the black hole. We have perturbed three different quantities—the velocity potential, the mass accretion rate and the relativistic Bernoulli’s constant to show that the acoustic metric obtained for these three cases are the same up to a conformal factor. By constructing the required causal structures, it has been demonstrated that the acoustic black holes are formed at the transonic points of the flow and the acoustic white holes are formed at the shock location. The corresponding acoustic surface gravity has been computed in terms of the relevant accretion variables and the background metric elements. We have performed a linear stability analysis of the background stationary flow.

A prescription for n-dimensional Vierbeins

International Nuclear Information System (INIS)

Bokhari, A.H.; Qadir, A.

1982-06-01

Recent developments in supergravity have brought the n-dimensional Vierbein formalism into prominence. Here we provide a prescription for writing down a Vierbein given an arbitrary (in general non-diagonal) metric tensor in a Riemannian or pseudo-Riemannian space. (author)

Metric reconstruction from Weyl scalars

Energy Technology Data Exchange (ETDEWEB)

Whiting, Bernard F; Price, Larry R [Department of Physics, PO Box 118440, University of Florida, Gainesville, FL 32611 (United States)

2005-08-07

The Kerr geometry has remained an elusive world in which to explore physics and delve into the more esoteric implications of general relativity. Following the discovery, by Kerr in 1963, of the metric for a rotating black hole, the most major advance has been an understanding of its Weyl curvature perturbations based on Teukolsky's discovery of separable wave equations some ten years later. In the current research climate, where experiments across the globe are preparing for the first detection of gravitational waves, a more complete understanding than concerns just the Weyl curvature is now called for. To understand precisely how comparatively small masses move in response to the gravitational waves they emit, a formalism has been developed based on a description of the whole spacetime metric perturbation in the neighbourhood of the emission region. Presently, such a description is not available for the Kerr geometry. While there does exist a prescription for obtaining metric perturbations once curvature perturbations are known, it has become apparent that there are gaps in that formalism which are still waiting to be filled. The most serious gaps include gauge inflexibility, the inability to include sources-which are essential when the emitting masses are considered-and the failure to describe the l = 0 and 1 perturbation properties. Among these latter properties of the perturbed spacetime, arising from a point mass in orbit, are the perturbed mass and axial component of angular momentum, as well as the very elusive Carter constant for non-axial angular momentum. A status report is given on recent work which begins to repair these deficiencies in our current incomplete description of Kerr metric perturbations.

Metric reconstruction from Weyl scalars

International Nuclear Information System (INIS)

Whiting, Bernard F; Price, Larry R

2005-01-01

The Kerr geometry has remained an elusive world in which to explore physics and delve into the more esoteric implications of general relativity. Following the discovery, by Kerr in 1963, of the metric for a rotating black hole, the most major advance has been an understanding of its Weyl curvature perturbations based on Teukolsky's discovery of separable wave equations some ten years later. In the current research climate, where experiments across the globe are preparing for the first detection of gravitational waves, a more complete understanding than concerns just the Weyl curvature is now called for. To understand precisely how comparatively small masses move in response to the gravitational waves they emit, a formalism has been developed based on a description of the whole spacetime metric perturbation in the neighbourhood of the emission region. Presently, such a description is not available for the Kerr geometry. While there does exist a prescription for obtaining metric perturbations once curvature perturbations are known, it has become apparent that there are gaps in that formalism which are still waiting to be filled. The most serious gaps include gauge inflexibility, the inability to include sources-which are essential when the emitting masses are considered-and the failure to describe the l = 0 and 1 perturbation properties. Among these latter properties of the perturbed spacetime, arising from a point mass in orbit, are the perturbed mass and axial component of angular momentum, as well as the very elusive Carter constant for non-axial angular momentum. A status report is given on recent work which begins to repair these deficiencies in our current incomplete description of Kerr metric perturbations

Fluctuating twistor-beam solutions and Pre-Quantum Kerr-Schild geometry

Energy Technology Data Exchange (ETDEWEB)

Burinskii, Alexander, E-mail: bur@ibrae.ac.r [Laboratory of Theoretical Physics, NSI Russian Academy of Sciences, B.Tulskaya 52, Moscow, 115191 (Russian Federation)

2010-04-01

Kerr-Schild (KS) geometry is based on a congruence of twistors which is determined by the Kerr theorem. We describe time-dependent KS solutions for electromagnetic excitations of black-holes taking into account the consistent back-reaction to metric. The exact solutions have the form of singular beam-like pulses supported on twistor null lines of the Kerr congruence. The twistor-beams have very strong back reaction to metric and BH horizon and produce a fluctuating KS geometry which takes an intermediate position between the Classical and Quantum gravity.

Fluctuating twistor-beam solutions and Pre-Quantum Kerr-Schild geometry

International Nuclear Information System (INIS)

Burinskii, Alexander

2010-01-01

Kerr-Schild (KS) geometry is based on a congruence of twistors which is determined by the Kerr theorem. We describe time-dependent KS solutions for electromagnetic excitations of black-holes taking into account the consistent back-reaction to metric. The exact solutions have the form of singular beam-like pulses supported on twistor null lines of the Kerr congruence. The twistor-beams have very strong back reaction to metric and BH horizon and produce a fluctuating KS geometry which takes an intermediate position between the Classical and Quantum gravity.

Non-commutative geometry inspired charged black holes

International Nuclear Information System (INIS)

Ansoldi, Stefano; Nicolini, Piero; Smailagic, Anais; Spallucci, Euro

2007-01-01

We find a new, non-commutative geometry inspired, solution of the coupled Einstein-Maxwell field equations describing a variety of charged, self-gravitating objects, including extremal and non-extremal black holes. The metric smoothly interpolates between de Sitter geometry, at short distance, and Reissner-Nordstrom geometry far away from the origin. Contrary to the ordinary Reissner-Nordstrom spacetime there is no curvature singularity in the origin neither 'naked' nor shielded by horizons. We investigate both Hawking process and pair creation in this new scenario

Code subspaces for LLM geometries

Science.gov (United States)

Berenstein, David; Miller, Alexandra

2018-03-01

We consider effective field theory around classical background geometries with a gauge theory dual, specifically those in the class of LLM geometries. These are dual to half-BPS states of N= 4 SYM. We find that the language of code subspaces is natural for discussing the set of nearby states, which are built by acting with effective fields on these backgrounds. This work extends our previous work by going beyond the strict infinite N limit. We further discuss how one can extract the topology of the state beyond N→∞ and find that, as before, uncertainty and entanglement entropy calculations provide a useful tool to do so. Finally, we discuss obstructions to writing down a globally defined metric operator. We find that the answer depends on the choice of reference state that one starts with. Therefore, within this setup, there is ambiguity in trying to write an operator that describes the metric globally.

Foundations of arithmetic differential geometry

CERN Document Server

Buium, Alexandru

2017-01-01

The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is "intrinsically curved"; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.

Shaping of arm configuration space by prescription of non-Euclidean metrics with applications to human motor control

Science.gov (United States)

Biess, Armin

2013-01-01

The study of the kinematic and dynamic features of human arm movements provides insights into the computational strategies underlying human motor control. In this paper a differential geometric approach to movement control is taken by endowing arm configuration space with different non-Euclidean metric structures to study the predictions of the generalized minimum-jerk (MJ) model in the resulting Riemannian manifold for different types of human arm movements. For each metric space the solution of the generalized MJ model is given by reparametrized geodesic paths. This geodesic model is applied to a variety of motor tasks ranging from three-dimensional unconstrained movements of a four degree of freedom arm between pointlike targets to constrained movements where the hand location is confined to a surface (e.g., a sphere) or a curve (e.g., an ellipse). For the latter speed-curvature relations are derived depending on the boundary conditions imposed (periodic or nonperiodic) and the compatibility with the empirical one-third power law is shown. Based on these theoretical studies and recent experimental findings, I argue that geodesics may be an emergent property of the motor system and that the sensorimotor system may shape arm configuration space by learning metric structures through sensorimotor feedback.

Finiteness of Ricci flat supersymmetric non-linear sigma-models

International Nuclear Information System (INIS)

Alvarez-Gaume, L.; Ginsparg, P.

1985-01-01

Combining the constraints of Kaehler differential geometry with the universality of the normal coordinate expansion in the background field method, we study the ultraviolet behavior of 2-dimensional supersymmetric non-linear sigma-models with target space an arbitrary riemannian manifold M. We show that the constraint of N=2 supersymmetry requires that all counterterms to the metric beyond one-loop order are cohomologically trivial. It follows that such supersymmetric non-linear sigma-models defined on locally symmetric spaces are super-renormalizable and that N=4 models are on-shell ultraviolet finite to all orders of perturbation theory. (orig.)

Space-time of class one

International Nuclear Information System (INIS)

Villasenor, R.F.; Bonilla, J.L.L.; Zuniga, G.O.; Matos, T.

1989-01-01

The authors study space-times embedded in E 5 (that means, pseudo-euclidean five-dimensional spaces) in the intrinsic rigidity case, i.e., when the second fundamental form b if can be determined by the internal geometry of the four-dimensional Riemannian space R 4 . They write down the Gauss and Codazzi equations determining the local isometric embedding of R 4 in E 5 and give some consequences of it. They prove that when there exists intrinsic rigidity, then b if is a linear combination of the metric and Ricci tensor; it is given some applications for the de Sitter and Einstein models

A Note on the Asymptotic Behavior of Parabolic Monge-Ampère Equations on Riemannian Manifolds

Directory of Open Access Journals (Sweden)

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.

Rainbows without unicorns: metric structures in theories with modified dispersion relations

International Nuclear Information System (INIS)

Lobo, Iarley P.; Loret, Niccolo; Nettel, Francisco

2017-01-01

Rainbow metrics are a widely used approach to the metric formalism for theories with modified dispersion relations. They have had a huge success in the quantum gravity phenomenology literature, since they allow one to introduce momentum-dependent space-time metrics into the description of systems with a modified dispersion relation. In this paper, we introduce the reader to some realizations of this general idea: the original rainbow metrics proposal, the momentum-space-inspired metric and a Finsler geometry approach. As the main result of this work we also present an alternative definition of a four-velocity dependent metric which allows one to handle the massless limit. This paper aims to highlight some of their properties and how to properly describe their relativistic realizations. (orig.)

Rainbows without unicorns: metric structures in theories with modified dispersion relations

Science.gov (United States)

Lobo, Iarley P.; Loret, Niccoló; Nettel, Francisco

2017-07-01

Rainbow metrics are a widely used approach to the metric formalism for theories with modified dispersion relations. They have had a huge success in the quantum gravity phenomenology literature, since they allow one to introduce momentum-dependent space-time metrics into the description of systems with a modified dispersion relation. In this paper, we introduce the reader to some realizations of this general idea: the original rainbow metrics proposal, the momentum-space-inspired metric and a Finsler geometry approach. As the main result of this work we also present an alternative definition of a four-velocity dependent metric which allows one to handle the massless limit. This paper aims to highlight some of their properties and how to properly describe their relativistic realizations.

Rainbows without unicorns: metric structures in theories with modified dispersion relations

Energy Technology Data Exchange (ETDEWEB)

Lobo, Iarley P. [Universita ' ' La Sapienza' ' , Dipartimento di Fisica, Rome (Italy); ICRANet, Pescara (Italy); CAPES Foundation, Ministry of Education of Brazil, Brasilia (Brazil); Universidade Federal da Paraiba, Departamento de Fisica, Joao Pessoa, PB (Brazil); INFN Sezione Roma 1 (Italy); Loret, Niccolo [Ruder Boskovic Institute, Division of Theoretical Physics, Zagreb (Croatia); Nettel, Francisco [Universita ' ' La Sapienza' ' , Dipartimento di Fisica, Rome (Italy); Universidad Nacional Autonoma de Mexico, Instituto de Ciencias Nucleares, Mexico (Mexico); INFN Sezione Roma 1 (Italy)

2017-07-15

Rainbow metrics are a widely used approach to the metric formalism for theories with modified dispersion relations. They have had a huge success in the quantum gravity phenomenology literature, since they allow one to introduce momentum-dependent space-time metrics into the description of systems with a modified dispersion relation. In this paper, we introduce the reader to some realizations of this general idea: the original rainbow metrics proposal, the momentum-space-inspired metric and a Finsler geometry approach. As the main result of this work we also present an alternative definition of a four-velocity dependent metric which allows one to handle the massless limit. This paper aims to highlight some of their properties and how to properly describe their relativistic realizations. (orig.)

Critique of information geometry

International Nuclear Information System (INIS)

Skilling, John

2014-01-01

As applied to probability, information geometry fails because probability distributions do not form a metric space. Probability theory rests on a compelling foundation of elementary symmetries, which also support information (aka minus entropy, Kullback-Leibler) H(p;q) as the unique measure of divergence from source probability distribution q to destination p. Because the only compatible connective H is from≠to asymmetric, H(p;q)≠H(q;p), there can be no compatible geometrical distance (which would necessarily be from=to symmetric). Hence there is no distance relationship compatible with the structure of probability theory. Metrics g and densities sqrt(det(g)) interpreted as prior probabilities follow from the definition of distance, and must fail likewise. Various metrics and corresponding priors have been proposed, Fisher's being the most popular, but all must behave unacceptably. This is illustrated with simple counter-examples

On construction of two-dimensional Riemannian manifolds embedded into enveloping Euclidean (pseudo-Euclidean) space

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)

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.)

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.)

Guide to Computational Geometry Processing

DEFF Research Database (Denmark)

Bærentzen, Jakob Andreas; Gravesen, Jens; Anton, François

be processed before it is useful. This Guide to Computational Geometry Processing reviews the algorithms for processing geometric data, with a practical focus on important techniques not covered by traditional courses on computer vision and computer graphics. This is balanced with an introduction...... to the theoretical and mathematical underpinnings of each technique, enabling the reader to not only implement a given method, but also to understand the ideas behind it, its limitations and its advantages. Topics and features: Presents an overview of the underlying mathematical theory, covering vector spaces......, metric space, affine spaces, differential geometry, and finite difference methods for derivatives and differential equations Reviews geometry representations, including polygonal meshes, splines, and subdivision surfaces Examines techniques for computing curvature from polygonal meshes Describes...

Lectures on discrete geometry

CERN Document Server

2002-01-01

Discrete geometry investigates combinatorial properties of configurations of geometric objects. To a working mathematician or computer scientist, it offers sophisticated results and techniques of great diversity and it is a foundation for fields such as computational geometry or combinatorial optimization. This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. Jiri Matousek is Professor of Com...

Donaldson invariants in algebraic geometry

International Nuclear Information System (INIS)

Goettsche, L.

2000-01-01

In these lectures I want to give an introduction to the relation of Donaldson invariants with algebraic geometry: Donaldson invariants are differentiable invariants of smooth compact 4-manifolds X, defined via moduli spaces of anti-self-dual connections. If X is an algebraic surface, then these moduli spaces can for a suitable choice of the metric be identified with moduli spaces of stable vector bundles on X. This can be used to compute Donaldson invariants via methods of algebraic geometry and has led to a lot of activity on moduli spaces of vector bundles and coherent sheaves on algebraic surfaces. We will first recall the definition of the Donaldson invariants via gauge theory. Then we will show the relation between moduli spaces of anti-self-dual connections and moduli spaces of vector bundles on algebraic surfaces, and how this makes it possible to compute Donaldson invariants via algebraic geometry methods. Finally we concentrate on the case that the number b + of positive eigenvalues of the intersection form on the second homology of the 4-manifold is 1. In this case the Donaldson invariants depend on the metric (or in the algebraic geometric case on the polarization) via a system of walls and chambers. We will study the change of the invariants under wall-crossing, and use this in particular to compute the Donaldson invariants of rational algebraic surfaces. (author)

Weyl-Invariant Extension of the Metric-Affine Gravity

International Nuclear Information System (INIS)

Vazirian, R.; Tanhayi, M. R.; Motahar, Z. A.

2015-01-01

Metric-affine geometry provides a nontrivial extension of the general relativity where the metric and connection are treated as the two independent fundamental quantities in constructing the spacetime (with nonvanishing torsion and nonmetricity). In this paper, we study the generic form of action in this formalism and then construct the Weyl-invariant version of this theory. It is shown that, in Weitzenböck space, the obtained Weyl-invariant action can cover the conformally invariant teleparallel action. Finally, the related field equations are obtained in the general case.

The odd side of torsion geometry

DEFF Research Database (Denmark)

Conti, Diego; Madsen, Thomas Bruun

2014-01-01

We introduce and study a notion of `Sasaki with torsion structure' (ST) as an odd-dimensional analogue of Kähler with torsion geometry (KT). These are normal almost contact metric manifolds that admit a unique compatible connection with 3-form torsion. Any odd-dimensional compact Lie group is sho...

Differential geometry

CERN Document Server

Ciarlet, Philippe G

2007-01-01

This book gives the basic notions of differential geometry, such as the metric tensor, the Riemann curvature tensor, the fundamental forms of a surface, covariant derivatives, and the fundamental theorem of surface theory in a selfcontained and accessible manner. Although the field is often considered a classical one, it has recently been rejuvenated, thanks to the manifold applications where it plays an essential role. The book presents some important applications to shells, such as the theory of linearly and nonlinearly elastic shells, the implementation of numerical methods for shells, and

Classification of digital affine noncommutative geometries

Science.gov (United States)

Majid, Shahn; Pachoł, Anna

2018-03-01

It is known that connected translation invariant n-dimensional noncommutative differentials dxi on the algebra k[x1, …, xn] of polynomials in n-variables over a field k are classified by commutative algebras V on the vector space spanned by the coordinates. These data also apply to construct differentials on the Heisenberg algebra "spacetime" with relations [xμ, xν] = λΘμν, where Θ is an antisymmetric matrix, as well as to Lie algebras with pre-Lie algebra structures. We specialise the general theory to the field k =F2 of two elements, in which case translation invariant metrics (i.e., with constant coefficients) are equivalent to making V a Frobenius algebra. We classify all of these and their quantum Levi-Civita bimodule connections for n = 2, 3, with partial results for n = 4. For n = 2, we find 3 inequivalent differential structures admitting 1, 2, and 3 invariant metrics, respectively. For n = 3, we find 6 differential structures admitting 0, 1, 2, 3, 4, 7 invariant metrics, respectively. We give some examples for n = 4 and general n. Surprisingly, not all our geometries for n ≥ 2 have zero quantum Riemann curvature. Quantum gravity is normally seen as a weighted "sum" over all possible metrics but our results are a step towards a deeper approach in which we must also "sum" over differential structures. Over F2 we construct some of our algebras and associated structures by digital gates, opening up the possibility of "digital geometry."

Some Progress in Conformal Geometry

Directory of Open Access Journals (Sweden)

Sun-Yung A. Chang

2007-12-01

Full Text Available This is a survey paper of our current research on the theory of partial differential equations in conformal geometry. Our intention is to describe some of our current works in a rather brief and expository fashion. We are not giving a comprehensive survey on the subject and references cited here are not intended to be complete. We introduce a bubble tree structure to study the degeneration of a class of Yamabe metrics on Bach flat manifolds satisfying some global conformal bounds on compact manifolds of dimension 4. As applications, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and diameter bound of the $sigma_2$-metrics in a class of conformal 4-manifolds. For conformally compact Einstein metrics we introduce an eigenfunction compactification. As a consequence we obtain some topological constraints in terms of renormalized volumes.

Nearly pseudo-Kähler manifolds and related special holonomies

CERN Document Server

Schäfer, Lars

2017-01-01

Developing and providing an overview of recent results on nearly Kähler geometry on pseudo-Riemannian manifolds, this monograph emphasizes the differences with the classical Riemannian geometry setting. The focal objects of the text are related to special holonomy and Killing spinors and have applications in high energy physics, such as supergravity and string theory. Before starting into the field, a self-contained introduction to the subject is given, aimed at students with a solid background in differential geometry. The book will therefore be accessible to masters and Ph.D. students who are beginning work on nearly Kähler geometry in pseudo-Riemannian signature, and also to non-experts interested in gaining an overview of the subject.  Moreover, a number of results and techniques are provided which will be helpful for differential geometers as well as for high energy physicists interested in the mathematical background of the geometric objects they need.

Essential equivalence of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) and steepest-entropy-ascent models of dissipation for nonequilibrium thermodynamics.

Science.gov (United States)

Montefusco, Alberto; Consonni, Francesco; Beretta, Gian Paolo

2015-04-01

By reformulating the steepest-entropy-ascent (SEA) dynamical model for nonequilibrium thermodynamics in the mathematical language of differential geometry, we compare it with the primitive formulation of the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC) model and discuss the main technical differences of the two approaches. In both dynamical models the description of dissipation is of the "entropy-gradient" type. SEA focuses only on the dissipative, i.e., entropy generating, component of the time evolution, chooses a sub-Riemannian metric tensor as dissipative structure, and uses the local entropy density field as potential. GENERIC emphasizes the coupling between the dissipative and nondissipative components of the time evolution, chooses two compatible degenerate structures (Poisson and degenerate co-Riemannian), and uses the global energy and entropy functionals as potentials. As an illustration, we rewrite the known GENERIC formulation of the Boltzmann equation in terms of the square root of the distribution function adopted by the SEA formulation. We then provide a formal proof that in more general frameworks, whenever all degeneracies in the GENERIC framework are related to conservation laws, the SEA and GENERIC models of the dissipative component of the dynamics are essentially interchangeable, provided of course they assume the same kinematics. As part of the discussion, we note that equipping the dissipative structure of GENERIC with the Leibniz identity makes it automatically SEA on metric leaves.

Characterizing the round sphere by mean distance

DEFF Research Database (Denmark)

Kokkendorff, Simon Lyngby

2008-01-01

We discuss the measure theoretic metric invariants extent, rendezvous number and mean distance of a general compact metric space X and relate these to classical metric invariants such as diameter and radius. In the final section we focus attention to the category of Riemannian manifolds. The main...

More on microstate geometries of 4d black holes

International Nuclear Information System (INIS)

Bianchi, M.; Morales, J.F.; Pieri, L.; Zinnato, N.

2017-01-01

We construct explicit examples of microstate geometries of four-dimensional black holes that lift to smooth horizon-free geometries in five dimensions. Solutions consist of half-BPS D-brane atoms distributed in ℝ 3 . Charges and positions of the D-brane centers are constrained by the bubble equations and boundary conditions ensuring the regularity of the metric and the match with the black hole geometry. In the case of three centers, we find that the moduli spaces of solutions includes disjoint one-dimensional components of (generically) finite volume.

More on microstate geometries of 4d black holes

Energy Technology Data Exchange (ETDEWEB)

Bianchi, M. [Università di Roma Tor Vergata and I.N.F.N, Dipartimento di Fisica,Via della Ricerca Scientifica, I-00133 Rome (Italy); Morales, J.F. [I.N.F.N. - Sezione di Roma 2 and Università di Roma Tor Vergata, Dipartimento di Fisica,Via della Ricerca Scientifica, I-00133 Roma (Italy); Pieri, L. [Università di Roma Tor Vergata and I.N.F.N, Dipartimento di Fisica,Via della Ricerca Scientifica, I-00133 Rome (Italy); Center for Research in String Theory, School of Physics and Astronomy,Queen Mary University of London, Mile End Road, London, E1 4NS (United Kingdom); Zinnato, N. [Università di Roma Tor Vergata and I.N.F.N, Dipartimento di Fisica,Via della Ricerca Scientifica, I-00133 Rome (Italy)

2017-05-29

We construct explicit examples of microstate geometries of four-dimensional black holes that lift to smooth horizon-free geometries in five dimensions. Solutions consist of half-BPS D-brane atoms distributed in ℝ{sup 3}. Charges and positions of the D-brane centers are constrained by the bubble equations and boundary conditions ensuring the regularity of the metric and the match with the black hole geometry. In the case of three centers, we find that the moduli spaces of solutions includes disjoint one-dimensional components of (generically) finite volume.

Real symplectic formulation of local special geometry

CERN Document Server

Ferrara, Sergio; Ferrara, Sergio; Macia, Oscar

2006-01-01

We consider a formulation of local special geometry in terms of Darboux special coordinates $P^I=(p^i,q_i)$, $I=1,...,2n$. A general formula for the metric is obtained which is manifestly $\\mathbf{Sp}(2n,\\mathbb{R})$ covariant. Unlike the rigid case the metric is not given by the Hessian of the real function $S(P)$ which is the Legendre transform of the imaginary part of the holomorphic prepotential. Rather it is given by an expression that contains $S$, its Hessian and the conjugate momenta $S_I=\\frac{\\partial S}{\\partial P^I}$. Only in the one-dimensional case ($n=1$) is the real (two-dimensional) metric proportional to the Hessian with an appropriate conformal factor.

Geometry in a dynamical system without space: Hyperbolic Geometry in Kuramoto Oscillator Systems

Science.gov (United States)

Engelbrecht, Jan; Chen, Bolun; Mirollo, Renato

Kuramoto oscillator networks have the special property that their time evolution is constrained to lie on 3D orbits of the Möbius group acting on the N-fold torus TN which explains the N - 3 constants of motion discovered by Watanabe and Strogatz. The dynamics for phase models can be further reduced to 2D invariant sets in T N - 1 which have a natural geometry equivalent to the unit disk Δ with hyperbolic metric. We show that the classic Kuramoto model with order parameter Z1 (the first moment of the oscillator configuration) is a gradient flow in this metric with a unique fixed point on each generic 2D invariant set, corresponding to the hyperbolic barycenter of an oscillator configuration. This gradient property makes the dynamics especially easy to analyze. We exhibit several new families of Kuramoto oscillator models which reduce to gradient flows in this metric; some of these have a richer fixed point structure including non-hyperbolic fixed points associated with fixed point bifurcations. Work Supported by NSF DMS 1413020.

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

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.

Functional integration over geometries

International Nuclear Information System (INIS)

Mottola, E.

1995-01-01

The geometric construction of the functional integral over coset spaces M/G is reviewed. The inner product on the cotangent space of infinitesimal deformations of M defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber G, the functional measure on the coset space M/G is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev--Popov determinant of the more traditional gauge fixed approach in non-abelian gauge theory. If the general construction is applied to the case where G is the group of coordinate reparameterizations of spacetime, the continuum functional integral over geometries, i.e. metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the metric in two and four dimensional spacetime. In two dimensions this approach generates the Polyakov--Liouville action of closed bosonic non-critical string theory. In four dimensions the corresponding effective action leads to novel conclusions on the importance of quantum effects in gravity in the far infrared, and in particular, a dramatic modification of the classical Einstein theory at cosmological distance scales, signaled first by the quantum instability of classical de Sitter spacetime. Finite volume scaling relations for the functional integral of quantum gravity in two and four dimensions are derived, and comparison with the discretized dynamical triangulation approach to the integration over geometries are discussed. Outstanding unsolved problems in both the continuum definition and the simplicial approach to the functional integral over geometries are highlighted

A note on post-Riemannian structures of spacetime

OpenAIRE

Hehl, Friedrich W.; Muench, Uwe

1997-01-01

A four-dimensional differentiable manifold is given with an arbitrary linear connection $\\Gamma_\\alpha^\\beta=\\Gamma_{i\\alpha}^\\beta dx^i$. Megged has claimed that he can define a metric $G_{\\alpha\\beta}$ by means of a certain integral equation such that the connection is compatible with the metric. We point out that Megged's implicite definition of his metric $G_{\\alpha\\beta}$ is equivalent to the assumption of a vanishing nonmetricity. Thus his result turns out to be trivial.

Electrodynamics and Spacetime Geometry: Foundations

Science.gov (United States)

Cabral, Francisco; Lobo, Francisco S. N.

2017-02-01

We explore the intimate connection between spacetime geometry and electrodynamics. This link is already implicit in the constitutive relations between the field strengths and excitations, which are an essential part of the axiomatic structure of electromagnetism, clearly formulated via integration theory and differential forms. We review the foundations of classical electromagnetism based on charge and magnetic flux conservation, the Lorentz force and the constitutive relations. These relations introduce the conformal part of the metric and allow the study of electrodynamics for specific spacetime geometries. At the foundational level, we discuss the possibility of generalizing the vacuum constitutive relations, by relaxing the fixed conditions of homogeneity and isotropy, and by assuming that the symmetry properties of the electro-vacuum follow the spacetime isometries. The implications of this extension are briefly discussed in the context of the intimate connection between electromagnetism and the geometry (and causal structure) of spacetime.

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.

Geometry of the TJ-II in Astra 6.0

International Nuclear Information System (INIS)

Lopez-Bruna, D.; Romero, J.A.; Castejon, F.

2006-01-01

One of the most exploited features of the TJ-II Heliac, a facility in the Laboratorio Nacional de Fusion (CIEMAT, Madrid), is its ability to explore plasmas in different magnetic configurations. For this reason, there are available libraries that provide the metrics and associated magnitudes for many among all possible configurations. On the other hand, the transport codes that can normally be used to perform transport calculations cannot dea properly with these geometries, which is especially delicate when there are induced plasma currents. In the present work we adopt ASTRA, a transport analysis shell, to study the approximations performed when calculations that impose axi-symmetry (as ASTRA does) are performed on magnetic configurations that are not really axi-symmetric. After describing how we obtain those TJ-II metric averages that must be set in ASTRA, we perform two comparisons: (i) we obtain the vacuum rotational transform as deduced from the metric coefficients but imposing axisymmetry, and compare the results with the rotational transform yielded by the existing libraries; and (ii) we build a ID transport code with TJ-II metrics so its results can be compared with those of ASTRA. In both cases, the differences found indicate that evaluating the evolution of the rotational transform under ohmic induction and transport evolution is acceptable assuming that the geometry itself does not evolve. (Author) 11 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.

Geometry of conics

CERN Document Server

Akopyan, A V

2007-01-01

The book is devoted to the properties of conics (plane curves of second degree) that can be formulated and proved using only elementary geometry. Starting with the well-known optical properties of conics, the authors move to less trivial results, both classical and contemporary. In particular, the chapter on projective properties of conics contains a detailed analysis of the polar correspondence, pencils of conics, and the Poncelet theorem. In the chapter on metric properties of conics the authors discuss, in particular, inscribed conics, normals to conics, and the Poncelet theorem for confoca

General relativity: An erfc metric

Science.gov (United States)

Plamondon, Réjean

2018-06-01

This paper proposes an erfc potential to incorporate in a symmetric metric. One key feature of this model is that it relies on the existence of an intrinsic physical constant σ, a star-specific proper length that scales all its surroundings. Based thereon, the new metric is used to study the space-time geometry of a static symmetric massive object, as seen from its interior. The analytical solutions to the Einstein equation are presented, highlighting the absence of singularities and discontinuities in such a model. The geodesics are derived in their second- and first-order differential formats. Recalling the slight impact of the new model on the classical general relativity tests in the solar system, a number of facts and open problems are briefly revisited on the basis of a heuristic definition of σ. A special attention is given to gravitational collapses and non-singular black holes.

Theory of liquid crystal elastomers and polymer networks : Connection between neoclassical theory and differential geometry.

Science.gov (United States)

Nguyen, Thanh-Son; Selinger, Jonathan V

2017-09-01

In liquid crystal elastomers and polymer networks, the orientational order of liquid crystals is coupled with elastic distortions of crosslinked polymers. Previous theoretical research has described these materials through two different approaches: a neoclassical theory based on the liquid crystal director and the deformation gradient tensor, and a geometric elasticity theory based on the difference between the actual metric tensor and a reference metric. Here, we connect those two approaches using a formalism based on differential geometry. Through this connection, we determine how both the director and the geometry respond to a change of temperature.

Euclidean geometry and its subgeometries

CERN Document Server

Specht, Edward John; Calkins, Keith G; Rhoads, Donald H

2015-01-01

In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties. There are over 300 exercises; solutions to many of the...

Thin shells joining local cosmic string geometries

Energy Technology Data Exchange (ETDEWEB)

Eiroa, Ernesto F. [Universidad de Buenos Aires, Ciudad Universitaria Pabellon I, Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Buenos Aires (Argentina); Instituto de Astronomia y Fisica del Espacio (IAFE, CONICET-UBA), Buenos Aires (Argentina); Rubin de Celis, Emilio; Simeone, Claudio [Universidad de Buenos Aires, Ciudad Universitaria Pabellon I, Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Buenos Aires (Argentina); Ciudad Universitaria Pabellon I, IFIBA-CONICET, Buenos Aires (Argentina)

2016-10-15

In this article we present a theoretical construction of spacetimes with a thin shell that joins two different local cosmic string geometries. We study two types of global manifolds, one representing spacetimes with a thin shell surrounding a cosmic string or an empty region with Minkowski metric, and the other corresponding to wormholes which are not symmetric across the throat located at the shell. We analyze the stability of the static configurations under perturbations preserving the cylindrical symmetry. For both types of geometries we find that the static configurations can be stable for suitable values of the parameters. (orig.)

Thin shells joining local cosmic string geometries

International Nuclear Information System (INIS)

Eiroa, Ernesto F.; Rubin de Celis, Emilio; Simeone, Claudio

2016-01-01

In this article we present a theoretical construction of spacetimes with a thin shell that joins two different local cosmic string geometries. We study two types of global manifolds, one representing spacetimes with a thin shell surrounding a cosmic string or an empty region with Minkowski metric, and the other corresponding to wormholes which are not symmetric across the throat located at the shell. We analyze the stability of the static configurations under perturbations preserving the cylindrical symmetry. For both types of geometries we find that the static configurations can be stable for suitable values of the parameters. (orig.)

A new perspective on relativity an odyssey in non-Euclidean geometries

CERN Document Server

Lavenda, Bernard H

2012-01-01

Starting off from noneuclidean geometries, apart from the method of Einstein's equations, this book derives and describes the phenomena of gravitation and diffraction. A historical account is presented, exposing the missing link in Einstein's construction of the theory of general relativity: the uniformly rotating disc, together with his failure to realize, that the Beltrami metric of hyperbolic geometry with constant curvature describes exactly the uniform acceleration observed.

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

Metric freeness and projectivity for classical and quantum normed modules

Energy Technology Data Exchange (ETDEWEB)

Helemskii, A Ya [M. V. Lomonosov Moscow State University, Moscow (Russian Federation)

2013-07-31

In functional analysis, there are several diverse approaches to the notion of projective module. We show that a certain general categorical scheme contains all basic versions as special cases. In this scheme, the notion of free object comes to the foreground, and, in the best categories, projective objects are precisely retracts of free ones. We are especially interested in the so-called metric version of projectivity and characterize the metrically free classical and quantum (= operator) normed modules. Informally speaking, so-called extremal projectivity, which was known earlier, is interpreted as a kind of 'asymptotical metric projectivity'. In addition, we answer the following specific question in the geometry of normed spaces: what is the structure of metrically projective modules in the simplest case of normed spaces? We prove that metrically projective normed spaces are precisely the subspaces of l{sub 1}(M) (where M is a set) that are denoted by l{sub 1}{sup 0}(M) and consist of finitely supported functions. Thus, in this case, projectivity coincides with freeness. Bibliography: 28 titles.

Anti-Kählerian Geometry on Lie Groups

Science.gov (United States)

Fernández-Culma, Edison Alberto; Godoy, Yamile

2018-03-01

Let G be a Lie group of even dimension and let ( g, J) be a left invariant anti-Kähler structure on G. In this article we study anti-Kähler structures considering the distinguished cases where the complex structure J is abelian or bi-invariant. We find that if G admits a left invariant anti-Kähler structure ( g, J) where J is abelian then the Lie algebra of G is unimodular and ( G, g) is a flat pseudo-Riemannian manifold. For the second case, we see that for any left invariant metric g for which J is an anti-isometry we obtain that the triple ( G, g, J) is an anti-Kähler manifold. Besides, given a left invariant anti-Hermitian structure on G we associate a covariant 3-tensor 𝜃 on its Lie algebra and prove that such structure is anti-Kähler if and only if 𝜃 is a skew-symmetric and pure tensor. From this tensor we classify the real 4-dimensional Lie algebras for which the corresponding Lie group has a left invariant anti-Kähler structure and study the moduli spaces of such structures (up to group isomorphisms that preserve the anti-Kähler structures).

Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry

International Nuclear Information System (INIS)

Cohl, H S; Kalnins, E G

2012-01-01

Due to the isotropy of d-dimensional hyperbolic space, there exists a spherically symmetric fundamental solution for its corresponding Laplace–Beltrami operator. The R-radius hyperboloid model of hyperbolic geometry with R > 0 represents a Riemannian manifold with negative-constant sectional curvature. We obtain a spherically symmetric fundamental solution of Laplace’s equation on this manifold in terms of its geodesic radius. We give several matching expressions for this fundamental solution including a definite integral over reciprocal powers of the hyperbolic sine, finite summation expressions over hyperbolic functions, Gauss hypergeometric functions and in terms of the associated Legendre function of the second kind with order and degree given by d/2 − 1 with real argument greater than unity. We also demonstrate uniqueness for a fundamental solution of Laplace’s equation on this manifold in terms of a vanishing decay at infinity. In rotationally invariant coordinate systems, we compute the azimuthal Fourier coefficients for a fundamental solution of Laplace’s equation on the R-radius hyperboloid. For d ⩾ 2, we compute the Gegenbauer polynomial expansion in geodesic polar coordinates for a fundamental solution of Laplace’s equation on this negative-constant curvature Riemannian manifold. In three dimensions, an addition theorem for the azimuthal Fourier coefficients of a fundamental solution for Laplace’s equation is obtained through comparison with its corresponding Gegenbauer expansion. (paper)

Cosmological implications of modified gravity induced by quantum metric fluctuations

Energy Technology Data Exchange (ETDEWEB)

Liu, Xing [Sun Yat-Sen University, School of Physics, Guangzhou (China); Sun Yat-Sen University, Yat Sen School, Guangzhou (China); Harko, Tiberiu [Babes-Bolyai University, Department of Physics, Cluj-Napoca (Romania); University College London, Department of Mathematics, London (United Kingdom); Liang, Shi-Dong [Sun Yat-Sen University, School of Physics, Guangzhou (China); Sun Yat-Sen University, State Key Laboratory of Optoelectronic Material and Technology, Guangdong Province Key Laboratory of Display Material and Technology, School of Physics, Guangzhou (China)

2016-08-15

We investigate the cosmological implications of modified gravities induced by the quantum fluctuations of the gravitational metric. If the metric can be decomposed as the sum of the classical and of a fluctuating part, of quantum origin, then the corresponding Einstein quantum gravity generates at the classical level modified gravity models with a non-minimal coupling between geometry and matter. As a first step in our study, after assuming that the expectation value of the quantum correction can be generally expressed in terms of an arbitrary second order tensor constructed from the metric and from the thermodynamic quantities characterizing the matter content of the Universe, we derive the (classical) gravitational field equations in their general form. We analyze in detail the cosmological models obtained by assuming that the quantum correction tensor is given by the coupling of a scalar field and of a scalar function to the metric tensor, and by a term proportional to the matter energy-momentum tensor. For each considered model we obtain the gravitational field equations, and the generalized Friedmann equations for the case of a flat homogeneous and isotropic geometry. In some of these models the divergence of the matter energy-momentum tensor is non-zero, indicating a process of matter creation, which corresponds to an irreversible energy flow from the gravitational field to the matter fluid, and which is direct consequence of the non-minimal curvature-matter coupling. The cosmological evolution equations of these modified gravity models induced by the quantum fluctuations of the metric are investigated in detail by using both analytical and numerical methods, and it is shown that a large variety of cosmological models can be constructed, which, depending on the numerical values of the model parameters, can exhibit both accelerating and decelerating behaviors. (orig.)

The Yamabe constant

International Nuclear Information System (INIS)

O Murchadha, N.

1991-01-01

The set of riemannian three-metrics with positive Yamabe constant defines the space of independent data for the gravitational field. The boundary of this set is investigated, and it is shown that metrics close to the boundary satisfy the positive-energy theorem. (Author) 18 refs

Predicting a contact's sensitivity to initial conditions using metrics of frictional coupling

International Nuclear Information System (INIS)

Flicek, Robert C.; Hills, David A.; Brake, Matthew Robert W.

2016-01-01

This paper presents a method for predicting how sensitive a frictional contact’s steady-state behavior is to its initial conditions. Previous research has proven that if a contact is uncoupled, i.e. if slip displacements do not influence the contact pressure distribution, then its steady-state response is independent of initial conditions, but if the contact is coupled, the steady-state response depends on initial conditions. In this paper, two metrics for quantifying coupling in discrete frictional systems are examined. These metrics suggest that coupling is dominated by material dissimilarity due to Dundurs’ composite material parameter β when β ≥ 0.2, but geometric mismatch becomes the dominant source of coupling for smaller values of β. Based on a large set of numerical simulations with different contact geometries, material combinations, and friction coefficients, a contact’s sensitivity to initial conditions is found to be correlated with the product of the coupling metric and the friction coefficient. For cyclic shear loading, this correlation is maintained for simulations with different contact geometries, material combinations, and friction coefficients. Furthermore, for cyclic bulk loading, the correlation is only maintained when the contact edge angle is held constant.

Singular value decomposition metrics show limitations of detector design in diffuse fluorescence tomography.

Science.gov (United States)

Leblond, Frederic; Tichauer, Kenneth M; Pogue, Brian W

2010-11-29

The spatial resolution and recovered contrast of images reconstructed from diffuse fluorescence tomography data are limited by the high scattering properties of light propagation in biological tissue. As a result, the image reconstruction process can be exceedingly vulnerable to inaccurate prior knowledge of tissue optical properties and stochastic noise. In light of these limitations, the optimal source-detector geometry for a fluorescence tomography system is non-trivial, requiring analytical methods to guide design. Analysis of the singular value decomposition of the matrix to be inverted for image reconstruction is one potential approach, providing key quantitative metrics, such as singular image mode spatial resolution and singular data mode frequency as a function of singular mode. In the present study, these metrics are used to analyze the effects of different sources of noise and model errors as related to image quality in the form of spatial resolution and contrast recovery. The image quality is demonstrated to be inherently noise-limited even when detection geometries were increased in complexity to allow maximal tissue sampling, suggesting that detection noise characteristics outweigh detection geometry for achieving optimal reconstructions.

Fourth-rank gravity and cosmology

International Nuclear Information System (INIS)

Marrakchi, A.L.; Tapia, V.

1992-07-01

We consider the consequences of describing the metric properties of space-time through a quartic line element. The associated ''metric'' is a fourth-rank tensor G μυλπ . In order to recover a Riemannian behaviour of the geometry it is necessary to have G μυλπ = g (μυ g λπ) . We construct a theory for the gravitational field based on the fourth-rank metric G μυλπ . In the absence of matter the fourth-rank metric becomes separable and the theory coincides with General Relativity. In the presence of matter we can maintain Riemmanianicity, but now gravitation couples, as compared to General Relativity, in a different way to matter. We develop a simple cosmological model based on a FRW metric with matter described by a perfect fluid. For the present time the field equations are compatible with k OBS = O and Ω OBS t CLAS approx. 10 20 t PLANCK approx. 10 -23 s. Our final and most important result is the fact that the entropy is an increasing function of time. When interpreted at the light of General Relativity the treatment is shown to be almost equivalent to that of the standard model of cosmology combined with the inflationary scenario. (author). 16 refs, 1 fig

Curve Matching with Applications in Medical Imaging

DEFF Research Database (Denmark)

Bauer, Martin; Bruveris, Martins; Harms, Philipp

2015-01-01

In the recent years, Riemannian shape analysis of curves and surfaces has found several applications in medical image analysis. In this paper we present a numerical discretization of second order Sobolev metrics on the space of regular curves in Euclidean space. This class of metrics has several...

Flat deformation theorem and symmetries in spacetime

International Nuclear Information System (INIS)

Llosa, Josep; Carot, Jaume

2009-01-01

The flat deformation theorem states that given a semi-Riemannian analytic metric g on a manifold, locally there always exists a two-form F, a scalar function c, and an arbitrarily prescribed scalar constraint depending on the point x of the manifold and on F and c, say Ψ(c, F, x) = 0, such that the deformed metric η = cg - εF 2 is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric g may be written in the extended Kerr-Schild form, namely η ab := ag ab - 2bk (a l b) where η is flat and k a , l a are two null covectors such that k a l a = -1; next we show how the symmetries of g are connected to those of η, more precisely; we show that if the original metric g admits a conformal Killing vector (including Killing vectors and homotheties), then the deformation may be carried out in a way such that the flat deformed metric η 'inherits' that symmetry.

Probability of stochastic processes and spacetime geometry

International Nuclear Information System (INIS)

Canessa, E.

2007-01-01

We made a first attempt to associate a probabilistic description of stochastic processes like birth-death processes with spacetime geometry in the Schwarzschild metrics on distance scales from the macro- to the micro-domains. We idealize an ergodic system in which system states communicate through a curved path composed of transition arrows where each arrow corresponds to a positive, analogous birth or death rate. (author)

Quantum group of isometries in classical and noncommutative geometry

International Nuclear Information System (INIS)

Goswami, D.

2007-04-01

We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold. Our formulation accommodates spectral triples which are not of type II. We give an explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in [7] as the universal quantum group of holomorphic isometries of the noncommutative torus. (author)

Scalar curvature in conformal geometry of Connes-Landi noncommutative manifolds

Science.gov (United States)

Liu, Yang

2017-11-01

We first propose a conformal geometry for Connes-Landi noncommutative manifolds and study the associated scalar curvature. The new scalar curvature contains its Riemannian counterpart as the commutative limit. Similar to the results on noncommutative two tori, the quantum part of the curvature consists of actions of the modular derivation through two local curvature functions. Explicit expressions for those functions are obtained for all even dimensions (greater than two). In dimension four, the one variable function shows striking similarity to the analytic functions of the characteristic classes appeared in the Atiyah-Singer local index formula, namely, it is roughly a product of the j-function (which defines the A ˆ -class of a manifold) and an exponential function (which defines the Chern character of a bundle). By performing two different computations for the variation of the Einstein-Hilbert action, we obtain deep internal relations between two local curvature functions. Straightforward verification for those relations gives a strong conceptual confirmation for the whole computational machinery we have developed so far, especially the Mathematica code hidden behind the paper.

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.

A new approach for gravity localization in six-dimensional geometries

International Nuclear Information System (INIS)

Santos, Victor Pereira do Nascimento; Almeida, Carlos Alberto Santos de

2011-01-01

Full text: The idea that spacetime may have more than four dimensions is old, originally presented as an attempt to unify Maxwell's theory of Electromagnetism with the brand-new gravitation theory of Einstein. Such extra dimensions are in principle unobservable to the energy scales currently available. However, its effects can be seen in short distance gravity experiments and in observations in cosmology. Also, it is used as a mechanism to explain the difference between the energy scales of the weak force and gravity, which is called the hierarchy problem. The current framework for the extra dimension scenario is consider the four-dimensional known universe as embedded in a higher dimensional space called bulk. The form of this bulk determines how we perceive gravity in our universe; then, the behaviour of gravitational field depends on the geometry of the bulk. Metric solutions were already presented for string-like defect, with and without matter sources, where was shown that the gravity Newtonian potential grows with the inverse cube of distance. Such correction arises from a very particular mass spectrum for the gravitational field, which already contains the orbital angular momentum contributions. In this work we study the behaviour of gravitational field in a extra-dimensional braneworld scenario, using non-factorizable geometries (which preserves Poincare symmetry) and setting suitable matter distributions in order to verify its localization, for several geometries. For such geometries it is possible to find explicit solutions for the tensor fluctuations of the metric. (author)

Observations on the Darboux coordinates for rigid special geometry

CERN Document Server

Ferrara, Sergio; Ferrara, Sergio; Macia, Oscar

2006-01-01

We exploit some relations which exist when (rigid) special geometry is formulated in real symplectic special coordinates $P^I=(p^\\Lambda,q_\\Lambda), I=1,...,2n$. The central role of the real $2n\\times 2n$ matrix $M(\\Re \\mathcal{F},\\Im \\mathcal{F})$, where $\\mathcal{F} = \\partial_\\Lambda\\partial_\\Sigma F$ and $F$ is the holomorphic prepotential, is elucidated in the real formalism. The property $M\\Omega M=\\Omega$ with $\\Omega$ being the invariant symplectic form is used to prove several identities in the Darboux formulation. In this setting the matrix $M$ coincides with the (negative of the) Hessian matrix $H(S)=\\frac{\\partial^2 S}{\\partial P^I\\partial P^J}$ of a certain hamiltonian real function $S(P)$, which also provides the metric of the special K\\"ahler manifold. When $S(P)=S(U+\\bar U)$ is regarded as a "K\\"ahler potential'' of a complex manifold with coordinates $U^I=\\frac12(P^I+iZ^I)$, then it provides a K\\"ahler metric of an hyperk\\"ahler manifold which describes the hypermultiplet geometry obtained by...

A Classical Based Derivation of Time Dilation Providing First Order Accuracy to Schwarzschild's Solution of Einstein's Field Equations

Science.gov (United States)

Austin, Rickey W.

In Einstein's theory of Special Relativity (SR), one method to derive relativistic kinetic energy is via applying the classical work-energy theorem to relativistic momentum. This approach starts with a classical based work-energy theorem and applies SR's momentum to the derivation. One outcome of this derivation is relativistic kinetic energy. From this derivation, it is rather straight forward to form a kinetic energy based time dilation function. In the derivation of General Relativity a common approach is to bypass classical laws as a starting point. Instead a rigorous development of differential geometry and Riemannian space is constructed, from which classical based laws are derived. This is in contrast to SR's approach of starting with classical laws and applying the consequences of the universal speed of light by all observers. A possible method to derive time dilation due to Newtonian gravitational potential energy (NGPE) is to apply SR's approach to deriving relativistic kinetic energy. It will be shown this method gives a first order accuracy compared to Schwarzschild's metric. The SR's kinetic energy and the newly derived NGPE derivation are combined to form a Riemannian metric based on these two energies. A geodesic is derived and calculations compared to Schwarzschild's geodesic for an orbiting test mass about a central, non-rotating, non-charged massive body. The new metric results in high accuracy calculations when compared to Einsteins General Relativity's prediction. The new method provides a candidate approach for starting with classical laws and deriving General Relativity effects. This approach mimics SR's method of starting with classical mechanics when deriving relativistic equations. As a compliment to introducing General Relativity, it provides a plausible scaffolding method from classical physics when teaching introductory General Relativity. A straight forward path from classical laws to General Relativity will be derived. This derivation

Open Problem: Kernel methods on manifolds and metric spaces

DEFF Research Database (Denmark)

Feragen, Aasa; Hauberg, Søren

2016-01-01

Radial kernels are well-suited for machine learning over general geodesic metric spaces, where pairwise distances are often the only computable quantity available. We have recently shown that geodesic exponential kernels are only positive definite for all bandwidths when the input space has strong...... linear properties. This negative result hints that radial kernel are perhaps not suitable over geodesic metric spaces after all. Here, however, we present evidence that large intervals of bandwidths exist where geodesic exponential kernels have high probability of being positive definite over finite...... datasets, while still having significant predictive power. From this we formulate conjectures on the probability of a positive definite kernel matrix for a finite random sample, depending on the geometry of the data space and the spread of the sample....

Quantification of the equivalence principle

International Nuclear Information System (INIS)

Epstein, K.J.

1978-01-01

Quantitative relationships illustrate Einstein's equivalence principle, relating it to Newton's ''fictitious'' forces arising from the use of noninertial frames, and to the form of the relativistic time dilatation in local Lorentz frames. The equivalence principle can be interpreted as the equivalence of general covariance to local Lorentz covariance, in a manner which is characteristic of Riemannian and pseudo-Riemannian geometries

Weyl metrics and wormholes

Energy Technology Data Exchange (ETDEWEB)

Gibbons, Gary W. [DAMTP, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA U.K. (United Kingdom); Volkov, Mikhail S., E-mail: gwg1@cam.ac.uk, E-mail: volkov@lmpt.univ-tours.fr [Laboratoire de Mathématiques et Physique Théorique, LMPT CNRS—UMR 7350, Université de Tours, Parc de Grandmont, Tours, 37200 France (France)

2017-05-01

We study solutions obtained via applying dualities and complexifications to the vacuum Weyl metrics generated by massive rods and by point masses. Rescaling them and extending to complex parameter values yields axially symmetric vacuum solutions containing singularities along circles that can be viewed as singular matter sources. These solutions have wormhole topology with several asymptotic regions interconnected by throats and their sources can be viewed as thin rings of negative tension encircling the throats. For a particular value of the ring tension the geometry becomes exactly flat although the topology remains non-trivial, so that the rings literally produce holes in flat space. To create a single ring wormhole of one metre radius one needs a negative energy equivalent to the mass of Jupiter. Further duality transformations dress the rings with the scalar field, either conventional or phantom. This gives rise to large classes of static, axially symmetric solutions, presumably including all previously known solutions for a gravity-coupled massless scalar field, as for example the spherically symmetric Bronnikov-Ellis wormholes with phantom scalar. The multi-wormholes contain infinite struts everywhere at the symmetry axes, apart from solutions with locally flat geometry.

Singularities and the geometry of spacetime

Science.gov (United States)

Hawking, Stephen

2014-11-01

The aim of this essay is to investigate certain aspects of the geometry of the spacetime manifold in the General Theory of Relativity with particular reference to the occurrence of singularities in cosmological solutions and their relation with other global properties. Section 2 gives a brief outline of Riemannian geometry. In Section 3, the General Theory of Relativity is presented in the form of two postulates and two requirements which are common to it and to the Special Theory of Relativity, and a third requirement, the Einstein field equations, which distinguish it from the Special Theory. There does not seem to be any alternative set of field equations which would not have some undeseriable features. Some exact solutions are described. In Section 4, the physical significance of curvature is investigated using the deviation equation for timelike and null curves. The Riemann tensor is decomposed into the Ricci tensor which represents the gravitational effect at a point of matter at that point and the Welyl tensor which represents the effect at a point of gravitational radiation and matter at other points. The two tensors are related by the Bianchi identities which are presented in a form analogous to the Maxwell equations. Some lemmas are given for the occurrence of conjugate points on timelike and null geodesics and their relation with the variation of timelike and null curves is established. Section 5 is concerned with properties of causal relations between points of spacetime. It is shown that these could be used to determine physically the manifold structure of spacetime if the strong causality assumption held. The concepts of a null horizon and a partial Cauchy surface are introduced and are used to prove a number of lemmas relating to the existence of a timelike curve of maximum length between two sets. In Section 6, the definition of a singularity of spacetime is given in terms of geodesic incompleteness. The various energy assumptions needed to prove

Eigenvalue pinching on spinc manifolds

Science.gov (United States)

Roos, Saskia

2017-02-01

We derive various pinching results for small Dirac eigenvalues using the classification of spinc and spin manifolds admitting nontrivial Killing spinors. For this, we introduce a notion of convergence for spinc manifolds which involves a general study on convergence of Riemannian manifolds with a principal S1-bundle. We also analyze the relation between the regularity of the Riemannian metric and the regularity of the curvature of the associated principal S1-bundle on spinc manifolds with Killing spinors.

Unified tractable model for downlink MIMO cellular networks using stochastic geometry

KAUST Repository

Afify, Laila H.

2016-07-26

Several research efforts are invested to develop stochastic geometry models for cellular networks with multiple antenna transmission and reception (MIMO). On one hand, there are models that target abstract outage probability and ergodic rate for simplicity. On the other hand, there are models that sacrifice simplicity to target more tangible performance metrics such as the error probability. Both types of models are completely disjoint in terms of the analytic steps to obtain the performance measures, which makes it challenging to conduct studies that account for different performance metrics. This paper unifies both techniques and proposes a unified stochastic-geometry based mathematical paradigm to account for error probability, outage probability, and ergodic rates in MIMO cellular networks. The proposed model is also unified in terms of the antenna configurations and leads to simpler error probability analysis compared to existing state-of-the-art models. The core part of the analysis is based on abstracting unnecessary information conveyed within the interfering signals by assuming Gaussian signaling. To this end, the accuracy of the proposed framework is verified against state-of-the-art models as well as system level simulations. We provide via this unified study insights on network design by reflecting system parameters effect on different performance metrics. © 2016 IEEE.

Geometry and analysis on manifolds in memory of professor Shoshichi Kobayashi

CERN Document Server

Mabuchi, Toshiki; Maeda, Yoshiaki; Noguchi, Junjiro; Weinstein, Alan

2015-01-01

This volume is dedicated to the memory of Shoshichi Kobayashi, and gathers contributions from distinguished researchers working on topics close to his research areas. The book is organized into three parts, with the first part presenting an overview of Professor Shoshichi Kobayashi’s career. This is followed by two expository course lectures (the second part) on recent topics in extremal Kähler metrics and value distribution theory, which will be helpful for graduate students in mathematics interested in new topics in complex geometry and complex analysis. Lastly, the third part of the volume collects authoritative research papers on differential geometry and complex analysis. Professor Shoshichi Kobayashi was a recognized international leader in the areas of differential and complex geometry. He contributed crucial ideas that are still considered fundamental in these fields. The book will be of interest to researchers in the fields of differential geometry, complex geometry, and several complex variables ...

Compactifications of deformed conifolds, branes and the geometry of qubits

Energy Technology Data Exchange (ETDEWEB)

Cvetič, M. [Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104 (United States); Center for Applied Mathematics and Theoretical Physics,University of Maribor, SI2000 Maribor (Slovenia); Gibbons, G.W. [Department of Physics and Astronomy,University of Pennsylvania, Philadelphia, PA 19104 (United States); DAMTP, Centre for Mathematical Sciences,Cambridge University, Wilberforce Road, Cambridge CB3 OWA (United Kingdom); Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350,Fédération Denis Poisson, Université François-Rabelais Tours,Parc de Grandmont, 37200 Tours (France); LE STUDIUM, Loire Valley Institute for Advanced Studies,Tours and Orleans (France); Pope, C.N. [DAMTP, Centre for Mathematical Sciences,Cambridge University, Wilberforce Road, Cambridge CB3 OWA (United Kingdom); George P. & Cynthia W. Mitchell Institute for Fundamental Physics and Astronomy,Texas A& M University, College Station, TX 77843-4242 (United States)

2016-01-22

We present three families of exact, cohomogeneity-one Einstein metrics in (2n+2) dimensions, which are generalizations of the Stenzel construction of Ricci-flat metrics to those with a positive cosmological constant. The first family of solutions are Fubini-Study metrics on the complex projective spaces ℂℙ{sup n+1}, written in a Stenzel form, whose principal orbits are the Stiefel manifolds V{sub 2}(ℝ{sup n+2})=SO(n+2)/SO(n) divided by ℤ{sub 2}. The second family are also Einstein-Kähler metrics, now on the Grassmannian manifolds G{sub 2}(ℝ{sup n+3})=SO(n+3)/((SO(n+1)×SO(2)), whose principal orbits are the Stiefel manifolds V{sub 2}(ℝ{sup n+2}) (with no ℤ{sub 2} factoring in this case). The third family are Einstein metrics on the product manifolds S{sup n+1}×S{sup n+1}, and are Kähler only for n=1. Some of these metrics are believed to play a role in studies of consistent string theory compactifications and in the context of the AdS/CFT correspondence. We also elaborate on the geometric approach to quantum mechanics based on the Kähler geometry of Fubini-Study metrics on ℂℙ{sup n+1}, and we apply the formalism to study the quantum entanglement of qubits.

Geometry and physics of pseudodifferential operators on manifolds

DEFF Research Database (Denmark)

Esposito, Giampiero; Napolitano, George M.

2015-01-01

A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: existence theorem for the function that generalizes the phase; analogue of Taylor's theorem; torsion and curvature terms in the symbolic...

Axisymmetric plasma equilibria in a Kerr metric

Science.gov (United States)

Elsässer, Klaus

2001-10-01

Plasma equilibria near a rotating black hole are considered within the multifluid description. An isothermal two-component plasma with electrons and positrons or ions is determined by four structure functions and the boundary conditions. These structure functions are the Bernoulli function and the toroidal canonical momentum per mass for each species. The quasi-neutrality assumption (no charge density, no toroidal current) allows to solve Maxwell's equations analytically for any axisymmetric stationary metric, and to reduce the fluid equations to one single scalar equation for the stream function \\chi of the positrons or ions, respectively. The basic smallness parameter is the ratio of the skin depth of electrons to the scale length of the metric and fluid quantities, and, in the case of an electron-ion plasma, the mass ratio m_e/m_i. The \\chi-equation can be solved by standard methods, and simple solutions for a Kerr geometry are available; they show characteristic flow patterns, depending on the structure functions and the boundary conditions.

Extended DBI massive gravity with generalized fiducial metric

Science.gov (United States)

Chullaphan, Tossaporn; Tannukij, Lunchakorn; Wongjun, Pitayuth

2015-06-01

We consider an extended model of DBI massive gravity by generalizing the fiducial metric to be an induced metric on the brane corresponding to a domain wall moving in five-dimensional Schwarzschild-Anti-de Sitter spacetime. The model admits all solutions of FLRW metric including flat, closed and open geometries while the original one does not. The background solutions can be divided into two branches namely self-accelerating branch and normal branch. For the self-accelerating branch, the graviton mass plays the role of cosmological constant to drive the late-time acceleration of the universe. It is found that the number degrees of freedom of gravitational sector is not correct similar to the original DBI massive gravity. There are only two propagating degrees of freedom from tensor modes. For normal branch, we restrict our attention to a particular class of the solutions which provides an accelerated expansion of the universe. It is found that the number of degrees of freedom in the model is correct. However, at least one of them is ghost degree of freedom which always present at small scale implying that the theory is not stable.

Extended DBI massive gravity with generalized fiducial metric

International Nuclear Information System (INIS)

Chullaphan, Tossaporn; Tannukij, Lunchakorn; Wongjun, Pitayuth

2015-01-01

We consider an extended model of DBI massive gravity by generalizing the fiducial metric to be an induced metric on the brane corresponding to a domain wall moving in five-dimensional Schwarzschild-Anti-de Sitter spacetime. The model admits all solutions of FLRW metric including flat, closed and open geometries while the original one does not. The background solutions can be divided into two branches namely self-accelerating branch and normal branch. For the self-accelerating branch, the graviton mass plays the role of cosmological constant to drive the late-time acceleration of the universe. It is found that the number degrees of freedom of gravitational sector is not correct similar to the original DBI massive gravity. There are only two propagating degrees of freedom from tensor modes. For normal branch, we restrict our attention to a particular class of the solutions which provides an accelerated expansion of the universe. It is found that the number of degrees of freedom in the model is correct. However, at least one of them is ghost degree of freedom which always present at small scale implying that the theory is not stable.

The appropriateness of using various Minkowskian metrics for representing cognitive configurations

OpenAIRE

G D Richardson

1981-01-01

Cognitive mapping has been a rapidly growing area of research concerned with how cognitive information about environments is represented, interpreted, and used. One area of research has been concerned with what geometry best represents cognitive spatial information. This paper further pursues this topic by examining which of three Minkowskian metrics (city-block, Euclidean, dominance) is most appropriate for representing cognitive configurations.

Diffeomorphometry and geodesic positioning systems for human anatomy.

Science.gov (United States)

Miller, Michael I; Younes, Laurent; Trouvé, Alain

2014-03-01

The Computational Anatomy project has largely been a study of large deformations within a Riemannian framework as an efficient point of view for generating metrics between anatomical configurations. This approach turns D'Arcy Thompson's comparative morphology of human biological shape and form into a metrizable space. Since the metric is constructed based on the geodesic length of the flows of diffeomorphisms connecting the forms, we call it diffeomorphometry . Just as importantly, since the flows describe algebraic group action on anatomical submanifolds and associated functional measurements, they become the basis for positioning information, which we term geodesic positioning . As well the geodesic connections provide Riemannian coordinates for locating forms in the anatomical orbit, which we call geodesic coordinates . These three components taken together - the metric, geodesic positioning of information, and geodesic coordinates - we term the geodesic positioning system . We illustrate via several examples in human and biological coordinate systems and machine learning of the statistical representation of shape and form.

A locally adaptive normal distribution

DEFF Research Database (Denmark)

Arvanitidis, Georgios; Hansen, Lars Kai; Hauberg, Søren

2016-01-01

entropy distribution under the given metric. The underlying metric is, however, non-parametric. We develop a maximum likelihood algorithm to infer the distribution parameters that relies on a combination of gradient descent and Monte Carlo integration. We further extend the LAND to mixture models......The multivariate normal density is a monotonic function of the distance to the mean, and its ellipsoidal shape is due to the underlying Euclidean metric. We suggest to replace this metric with a locally adaptive, smoothly changing (Riemannian) metric that favors regions of high local density...

Entropy Production and Equilibrium Conditions of General-Covariant Spin Systems

Directory of Open Access Journals (Sweden)

Wolfgang Muschik

2015-12-01

Full Text Available In generalizing the special-relativistic one-component version of Eckart’s continuum thermodynamics to general-relativistic space-times with Riemannian or post-Riemannian geometry as presented by Schouten (Schouten, J.A. Ricci-Calculus, 1954 and Blagojevic (Blagojevic, M. Gauge Theories of Gravitation, 2013 we consider the entropy production and other thermodynamical quantities, such as the entropy flux and the Gibbs fundamental equation. We discuss equilibrium conditions in gravitational theories, which are based on such geometries. In particular, thermodynamic implications of the non-symmetry of the energy-momentum tensor and the related spin balance equations are investigated, also for the special case of general relativity.

On the Chern-Gauss-Bonnet theorem for the noncommutative 4-sphere

Science.gov (United States)

Arnlind, Joakim; Wilson, Mitsuru

2017-01-01

We construct a differential calculus over the noncommutative 4-sphere in the framework of pseudo-Riemannian calculi, and show that for every metric in a conformal class of perturbations of the round metric, there exists a unique metric and torsion-free connection. Furthermore, we find a localization of the projective module corresponding to the space of vector fields, which allows us to formulate a Chern-Gauss-Bonnet type theorem for the noncommutative 4-sphere.

On geodesics in low regularity

Science.gov (United States)

Sämann, Clemens; Steinbauer, Roland

2018-02-01

We consider geodesics in both Riemannian and Lorentzian manifolds with metrics of low regularity. We discuss existence of extremal curves for continuous metrics and present several old and new examples that highlight their subtle interrelation with solutions of the geodesic equations. Then we turn to the initial value problem for geodesics for locally Lipschitz continuous metrics and generalize recent results on existence, regularity and uniqueness of solutions in the sense of Filippov.

Geometric Theory of Heat from Souriau Lie Groups Thermodynamics and Koszul Hessian Geometry: Applications in Information Geometry for Exponential Families

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.

Triangulation in Friedmann's cosmological model

International Nuclear Information System (INIS)

Fagundes, H.V.

1977-01-01

In Friedmann's model, physical 3-space has a curvature K = constant. In the cases of greatest interest (K different from 0) triangulation for the measurement of great distances should be based on non-Euclidean geometries: Riemannian (or doubly elliptic) geometry for a closed universe and Bolyai-Lobatchevsky's (or hiperbolic) geometry for an open universe [pt

Engineering graphics theoretical foundations of engineering geometry for design

CERN Document Server

Brailov, Aleksandr Yurievich

2016-01-01

This professional treatise on engineering graphics emphasizes engineering geometry as the theoretical foundation for communication of design ideas with real world structures and products. It considers each theoretical notion of engineering geometry as a complex solution of direct- and inverse-problems of descriptive geometry and each solution of basic engineering problems presented is accompanied by construction of biunique two- and three-dimension models of geometrical images. The book explains the universal structure of formal algorithms of the solutions of positional, metric, and axonometric problems, as well as the solutions of problems of construction in developing a curvilinear surface. The book further characterizes and explains the added laws of projective connections to facilitate construction of geometrical images in any of eight octants. Laws of projective connections allow constructing the complex drawing of a geometrical image in the American system of measurement and the European system of measu...

Analog geometry in an expanding fluid from AdS/CFT perspective

Science.gov (United States)

Bilić, Neven; Domazet, Silvije; Tolić, Dijana

2015-04-01

The dynamics of an expanding hadron fluid at temperatures below the chiral transition is studied in the framework of AdS/CFT correspondence. We establish a correspondence between the asymptotic AdS geometry in the 4 + 1 dimensional bulk with the analog spacetime geometry on its 3 + 1 dimensional boundary with the background fluid undergoing a spherical Bjorken type expansion. The analog metric tensor on the boundary depends locally on the soft pion dispersion relation and the four-velocity of the fluid. The AdS/CFT correspondence provides a relation between the pion velocity and the critical temperature of the chiral phase transition.

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

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 $X$....

Geometry and dynamics of a tidally deformed black hole

International Nuclear Information System (INIS)

Poisson, Eric; Vlasov, Igor

2010-01-01

The metric of a nonrotating black hole deformed by a tidal interaction is calculated and expressed as an expansion in the strength of the tidal coupling. The expansion parameter is the inverse length scale R -1 , where R is the radius of curvature of the external spacetime in which the black hole moves. The expansion begins at order R -2 , and it is carried out through order R -4 . The metric is parametrized by a number of tidal multipole moments, which specify the black hole's tidal environment. The tidal moments are freely-specifiable functions of time that are related to the Weyl tensor of the external spacetime. At order R -2 the metric involves the tidal quadrupole moments E ab and B ab . At order R -3 it involves the time derivative of the quadrupole moments and the tidal octupole moments E abc and B abc . At order R -4 the metric involves the second time derivative of the quadrupole moments, the first time derivative of the octupole moments, the tidal hexadecapole moments E abcd and B abcd , and bilinear combinations of the quadrupole moments. The metric is presented in a light-cone coordinate system that possesses a clear geometrical meaning: The advanced-time coordinate v is constant on past light cones that converge toward the black hole; the angles θ and φ are constant on the null generators of each light cone; and the radial coordinate r is an affine parameter on each generator, which decreases as the light cones converge toward the black hole. The coordinates are well-behaved on the black-hole horizon, and they are adjusted so that the coordinate description of the horizon is the same as in the Schwarzschild geometry: r=2M+O(R -5 ). At the order of accuracy maintained in this work, the horizon is a stationary null hypersurface foliated by apparent horizons; it is an isolated horizon in the sense of Ashtekar and Krishnan. As an application of our results we examine the induced geometry and dynamics of the horizon, and calculate the rate at which the

Quantum κ-deformed differential geometry and field theory

Science.gov (United States)

Mercati, Flavio

2016-03-01

I introduce in κ-Minkowski noncommutative spacetime the basic tools of quantum differential geometry, namely bicovariant differential calculus, Lie and inner derivatives, the integral, the Hodge-∗ and the metric. I show the relevance of these tools for field theory with an application to complex scalar field, for which I am able to identify a vector-valued four-form which generalizes the energy-momentum tensor. Its closedness is proved, expressing in a covariant form the conservation of energy-momentum.

Spatial reorientation by geometry in bumblebees.

Directory of Open Access Journals (Sweden)

Valeria Anna Sovrano

Full Text Available Human and non-human animals are capable of using basic geometric information to reorient in an environment. Geometric information includes metric properties associated with spatial surfaces (e.g., short vs. long wall and left-right directionality or 'sense' (e.g. a long wall to the left of a short wall. However, it remains unclear whether geometric information is encoded by explicitly computing the layout of surface geometry or by matching images of the environment. View-based spatial encoding is generally thought to hold for insect navigation and, very recently, evidence for navigation by geometry has been reported in ants but only in a condition which does not allow the animals to use features located far from the goal. In this study we tested the spatial reorientation abilities of bumblebees (Bombus terrestris. After spatial disorientation, by passive rotation both clockwise and anticlockwise, bumblebees had to find one of the four exit holes located in the corners of a rectangular enclosure. Bumblebees systematically confused geometrically equivalent exit corners (i.e. corners with the same geometric arrangement of metric properties and sense, for example a short wall to the left of a long wall. However, when one wall of the enclosure was a different colour, bumblebees appeared to combine this featural information (either near or far from the goal with geometric information to find the correct exit corner. Our results show that bumblebees are able to use both geometric and featural information to reorient themselves, even when features are located far from the goal.

Sharp metric obstructions for quasi-Einstein metrics

Science.gov (United States)

Case, Jeffrey S.

2013-02-01

Using the tractor calculus to study smooth metric measure spaces, we adapt results of Gover and Nurowski to give sharp metric obstructions to the existence of quasi-Einstein metrics on suitably generic manifolds. We do this by introducing an analogue of the Weyl tractor W to the setting of smooth metric measure spaces. The obstructions we obtain can be realized as tensorial invariants which are polynomial in the Riemann curvature tensor and its divergence. By taking suitable limits of their tensorial forms, we then find obstructions to the existence of static potentials, generalizing to higher dimensions a result of Bartnik and Tod, and to the existence of potentials for gradient Ricci solitons.

Multivariate tensor-based morphometry on surfaces: application to mapping ventricular abnormalities in HIV/AIDS.

Science.gov (United States)

Wang, Yalin; Zhang, Jie; Gutman, Boris; Chan, Tony F; Becker, James T; Aizenstein, Howard J; Lopez, Oscar L; Tamburo, Robert J; Toga, Arthur W; Thompson, Paul M

2010-02-01

Here we developed a new method, called multivariate tensor-based surface morphometry (TBM), and applied it to study lateral ventricular surface differences associated with HIV/AIDS. Using concepts from differential geometry and the theory of differential forms, we created mathematical structures known as holomorphic one-forms, to obtain an efficient and accurate conformal parameterization of the lateral ventricular surfaces in the brain. The new meshing approach also provides a natural way to register anatomical surfaces across subjects, and improves on prior methods as it handles surfaces that branch and join at complex 3D junctions. To analyze anatomical differences, we computed new statistics from the Riemannian surface metrics-these retain multivariate information on local surface geometry. We applied this framework to analyze lateral ventricular surface morphometry in 3D MRI data from 11 subjects with HIV/AIDS and 8 healthy controls. Our method detected a 3D profile of surface abnormalities even in this small sample. Multivariate statistics on the local tensors gave better effect sizes for detecting group differences, relative to other TBM-based methods including analysis of the Jacobian determinant, the largest and smallest eigenvalues of the surface metric, and the pair of eigenvalues of the Jacobian matrix. The resulting analysis pipeline may improve the power of surface-based morphometry studies of the brain. Copyright (c) 2009 Elsevier Inc. All rights reserved.

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

On the compatible weakly nonlocal Poisson brackets of hydrodynamic type

Directory of Open Access Journals (Sweden)

Andrei Ya. Maltsev

2002-01-01

of hydrodynamic type (Ferapontov brackets and the corresponding integrable hierarchies. We show that, under the requirement of the nondegeneracy of the corresponding “first” pseudo-Riemannian metric g(0 νμ and also some nondegeneracy requirement for the nonlocal part, it is possible to introduce a “canonical” set of “integrable hierarchies” based on the Casimirs, momentum functional and some “canonical Hamiltonian functions.” We prove also that all the “higher” “positive” Hamiltonian operators and the “negative” symplectic forms have the weakly nonlocal form in this case. The same result is also true for “negative” Hamiltonian operators and “positive” symplectic structures in the case when both pseudo-Riemannian metrics g(0 νμ and g(1 νμ are nondegenerate.

Prototypic Development and Evaluation of a Medium Format Metric Camera

Science.gov (United States)

Hastedt, H.; Rofallski, R.; Luhmann, T.; Rosenbauer, R.; Ochsner, D.; Rieke-Zapp, D.

2018-05-01

Engineering applications require high-precision 3D measurement techniques for object sizes that vary between small volumes (2-3 m in each direction) and large volumes (around 20 x 20 x 1-10 m). The requested precision in object space (1σ RMS) is defined to be within 0.1-0.2 mm for large volumes and less than 0.01 mm for small volumes. In particular, focussing large volume applications the availability of a metric camera would have different advantages for several reasons: 1) high-quality optical components and stabilisations allow for a stable interior geometry of the camera itself, 2) a stable geometry leads to a stable interior orientation that enables for an a priori camera calibration, 3) a higher resulting precision can be expected. With this article the development and accuracy evaluation of a new metric camera, the ALPA 12 FPS add|metric will be presented. Its general accuracy potential is tested against calibrated lengths in a small volume test environment based on the German Guideline VDI/VDE 2634.1 (2002). Maximum length measurement errors of less than 0.025 mm are achieved with different scenarios having been tested. The accuracy potential for large volumes is estimated within a feasibility study on the application of photogrammetric measurements for the deformation estimation on a large wooden shipwreck in the German Maritime Museum. An accuracy of 0.2 mm-0.4 mm is reached for a length of 28 m (given by a distance from a lasertracker network measurement). All analyses have proven high stabilities of the interior orientation of the camera and indicate the applicability for a priori camera calibration for subsequent 3D measurements.

PROTOTYPIC DEVELOPMENT AND EVALUATION OF A MEDIUM FORMAT METRIC CAMERA

Directory of Open Access Journals (Sweden)

H. Hastedt

2018-05-01

Full Text Available Engineering applications require high-precision 3D measurement techniques for object sizes that vary between small volumes (2–3 m in each direction and large volumes (around 20 x 20 x 1–10 m. The requested precision in object space (1σ RMS is defined to be within 0.1–0.2 mm for large volumes and less than 0.01 mm for small volumes. In particular, focussing large volume applications the availability of a metric camera would have different advantages for several reasons: 1 high-quality optical components and stabilisations allow for a stable interior geometry of the camera itself, 2 a stable geometry leads to a stable interior orientation that enables for an a priori camera calibration, 3 a higher resulting precision can be expected. With this article the development and accuracy evaluation of a new metric camera, the ALPA 12 FPS add|metric will be presented. Its general accuracy potential is tested against calibrated lengths in a small volume test environment based on the German Guideline VDI/VDE 2634.1 (2002. Maximum length measurement errors of less than 0.025 mm are achieved with different scenarios having been tested. The accuracy potential for large volumes is estimated within a feasibility study on the application of photogrammetric measurements for the deformation estimation on a large wooden shipwreck in the German Maritime Museum. An accuracy of 0.2 mm–0.4 mm is reached for a length of 28 m (given by a distance from a lasertracker network measurement. All analyses have proven high stabilities of the interior orientation of the camera and indicate the applicability for a priori camera calibration for subsequent 3D measurements.

Metric dimensional reduction at singularities with implications to Quantum Gravity

International Nuclear Information System (INIS)

Stoica, Ovidiu Cristinel

2014-01-01

A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimensional reduction would still be desirable. The main contribution of this article is to show that dimensional reduction effects are due to General Relativity at singularities, and do not need to be postulated ad-hoc. Recent advances in understanding the geometry of singularities do not require modification of General Relativity, being just non-singular extensions of its mathematics to the limit cases. They turn out to work fine for some known types of cosmological singularities (black holes and FLRW Big-Bang), allowing a choice of the fundamental geometric invariants and physical quantities which remain regular. The resulting equations are equivalent to the standard ones outside the singularities. One consequence of this mathematical approach to the singularities in General Relativity is a special, (geo)metric type of dimensional reduction: at singularities, the metric tensor becomes degenerate in certain spacetime directions, and some properties of the fields become independent of those directions. Effectively, it is like one or more dimensions of spacetime just vanish at singularities. This suggests that it is worth exploring the possibility that the geometry of singularities leads naturally to the spontaneous dimensional reduction needed by Quantum Gravity. - Highlights: • The singularities we introduce are described by finite geometric/physical objects. • Our singularities are accompanied by dimensional reduction effects. • They affect the metric, the measure, the topology, the gravitational DOF (Weyl = 0). • Effects proposed in other approaches to Quantum Gravity are obtained naturally. • The geometric dimensional reduction obtained

Poincare ball embeddings of the optical geometry

International Nuclear Information System (INIS)

Abramowicz, M A; Bengtsson, I; Karas, V; Rosquist, K

2002-01-01

It is shown that the optical geometry of the Reissner-Nordstroem exterior metric can be embedded in a hyperbolic space all the way down to its outer horizon. The adopted embedding procedure removes a breakdown of flat-space embeddings which occurs outside the horizon, at and below the Buchdahl-Bondi limit (R/M=9/4 in the Schwarzschild case). In particular, the horizon can be captured in the optical geometry embedding diagram. Moreover, by using the compact Poincare ball representation of the hyperbolic space, the embedding diagram can cover the whole extent of radius from spatial infinity down to the horizon. Attention is drawn to the advantages of such embeddings in an appropriately curved space: this approach gives compact embeddings and it clearly distinguishes the case of an extremal black hole from a non-extremal one in terms of the topology of the embedded horizon

Metrical connection in space-time, Newton's and Hubble's laws

International Nuclear Information System (INIS)

Maeder, A.

1978-01-01

The theory of gravitation in general relativity is not scale invariant. Here, we follow Dirac's proposition of a scale invariant theory of gravitation (i.e. a theory in which the equations keep their form when a transformation of scale is made). We examine some concepts of Weyl's geometry, like the metrical connection, the scale transformations and invariance, and we discuss their consequences for the equation of the geodetic motion and for its Newtonian limit. Under general conditions, we show that the only non-vanishing component of the coefficient of metrical connection may be identified with Hubble's constant. In this framework, the equivalent to the Newtonian approximation for the equation of motion contains an additional acceleration term Hdr vector /dt, which produces an expansion of gravitational systems. The velocity of this expansion is shown to increase linearly with the distance between interacting objects. The relative importance of this new expansion term to the Newtonian one varies like (2rhosub(c)/rho)sup(1/2), where rhosub(c) is the critical density of the Einsteinde Sitter model and rho is the mean density of the considered gravitational configuration. Thus, this 'generalized expansion' is important essentially for systems of mean density not too much above the critical density. Finally, our main conclusion is that in the integrable Weyl geometry, Hubble's law - like Newton's law - would appear as an intrinsic property of gravitation, being only the most visible manifestation of a general effect characterizing the gravitational interaction. (orig.) [de

On Geodesic Exponential Kernels

DEFF Research Database (Denmark)

Feragen, Aasa; Lauze, François; Hauberg, Søren

2015-01-01

This extended abstract summarizes work presented at CVPR 2015 [1]. Standard statistics and machine learning tools require input data residing in a Euclidean space. However, many types of data are more faithfully represented in general nonlinear metric spaces or Riemannian manifolds, e.g. shapes, ......, symmetric positive definite matrices, human poses or graphs. The underlying metric space captures domain specific knowledge, e.g. non-linear constraints, which is available a priori. The intrinsic geodesic metric encodes this knowledge, often leading to improved statistical models....

Scale-invariant gravity: spacetime recovered

International Nuclear Information System (INIS)

Kelleher, Bryan

2004-01-01

The configuration space of general relativity is superspace-the space of all Riemannian 3-metrics modulo diffeomorphisms. However, it has been argued that the configuration space for gravity should be conformal superspace-the space of all Riemannian 3-metrics modulo diffeomorphisms and conformal transformations. Recently a manifestly three-dimensional theory was constructed with conformal superspace as the configuration space. Here a fully four-dimensional action is constructed so as to be invariant under conformal transformations of the 4-metric using general relativity as a guide. This action is then decomposed to a (3 + 1)-dimensional form and from this to its Jacobi form. The surprising thing is that the new theory turns out to be precisely the original three-dimensional theory. The physical data are identified and used to find the physical representation of the theory. In this representation the theory is extremely similar to general relativity. The clarity of the four-dimensional picture should prove very useful for comparing the theory with those aspects of general relativity which are usually treated in the four-dimensional framework

$\\eta$-metric structures

OpenAIRE

Gaba, Yaé Ulrich

2017-01-01

In this paper, we discuss recent results about generalized metric spaces and fixed point theory. We introduce the notion of $\\eta$-cone metric spaces, give some topological properties and prove some fixed point theorems for contractive type maps on these spaces. In particular we show that theses $\\eta$-cone metric spaces are natural generalizations of both cone metric spaces and metric type spaces.

Bilevel Optimization for Scene Segmentation of LiDAR Point Cloud

Directory of Open Access Journals (Sweden)

LI Minglei

2018-02-01

Full Text Available The segmentation of point clouds obtained by light detection and ranging (LiDAR systems is a critical step for many tasks,such as data organization,reconstruction and information extraction.In this paper,we propose a bilevel progressive optimization algorithm based on the local differentiability.First,we define the topological relation and distance metric of points in the framework of Riemannian geometry,and in the point-based level using k-means method generates over-segmentation results,e.g.super voxels.Then these voxels are formulated as nodes which consist a minimal spanning tree.High level features are extracted from voxel structures,and a graph-based optimization method is designed to yield the final adaptive segmentation results.The implementation experiments on real data demonstrate that our method is efficient and superior to state-of-the-art methods.

Real analysis and applications

CERN Document Server

Botelho, Fabio Silva

2018-01-01

This textbook introduces readers to real analysis in one and n dimensions. It is divided into two parts: Part I explores real analysis in one variable, starting with key concepts such as the construction of the real number system, metric spaces, and real sequences and series. In turn, Part II addresses the multi-variable aspects of real analysis. Further, the book presents detailed, rigorous proofs of the implicit theorem for the vectorial case by applying the Banach fixed-point theorem and the differential forms concept to surfaces in Rn. It also provides a brief introduction to Riemannian geometry. With its rigorous, elegant proofs, this self-contained work is easy to read, making it suitable for undergraduate and beginning graduate students seeking a deeper understanding of real analysis and applications, and for all those looking for a well-founded, detailed approach to real analysis.

New directions in Dirichlet forms

CERN Document Server

Jost, Jürgen; Mosco, Umberto; Rockner, Michael; Sturm, Karl-Theodor

1998-01-01

The theory of Dirichlet forms brings together methods and insights from the calculus of variations, stochastic analysis, partial differential and difference equations, potential theory, Riemannian geometry and more. This book features contributions by leading experts and provides up-to-date, authoritative accounts on exciting developments in the field and on new research perspectives. Topics covered include the following: stochastic analysis on configuration spaces, specifically a mathematically rigorous approach to the stochastic dynamics of Gibbs measures and infinite interacting particle systems; subelliptic PDE, homogenization, and fractals; geometric aspects of Dirichlet forms on metric spaces and function theory on such spaces; generalized harmonic maps as nonlinear analogues of Dirichlet forms, with an emphasis on non-locally compact situations; and a stochastic approach based on Brownian motion to harmonic maps and their regularity. Various new connections between the topics are featured, and it is de...

Reconstructing 1/2 BPS space-time metrics from matrix models and spin chains

International Nuclear Information System (INIS)

Vazquez, Samuel E.

2007-01-01

Using the anti-de Sitter/conformal field theories (AdS/CFT) correspondence, we address the question of how to measure complicated space-time metrics using gauge theory probes. In particular, we consider the case of the 1/2 Bogomol'nyi-Prasad-Sommerfield geometries of type IIB supergravity. These geometries are classified by certain droplets in a two-dimensional spacelike hypersurface. We show how to reconstruct the full metric inside these droplets using the one-loop N=4 super Yang-Mills theory dilatation operator. This is done by considering long operators in the SU(2) sector, which are dual to fast rotating strings on the droplets. We develop new powerful techniques for large N complex matrix models that allow us to construct the Hamiltonian for these strings. We find that the Hamiltonian can be mapped to a dynamical spin chain. That is, the length of the chain is not fixed. Moreover, all of these spin chains can be explicitly constructed using an interesting algebra which is derived from the matrix model. Our techniques work for general droplet configurations. As an example, we study a single elliptical droplet and the hypotrochoid

On the near horizon rotating black hole geometries with NUT charges

Energy Technology Data Exchange (ETDEWEB)

Galajinsky, Anton; Orekhov, Kirill [Tomsk Polytechnic University, Laboratory of Mathematical Physics, Tomsk (Russian Federation)

2016-09-15

The near horizon geometries are usually constructed by implementing a specific limit to a given extreme black hole configuration. Their salient feature is that the isometry group includes the conformal subgroup SO(2, 1). In this work, we turn the logic around and use the conformal invariants for constructing Ricci-flat metrics in d = 4 and d = 5 where the vacuum Einstein equations reduce to a coupled set of ordinary differential equations. In four dimensions the analysis can be carried out in full generality and the resulting metric describes the d = 4 near horizon Kerr-NUT black hole. In five dimensions we choose a specific ansatz whose structure is similar to the d = 5 near horizon Myers-Perry black hole. A Ricci-flat metric involving five arbitrary parameters is constructed. A particular member of this family, which is characterized by three parameters, seems to be a natural candidate to describe the d = 5 near horizon Myers- Perry black hole with a NUT charge. (orig.)

On the near horizon rotating black hole geometries with NUT charges

International Nuclear Information System (INIS)

Galajinsky, Anton; Orekhov, Kirill

2016-01-01

The near horizon geometries are usually constructed by implementing a specific limit to a given extreme black hole configuration. Their salient feature is that the isometry group includes the conformal subgroup SO(2, 1). In this work, we turn the logic around and use the conformal invariants for constructing Ricci-flat metrics in d = 4 and d = 5 where the vacuum Einstein equations reduce to a coupled set of ordinary differential equations. In four dimensions the analysis can be carried out in full generality and the resulting metric describes the d = 4 near horizon Kerr-NUT black hole. In five dimensions we choose a specific ansatz whose structure is similar to the d = 5 near horizon Myers-Perry black hole. A Ricci-flat metric involving five arbitrary parameters is constructed. A particular member of this family, which is characterized by three parameters, seems to be a natural candidate to describe the d = 5 near horizon Myers- Perry black hole with a NUT charge. (orig.)

Selecting landscape metrics as indicators of spatial heterogeneity-A comparison among Greek landscapes

Science.gov (United States)

Plexida, Sofia G.; Sfougaris, Athanassios I.; Ispikoudis, Ioannis P.; Papanastasis, Vasilios P.

2014-02-01

This paper investigates the spatial heterogeneity of three landscapes along an altitudinal gradient and different human land use. The main aim was the identification of appropriate landscape indicators using different extents. ASTER image was used to create a land cover map consisting of three landscapes which differed in altitude and land use. A number of landscape metrics quantifying patch complexity, configuration, diversity and connectivity were derived from the thematic map at the landscape level. There were significant differences among the three landscapes regarding these four aspects of landscape heterogeneity. The analysis revealed a specific pattern of land use where lowlands are being increasingly utilized by humans (percentage of agricultural land = 65.84%) characterized by physical connectedness (high values of Patch Cohesion Index) and relatively simple geometries (low values of fractal dimension index). The landscape pattern of uplands was found to be highly diverse based upon the Shannon Diversity index. After selecting the scale (600 ha) where metrics values stabilized, it was shown that metrics were more correlated at the small scale of 60 ha. From the original 24 metrics, 14 individual metrics with high Spearman correlation coefficient and Variance Inflation Factor criterion were eliminated, leaving 10 representative metrics for subsequent analysis. Data reduction analysis showed that Patch Density, Area-Weighted Mean Fractal Dimension Index and Patch Cohesion Index are suitable to describe landscape patterns irrespective of the scale. A systematic screening of these metrics could enhance a deeper understanding of the results obtained by them and contribute to a sustainable landscape management of Mediterranean landscapes.

The Extended HANDS Characterization and Analysis of Metric Biases

Science.gov (United States)

Kelecy, T.; Knox, R.; Cognion, R.

The Extended High Accuracy Network Determination System (Extended HANDS) consists of a network of low cost, high accuracy optical telescopes designed to support space surveillance and development of space object characterization technologies. Comprising off-the-shelf components, the telescopes are designed to provide sub arc-second astrometric accuracy. The design and analysis team are in the process of characterizing the system through development of an error allocation tree whose assessment is supported by simulation, data analysis, and calibration tests. The metric calibration process has revealed 1-2 arc-second biases in the right ascension and declination measurements of reference satellite position, and these have been observed to have fairly distinct characteristics that appear to have some dependence on orbit geometry and tracking rates. The work presented here outlines error models developed to aid in development of the system error budget, and examines characteristic errors (biases, time dependence, etc.) that might be present in each of the relevant system elements used in the data collection and processing, including the metric calibration processing. The relevant reference frames are identified, and include the sensor (CCD camera) reference frame, Earth-fixed topocentric frame, topocentric inertial reference frame, and the geocentric inertial reference frame. The errors modeled in each of these reference frames, when mapped into the topocentric inertial measurement frame, reveal how errors might manifest themselves through the calibration process. The error analysis results that are presented use satellite-sensor geometries taken from periods where actual measurements were collected, and reveal how modeled errors manifest themselves over those specific time periods. These results are compared to the real calibration metric data (right ascension and declination residuals), and sources of the bias are hypothesized. In turn, the actual right ascension and

Einstein solvmanifolds and the pre-Einstein derivation

OpenAIRE

Nikolayevsky, Y.

2008-01-01

An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre...

Geometry of the TJ-II in Astra 6.0; Geometria del TJ-II en Astra 6.0

Energy Technology Data Exchange (ETDEWEB)

Lopez-Bruna, D.; Romero, J.A.; Castejon, F.

2006-07-01

One of the most exploited features of the TJ-II Heliac, a facility in the Laboratorio Nacional de Fusion (CIEMAT, Madrid), is its ability to explore plasmas in different magnetic configurations. For this reason, there are available libraries that provide the metrics and associated magnitudes for many among all possible configurations. On the other hand, the transport codes that can normally be used to perform transport calculations cannot dea properly with these geometries, which is especially delicate when there are induced plasma currents. In the present work we adopt ASTRA, a transport analysis shell, to study the approximations performed when calculations that impose axi-symmetry (as ASTRA does) are performed on magnetic configurations that are not really axi-symmetric. After describing how we obtain those TJ-II metric averages that must be set in ASTRA, we perform two comparisons: (i) we obtain the vacuum rotational transform as deduced from the metric coefficients but imposing axisymmetry, and compare the results with the rotational transform yielded by the existing libraries; and (ii) we build a ID transport code with TJ-II metrics so its results can be compared with those of ASTRA. In both cases, the differences found indicate that evaluating the evolution of the rotational transform under ohmic induction and transport evolution is acceptable assuming that the geometry itself does not evolve. (Author) 11 refs.

(Ln-bar, g)-spaces. General relativity over V4-bar - spaces

International Nuclear Information System (INIS)

Manoff, S.; Kolarov, A.; Dimitrov, B.

1998-01-01

The results from the considerations of differentiable manifolds with contravariant and covariant affine connections and metrics are specialized for the case of (L n bar, g)-spaces with metric transport (∇ ξ g = 0 for all ξ is T (M), g ij;k = 0 and f j i = e φ · g j i (the s.c. (pseudo)Riemannian spaces with contravariant and covariant symmetric affine connections). Einstein's theory of gravitation is considered in (pseudo)Riemannian spaces with different (not only by sign) contravariant and covariant affine connections ((V n bar)-spaces, n = 4). The Euler-Lagrange equations and the corresponding energy-momentum tensors (EMT-s) are obtained and compared with the Einstein equations and the EMT-s in V 4 -spaces. The geodesic and autoparallel equations in V 4 bar -spaces are found as different equations in contrast to the case of V 4 -spaces

Average methods and their applications in Differential Geometry I

OpenAIRE

Vincze, Csaba

2013-01-01

In Minkowski geometry the metric features are based on a compact convex body containing the origin in its interior. This body works as a unit ball with its boundary formed by the unit vectors. Using one-homogeneous extension we have a so-called Minkowski functional to measure the lenght of vectors. The half of its square is called the energy function. Under some regularity conditions we can introduce an average Euclidean inner product by integrating the Hessian matrix of the energy function o...

On ruled surface in 3-dimensional almost contact metric manifold

Science.gov (United States)

Karacan, Murat Kemal; Yuksel, Nural; Ikiz, Hasibe

In this paper, we study ruled surface in 3-dimensional almost contact metric manifolds by using surface theory defined by Gök [Surfaces theory in contact geometry, PhD thesis (2010)]. We also studied the theory of curves using cross product defined by Camcı. In this study, we obtain the distribution parameters of the ruled surface and then some results and theorems are presented with special cases. Moreover, some relationships among asymptotic curve and striction line of the base curve of the ruled surface have been found.

Geodesic deviation and Minikowski space

International Nuclear Information System (INIS)

Barraco, D.; Kozameh, C.; Newman, E.T.; Tod, P.

1990-01-01

The authors study the properties of the solution space of local surface-forming null sub-congruences in the neighborhood of a given null geodesic in a pseudo-Riemannian space-time. This solution space is a three-dimensional manifold, naturally endowed with a conformal Minkowski metric

Harmonic mapping character of Rosen's bimetric theory of gravity and the geometry of its harmonic mapping space

International Nuclear Information System (INIS)

Stoeger, W.R.; Whitman, A.P.; Knill, R.J.

1985-01-01

After showing that Rosen's bimetric theory of gravity is a harmonic map, the geometry of the ten-dimensional harmonic mapping space (HMS), and of its nine-dimensional symmetric submanifolds, which are the leaves of the codimension one foliation of the HMS, is detailed. Both structures are global affinely symmetric spaces. For each, the metric, connections, and Riemann, Ricci, and scalar curvatures are given. The Killing vectors in each case are also worked out and related to the ''conserved quantities'' naturally associated with the harmonic mapping character of the theory. The structure of the Rosen HMS is very much like that determined by the DeWitt metric on the six-dimensional Wheeler superspace of all positive definite three-dimensional metrics. It is clear that a slight modification of the Rosen HMS metric will yield the corresponding metric on the space of all four-dimensional metrics of Lorentz signature. Finally, interesting avenues of further research are indicated, particularly with respect to the structure and comparison of Lagrangian-based gravitational theories which are similar to Einstein's general relativity

Geometry of the 2+1 black hole

International Nuclear Information System (INIS)

Banados, M.; Henneaux, M.; Teitelboim, C.; Zanelli, J.

1993-01-01

The geometry of the spinning black holes of standard Einstein theory in 2+1 dimensions, with a negative cosmological constant, and without couplings to matter, is analyzed in detail. It is shown that the black hole arises from identifications of points of anti--de Sitter space by a discrete subgroup of SO(2,2). The generic black hole is a smooth manifold in the metric sense. The surface r=0 is not a curvature singularity but, rather, a singularity in the causal structure. Continuing past it would introduce closed timelike lines. However, simple examples show the regularity of the metric at r=0 to be unstable: couplings to matter bring in a curvature singularity there. Kruskal coordinates and Penrose diagrams are exhibited. Special attention is given to the limiting cases of (i) the spinless hole of zero mass, which differs from anti--de Sitter space and plays the role of the vacuum, and (ii) the spinning hole of maximal angular momentum. A thorough classification of the elements of the Lie algebra of SO(2,2) is given in an appendix

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.

Inferring imagined speech using EEG signals: a new approach using Riemannian manifold features

Science.gov (United States)

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.

Hypermultiplets and hypercomplex geometry from six to three dimensions

International Nuclear Information System (INIS)

Rosseel, Jan; Proeyen, Antoine van

2004-01-01

The formulation of hypermultiplets that has been developed for five-dimensional matter multiplets is by dimensional reductions translated into the appropriate spinor language for six and four dimensions. We also treat the theories without actions that have the geometrical structure of hypercomplex geometry. The latter is the generalization of hyper-Kaehler geometry that does not require a Hermitian metric and hence corresponds to field equations without action. The translation tables of this paper allow the direct application of superconformal tensor calculus for the hypermultiplets using the available Weyl multiplets in six and four dimensions. Furthermore, the hypermultiplets in three dimensions that result from reduction of vector multiplets in four dimensions are considered, leading to a superconformal formulation of the c-map and an expression for the main geometric quantities of the hyper-Kaehler manifolds in the image of this map

Semantic metrics

OpenAIRE

Hu, Bo; Kalfoglou, Yannis; Dupplaw, David; Alani, Harith; Lewis, Paul; Shadbolt, Nigel

2006-01-01

In the context of the Semantic Web, many ontology-related operations, e.g. ontology ranking, segmentation, alignment, articulation, reuse, evaluation, can be boiled down to one fundamental operation: computing the similarity and/or dissimilarity among ontological entities, and in some cases among ontologies themselves. In this paper, we review standard metrics for computing distance measures and we propose a series of semantic metrics. We give a formal account of semantic metrics drawn from a...

Metric modular spaces

CERN Document Server

Chistyakov, Vyacheslav

2015-01-01

Aimed toward researchers and graduate students familiar with elements of functional analysis, linear algebra, and general topology; this book contains a general study of modulars, modular spaces, and metric modular spaces. Modulars may be thought of as generalized velocity fields and serve two important purposes: generate metric spaces in a unified manner and provide a weaker convergence, the modular convergence, whose topology is non-metrizable in general. Metric modular spaces are extensions of metric spaces, metric linear spaces, and classical modular linear spaces. The topics covered include the classification of modulars, metrizability of modular spaces, modular transforms and duality between modular spaces, metric  and modular topologies. Applications illustrated in this book include: the description of superposition operators acting in modular spaces, the existence of regular selections of set-valued mappings, new interpretations of spaces of Lipschitzian and absolutely continuous mappings, the existe...

Next-Generation Metrics: Responsible Metrics & Evaluation for Open Science

Energy Technology Data Exchange (ETDEWEB)

Wilsdon, J.; Bar-Ilan, J.; Peters, I.; Wouters, P.

2016-07-01

Metrics evoke a mixed reaction from the research community. A commitment to using data to inform decisions makes some enthusiastic about the prospect of granular, real-time analysis o of research and its wider impacts. Yet we only have to look at the blunt use of metrics such as journal impact factors, h-indices and grant income targets, to be reminded of the pitfalls. Some of the most precious qualities of academic culture resist simple quantification, and individual indicators often struggle to do justice to the richness and plurality of research. Too often, poorly designed evaluation criteria are “dominating minds, distorting behaviour and determining careers (Lawrence, 2007).” Metrics hold real power: they are constitutive of values, identities and livelihoods. How to exercise that power to more positive ends has been the focus of several recent and complementary initiatives, including the San Francisco Declaration on Research Assessment (DORA1), the Leiden Manifesto2 and The Metric Tide3 (a UK government review of the role of metrics in research management and assessment). Building on these initiatives, the European Commission, under its new Open Science Policy Platform4, is now looking to develop a framework for responsible metrics for research management and evaluation, which can be incorporated into the successor framework to Horizon 2020. (Author)

A note on axial symmetries

International Nuclear Information System (INIS)

Beetle, Christopher; Wilder, Shawn

2015-01-01

This note describes how to characterize and normalize an axial Killing field on a general Riemannian geometry or four-dimensional Lorentzian geometry. No global assumptions are necessary, such as that the orbits of the Killing field all have period 2π. Rather, any Killing field that vanishes at at least one point necessarily has the expected global properties. (note)

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)

On transverse exponential stability and its use in incremental stability, observer and synchronization

NARCIS (Netherlands)

Andrieu, Vincent; Jayawardhana, Bayu; Praly, Laurent

2013-01-01

We study the relation between the exponential stability of an invariant manifold and the existence of a Riemannian metric for which the flow is “transversally” contracting. More precisely, we investigate how the following properties are related to each other: i). A manifold is “transversally”

Geometric analysis

CERN Document Server

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.

Synoptic evaluation of scale-dependent metrics for hydrographic line feature geometry

Science.gov (United States)

Stanislawski, Larry V.; Buttenfield, Barbara P.; Raposo, Paulo; Cameron, Madeline; Falgout, Jeff T.

2015-01-01

conterminous United States and compared to topographic metrics. A concurrent processing workflow is implemented using a Linux high-performance computing cluster to simultaneously process multiple subbasins, and thereby complete the work in a fraction of the time required for a single-process environment. In addition, similar metrics are generated for several levels of simplification of the hydrographic features to quantify the effects of simplification over the various landscape conditions. Objectives of this exploratory investigation are to quantify geometric characteristics of linear hydrographic features over the various terrain conditions within the conterminous United States and thereby illuminate relations between stream geomorphological conditions and cartographic representation. The synoptic view of these characteristics over regional watersheds that is afforded through concurrent processing, in conjunction with terrain conditions, may reveal patterns for classifying cartographic stream features into stream geomorphological classes. Furthermore, the synoptic measurement of the amount of change in geometric characteristics caused by the several levels of simplification can enable estimation of tolerance values that appropriately control simplification-induced geometric change of the cartographic features within the various geomorphological classes in the country. Hence, these empirically derived rules or relations could help generate multiscale-representations of features through automated generalization that adequately maintain surface drainage variations and patterns reflective of the natural stream geomorphological conditions across the country.

Extended differential geometry as a basis for physical field theory

International Nuclear Information System (INIS)

Bruce, M.H.

1984-01-01

An extension of Riemann differential geometry is considered as a broadened uniform basis for physical field theory. The requirements for such a theory are set and interpreted as a generalized Ricci calculus capable of supporting certain physical affine motions and metric constraints. Both tensor and spinor languages are considered and a variational calculus is formulated within the geometry. The dominant emergent feature is the replacement of ordinary derivatives by generalized differential operators involving the usual Christoffel symbols as well as more general connection parameters. Then the Euler-Lagrange equations with constraints may be regarded as a general differential geometry and an action principle is formulated to give equations of motion in terms of generalized momentum operations. A cononical momentum tensor is employed which yields, by a generalized boundary variations of the action a set of conservation laws. The formulation is then applied to such diverse topics as the generalizing of the Dirac equation, the Lorentz and radiation terms for a charged particle, the relativistic rotator, and considerations on a geometric origin for the the Einstein energy density tensor

Conformal maps between pseudo-Finsler spaces

Science.gov (United States)

Voicu, Nicoleta

The paper aims to initiate a systematic study of conformal mappings between Finsler spacetimes and, more generally, between pseudo-Finsler spaces. This is done by extending several results in pseudo-Riemannian geometry which are necessary for field-theoretical applications and by proposing a technique that reduces some problems involving pseudo-Finslerian conformal vector fields to their pseudo-Riemannian counterparts. Also, we point out, by constructing classes of examples, that conformal groups of flat (locally Minkowskian) pseudo-Finsler spaces can be much richer than both flat Finslerian and pseudo-Euclidean conformal groups.

Arbitrariness of geometry and the aether

International Nuclear Information System (INIS)

Browne, P.F.

1976-01-01

As emphasized by Milne, an observer ultimately depends on the transmission and reception of light signals for the measurement of natural lengths and periods remote from his world point. The laws of geometry which are obeyed when these lengths and periods are plotted on a space--time depend, inevitably, on assumptions concerning the dependence of light velocity on the spatial and temporal coordinates. A convention regarding light velocity fixes the geometry, and conversely. However, the convention of flat space--time implies nonintegrable ''radar distances'' unless the concept of coordinate-dependent units of measure is employed. Einstein's space--time has the advantage of admitting a special reference system R with respect to which the aether fluid is at rest and the total gravitational field vanishes. A holonomic transformation from R to another reference system R belonging to the same space--time introduces a nonpermanent gravitational field and holonomic aether motion. A nonholonomic transformation from R to a reference system R* which belongs to a different space--time introduces a permanent gravitational field and nonholonomic aether motion. The arbitrariness of geometry is expressed by extending covariance to include the latter transformation. By means of a nonholonomic (or units) transformation it is possible, with the aid of the principle of equivalence, to obtain the Schwarzschild and de Sitter metrics from the Newtonian fields that would arise in a flat space--time description. Some light is thrown on the interpretation of cosmological models

Baby universe metric equivalent to an interior black-hole metric

International Nuclear Information System (INIS)

Gonzalez-Diaz, P.F.

1991-01-01

It is shown that the maximally extended metric corresponding to a large wormhole is the unique possible wormhole metric whose baby universe sector is conformally equivalent ot the maximal inextendible Kruskal metric corresponding to the interior region of a Schwarzschild black hole whose gravitational radius is half the wormhole neck radius. The physical implications of this result in the black hole evaporation process are discussed. (orig.)

On Kähler–Norden manifolds

Indian Academy of Sciences (India)

Abstract. This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler–Norden manifolds using the theory of Tachibana operators is presented.

1/4-pinched contact sphere theorem

DEFF Research Database (Denmark)

Ge, Jian; Huang, Yang

2016-01-01

Given a closed contact 3-manifold with a compatible Riemannian metric, we show that if the sectional curvature is 1/4-pinched, then the contact structure is universally tight. This result improves the Contact Sphere Theorem in [EKM12], where a 4/9-pinching constant was imposed. Some tightness...

Einstein-Gauss-Bonnet metrics: black holes, black strings and a staticity theorem

International Nuclear Information System (INIS)

Bogdanos, C.; Charmousis, C.; Gouteraux, B.; Zegers, R.

2009-01-01

We find the general solution of the 6-dimensional Einstein-Gauss-Bonnet equations in a large class of space and time-dependent warped geometries. Several distinct families of solutions are found, some of which include black string metrics, space and time-dependent solutions and black holes with exotic horizons. Among these, some are shown to verify a Birkhoff type staticity theorem, although here, the usual assumption of maximal symmetry on the horizon is relaxed, allowing exotic horizon geometries. We provide explicit examples of such static exotic black holes, including ones whose horizon geometry is that of a Bergman space. We find that the situation is very different from higher-dimensional general relativity, where Einstein spaces are admissible black hole horizons and the associated black hole potential is not even affected. In Einstein-Gauss-Bonnet theory, on the contrary, the non-trivial Weyl tensor of such exotic horizons is exposed to the bulk dynamics through the higher order Gauss-Bonnet term, severely constraining the allowed horizon geometries and adding a novel charge-like parameter to the black hole potential. The latter is related to the Euler characteristic of the four-dimensional horizon and provides, in some cases, additional black hole horizons.

WIMT in Gullstraend-Painleve and Reissner-Nordstroem metrics: induced stable gravito-magnetic monopoles

Energy Technology Data Exchange (ETDEWEB)

Romero, Jesus Martin [Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET), Instituto de Investigaciones Fisicas de Mar del Plata (IFIMAR), Mar del Plata (Argentina); Bellini, Mauricio [Universidad Nacional de Mar del Plata, Departamento de Fisica, Facultad de Ciencias Exactas y Naturales, Mar del Plata (Argentina); Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET), Instituto de Investigaciones Fisicas de Mar del Plata (IFIMAR), Mar del Plata (Argentina)

2015-05-15

The aim of this work is to apply Weitzeboeck Induced Matter Theory (WIMT) to Gullstraend-Painleve and Reissner-Nordstroem metrics in the framework of WIMT. This is a newly developed method that extends Induced Matter Theory from a curved 5D manifold using the Weitzeboeck's geometry, using the fact that the Riemann-Weitzenboeck curvature tensor is always null. We obtain the presence of currents whose interpretation can lead to the presence of stable gravito-magnetic monopoles. (orig.)

WIMT in Gullstraend-Painleve and Reissner-Nordstroem metrics: induced stable gravito-magnetic monopoles

International Nuclear Information System (INIS)

Romero, Jesus Martin; Bellini, Mauricio

2015-01-01

The aim of this work is to apply Weitzeboeck Induced Matter Theory (WIMT) to Gullstraend-Painleve and Reissner-Nordstroem metrics in the framework of WIMT. This is a newly developed method that extends Induced Matter Theory from a curved 5D manifold using the Weitzeboeck's geometry, using the fact that the Riemann-Weitzenboeck curvature tensor is always null. We obtain the presence of currents whose interpretation can lead to the presence of stable gravito-magnetic monopoles. (orig.)

Stochastic Geometry and Quantum Gravity: Some Rigorous Results

Science.gov (United States)

Zessin, H.

The aim of these lectures is a short introduction into some recent developments in stochastic geometry which have one of its origins in simplicial gravity theory (see Regge Nuovo Cimento 19: 558-571, 1961). The aim is to define and construct rigorously point processes on spaces of Euclidean simplices in such a way that the configurations of these simplices are simplicial complexes. The main interest then is concentrated on their curvature properties. We illustrate certain basic ideas from a mathematical point of view. An excellent representation of this area can be found in Schneider and Weil (Stochastic and Integral Geometry, Springer, Berlin, 2008. German edition: Stochastische Geometrie, Teubner, 2000). In Ambjørn et al. (Quantum Geometry Cambridge University Press, Cambridge, 1997) you find a beautiful account from the physical point of view. More recent developments in this direction can be found in Ambjørn et al. ("Quantum gravity as sum over spacetimes", Lect. Notes Phys. 807. Springer, Heidelberg, 2010). After an informal axiomatic introduction into the conceptual foundations of Regge's approach the first lecture recalls the concepts and notations used. It presents the fundamental zero-infinity law of stochastic geometry and the construction of cluster processes based on it. The second lecture presents the main mathematical object, i.e. Poisson-Delaunay surfaces possessing an intrinsic random metric structure. The third and fourth lectures discuss their ergodic behaviour and present the two-dimensional Regge model of pure simplicial quantum gravity. We terminate with the formulation of basic open problems. Proofs are given in detail only in a few cases. In general the main ideas are developed. Sufficiently complete references are given.

Properties of C-metric spaces

Science.gov (United States)

Croitoru, Anca; Apreutesei, Gabriela; Mastorakis, Nikos E.

2017-09-01

The subject of this paper belongs to the theory of approximate metrics [23]. An approximate metric on X is a real application defined on X × X that satisfies only a part of the metric axioms. In a recent paper [23], we introduced a new type of approximate metric, named C-metric, that is an application which satisfies only two metric axioms: symmetry and triangular inequality. The remarkable fact in a C-metric space is that a topological structure induced by the C-metric can be defined. The innovative idea of this paper is that we obtain some convergence properties of a C-metric space in the absence of a metric. In this paper we investigate C-metric spaces. The paper is divided into four sections. Section 1 is for Introduction. In Section 2 we recall some concepts and preliminary results. In Section 3 we present some properties of C-metric spaces, such as convergence properties, a canonical decomposition and a C-fixed point theorem. Finally, in Section 4 some conclusions are highlighted.

Learning Low-Dimensional Metrics

OpenAIRE

Jain, Lalit; Mason, Blake; Nowak, Robert

2017-01-01

This paper investigates the theoretical foundations of metric learning, focused on three key questions that are not fully addressed in prior work: 1) we consider learning general low-dimensional (low-rank) metrics as well as sparse metrics; 2) we develop upper and lower (minimax)bounds on the generalization error; 3) we quantify the sample complexity of metric learning in terms of the dimension of the feature space and the dimension/rank of the underlying metric;4) we also bound the accuracy ...

Phase transition and thermodynamical geometry for Schwarzschild AdS black hole in AdS_5×S"5 spacetime

International Nuclear Information System (INIS)

Zhang, Jia-Lin; Cai, Rong-Gen; Yu, Hongwei

2015-01-01

We study the thermodynamics and thermodynamic geometry of a five-dimensional Schwarzschild AdS black hole in AdS_5×S"5 spacetime by treating the cosmological constant as the number of colors in the boundary gauge theory and its conjugate quantity as the associated chemical potential. It is found that the chemical potential is always negative in the stable branch of black hole thermodynamics and it has a chance to be positive, but appears in the unstable branch. We calculate the scalar curvatures of the thermodynamical Weinhold metric, Ruppeiner metric and Quevedo metric, respectively and we find that the scalar curvature in the Weinhold metric is always vanishing, while in the Ruppeiner metric the divergence of the scalar curvature is related to the divergence of the heat capacity with fixed chemical potential, and in the Quevedo metric the divergence of the scalar curvature is related to the divergence of the heat capacity with fixed number of colors and to the vanishing of the heat capacity with fixed chemical potential.

Scale space representations locally adapted to the geometry of base and target manifold

NARCIS (Netherlands)

Florack, L.M.J.

2010-01-01

We generalize the Gaussian multi-resolution image paradigm for a Euclidean domain to general Riemannian base manifolds and also account for the codomain by considering the extension into a fibre bundle structure. We elaborate on aspects of parametrization and gauge, as these are important in

Scalar-metric and scalar-metric-torsion gravitational theories

International Nuclear Information System (INIS)

Aldersley, S.J.

1977-01-01

The techniques of dimensional analysis and of the theory of tensorial concomitants are employed to study field equations in gravitational theories which incorporate scalar fields of the Brans-Dicke type. Within the context of scalar-metric gravitational theories, a uniqueness theorem for the geometric (or gravitational) part of the field equations is proven and a Lagrangian is determined which is uniquely specified by dimensional analysis. Within the context of scalar-metric-torsion gravitational theories a uniqueness theorem for field Lagrangians is presented and the corresponding Euler-Lagrange equations are given. Finally, an example of a scalar-metric-torsion theory is presented which is similar in many respects to the Brans-Dicke theory and the Einstein-Cartan theory

Prescribed curvature tensor in locally conformally flat manifolds

Science.gov (United States)

Pina, Romildo; Pieterzack, Mauricio

2018-01-01

A global existence theorem for the prescribed curvature tensor problem in locally conformally flat manifolds is proved for a special class of tensors R. Necessary and sufficient conditions for the existence of a metric g ¯ , conformal to Euclidean g, are determined such that R ¯ = R, where R ¯ is the Riemannian curvature tensor of the metric g ¯ . The solution to this problem is given explicitly for special cases of the tensor R, including the case where the metric g ¯ is complete on Rn. Similar problems are considered for locally conformally flat manifolds.

METRIC context unit architecture

Energy Technology Data Exchange (ETDEWEB)

Simpson, R.O.

1988-01-01

METRIC is an architecture for a simple but powerful Reduced Instruction Set Computer (RISC). Its speed comes from the simultaneous processing of several instruction streams, with instructions from the various streams being dispatched into METRIC's execution pipeline as they become available for execution. The pipeline is thus kept full, with a mix of instructions for several contexts in execution at the same time. True parallel programming is supported within a single execution unit, the METRIC Context Unit. METRIC's architecture provides for expansion through the addition of multiple Context Units and of specialized Functional Units. The architecture thus spans a range of size and performance from a single-chip microcomputer up through large and powerful multiprocessors. This research concentrates on the specification of the METRIC Context Unit at the architectural level. Performance tradeoffs made during METRIC's design are discussed, and projections of METRIC's performance are made based on simulation studies.

On generalized de Rham-Hodge complexes, the related characteristic Chern classes and some applications to integrable multi-dimensional differential systems on Riemannian manifolds

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)

Geometries

CERN Document Server

Sossinsky, A B

2012-01-01

The book is an innovative modern exposition of geometry, or rather, of geometries; it is the first textbook in which Felix Klein's Erlangen Program (the action of transformation groups) is systematically used as the basis for defining various geometries. The course of study presented is dedicated to the proposition that all geometries are created equal--although some, of course, remain more equal than others. The author concentrates on several of the more distinguished and beautiful ones, which include what he terms "toy geometries", the geometries of Platonic bodies, discrete geometries, and classical continuous geometries. The text is based on first-year semester course lectures delivered at the Independent University of Moscow in 2003 and 2006. It is by no means a formal algebraic or analytic treatment of geometric topics, but rather, a highly visual exposition containing upwards of 200 illustrations. The reader is expected to possess a familiarity with elementary Euclidean geometry, albeit those lacking t...

Algebraic structure of Robinson–Trautman and Kundt geometries in arbitrary dimension

International Nuclear Information System (INIS)

Podolský, J; Švarc, R

2015-01-01

We investigate the Weyl tensor algebraic structure of a fully general family of D-dimensional geometries that admit a non-twisting and shear-free null vector field k. From the coordinate components of the curvature tensor we explicitly derive all Weyl scalars of various boost weights. This enables us to give a complete algebraic classification of the metrics in the case when the optically privileged null direction k is a (multiple) Weyl aligned null direction (WAND). No field equations are applied, so the results are valid not only in Einstein's gravity, including its extension to higher dimensions, but also in any metric gravitation theory that admits non-twisting and shear-free spacetimes. We prove that all such geometries are of type I(b), or more special, and we derive surprisingly simple necessary and sufficient conditions under which k is a double, triple or quadruple WAND. All possible algebraically special types, including the refinement to subtypes, are thus identified, namely II(a), II(b), II(c), II(d), III(a), III(b), N, O, II i , III i , D, D(a), D(b), D(c), D(d), and their combinations. Some conditions are identically satisfied in four dimensions. We discuss both important subclasses, namely the Kundt family of geometries with the vanishing expansion (Θ=0) and the Robinson–Trautman family (Θ ≠ 0, and in particular Θ=1/r). Finally, we apply Einstein's field equations and obtain a classification of all Robinson–Trautman vacuum spacetimes. This reveals fundamental algebraic differences in the D>4 and D=4 cases, namely that in higher dimensions there only exist such spacetimes of types D(a) ≡ D(abd), D(c) ≡ D(bcd) and O. (paper)

Koszul information geometry and Souriau Lie group thermodynamics

Energy Technology Data Exchange (ETDEWEB)

Barbaresco, Frédéric, E-mail: frederic.barbaresco@thalesgroup.com

2015-01-13

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from 'Characteristic Functions', was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of 'Information Geometry' theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean 'Moment map' by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author in [10][11].

Geometry and project: the example of Piazza della Vittoria in Genoa

Directory of Open Access Journals (Sweden)

Luisa Cogorno

2012-06-01

Full Text Available The study on the Genoese architectures in the thirties of Piazza della Vittoria  highlights the paramount  role of geometry for the project. The square, created by architect Marcello Piacentini between 1927 and 1930, stands opposite the railway station Genoa Brignole, pursuing a very logical representation: a new, very wide square surrounded by arcades ... the new civic-wordly heart of the  city (M. Piacentini. The methodological tool applied for the critical reading  was the  integrated relief: perceptual and historical analysis, relief and graphic representation of the whole and of the details, reconstruction of the compositional logic, study of materials used; the different phases showed the trasversality of the role of geometry –verified by the metric survey and by the compositive reading of the proportions- and the presence of dimensional canons unusual for the city.

Metrics to describe the effects of landscape pattern on hydrology in a lotic peatland

Science.gov (United States)

Yuan, J.; Cohen, M. J.; Kaplan, D. A.; Acharya, S.; Larsen, L.; Nungesser, M.

2013-12-01

Strong reciprocal interactions exist between landscape patterns and ecological processes. Hydrology is the dominant abiotic driver of ecological processes in wetlands, particularly flowing wetlands, but is both the control on and controlled by the geometry of vegetation patterning. Landscape metrics are widely used to quantitatively link pattern and process. Our goal here was to use several candidate spatial pattern metrics to predict the effects of wetland vegetation pattern on hydrologic regime, specifically hydroperiod, in the ridge-slough patterned landscape of the Everglades. The metrics focus on the capacity for longitudinally connected flow, and thus the ability of this low-gradient patterned landscape to route water from upstream. We first explored flow friction cost (FFC), a weighted spatial distance procedure wherein ridges have a high flow cost than sloughs by virtue of their elevation and vegetation structure, to evaluate water movement through different landscape configurations. We also investigated existing published flow metrics, specifically the Directional Connectivity Index (DCI) and Landscape Discharge Competence (LDC), that seek to quantify connectivity, one of the sentinel targets of ecological restoration. Hydroperiod was estimated using a numerical hydrologic model (SWIFT 2D) in real and synthetic landscapes with varying vegetation properties ( patch anisotropy, ridge density). Synthetic landscapes were constrained by the geostatistical properties of the best conserved patterned, and contained five anisotropy levels and seven ridge density levels. These were used to construct the relationship between landscape metrics and hydroperiod. Then, using historical images from 1940 to 2004, we applied the metrics toback-cast hydroperiod. Current vegetation maps were used to test scale dependency for each metric. Our results suggest that both FFC and DCI are good predictors of hydroperiod under free flowing conditions, and that they can be used

State-space Manifold and Rotating Black Holes

CERN Document Server

Bellucci, Stefano

2010-01-01

We study a class of fluctuating higher dimensional black hole configurations obtained in string theory/ $M$-theory compactifications. We explore the intrinsic Riemannian geometric nature of Gaussian fluctuations arising from the Hessian of the coarse graining entropy, defined over an ensemble of brane microstates. It has been shown that the state-space geometry spanned by the set of invariant parameters is non-degenerate, regular and has a negative scalar curvature for the rotating Myers-Perry black holes, Kaluza-Klein black holes, supersymmetric $AdS_5$ black holes, $D_1$-$D_5$ configurations and the associated BMPV black holes. Interestingly, these solutions demonstrate that the principal components of the state-space metric tensor admit a positive definite form, while the off diagonal components do not. Furthermore, the ratio of diagonal components weakens relatively faster than the off diagonal components, and thus they swiftly come into an equilibrium statistical configuration. Novel aspects of the scali...

Geometric flows and (some of) their physical applications

CERN Document Server

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...

Gravity, two times, tractors, Weyl invariance, and six-dimensional quantum mechanics

International Nuclear Information System (INIS)

Bonezzi, R.; Latini, E.; Waldron, A.

2010-01-01

Fefferman and Graham showed some time ago that four-dimensional conformal geometries could be analyzed in terms of six-dimensional, ambient, Riemannian geometries admitting a closed homothety. Recently, it was shown how conformal geometry provides a description of physics manifestly invariant under local choices of unit systems. Strikingly, Einstein's equations are then equivalent to the existence of a parallel scale tractor (a six-component vector subject to a certain first order covariant constancy condition at every point in four-dimensional spacetime). These results suggest a six-dimensional description of four-dimensional physics, a viewpoint promulgated by the 2 times physics program of Bars. The Fefferman-Graham construction relies on a triplet of operators corresponding, respectively, to a curved six-dimensional light cone, the dilation generator and the Laplacian. These form an sp(2) algebra which Bars employs as a first class algebra of constraints in a six-dimensional gauge theory. In this article four-dimensional gravity is recast in terms of six-dimensional quantum mechanics by melding the 2 times and tractor approaches. This parent formulation of gravity is built from an infinite set of six-dimensional fields. Successively integrating out these fields yields various novel descriptions of gravity including a new four-dimensional one built from a scalar doublet, a tractor-vector multiplet and a conformal class of metrics.

Fault Management Metrics

Science.gov (United States)

Johnson, Stephen B.; Ghoshal, Sudipto; Haste, Deepak; Moore, Craig

2017-01-01

This paper describes the theory and considerations in the application of metrics to measure the effectiveness of fault management. Fault management refers here to the operational aspect of system health management, and as such is considered as a meta-control loop that operates to preserve or maximize the system's ability to achieve its goals in the face of current or prospective failure. As a suite of control loops, the metrics to estimate and measure the effectiveness of fault management are similar to those of classical control loops in being divided into two major classes: state estimation, and state control. State estimation metrics can be classified into lower-level subdivisions for detection coverage, detection effectiveness, fault isolation and fault identification (diagnostics), and failure prognosis. State control metrics can be classified into response determination effectiveness and response effectiveness. These metrics are applied to each and every fault management control loop in the system, for each failure to which they apply, and probabilistically summed to determine the effectiveness of these fault management control loops to preserve the relevant system goals that they are intended to protect.

Completion of a Dislocated Metric Space

Directory of Open Access Journals (Sweden)

P. Sumati Kumari

2015-01-01

Full Text Available We provide a construction for the completion of a dislocated metric space (abbreviated d-metric space; we also prove that the completion of the metric associated with a d-metric coincides with the metric associated with the completion of the d-metric.

Metrics with vanishing quantum corrections

International Nuclear Information System (INIS)

Coley, A A; Hervik, S; Gibbons, G W; Pope, C N

2008-01-01

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor T μν (g αβ , ∂ τ g αβ , ∂ τ ∂ σ g αβ , ...,) constructed from sums of terms, the involving contractions of the metric and powers of arbitrary covariant derivatives of the curvature tensor. A classical solution, such as an Einstein metric, is called universal if, when evaluated on that Einstein metric, T μν is a multiple of the metric. A Ricci flat classical solution is called strongly universal if, when evaluated on that Ricci flat metric, T μν vanishes. It is well known that pp-waves in four spacetime dimensions are strongly universal. We focus attention on a natural generalization; Einstein metrics with holonomy Sim(n - 2) in which all scalar invariants are zero or constant. In four dimensions we demonstrate that the generalized Ghanam-Thompson metric is weakly universal and that the Goldberg-Kerr metric is strongly universal; indeed, we show that universality extends to all four-dimensional Sim(2) Einstein metrics. We also discuss generalizations to higher dimensions

Geometry

CERN Document Server

Prasolov, V V

2015-01-01

This book provides a systematic introduction to various geometries, including Euclidean, affine, projective, spherical, and hyperbolic geometries. Also included is a chapter on infinite-dimensional generalizations of Euclidean and affine geometries. A uniform approach to different geometries, based on Klein's Erlangen Program is suggested, and similarities of various phenomena in all geometries are traced. An important notion of duality of geometric objects is highlighted throughout the book. The authors also include a detailed presentation of the theory of conics and quadrics, including the theory of conics for non-Euclidean geometries. The book contains many beautiful geometric facts and has plenty of problems, most of them with solutions, which nicely supplement the main text. With more than 150 figures illustrating the arguments, the book can be recommended as a textbook for undergraduate and graduate-level courses in geometry.

Remarks on G-Metric Spaces

Directory of Open Access Journals (Sweden)

Bessem Samet

2013-01-01

Full Text Available In 2005, Mustafa and Sims (2006 introduced and studied a new class of generalized metric spaces, which are called G-metric spaces, as a generalization of metric spaces. We establish some useful propositions to show that many fixed point theorems on (nonsymmetric G-metric spaces given recently by many authors follow directly from well-known theorems on metric spaces. Our technique can be easily extended to other results as shown in application.

On the conformal equivalence of harmonic maps and exponentially harmonic maps

International Nuclear Information System (INIS)

Hong Minchun.

1991-06-01

Suppose that (M,g) and (N,h) are compact smooth Riemannian manifolds without boundaries. For m = dim M ≥3, and Φ: (M,g) → (N,h) is exponentially harmonic, there exists a smooth metric g-tilde conformally equivalent to g such that Φ: (M,g-tilde) → (N,h) is harmonic. (author). 7 refs

Critical geometry of a thermal big bang

Science.gov (United States)

Afshordi, Niayesh; Magueijo, João

2016-11-01

We explore the space of scalar-tensor theories containing two nonconformal metrics, and find a discontinuity pointing to a "critical" cosmological solution. Due to the different maximal speeds of propagation for matter and gravity, the cosmological fluctuations start off inside the horizon even without inflation, and will more naturally have a thermal origin (since there is never vacuum domination). The critical model makes an unambiguous, nontuned prediction for the spectral index of the scalar fluctuations: nS=0.96478 (64 ) . Considering also that no gravitational waves are produced, we have unveiled the most predictive model on offer. The model has a simple geometrical interpretation as a probe 3-brane embedded in an E AdS2×E3 geometry.

Common fixed point theorems for fuzzy mappings in metric space under φ-contraction condition

International Nuclear Information System (INIS)

Abu-Donia, H.M.

2007-01-01

Some common fixed point theorems for multi-valued mappings under φ-contraction condition have been studied by Rashwan [Rashwan RA, Ahmed MA. Fixed points for φ-contraction type multivalued mappings. J Indian Acad Math 1995;17(2):194-204]. Butnariu [Butnariu D. Fixed point for fuzzy mapping. Fuzzy Sets Syst 1982;7:191-207] and Helipern [Hilpern S. Fuzzy mapping and fixed point theorem. J Math Anal Appl 1981;83:566-9] also, discussed some fixed point theorems for fuzzy mappings in the category of metric spaces. In this paper, we discussed some common fixed point theorems for fuzzy mappings in metric space under φ-contraction condition. Our investigation are related to the fuzzy form of Hausdorff metric which is a basic tool for computing Hausdorff dimensions. These dimensions help in understanding ε ∞ -space [El-Naschie MS. On the unification of the fundamental forces and complex time in the ε ∞ -space. Chaos, Solitons and Fractals 2000;11:1149-62] and are used in high energy physics [El-Naschie MS. Wild topology hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons and Fractals 2002;13:1935-45

Common fixed point theorems for fuzzy mappings in metric space under {phi}-contraction condition

Energy Technology Data Exchange (ETDEWEB)

Abu-Donia, H.M. [Department of Mathematics, Faculty of Science, Zagazig University, Zagazig (Egypt)

2007-10-15

Some common fixed point theorems for multi-valued mappings under {phi}-contraction condition have been studied by Rashwan [Rashwan RA, Ahmed MA. Fixed points for {phi}-contraction type multivalued mappings. J Indian Acad Math 1995;17(2):194-204]. Butnariu [Butnariu D. Fixed point for fuzzy mapping. Fuzzy Sets Syst 1982;7:191-207] and Helipern [Hilpern S. Fuzzy mapping and fixed point theorem. J Math Anal Appl 1981;83:566-9] also, discussed some fixed point theorems for fuzzy mappings in the category of metric spaces. In this paper, we discussed some common fixed point theorems for fuzzy mappings in metric space under {phi}-contraction condition. Our investigation are related to the fuzzy form of Hausdorff metric which is a basic tool for computing Hausdorff dimensions. These dimensions help in understanding {epsilon} {sup {infinity}}-space [El-Naschie MS. On the unification of the fundamental forces and complex time in the {epsilon} {sup {infinity}}-space. Chaos, Solitons and Fractals 2000;11:1149-62] and are used in high energy physics [El-Naschie MS. Wild topology hyperbolic geometry and fusion algebra of high energy particle physics. Chaos, Solitons and Fractals 2002;13:1935-45].

Metric-adjusted skew information

DEFF Research Database (Denmark)

Liang, Cai; Hansen, Frank

2010-01-01

on a bipartite system and proved superadditivity of the Wigner-Yanase-Dyson skew informations for such states. We extend this result to the general metric-adjusted skew information. We finally show that a recently introduced extension to parameter values 1 ...We give a truly elementary proof of the convexity of metric-adjusted skew information following an idea of Effros. We extend earlier results of weak forms of superadditivity to general metric-adjusted skew information. Recently, Luo and Zhang introduced the notion of semi-quantum states...... of (unbounded) metric-adjusted skew information....

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

Euclidean scalar Green's functions near the black hole and black brane horizons

International Nuclear Information System (INIS)

Haba, Z

2009-01-01

We discuss approximations of the Riemannian geometry near the horizon. If a (D + 1)-dimensional manifold N has a bifurcate Killing horizon then we approximate N by a product of the two-dimensional Rindler space R 2 and a (D - 1)-dimensional Riemannian manifold M. We obtain approximate formulae for scalar Green's functions. We study the behavior of the Green's functions near the horizon and their dimensional reduction. We show that if M is compact then the Green's function near the horizon can be approximated by the Green's function of the two-dimensional quantum field theory. The correction term is exponentially small away from the horizon. We extend the results to black brane solutions of supergravity in 10 and 11 dimensions. The near-horizon geometry can be approximated by N=AdS p xS q . We discuss the Euclidean Green's functions on N and their behavior near the horizon.

Software metrics: Software quality metrics for distributed systems. [reliability engineering

Science.gov (United States)

Post, J. V.

1981-01-01

Software quality metrics was extended to cover distributed computer systems. Emphasis is placed on studying embedded computer systems and on viewing them within a system life cycle. The hierarchy of quality factors, criteria, and metrics was maintained. New software quality factors were added, including survivability, expandability, and evolvability.

Surface geometry of a rotating black hole in a magnetic field

International Nuclear Information System (INIS)

Kulkarni, R.; Dadhich, N.

1986-01-01

We study the intrinsic geometry of the surface of a rotating black hole in a uniform magnetic field, using a metric discovered by Ernst and Wild. Rotating black holes are analogous to material rotating bodies according to Smarr since black holes also tend to become more oblate on being spun up. Our study shows that the presence of a strong magnetic field ensures that a black hole actually becomes increasingly prolate on being spun up. Studying the intrinsic geometry of the black-hole surface also gives rise to an interesting embedding problem. Smarr shows that a Kerr black hole cannot be globally isometrically embedded in R 3 if its specific angular momentum a exceeds (√3 /2)mapprox.0.866. . .m. We show that in the presence of a magnetic field of strength B, satisfying 2- √3 2 m 2 3 for all values of the angular momentum

A Maxwell-vector p-wave holographic superconductor in a particular background AdS black hole metric

Directory of Open Access Journals (Sweden)

Dan Wen

2018-05-01

Full Text Available We study the p-wave holographic superconductor for AdS black holes with planar event horizon topology for a particular Lovelock gravity, in which the action is characterized by a self-interacting scalar field nonminimally coupled to the gravity theory which is labeled by an integer k. As the Lovelock theory of gravity is the most general metric theory of gravity based on the fundamental assumptions of general relativity, it is a desirable theory to describe the higher dimensional spacetime geometry. The present work is devoted to studying the properties of the p-wave holographic superconductor by including a Maxwell field which nonminimally couples to a complex vector field in a higher dimensional background metric. In the probe limit, we find that the critical temperature decreases with the increase of the index k of the background black hole metric, which shows that a larger k makes it harder for the condensation to form. We also observe that the index k affects the conductivity and the gap frequency of the holographic superconductors.

A Maxwell-vector p-wave holographic superconductor in a particular background AdS black hole metric

Science.gov (United States)

Wen, Dan; Yu, Hongwei; Pan, Qiyuan; Lin, Kai; Qian, Wei-Liang

2018-05-01

We study the p-wave holographic superconductor for AdS black holes with planar event horizon topology for a particular Lovelock gravity, in which the action is characterized by a self-interacting scalar field nonminimally coupled to the gravity theory which is labeled by an integer k. As the Lovelock theory of gravity is the most general metric theory of gravity based on the fundamental assumptions of general relativity, it is a desirable theory to describe the higher dimensional spacetime geometry. The present work is devoted to studying the properties of the p-wave holographic superconductor by including a Maxwell field which nonminimally couples to a complex vector field in a higher dimensional background metric. In the probe limit, we find that the critical temperature decreases with the increase of the index k of the background black hole metric, which shows that a larger k makes it harder for the condensation to form. We also observe that the index k affects the conductivity and the gap frequency of the holographic superconductors.

Transverse force on a moving vortex with the acoustic geometry

International Nuclear Information System (INIS)

Zhang Pengming; Cao Liming; Duan Yishi; Zhong Chengkui

2004-01-01

We consider the transverse force on a moving vortex with the acoustic metric using the phi-mapping topological current theory. In the frame of effective space-time geometry the vortex appear naturally by virtue of the vortex tensor in the Lorentz space-time and we show that it is just the vortex derived with the order parameter in the condensed matter. With the usual Lagrangian we obtain the equation of motion for the vortex. At last, we show that the transverse force on the moving vortex in our equation is just the usual Magnus force in a simple model

The metric system: An introduction

Science.gov (United States)

Lumley, Susan M.

On 13 Jul. 1992, Deputy Director Duane Sewell restated the Laboratory's policy on conversion to the metric system which was established in 1974. Sewell's memo announced the Laboratory's intention to continue metric conversion on a reasonable and cost effective basis. Copies of the 1974 and 1992 Administrative Memos are contained in the Appendix. There are three primary reasons behind the Laboratory's conversion to the metric system. First, Public Law 100-418, passed in 1988, states that by the end of fiscal year 1992 the Federal Government must begin using metric units in grants, procurements, and other business transactions. Second, on 25 Jul. 1991, President George Bush signed Executive Order 12770 which urged Federal agencies to expedite conversion to metric units. Third, the contract between the University of California and the Department of Energy calls for the Laboratory to convert to the metric system. Thus, conversion to the metric system is a legal requirement and a contractual mandate with the University of California. Public Law 100-418 and Executive Order 12770 are discussed in more detail later in this section, but first they examine the reasons behind the nation's conversion to the metric system. The second part of this report is on applying the metric system.

The metric system: An introduction

Energy Technology Data Exchange (ETDEWEB)

Lumley, S.M.

1995-05-01

On July 13, 1992, Deputy Director Duane Sewell restated the Laboratory`s policy on conversion to the metric system which was established in 1974. Sewell`s memo announced the Laboratory`s intention to continue metric conversion on a reasonable and cost effective basis. Copies of the 1974 and 1992 Administrative Memos are contained in the Appendix. There are three primary reasons behind the Laboratory`s conversion to the metric system. First, Public Law 100-418, passed in 1988, states that by the end of fiscal year 1992 the Federal Government must begin using metric units in grants, procurements, and other business transactions. Second, on July 25, 1991, President George Bush signed Executive Order 12770 which urged Federal agencies to expedite conversion to metric units. Third, the contract between the University of California and the Department of Energy calls for the Laboratory to convert to the metric system. Thus, conversion to the metric system is a legal requirement and a contractual mandate with the University of California. Public Law 100-418 and Executive Order 12770 are discussed in more detail later in this section, but first they examine the reasons behind the nation`s conversion to the metric system. The second part of this report is on applying the metric system.

A tensor formulation of the equation of transfer for spherically symmetric flows. [radiative transfer in seven dimensional Riemannian space

Science.gov (United States)

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.

Information Fusion and Control in Hierarchical Systems

Science.gov (United States)

2013-05-01

this complicates the analysis without bringing any additional insights. Therefore, for simplicity, we hence- forth assume a threshold value of 1 in our... complicated expression of Γkij which prevents solution of the differential equations (6.14). 6.3.4 Curvatures and information In the mathematical field of...P. Petersen, Riemannian geometry, Springer, New York, 1998. [77] Z. Yang, J. Laaksonen, Principal whitened gradient for information geometry, Neural

Evaluation of dose-volume metrics for microbeam radiation therapy dose distributions in head phantoms of various sizes using Monte Carlo simulations

Science.gov (United States)

Anderson, Danielle; Siegbahn, E. Albert; Fallone, B. Gino; Serduc, Raphael; Warkentin, Brad

2012-05-01

This work evaluates four dose-volume metrics applied to microbeam radiation therapy (MRT) using simulated dosimetric data as input. We seek to improve upon the most frequently used MRT metric, the peak-to-valley dose ratio (PVDR), by analyzing MRT dose distributions from a more volumetric perspective. Monte Carlo simulations were used to calculate dose distributions in three cubic head phantoms: a 2 cm mouse head, an 8 cm cat head and a 16 cm dog head. The dose distribution was calculated for a 4 × 4 mm2 microbeam array in each phantom, as well as a 16 × 16 mm2 array in the 8 cm cat head, and a 32 × 32 mm2 array in the 16 cm dog head. Microbeam widths of 25, 50 and 75 µm and center-to-center spacings of 100, 200 and 400 µm were considered. The metrics calculated for each simulation were the conventional PVDR, the peak-to-mean valley dose ratio (PMVDR), the mean dose and the percentage volume below a threshold dose. The PVDR ranged between 3 and 230 for the 2 cm mouse phantom, and between 2 and 186 for the 16 cm dog phantom depending on geometry. The corresponding ranges for the PMVDR were much smaller, being 2-49 (mouse) and 2-46 (dog), and showed a slightly weaker dependence on phantom size and array size. The ratio of the PMVDR to the PVDR varied from 0.21 to 0.79 for the different collimation configurations, indicating a difference between the geometric dependence on outcome that would be predicted by these two metrics. For unidirectional irradiation, the mean lesion dose was 102%, 79% and 42% of the mean skin dose for the 2 cm mouse, 8 cm cat and 16 cm dog head phantoms, respectively. However, the mean lesion dose recovered to 83% of the mean skin dose in the 16 cm dog phantom in intersecting cross-firing regions. The percentage volume below a 10% dose threshold was highly dependent on geometry, with ranges for the different collimation configurations of 2-87% and 33-96% for the 2 cm mouse and 16 cm dog heads, respectively. The results of this study

Evaluation of dose-volume metrics for microbeam radiation therapy dose distributions in head phantoms of various sizes using Monte Carlo simulations

International Nuclear Information System (INIS)

Anderson, Danielle; Fallone, B Gino; Warkentin, Brad; Siegbahn, E Albert; Serduc, Raphael

2012-01-01

This work evaluates four dose-volume metrics applied to microbeam radiation therapy (MRT) using simulated dosimetric data as input. We seek to improve upon the most frequently used MRT metric, the peak-to-valley dose ratio (PVDR), by analyzing MRT dose distributions from a more volumetric perspective. Monte Carlo simulations were used to calculate dose distributions in three cubic head phantoms: a 2 cm mouse head, an 8 cm cat head and a 16 cm dog head. The dose distribution was calculated for a 4 × 4 mm 2 microbeam array in each phantom, as well as a 16 × 16 mm 2 array in the 8 cm cat head, and a 32 × 32 mm 2 array in the 16 cm dog head. Microbeam widths of 25, 50 and 75 µm and center-to-center spacings of 100, 200 and 400 µm were considered. The metrics calculated for each simulation were the conventional PVDR, the peak-to-mean valley dose ratio (PMVDR), the mean dose and the percentage volume below a threshold dose. The PVDR ranged between 3 and 230 for the 2 cm mouse phantom, and between 2 and 186 for the 16 cm dog phantom depending on geometry. The corresponding ranges for the PMVDR were much smaller, being 2–49 (mouse) and 2–46 (dog), and showed a slightly weaker dependence on phantom size and array size. The ratio of the PMVDR to the PVDR varied from 0.21 to 0.79 for the different collimation configurations, indicating a difference between the geometric dependence on outcome that would be predicted by these two metrics. For unidirectional irradiation, the mean lesion dose was 102%, 79% and 42% of the mean skin dose for the 2 cm mouse, 8 cm cat and 16 cm dog head phantoms, respectively. However, the mean lesion dose recovered to 83% of the mean skin dose in the 16 cm dog phantom in intersecting cross-firing regions. The percentage volume below a 10% dose threshold was highly dependent on geometry, with ranges for the different collimation configurations of 2–87% and 33–96% for the 2 cm mouse and 16 cm dog heads, respectively. The results of this

Attack-Resistant Trust Metrics

Science.gov (United States)

Levien, Raph

The Internet is an amazingly powerful tool for connecting people together, unmatched in human history. Yet, with that power comes great potential for spam and abuse. Trust metrics are an attempt to compute the set of which people are trustworthy and which are likely attackers. This chapter presents two specific trust metrics developed and deployed on the Advogato Website, which is a community blog for free software developers. This real-world experience demonstrates that the trust metrics fulfilled their goals, but that for good results, it is important to match the assumptions of the abstract trust metric computation to the real-world implementation.

Lie symmetries for systems of evolution equations

Science.gov (United States)

Paliathanasis, Andronikos; Tsamparlis, Michael

2018-01-01

The Lie symmetries for a class of systems of evolution equations are studied. The evolution equations are defined in a bimetric space with two Riemannian metrics corresponding to the space of the independent and dependent variables of the differential equations. The exact relation of the Lie symmetries with the collineations of the bimetric space is determined.

Formal matched asymptotics for degenerate Ricci flow neckpinches

International Nuclear Information System (INIS)

Angenent, Sigurd B; Isenberg, James; Knopf, Dan

2011-01-01

Gu and Zhu (2008 Commun. Anal. Geom. 16 467–94) have shown that type-II Ricci flow singularities develop from nongeneric rotationally symmetric Riemannian metrics on S n+1 (n≥2). In this paper, we describe and provide plausibility arguments for a detailed asymptotic profile and rate of curvature blow-up that we predict such solutions exhibit

Symmetries of the dual metrics

International Nuclear Information System (INIS)

Baleanu, D.

1998-01-01

The geometric duality between the metric g μν and a Killing tensor K μν is studied. The conditions were found when the symmetries of the metric g μν and the dual metric K μν are the same. Dual spinning space was constructed without introduction of torsion. The general results are applied to the case of Kerr-Newmann metric

Overview of journal metrics

Directory of Open Access Journals (Sweden)

Kihong Kim

2018-02-01

Full Text Available Various kinds of metrics used for the quantitative evaluation of scholarly journals are reviewed. The impact factor and related metrics including the immediacy index and the aggregate impact factor, which are provided by the Journal Citation Reports, are explained in detail. The Eigenfactor score and the article influence score are also reviewed. In addition, journal metrics such as CiteScore, Source Normalized Impact per Paper, SCImago Journal Rank, h-index, and g-index are discussed. Limitations and problems that these metrics have are pointed out. We should be cautious to rely on those quantitative measures too much when we evaluate journals or researchers.

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.)

The Graph, Geometry and Symmetries of the Genetic Code with Hamming Metric

Directory of Open Access Journals (Sweden)

Reijer Lenstra

2015-07-01

Full Text Available The similarity patterns of the genetic code result from similar codons encoding similar messages. We develop a new mathematical model to analyze these patterns. The physicochemical characteristics of amino acids objectively quantify their differences and similarities; the Hamming metric does the same for the 64 codons of the codon set. (Hamming distances equal the number of different codon positions: AAA and AAC are at 1-distance; codons are maximally at 3-distance. The CodonPolytope, a 9-dimensional geometric object, is spanned by 64 vertices that represent the codons and the Euclidian distances between these vertices correspond one-to-one with intercodon Hamming distances. The CodonGraph represents the vertices and edges of the polytope; each edge equals a Hamming 1-distance. The mirror reflection symmetry group of the polytope is isomorphic to the largest permutation symmetry group of the codon set that preserves Hamming distances. These groups contain 82,944 symmetries. Many polytope symmetries coincide with the degeneracy and similarity patterns of the genetic code. These code symmetries are strongly related with the face structure of the polytope with smaller faces displaying stronger code symmetries. Splitting the polytope stepwise into smaller faces models an early evolution of the code that generates this hierarchy of code symmetries. The canonical code represents a class of 41,472 codes with equivalent symmetries; a single class among an astronomical number of symmetry classes comprising all possible codes.

Holographic Spherically Symmetric Metrics

Science.gov (United States)

Petri, Michael

The holographic principle (HP) conjectures, that the maximum number of degrees of freedom of any realistic physical system is proportional to the system's boundary area. The HP has its roots in the study of black holes. It has recently been applied to cosmological solutions. In this article we apply the HP to spherically symmetric static space-times. We find that any regular spherically symmetric object saturating the HP is subject to tight constraints on the (interior) metric, energy-density, temperature and entropy-density. Whenever gravity can be described by a metric theory, gravity is macroscopically scale invariant and the laws of thermodynamics hold locally and globally, the (interior) metric of a regular holographic object is uniquely determined up to a constant factor and the interior matter-state must follow well defined scaling relations. When the metric theory of gravity is general relativity, the interior matter has an overall string equation of state (EOS) and a unique total energy-density. Thus the holographic metric derived in this article can serve as simple interior 4D realization of Mathur's string fuzzball proposal. Some properties of the holographic metric and its possible experimental verification are discussed. The geodesics of the holographic metric describe an isotropically expanding (or contracting) universe with a nearly homogeneous matter-distribution within the local Hubble volume. Due to the overall string EOS the active gravitational mass-density is zero, resulting in a coasting expansion with Ht = 1, which is compatible with the recent GRB-data.

Latent geometry of bipartite networks

Science.gov (United States)

Kitsak, Maksim; Papadopoulos, Fragkiskos; Krioukov, Dmitri

2017-03-01

Despite the abundance of bipartite networked systems, their organizing principles are less studied compared to unipartite networks. Bipartite networks are often analyzed after projecting them onto one of the two sets of nodes. As a result of the projection, nodes of the same set are linked together if they have at least one neighbor in common in the bipartite network. Even though these projections allow one to study bipartite networks using tools developed for unipartite networks, one-mode projections lead to significant loss of information and artificial inflation of the projected network with fully connected subgraphs. Here we pursue a different approach for analyzing bipartite systems that is based on the observation that such systems have a latent metric structure: network nodes are points in a latent metric space, while connections are more likely to form between nodes separated by shorter distances. This approach has been developed for unipartite networks, and relatively little is known about its applicability to bipartite systems. Here, we fully analyze a simple latent-geometric model of bipartite networks and show that this model explains the peculiar structural properties of many real bipartite systems, including the distributions of common neighbors and bipartite clustering. We also analyze the geometric information loss in one-mode projections in this model and propose an efficient method to infer the latent pairwise distances between nodes. Uncovering the latent geometry underlying real bipartite networks can find applications in diverse domains, ranging from constructing efficient recommender systems to understanding cell metabolism.

Intelligent Patching of Conceptual Geometry for CFD Analysis

Science.gov (United States)

Li, Wu

2010-01-01

The iPatch computer code for intelligently patching surface grids was developed to convert conceptual geometry to computational fluid dynamics (CFD) geometry (see figure). It automatically uses bicubic B-splines to extrapolate (if necessary) each surface in a conceptual geometry so that all the independently defined geometric components (such as wing and fuselage) can be intersected to form a watertight CFD geometry. The software also computes the intersection curves of surface patches at any resolution (up to 10.4 accuracy) specified by the user, and it writes the B-spline surface patches, and the corresponding boundary points, for the watertight CFD geometry in the format that can be directly used by the grid generation tool VGRID. iPatch requires that input geometry be in PLOT3D format where each component surface is defined by a rectangular grid {(x(i,j), y(i,j), z(i,j)):1less than or equal to i less than or equal to m, 1 less than or equal to j less than or equal to n} that represents a smooth B-spline surface. All surfaces in the PLOT3D file conceptually represent a watertight geometry of components of an aircraft on the half-space y greater than or equal to 0. Overlapping surfaces are not allowed, but could be fixed by a utility code "fixp3d". The fixp3d utility code first finds the two grid lines on the two surface grids that are closest to each other in Hausdorff distance (a metric to measure the discrepancies of two sets); then uses one of the grid lines as the transition line, extending grid lines on one grid to the other grid to form a merged grid. Any two connecting surfaces shall have a "visually" common boundary curve, or can be described by an intersection relationship defined in a geometry specification file. The intersection of two surfaces can be at a conceptual level. However, the intersection is directional (along either i or j index direction), and each intersecting grid line (or its spine extrapolation) on the first surface should intersect

Quantum cosmology of a Bianchi III LRS geometry coupled to a source free electromagnetic field

Science.gov (United States)

Karagiorgos, A.; Pailas, T.; Dimakis, N.; Terzis, Petros A.; Christodoulakis, T.

2018-03-01

We consider a Bianchi type III axisymmetric geometry in the presence of an electromagnetic field. A first result at the classical level is that the symmetry of the geometry need not be applied on the electromagnetic tensor Fμν the algebraic restrictions, implied by the Einstein field equations to the stress energy tensor Tμν, suffice to reduce the general Fμν to the appropriate form. The classical solution thus found contains a time dependent electric and a constant magnetic charge. The solution is also reachable from the corresponding mini-superspace action, which is strikingly similar to the Reissner-Nordstr{öm one. This points to a connection between the black hole geometry and the cosmological solution here found, which is the analog of the known correlation between the Schwarzschild and the Kantowski-Sachs metrics. The configuration space is drastically modified by the presence of the magnetic charge from a 3D flat to a 3D pp wave geometry. We map the emerging linear and quadratic classical integrals of motion, to quantum observables. Along with the Wheeler-DeWitt equation these observables provide unique, up to constants, wave functions. The employment of a Bohmian interpretation of these quantum states results in deterministic (semi-classical) geometries most of which are singularity free.

Rational Thinking and Reasonable Thinking in Physics

Directory of Open Access Journals (Sweden)

Isaeva E. A.

2008-04-01

Full Text Available The usual concept of space and time, based on Aristotle's principle of contemplation of the world and of the absoluteness of time, is a product of rational thinking. At the same time, in philosophy, rational thinking differs from reasonable thinking; the aim of logic is to distinguish finite forms from infinite forms. Agreeing that space and time are things of infinity in this work, we shall show that, with regard to these two things, it is necessary to apply reasonable thinking. Spaces with non-Euclidean geometry, for example Riemannian and Finslerian spaces, in particular, the space of the General Theory of the Relativity (four-dimensional pseudo-Riemannian geometry and also the concept of multi-dimensional space-time are products of reasonable thinking. Consequently, modern physical experiment not dealing with daily occurrences (greater speeds than a low speed to the velocity of light, strong fields, singularities, etc. can be covered only by reasonable thinking.

The many faces of Maxwell, Dirac and Einstein equations a Clifford bundle approach

CERN Document Server

Rodrigues, Jr, Waldyr A

2016-01-01

This book is an exposition of the algebra and calculus of differential forms, of the Clifford and Spin-Clifford bundle formalisms, and of vistas to a formulation of important concepts of differential geometry indispensable for an in-depth understanding of space-time physics. The formalism discloses the hidden geometrical nature of spinor fields. Maxwell, Dirac and Einstein fields are shown to have representatives by objects of the same mathematical nature, namely sections of an appropriate Clifford bundle. This approach reveals unity in diversity and suggests relationships that are hidden in the standard formalisms and opens new paths for research. This thoroughly revised second edition also adds three new chapters: on the Clifford bundle approach to the Riemannian or semi-Riemannian differential geometry of branes; on Komar currents in the context of the General Relativity theory; and an analysis of the similarities and main differences between Dirac, Majorana and ELKO spinor fields. The exercises with solut...

Rational Thinking and Reasonable Thinking in Physics

Directory of Open Access Journals (Sweden)

Isaeva E. A.

2008-04-01

Full Text Available The usual concept of space and time, based on Aristotle’s principle of contemplation of the world and of the absoluteness of time, is a product of rational thinking. At the same time, in philosophy, rational thinking differs from reasonable thinking; the aim of logic is to distinguish finite forms from infinite forms. Agreeing that space and time are things of infinity in this work, we shall show that, with regard to these two things, it is necessary to apply reasonable thinking. Spaces with non-Euclidean geometry, for example Riemannian and Finslerian spaces, in particular, the space of the General Theory of the Relativity (four-dimensional pseudo-Riemannian geometry and also the concept of multi-dimensional space-time are products of reasonable thinking. Consequently, modern physical experiment not dealing with daily occurrences (greater speeds than a low speed to the velocity of light, strong fields, singularities, etc. can be covered only by reasonable thinking.

Model-Based Referenceless Quality Metric of 3D Synthesized Images Using Local Image Description.

Science.gov (United States)

Gu, Ke; Jakhetiya, Vinit; Qiao, Jun-Fei; Li, Xiaoli; Lin, Weisi; Thalmann, Daniel

2017-07-28

New challenges have been brought out along with the emerging of 3D-related technologies such as virtual reality (VR), augmented reality (AR), and mixed reality (MR). Free viewpoint video (FVV), due to its applications in remote surveillance, remote education, etc, based on the flexible selection of direction and viewpoint, has been perceived as the development direction of next-generation video technologies and has drawn a wide range of researchers' attention. Since FVV images are synthesized via a depth image-based rendering (DIBR) procedure in the "blind" environment (without reference images), a reliable real-time blind quality evaluation and monitoring system is urgently required. But existing assessment metrics do not render human judgments faithfully mainly because geometric distortions are generated by DIBR. To this end, this paper proposes a novel referenceless quality metric of DIBR-synthesized images using the autoregression (AR)-based local image description. It was found that, after the AR prediction, the reconstructed error between a DIBR-synthesized image and its AR-predicted image can accurately capture the geometry distortion. The visual saliency is then leveraged to modify the proposed blind quality metric to a sizable margin. Experiments validate the superiority of our no-reference quality method as compared with prevailing full-, reduced- and no-reference models.

Metric regularity and subdifferential calculus

International Nuclear Information System (INIS)

Ioffe, A D

2000-01-01

The theory of metric regularity is an extension of two classical results: the Lyusternik tangent space theorem and the Graves surjection theorem. Developments in non-smooth analysis in the 1980s and 1990s paved the way for a number of far-reaching extensions of these results. It was also well understood that the phenomena behind the results are of metric origin, not connected with any linear structure. At the same time it became clear that some basic hypotheses of the subdifferential calculus are closely connected with the metric regularity of certain set-valued maps. The survey is devoted to the metric theory of metric regularity and its connection with subdifferential calculus in Banach spaces

Context-dependent ATC complexity metric

NARCIS (Netherlands)

Mercado Velasco, G.A.; Borst, C.

2015-01-01

Several studies have investigated Air Traffic Control (ATC) complexity metrics in a search for a metric that could best capture workload. These studies have shown how daunting the search for a universal workload metric (one that could be applied in different contexts: sectors, traffic patterns,

Instanton strings and hyper-Kaehler geometry

International Nuclear Information System (INIS)

Dijkgraaf, Robbert

1999-01-01

We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N = (4, 4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS 3 . We derive a precise map relating the moduli of the K3 type 1113 string compactification to the moduli of these conformal field theories and the corresponding classical hyper-Kahler geometry. We conclude that in the absence of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a self-dual NS B-field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyper-Kaehler manifolds

Surface geometry of 5D black holes and black rings

International Nuclear Information System (INIS)

Frolov, Valeri P.; Goswami, Rituparno

2007-01-01

We discuss geometrical properties of the horizon surface of five-dimensional rotating black holes and black rings. Geometrical invariants characterizing these 3D geometries are calculated. We obtain a global embedding of the 5D rotating black horizon surface into a flat space. We also describe the Kaluza-Klein reduction of the black ring solution (along the direction of its rotation) which, though it is nakedly singular, relates this solution to the 4D metric of a static black hole distorted by the presence of external scalar (dilaton) and vector ('electromagnetic') fields. The properties of the reduced black hole horizon and its embedding in E 3 are briefly discussed

DLA Energy Biofuel Feedstock Metrics Study

Science.gov (United States)

2012-12-11

moderately/highly in- vasive  Metric 2: Genetically modified organism ( GMO ) hazard, Yes/No and Hazard Category  Metric 3: Species hybridization...4– biofuel distribution Stage # 5– biofuel use Metric 1: State inva- siveness ranking Yes Minimal Minimal No No Metric 2: GMO hazard Yes...may utilize GMO microbial or microalgae species across the applicable biofuel life cycles (stages 1–3). The following consequence Metrics 4–6 then

Self-adjointness of the Gaffney Laplacian on Vector Bundles

International Nuclear Information System (INIS)

Bandara, Lashi; Milatovic, Ognjen

2015-01-01

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator

Self-adjointness of the Gaffney Laplacian on Vector Bundles

Energy Technology Data Exchange (ETDEWEB)

Bandara, Lashi, E-mail: lashi.bandara@chalmers.se [Chalmers University of Technology and University of Gothenburg, Mathematical Sciences (Sweden); Milatovic, Ognjen, E-mail: omilatov@unf.edu [University of North Florida, Department of Mathematics and Statistics (United States)

2015-12-15

We study the Gaffney Laplacian on a vector bundle equipped with a compatible metric and connection over a Riemannian manifold that is possibly geodesically incomplete. Under the hypothesis that the Cauchy boundary is polar, we demonstrate the self-adjointness of this Laplacian. Furthermore, we show that negligible boundary is a necessary and sufficient condition for the self-adjointness of this operator.

Discrete causal theory emergent spacetime and the causal metric hypothesis

CERN Document Server

Dribus, Benjamin F

2017-01-01

This book evaluates and suggests potentially critical improvements to causal set theory, one of the best-motivated approaches to the outstanding problems of fundamental physics. Spacetime structure is of central importance to physics beyond general relativity and the standard model. The causal metric hypothesis treats causal relations as the basis of this structure. The book develops the consequences of this hypothesis under the assumption of a fundamental scale, with smooth spacetime geometry viewed as emergent. This approach resembles causal set theory, but differs in important ways; for example, the relative viewpoint, emphasizing relations between pairs of events, and relationships between pairs of histories, is central. The book culminates in a dynamical law for quantum spacetime, derived via generalized path summation.

Relative-locality distant observers and the phenomenology of momentum-space geometry

International Nuclear Information System (INIS)

Amelino-Camelia, Giovanni; Rosati, Giacomo; Trevisan, Gabriele; Arzano, Michele; Kowalski-Glikman, Jerzy

2012-01-01

We study the translational invariance of the relative-locality framework proposed in Amelino-Camelia et al (2011 Phys. Rev. D 84 084010), which had been previously established only for the case of a single interaction. We provide an explicit example of boundary conditions at endpoints of worldlines, which indeed ensures the desired translational invariance for processes involving several interactions, even when some of the interactions are causally connected (particle exchange). We illustrate the properties of the associated relativistic description of distant observers within the example of a κ-Poincare-inspired momentum-space geometry, with de Sitter metric and parallel transport governed by a non-metric and torsionful connection. We find that in such a theory, simultaneously emitted massless particles do not reach simultaneously a distant detector, as expected in light of the findings of Freidel and Smolin (2011 arXiv:1103.5626) on the implications of non-metric connections. We also show that the theory admits a free-particle limit, where the relative-locality results of Amelino-Camelia et al (2011 Phys. Lett. B 700 150) are reproduced. We establish that the torsion of the κ-Poincare connection introduces a small (but observably large) dependence of the time of detection, for simultaneously emitted particles, on some properties of the interactions producing the particles at the source. (paper)

Relative-locality distant observers and the phenomenology of momentum-space geometry

Science.gov (United States)

Amelino-Camelia, Giovanni; Arzano, Michele; Kowalski-Glikman, Jerzy; Rosati, Giacomo; Trevisan, Gabriele

2012-04-01

We study the translational invariance of the relative-locality framework proposed in Amelino-Camelia et al (2011 Phys. Rev. D 84 084010), which had been previously established only for the case of a single interaction. We provide an explicit example of boundary conditions at endpoints of worldlines, which indeed ensures the desired translational invariance for processes involving several interactions, even when some of the interactions are causally connected (particle exchange). We illustrate the properties of the associated relativistic description of distant observers within the example of a κ-Poincaré-inspired momentum-space geometry, with de Sitter metric and parallel transport governed by a non-metric and torsionful connection. We find that in such a theory, simultaneously emitted massless particles do not reach simultaneously a distant detector, as expected in light of the findings of Freidel and Smolin (2011 arXiv:1103.5626) on the implications of non-metric connections. We also show that the theory admits a free-particle limit, where the relative-locality results of Amelino-Camelia et al (2011 Phys. Lett. B 700 150) are reproduced. We establish that the torsion of the κ-Poincaré connection introduces a small (but observably large) dependence of the time of detection, for simultaneously emitted particles, on some properties of the interactions producing the particles at the source.

Symmetries of Taub-NUT dual metrics

International Nuclear Information System (INIS)

Baleanu, D.; Codoban, S.

1998-01-01

Recently geometric duality was analyzed for a metric which admits Killing tensors. An interesting example arises when the manifold has Killing-Yano tensors. The symmetries of the dual metrics in the case of Taub-NUT metric are investigated. Generic and non-generic symmetries of dual Taub-NUT metric are analyzed

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.)

Metric learning

CERN Document Server

Bellet, Aurelien; Sebban, Marc

2015-01-01

Similarity between objects plays an important role in both human cognitive processes and artificial systems for recognition and categorization. How to appropriately measure such similarities for a given task is crucial to the performance of many machine learning, pattern recognition and data mining methods. This book is devoted to metric learning, a set of techniques to automatically learn similarity and distance functions from data that has attracted a lot of interest in machine learning and related fields in the past ten years. In this book, we provide a thorough review of the metric learnin

Technical Privacy Metrics: a Systematic Survey

OpenAIRE

Wagner, Isabel; Eckhoff, David

2018-01-01

The file attached to this record is the author's final peer reviewed version The goal of privacy metrics is to measure the degree of privacy enjoyed by users in a system and the amount of protection offered by privacy-enhancing technologies. In this way, privacy metrics contribute to improving user privacy in the digital world. The diversity and complexity of privacy metrics in the literature makes an informed choice of metrics challenging. As a result, instead of using existing metrics, n...

On Information Metrics for Spatial Coding.

Science.gov (United States)

Souza, Bryan C; Pavão, Rodrigo; Belchior, Hindiael; Tort, Adriano B L

2018-04-01

The hippocampal formation is involved in navigation, and its neuronal activity exhibits a variety of spatial correlates (e.g., place cells, grid cells). The quantification of the information encoded by spikes has been standard procedure to identify which cells have spatial correlates. For place cells, most of the established metrics derive from Shannon's mutual information (Shannon, 1948), and convey information rate in bits/s or bits/spike (Skaggs et al., 1993, 1996). Despite their widespread use, the performance of these metrics in relation to the original mutual information metric has never been investigated. In this work, using simulated and real data, we find that the current information metrics correlate less with the accuracy of spatial decoding than the original mutual information metric. We also find that the top informative cells may differ among metrics, and show a surrogate-based normalization that yields comparable spatial information estimates. Since different information metrics may identify different neuronal populations, we discuss current and alternative definitions of spatially informative cells, which affect the metric choice. Copyright © 2018 IBRO. Published by Elsevier Ltd. All rights reserved.

Generalized Painleve-Gullstrand metrics

Energy Technology Data Exchange (ETDEWEB)

Lin Chunyu [Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan (China)], E-mail: l2891112@mail.ncku.edu.tw; Soo Chopin [Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan (China)], E-mail: cpsoo@mail.ncku.edu.tw

2009-02-02

An obstruction to the implementation of spatially flat Painleve-Gullstrand (PG) slicings is demonstrated, and explicitly discussed for Reissner-Nordstroem and Schwarzschild-anti-deSitter spacetimes. Generalizations of PG slicings which are not spatially flat but which remain regular at the horizons are introduced. These metrics can be obtained from standard spherically symmetric metrics by physical Lorentz boosts. With these generalized PG metrics, problematic contributions to the imaginary part of the action in the Parikh-Wilczek derivation of Hawking radiation due to the obstruction can be avoided.

Geometry and Mechanics of Chiral Pod Opening

Science.gov (United States)

Sharon, Eran; Armon, Shahaf; Efrati, Efi; Kupferman, Raz

2012-02-01

We study the geometry and mechanics that drive the opening of Bauhinia seeds pods. The pod valve wall consists of two fibrous layers oriented at ± 45^o with respect to the pod axis. Upon drying, each of the layers shrinks uniaxially, perpendicularly to the fibers orientation. This active deformation turn the valve into an incompatible sheet with reference saddle-like curvature tensor and a flat (Euclidean) reference metric. These two intrinsic properties are incompatible. The shape is, therefore, selected by a stretching-bending competition. Strips cut from the valve tissue and from synthetic model material adopt various helical configurations. We provide analytical expressions for these configurations in the bending and stretching dominated regimes. Surface measurements show the transition from minimal surfaces (narrow limit) to cylindrical ones (wide limit). Finally, we show how plants use these mechanical principles using different tissue architectures.

Kerr metric in the deSitter background

International Nuclear Information System (INIS)

Vaidya, P.C.

1984-01-01

In addition to the Kerr metric with cosmological constant Λ several other metrics are presented giving a Kerr-like solution of Einstein's equations in the background of deSitter universe. A new metric of what may be termed as rotating deSitter space-time devoid of matter but containing null fluid with twisting null rays, has been presented. This metric reduces to the standard deSitter metric when the twist in the rays vanishes. Kerr metric in this background is the immediate generalization of Schwarzschild's exterior metric with cosmological constant. (author)

Introducing geometry concept based on history of Islamic geometry

Science.gov (United States)

Maarif, S.; Wahyudin; Raditya, A.; Perbowo, K. S.

2018-01-01

Geometry is one of the areas of mathematics interesting to discuss. Geometry also has a long history in mathematical developments. Therefore, it is important integrated historical development of geometry in the classroom to increase’ knowledge of how mathematicians earlier finding and constructing a geometric concept. Introduction geometrical concept can be started by introducing the Muslim mathematician who invented these concepts so that students can understand in detail how a concept of geometry can be found. However, the history of mathematics development, especially history of Islamic geometry today is less popular in the world of education in Indonesia. There are several concepts discovered by Muslim mathematicians that should be appreciated by the students in learning geometry. Great ideas of mathematicians Muslim can be used as study materials to supplement religious character values taught by Muslim mathematicians. Additionally, by integrating the history of geometry in teaching geometry are expected to improve motivation and geometrical understanding concept.

Kerr metric in cosmological background

Energy Technology Data Exchange (ETDEWEB)

Vaidya, P C [Gujarat Univ., Ahmedabad (India). Dept. of Mathematics

1977-06-01

A metric satisfying Einstein's equation is given which in the vicinity of the source reduces to the well-known Kerr metric and which at large distances reduces to the Robertson-Walker metric of a nomogeneous cosmological model. The radius of the event horizon of the Kerr black hole in the cosmological background is found out.

Nonperturbative quantum geometries

International Nuclear Information System (INIS)

Jacobson, T.; California Univ., Santa Barbara; Smolin, L.; California Univ., Santa Barbara

1988-01-01

Using the self-dual representation of quantum general relativity, based on Ashtekar's new phase space variables, we present an infinite dimensional family of quantum states of the gravitational field which are exactly annihilated by the hamiltonian constraint. These states are constructed from Wilson loops for Ashtekar's connection (which is the spatial part of the left handed spin connection). We propose a new regularization procedure which allows us to evaluate the action of the hamiltonian constraint on these states. Infinite linear combinations of these states which are formally annihilated by the diffeomorphism constraints as well are also described. These are explicit examples of physical states of the gravitational field - and for the compact case are exact zero eigenstates of the hamiltonian of quantum general relativity. Several different approaches to constructing diffeomorphism invariant states in the self dual representation are also described. The physical interpretation of the states described here is discussed. However, as we do not yet know the physical inner product, any interpretation is at this stage speculative. Nevertheless, this work suggests that quantum geometry at Planck scales might be much simpler when explored in terms of the parallel transport of left-handed spinors than when explored in terms of the three metric. (orig.)

Two classes of metric spaces

Directory of Open Access Journals (Sweden)

Isabel Garrido

2016-04-01

Full Text Available The class of metric spaces (X,d known as small-determined spaces, introduced by Garrido and Jaramillo, are properly defined by means of some type of real-valued Lipschitz functions on X. On the other hand, B-simple metric spaces introduced by Hejcman are defined in terms of some kind of bornologies of bounded subsets of X. In this note we present a common framework where both classes of metric spaces can be studied which allows us to see not only the relationships between them but also to obtain new internal characterizations of these metric properties.

Smooth and Energy Saving Gait Planning for Humanoid Robot Using Geodesics

Directory of Open Access Journals (Sweden)

Liandong Zhang

2012-01-01

Full Text Available A novel gait planning method using geodesics for humanoid robot is given in this paper. Both the linear inverted pendulum model and the exact Single Support Phase (SSP are studied in our energy optimal gait planning based on geodesics. The kinetic energy of a 2-dimension linear inverted pendulum is obtained at first. We regard the kinetic energy as the Riemannian metric and the geodesic on this metric is studied and this is the shortest line between two points on the Riemannian surface. This geodesic is the optimal kinetic energy gait for the COG because the kinetic energy along geodesic is invariant according to the geometric property of geodesics and the walking is smooth and energy saving. Then the walking in Single Support Phase is studied and the energy optimal gait for the swing leg is obtained using our geodesics method. Finally, experiments using state-of-the-art method and using our geodesics optimization method are carried out respectively and the corresponding currents of the joint motors are recorded. With the currents comparing results, the feasibility of this new gait planning method is verified.

Geometry through history Euclidean, hyperbolic, and projective geometries

CERN Document Server

Dillon, Meighan I

2018-01-01

Presented as an engaging discourse, this textbook invites readers to delve into the historical origins and uses of geometry. The narrative traces the influence of Euclid’s system of geometry, as developed in his classic text The Elements, through the Arabic period, the modern era in the West, and up to twentieth century mathematics. Axioms and proof methods used by mathematicians from those periods are explored alongside the problems in Euclidean geometry that lead to their work. Students cultivate skills applicable to much of modern mathematics through sections that integrate concepts like projective and hyperbolic geometry with representative proof-based exercises. For its sophisticated account of ancient to modern geometries, this text assumes only a year of college mathematics as it builds towards its conclusion with algebraic curves and quaternions. Euclid’s work has affected geometry for thousands of years, so this text has something to offer to anyone who wants to broaden their appreciation for the...

On characterizations of quasi-metric completeness

Energy Technology Data Exchange (ETDEWEB)

Dag, H.; Romaguera, S.; Tirado, P.

2017-07-01

Hu proved in [4] that a metric space (X, d) is complete if and only if for any closed subspace C of (X, d), every Banach contraction on C has fixed point. Since then several authors have investigated the problem of characterizing the metric completeness by means of fixed point theorems. Recently this problem has been studied in the more general context of quasi-metric spaces for different notions of completeness. Here we present a characterization of a kind of completeness for quasi-metric spaces by means of a quasi-metric versions of Hu’s theorem. (Author)

Riemannian foliations on quaternion CR-submanifolds of an almost ...

Indian Academy of Sciences (India)

Department of Mathematics and Computer Science, Petroleum-Gas University of Ploieşti, Bulevardul Bucures¸ti, Nr. 39, Ploieşti 100680, Romania; Research Center in Geometry, Topology and Algebra, Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei, Nr. 14, Sector 1, Bucharest 70109, ...

Topological sigma B model in 4-dimensions

International Nuclear Information System (INIS)

Jun, Hyun-Keun; Park, Jae-Suk

2008-01-01

We propose a 4-dimensional version of topological sigma B-model, governing maps from a smooth compact 4-manifold M to a Calabi-Yau target manifold X. The theory depends on complex structure of X, while is independent of Kaehler metric of X. The theory is also a 4-dimensional topological field theory in the sense that the theory is independent of variation of Riemannian metric of the source 4-manifold M, potentially leading to new smooth invariant of 4-manifolds. We argue that the theory also comes with a topological family parametrized by the extended moduli space of complex structures.

Noncommutative geometry and its application to the standard model

Energy Technology Data Exchange (ETDEWEB)

Martinetti, Pierre [Georg-August Universitaet, Goettingen (Germany)

2009-07-01

We give an overview of the description of the standard model of particle physics minimally coupled to gravity within the framework of noncommutative geometry. Especially we study in detail the metric structure of spacetime that emerges from the spectral triple recently proposed by Chamseddine, Connes and Marcolli. Within this framework points of spacetime acquire an internal structure inherited from the gauge group of the standard model. A distance is defined on this generalized spacetime which is fully encoded by the Yang-Mills gauge fields together with the Higgs field. We focus on some explicit examples, underlying the link between this distance and other distances well known by physicists and mathematicians, such has the Carnot-Caratheodory horizontal distance or the Monge-Kantorovitch transport distance.

Engineering performance metrics

Science.gov (United States)

Delozier, R.; Snyder, N.

1993-03-01

Implementation of a Total Quality Management (TQM) approach to engineering work required the development of a system of metrics which would serve as a meaningful management tool for evaluating effectiveness in accomplishing project objectives and in achieving improved customer satisfaction. A team effort was chartered with the goal of developing a system of engineering performance metrics which would measure customer satisfaction, quality, cost effectiveness, and timeliness. The approach to developing this system involved normal systems design phases including, conceptual design, detailed design, implementation, and integration. The lessons teamed from this effort will be explored in this paper. These lessons learned may provide a starting point for other large engineering organizations seeking to institute a performance measurement system accomplishing project objectives and in achieving improved customer satisfaction. To facilitate this effort, a team was chartered to assist in the development of the metrics system. This team, consisting of customers and Engineering staff members, was utilized to ensure that the needs and views of the customers were considered in the development of performance measurements. The development of a system of metrics is no different than the development of any type of system. It includes the steps of defining performance measurement requirements, measurement process conceptual design, performance measurement and reporting system detailed design, and system implementation and integration.

Image-based reconstruction of three-dimensional myocardial infarct geometry for patient-specific modeling of cardiac electrophysiology

Energy Technology Data Exchange (ETDEWEB)

Ukwatta, Eranga, E-mail: eukwatt1@jhu.edu; Arevalo, Hermenegild; Pashakhanloo, Farhad; Prakosa, Adityo; Vadakkumpadan, Fijoy [Institute for Computational Medicine, Johns Hopkins University, Baltimore, Maryland 21205 and Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21205 (United States); Rajchl, Martin [Department of Computing, Imperial College London, London SW7 2AZ (United Kingdom); White, James [Stephenson Cardiovascular MR Centre, University of Calgary, Calgary, Alberta T2N 2T9 (Canada); Herzka, Daniel A.; McVeigh, Elliot [Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21205 (United States); Lardo, Albert C. [Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21205 and Division of Cardiology, Johns Hopkins Institute of Medicine, Baltimore, Maryland 21224 (United States); Trayanova, Natalia A. [Institute for Computational Medicine, Johns Hopkins University, Baltimore, Maryland 21205 (United States); Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21205 (United States); Department of Biomedical Engineering, Johns Hopkins Institute of Medicine, Baltimore, Maryland 21205 (United States)

2015-08-15

Purpose: Accurate three-dimensional (3D) reconstruction of myocardial infarct geometry is crucial to patient-specific modeling of the heart aimed at providing therapeutic guidance in ischemic cardiomyopathy. However, myocardial infarct imaging is clinically performed using two-dimensional (2D) late-gadolinium enhanced cardiac magnetic resonance (LGE-CMR) techniques, and a method to build accurate 3D infarct reconstructions from the 2D LGE-CMR images has been lacking. The purpose of this study was to address this need. Methods: The authors developed a novel methodology to reconstruct 3D infarct geometry from segmented low-resolution (Lo-res) clinical LGE-CMR images. Their methodology employed the so-called logarithm of odds (LogOdds) function to implicitly represent the shape of the infarct in segmented image slices as LogOdds maps. These 2D maps were then interpolated into a 3D image, and the result transformed via the inverse of LogOdds to a binary image representing the 3D infarct geometry. To assess the efficacy of this method, the authors utilized 39 high-resolution (Hi-res) LGE-CMR images, including 36 in vivo acquisitions of human subjects with prior myocardial infarction and 3 ex vivo scans of canine hearts following coronary ligation to induce infarction. The infarct was manually segmented by trained experts in each slice of the Hi-res images, and the segmented data were downsampled to typical clinical resolution. The proposed method was then used to reconstruct 3D infarct geometry from the downsampled images, and the resulting reconstructions were compared with the manually segmented data. The method was extensively evaluated using metrics based on geometry as well as results of electrophysiological simulations of cardiac sinus rhythm and ventricular tachycardia in individual hearts. Several alternative reconstruction techniques were also implemented and compared with the proposed method. Results: The accuracy of the LogOdds method in reconstructing 3D

Brand metrics that matter

NARCIS (Netherlands)

Muntinga, D.; Bernritter, S.

2017-01-01

Het merk staat steeds meer centraal in de organisatie. Het is daarom essentieel om de gezondheid, prestaties en ontwikkelingen van het merk te meten. Het is echter een uitdaging om de juiste brand metrics te selecteren. Een enorme hoeveelheid metrics vraagt de aandacht van merkbeheerders. Maar welke

Privacy Metrics and Boundaries

NARCIS (Netherlands)

L-F. Pau (Louis-François)

2005-01-01

textabstractThis paper aims at defining a set of privacy metrics (quantitative and qualitative) in the case of the relation between a privacy protector ,and an information gatherer .The aims with such metrics are: -to allow to assess and compare different user scenarios and their differences; for

2D Affine and Projective Shape Analysis.

Science.gov (United States)

Bryner, Darshan; Klassen, Eric; Huiling Le; Srivastava, Anuj

2014-05-01

Current techniques for shape analysis tend to seek invariance to similarity transformations (rotation, translation, and scale), but certain imaging situations require invariance to larger groups, such as affine or projective groups. Here we present a general Riemannian framework for shape analysis of planar objects where metrics and related quantities are invariant to affine and projective groups. Highlighting two possibilities for representing object boundaries-ordered points (or landmarks) and parameterized curves-we study different combinations of these representations (points and curves) and transformations (affine and projective). Specifically, we provide solutions to three out of four situations and develop algorithms for computing geodesics and intrinsic sample statistics, leading up to Gaussian-type statistical models, and classifying test shapes using such models learned from training data. In the case of parameterized curves, we also achieve the desired goal of invariance to re-parameterizations. The geodesics are constructed by particularizing the path-straightening algorithm to geometries of current manifolds and are used, in turn, to compute shape statistics and Gaussian-type shape models. We demonstrate these ideas using a number of examples from shape and activity recognition.

Architectural geometry

KAUST Repository

Pottmann, Helmut

2014-11-26

Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.

Architectural geometry

KAUST Repository

Pottmann, Helmut; Eigensatz, Michael; Vaxman, Amir; Wallner, Johannes

2014-01-01

Around 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.

Introduction to vector and tensor analysis

CERN Document Server

Wrede, Robert C

1972-01-01

A broad introductory treatment, this volume examines general Cartesian coordinates, the cross product, Einstein's special theory of relativity, bases in general coordinate systems, maxima and minima of functions of two variables, line integrals, integral theorems, fundamental notions in n-space, Riemannian geometry, algebraic properties of the curvature tensor, and more. 1963 edition.

Curvature in mathematics and physics

CERN Document Server

Sternberg, Shlomo

2012-01-01

This original Dover textbook is based on an advanced undergraduate course taught by the author for more than 50 years. It introduces semi-Riemannian geometry and its principal physical application, Einstein's theory of general relativity, using the Cartan exterior calculus as a principal tool. Prerequisites include linear algebra and advanced calculus. 2012 edition.

Two lectures on D-geometry and noncommutative geometry

International Nuclear Information System (INIS)

Douglas, M.R.

1999-01-01

This is a write-up of lectures given at the 1998 Spring School at the Abdus Salam ICTP. We give a conceptual introduction to D-geometry, the study of geometry as seen by D-branes in string theory, and to noncommutative geometry as it has appeared in D-brane and Matrix theory physics. (author)

Twistor geometry

NARCIS (Netherlands)

van den Broek, P.M.

1984-01-01

The aim of this paper is to give a detailed exposition of the relation between the geometry of twistor space and the geometry of Minkowski space. The paper has a didactical purpose; no use has been made of differential geometry and cohomology.

Moving from gamma passing rates to patient DVH-based QA metrics in pretreatment dose QA

Energy Technology Data Exchange (ETDEWEB)

Zhen, Heming; Nelms, Benjamin E.; Tome, Wolfgang A. [Department of Medical Physics, University of Wisconsin, Madison, Wisconsin 53705 (United States); Department of Human Oncology, University of Wisconsin, Madison, Wisconsin 53792 and Canis Lupus LLC, Merrimac, Wisconsin 53561 (United States); Department of Medical Physics, University of Wisconsin, Madison, Wisconsin 53705 and Department of Human Oncology, University of Wisconsin, Madison, Wisconsin 53792 (United States)

2011-10-15

Purpose: The purpose of this work is to explore the usefulness of the gamma passing rate metric for per-patient, pretreatment dose QA and to validate a novel patient-dose/DVH-based method and its accuracy and correlation. Specifically, correlations between: (1) gamma passing rates for three 3D dosimeter detector geometries vs clinically relevant patient DVH-based metrics; (2) Gamma passing rates of whole patient dose grids vs DVH-based metrics, (3) gamma passing rates filtered by region of interest (ROI) vs DVH-based metrics, and (4) the capability of a novel software algorithm that estimates corrected patient Dose-DVH based on conventional phan-tom QA data are analyzed. Methods: Ninety six unique ''imperfect'' step-and-shoot IMRT plans were generated by applying four different types of errors on 24 clinical Head/Neck patients. The 3D patient doses as well as the dose to a cylindrical QA phantom were then recalculated using an error-free beam model to serve as a simulated measurement for comparison. Resulting deviations to the planned vs simulated measured DVH-based metrics were generated, as were gamma passing rates for a variety of difference/distance criteria covering: dose-in-phantom comparisons and dose-in-patient comparisons, with the in-patient results calculated both over the whole grid and per-ROI volume. Finally, patient dose and DVH were predicted using the conventional per-beam planar data as input into a commercial ''planned dose perturbation'' (PDP) algorithm, and the results of these predicted DVH-based metrics were compared to the known values. Results: A range of weak to moderate correlations were found between clinically relevant patient DVH metrics (CTV-D95, parotid D{sub mean}, spinal cord D1cc, and larynx D{sub mean}) and both 3D detector and 3D patient gamma passing rate (3%/3 mm, 2%/2 mm) for dose-in-phantom along with dose-in-patient for both whole patient volume and filtered per-ROI. There was

Geometry

Indian Academy of Sciences (India)

. In the previous article we looked at the origins of synthetic and analytic geometry. More practical minded people, the builders and navigators, were studying two other aspects of geometry- trigonometry and integral calculus. These are actually ...

Cyber threat metrics.

Energy Technology Data Exchange (ETDEWEB)

Frye, Jason Neal; Veitch, Cynthia K.; Mateski, Mark Elliot; Michalski, John T.; Harris, James Mark; Trevino, Cassandra M.; Maruoka, Scott

2012-03-01

Threats are generally much easier to list than to describe, and much easier to describe than to measure. As a result, many organizations list threats. Fewer describe them in useful terms, and still fewer measure them in meaningful ways. This is particularly true in the dynamic and nebulous domain of cyber threats - a domain that tends to resist easy measurement and, in some cases, appears to defy any measurement. We believe the problem is tractable. In this report we describe threat metrics and models for characterizing threats consistently and unambiguously. The purpose of this report is to support the Operational Threat Assessment (OTA) phase of risk and vulnerability assessment. To this end, we focus on the task of characterizing cyber threats using consistent threat metrics and models. In particular, we address threat metrics and models for describing malicious cyber threats to US FCEB agencies and systems.

Fixed point theory in metric type spaces

CERN Document Server

Agarwal, Ravi P; O’Regan, Donal; Roldán-López-de-Hierro, Antonio Francisco

2015-01-01

Written by a team of leading experts in the field, this volume presents a self-contained account of the theory, techniques and results in metric type spaces (in particular in G-metric spaces); that is, the text approaches this important area of fixed point analysis beginning from the basic ideas of metric space topology. The text is structured so that it leads the reader from preliminaries and historical notes on metric spaces (in particular G-metric spaces) and on mappings, to Banach type contraction theorems in metric type spaces, fixed point theory in partially ordered G-metric spaces, fixed point theory for expansive mappings in metric type spaces, generalizations, present results and techniques in a very general abstract setting and framework. Fixed point theory is one of the major research areas in nonlinear analysis. This is partly due to the fact that in many real world problems fixed point theory is the basic mathematical tool used to establish the existence of solutions to problems which arise natur...

Deep Transfer Metric Learning.

Science.gov (United States)

Junlin Hu; Jiwen Lu; Yap-Peng Tan; Jie Zhou

2016-12-01

Conventional metric learning methods usually assume that the training and test samples are captured in similar scenarios so that their distributions are assumed to be the same. This assumption does not hold in many real visual recognition applications, especially when samples are captured across different data sets. In this paper, we propose a new deep transfer metric learning (DTML) method to learn a set of hierarchical nonlinear transformations for cross-domain visual recognition by transferring discriminative knowledge from the labeled source domain to the unlabeled target domain. Specifically, our DTML learns a deep metric network by maximizing the inter-class variations and minimizing the intra-class variations, and minimizing the distribution divergence between the source domain and the target domain at the top layer of the network. To better exploit the discriminative information from the source domain, we further develop a deeply supervised transfer metric learning (DSTML) method by including an additional objective on DTML, where the output of both the hidden layers and the top layer are optimized jointly. To preserve the local manifold of input data points in the metric space, we present two new methods, DTML with autoencoder regularization and DSTML with autoencoder regularization. Experimental results on face verification, person re-identification, and handwritten digit recognition validate the effectiveness of the proposed methods.

Energy functionals for Calabi-Yau metrics

International Nuclear Information System (INIS)

Headrick, M; Nassar, A

2013-01-01

We identify a set of ''energy'' functionals on the space of metrics in a given Kähler class on a Calabi-Yau manifold, which are bounded below and minimized uniquely on the Ricci-flat metric in that class. Using these functionals, we recast the problem of numerically solving the Einstein equation as an optimization problem. We apply this strategy, using the ''algebraic'' metrics (metrics for which the Kähler potential is given in terms of a polynomial in the projective coordinates), to the Fermat quartic and to a one-parameter family of quintics that includes the Fermat and conifold quintics. We show that this method yields approximations to the Ricci-flat metric that are exponentially accurate in the degree of the polynomial (except at the conifold point, where the convergence is polynomial), and therefore orders of magnitude more accurate than the balanced metrics, previously studied as approximations to the Ricci-flat metric. The method is relatively fast and easy to implement. On the theoretical side, we also show that the functionals can be used to give a heuristic proof of Yau's theorem

Environmental forcing metrics to quantify short-term foredune morphodynamics

Science.gov (United States)

Spore, N.; Conery, I.; Brodie, K. L.; Palmsten, M.

2016-12-01

Coastal foredunes evolve continuously due to competing aeolian and hydrodynamic processes. Onshore to shore-parallel winds transport sand to the dune while storm-driven surge and wave runup remove sand from the dune. Dune-growth requires periods of time when the wind exceeds a threshold velocity to initiate transport and the relative geometry of the dry beach to the wind direction to create large fetches. This study aims to derive an aeolian transport potential (ATP) metric from the precipitation, available fetch (a function of wind angle and dry-beach width), and a threshold wind speed to initiate transport. ATP is then combined with a hydrodynamic transport potential (HTP) metric, defined as the number of hours of wave impact to the foredune or upper beach, to assess the time-dependent magnitude of the forcing factors affecting morphological evolution of the foredune between monthly terrestrial lidar surveys.This study focuses on two distinctly different dune fields and their frontal or incipient dune ridges in Duck, NC at the USACE Field Research Facility (FRF): (1) an undisturbed, tall and narrow recently impacted dune with a near vertical face; and (2) an undisturbed, shorter and wider dune with gentler and more hummocky slopes. The two sites are separated by dune toe position were extracted from the DEMs and compared to the ATP and HTP values between the surveys to evaluate the dominant factors affecting sediment flux to the system.

Magnetic geometry and physics of advanced divertors: The X-divertor and the snowflake

International Nuclear Information System (INIS)

Kotschenreuther, Mike; Valanju, Prashant; Covele, Brent; Mahajan, Swadesh

2013-01-01

Advanced divertors are magnetic geometries where a second X-point is added in the divertor region to address the serious challenges of burning plasma power exhaust. Invoking physical arguments, numerical work, and detailed model magnetic field analysis, we investigate the magnetic field structure of advanced divertors in the physically relevant region for power exhaust—the scrape-off layer. A primary result of our analysis is the emergence of a physical “metric,” the Divertor Index DI, which quantifies the flux expansion increase as one goes from the main X-point to the strike point. It clearly separates three geometries with distinct consequences for divertor physics—the Standard Divertor (DI = 1), and two advanced geometries—the X-Divertor (XD, DI > 1) and the Snowflake (DI < 1). The XD, therefore, cannot be classified as one variant of the Snowflake. By this measure, recent National Spherical Torus Experiment and DIIID experiments are X-Divertors, not Snowflakes

Magnetic geometry and physics of advanced divertors: The X-divertor and the snowflake

Energy Technology Data Exchange (ETDEWEB)

Kotschenreuther, Mike; Valanju, Prashant; Covele, Brent; Mahajan, Swadesh [Institute for Fusion Studies, The University of Texas at Austin, Austin, Texas 78712 (United States)

2013-10-15

Advanced divertors are magnetic geometries where a second X-point is added in the divertor region to address the serious challenges of burning plasma power exhaust. Invoking physical arguments, numerical work, and detailed model magnetic field analysis, we investigate the magnetic field structure of advanced divertors in the physically relevant region for power exhaust—the scrape-off layer. A primary result of our analysis is the emergence of a physical “metric,” the Divertor Index DI, which quantifies the flux expansion increase as one goes from the main X-point to the strike point. It clearly separates three geometries with distinct consequences for divertor physics—the Standard Divertor (DI = 1), and two advanced geometries—the X-Divertor (XD, DI > 1) and the Snowflake (DI < 1). The XD, therefore, cannot be classified as one variant of the Snowflake. By this measure, recent National Spherical Torus Experiment and DIIID experiments are X-Divertors, not Snowflakes.

Spectrum-efficient multi-channel design for coexisting IEEE 802.15.4 networks: A stochastic geometry approach

KAUST Repository

Elsawy, Hesham

2014-07-01

For networks with random topologies (e.g., wireless ad-hoc and sensor networks) and dynamically varying channel gains, choosing the long term operating parameters that optimize the network performance metrics is very challenging. In this paper, we use stochastic geometry analysis to develop a novel framework to design spectrum-efficient multi-channel random wireless networks based on the IEEE 802.15.4 standard. The proposed framework maximizes both spatial and time domain frequency utilization under channel gain uncertainties to minimize the number of frequency channels required to accommodate a certain population of coexisting IEEE 802.15.4 networks. The performance metrics are the outage probability and the self admission failure probability. We relax the single channel assumption that has been used traditionally in the stochastic geometry analysis. We show that the intensity of the admitted networks does not increase linearly with the number of channels and the rate of increase of the intensity of the admitted networks decreases with the number of channels. By using graph theory, we obtain the minimum required number of channels to accommodate a certain intensity of coexisting networks under a self admission failure probability constraint. To this end, we design a superframe structure for the coexisting IEEE 802.15.4 networks and a method for time-domain interference alignment. © 2002-2012 IEEE.

Phase space descriptions for simplicial 4D geometries

International Nuclear Information System (INIS)

Dittrich, Bianca; Ryan, James P

2011-01-01

Starting from the canonical phase space for discretized (4D) BF theory, we implement a canonical version of the simplicity constraints and construct phase spaces for simplicial geometries. Our construction allows us to study the connection between different versions of Regge calculus and approaches using connection variables, such as loop quantum gravity. We find that on a fixed triangulation the (gauge invariant) phase space associated with loop quantum gravity is genuinely larger than the one for length and even area Regge calculus. Rather, it corresponds to the phase space of area-angle Regge calculus, as defined in [1] (prior to the imposition of gluing constraints, which ensure the metricity of the triangulation). Finally, we show that for a subclass of triangulations one can construct first-class Hamiltonian and diffeomorphism constraints leading to flat 4D spacetimes.

Regge calculus from discontinuous metrics

International Nuclear Information System (INIS)

Khatsymovsky, V.M.

2003-01-01

Regge calculus is considered as a particular case of the more general system where the linklengths of any two neighbouring 4-tetrahedra do not necessarily coincide on their common face. This system is treated as that one described by metric discontinuous on the faces. In the superspace of all discontinuous metrics the Regge calculus metrics form some hypersurface defined by continuity conditions. Quantum theory of the discontinuous metric system is assumed to be fixed somehow in the form of quantum measure on (the space of functionals on) the superspace. The problem of reducing this measure to the Regge hypersurface is addressed. The quantum Regge calculus measure is defined from a discontinuous metric measure by inserting the δ-function-like phase factor. The requirement that continuity conditions be imposed in a 'face-independent' way fixes this factor uniquely. The term 'face-independent' means that this factor depends only on the (hyper)plane spanned by the face, not on it's form and size. This requirement seems to be natural from the viewpoint of existence of the well-defined continuum limit maximally free of lattice artefacts

Numerical Calabi-Yau metrics

International Nuclear Information System (INIS)

Douglas, Michael R.; Karp, Robert L.; Lukic, Sergio; Reinbacher, Rene

2008-01-01

We develop numerical methods for approximating Ricci flat metrics on Calabi-Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results

Metrics for Evaluation of Student Models

Science.gov (United States)

Pelanek, Radek

2015-01-01

Researchers use many different metrics for evaluation of performance of student models. The aim of this paper is to provide an overview of commonly used metrics, to discuss properties, advantages, and disadvantages of different metrics, to summarize current practice in educational data mining, and to provide guidance for evaluation of student…

Molecular geometry

CERN Document Server

Rodger, Alison

1995-01-01

Molecular Geometry discusses topics relevant to the arrangement of atoms. The book is comprised of seven chapters that tackle several areas of molecular geometry. Chapter 1 reviews the definition and determination of molecular geometry, while Chapter 2 discusses the unified view of stereochemistry and stereochemical changes. Chapter 3 covers the geometry of molecules of second row atoms, and Chapter 4 deals with the main group elements beyond the second row. The book also talks about the complexes of transition metals and f-block elements, and then covers the organometallic compounds and trans

Stabilization of rotational motion with application to spacecraft attitude control

DEFF Research Database (Denmark)

Wisniewski, Rafal

2000-01-01

for global stabilization of a rotary motion. Along with a model of the system formulated in the Hamilton's canonical from the algorithm uses information about a required potential energy and a dissipation term. The control action is the sum of the gradient of the potential energy and the dissipation force......The objective of this paper is to develop a control scheme for stabilization of a hamiltonian system. The method generalizes the results available in the literature on motion control in the Euclidean space to an arbitrary differrential manifol equipped with a metric. This modification is essencial...... on a Riemannian manifold. The Lyapnov stability theory is adapted and reformulated to fit to the new framework of Riemannian manifolds. Toillustrate the results a spacecraft attitude control problem is considered. Firstly, a global canonical representation for the spacecraft motion is found, then three spacecraft...

Stabilization of rotational motion with application to spacecraft attitude control

DEFF Research Database (Denmark)

Wisniewski, Rafal

2001-01-01

for global stabilization of a rotary motion. Along with a model of the system formulated in the Hamilton's canonical from the algorithm uses information about a required potential energy and a dissipation term. The control action is the sum of the gradient of the potential energy and the dissipation force......The objective of this paper is to develop a control scheme for stabilization of a hamiltonian system. The method generalizes the results available in the literature on motion control in the Euclidean space to an arbitrary differrential manifol equipped with a metric. This modification is essencial...... on a Riemannian manifold. The Lyapnov stability theory is adapted and reformulated to fit to the new framework of Riemannian manifolds. Toillustrate the results a spacecraft attitude control problem is considered. Firstly, a global canonical representation for the spacecraft motion is found, then three spacecraft...

Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

CERN Document Server

Vacaru, Sergiu I

2015-01-01

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost K\\"ahler - Ricci geometric evolution equations. The second goal is to study how fixed Lie algebroid, i.e. Ricci soliton, configurations can be constructed for Riemannian manifolds and/or (co) tangent bundles endowed with nonholonomic distributions modelling (generalized) Einstein or Finsler - Cartan spaces. Finally, there are provided some examples of generic off-diagonal solutions for Lie algebroid type Ricci solitons and (effective) Einstein and Lagrange-Finsler algebro...

The decomposition of deformation: New metrics to enhance shape analysis in medical imaging.

Science.gov (United States)

Varano, Valerio; Piras, Paolo; Gabriele, Stefano; Teresi, Luciano; Nardinocchi, Paola; Dryden, Ian L; Torromeo, Concetta; Puddu, Paolo E

2018-05-01

In landmarks-based Shape Analysis size is measured, in most cases, with Centroid Size. Changes in shape are decomposed in affine and non affine components. Furthermore the non affine component can be in turn decomposed in a series of local deformations (partial warps). If the extent of deformation between two shapes is small, the difference between Centroid Size and m-Volume increment is barely appreciable. In medical imaging applied to soft tissues bodies can undergo very large deformations, involving large changes in size. The cardiac example, analyzed in the present paper, shows changes in m-Volume that can reach the 60%. We show here that standard Geometric Morphometrics tools (landmarks, Thin Plate Spline, and related decomposition of the deformation) can be generalized to better describe the very large deformations of biological tissues, without losing a synthetic description. In particular, the classical decomposition of the space tangent to the shape space in affine and non affine components is enriched to include also the change in size, in order to give a complete description of the tangent space to the size-and-shape space. The proposed generalization is formulated by means of a new Riemannian metric describing the change in size as change in m-Volume rather than change in Centroid Size. This leads to a redefinition of some aspects of the Kendall's size-and-shape space without losing Kendall's original formulation. This new formulation is discussed by means of simulated examples using 2D and 3D platonic shapes as well as a real example from clinical 3D echocardiographic data. We demonstrate that our decomposition based approaches discriminate very effectively healthy subjects from patients affected by Hypertrophic Cardiomyopathy. Copyright © 2018 Elsevier B.V. All rights reserved.

Multivariate Tensor-based Brain Anatomical Surface Morphometry via Holomorphic One-Forms

OpenAIRE

Wang, Yalin; Chan, Tony F.; Toga, Arthur W.; Thompson, Paul M.

2009-01-01

Here we introduce multivariate tensor-based surface morphometry using holomorphic one-forms to study brain anatomy. We computed new statistics from the Riemannian metric tensors that retain the full information in the deformation tensor fields. We introduce two different holomorphic one-forms that induce different surface conformal parameterizations. We applied this framework to 3D MRI data to analyze hippocampal surface morphometry in Alzheimer’s Disease (AD; 26 subjects), lateral ventricula...

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.

Issues in Benchmark Metric Selection

Science.gov (United States)

Crolotte, Alain

It is true that a metric can influence a benchmark but will esoteric metrics create more problems than they will solve? We answer this question affirmatively by examining the case of the TPC-D metric which used the much debated geometric mean for the single-stream test. We will show how a simple choice influenced the benchmark and its conduct and, to some extent, DBMS development. After examining other alternatives our conclusion is that the “real” measure for a decision-support benchmark is the arithmetic mean.

A contest of sensors in close range 3D imaging: performance evaluation with a new metric test object

Directory of Open Access Journals (Sweden)

M. Hess

2014-06-01

Full Text Available An independent means of 3D image quality assessment is introduced, addressing non-professional users of sensors and freeware, which is largely characterized as closed-sourced and by the absence of quality metrics for processing steps, such as alignment. A performance evaluation of commercially available, state-of-the-art close range 3D imaging technologies is demonstrated with the help of a newly developed Portable Metric Test Artefact. The use of this test object provides quality control by a quantitative assessment of 3D imaging sensors. It will enable users to give precise specifications which spatial resolution and geometry recording they expect as outcome from their 3D digitizing process. This will lead to the creation of high-quality 3D digital surrogates and 3D digital assets. The paper is presented in the form of a competition of teams, and a possible winner will emerge.

Robustness of climate metrics under climate policy ambiguity

International Nuclear Information System (INIS)

Ekholm, Tommi; Lindroos, Tomi J.; Savolainen, Ilkka

2013-01-01

Highlights: • We assess the economic impacts of using different climate metrics. • The setting is cost-efficient scenarios for three interpretations of the 2C target. • With each target setting, the optimal metric is different. • Therefore policy ambiguity prevents the selection of an optimal metric. • Robust metric values that perform well with multiple policy targets however exist. -- Abstract: A wide array of alternatives has been proposed as the common metrics with which to compare the climate impacts of different emission types. Different physical and economic metrics and their parameterizations give diverse weights between e.g. CH 4 and CO 2 , and fixing the metric from one perspective makes it sub-optimal from another. As the aims of global climate policy involve some degree of ambiguity, it is not possible to determine a metric that would be optimal and consistent with all policy aims. This paper evaluates the cost implications of using predetermined metrics in cost-efficient mitigation scenarios. Three formulations of the 2 °C target, including both deterministic and stochastic approaches, shared a wide range of metric values for CH 4 with which the mitigation costs are only slightly above the cost-optimal levels. Therefore, although ambiguity in current policy might prevent us from selecting an optimal metric, it can be possible to select robust metric values that perform well with multiple policy targets

Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology & Symplectic Geometry, Noncommutative Geometry and Physics

CERN Document Server

Eliashberg, Yakov; Maeda, Yoshiaki; Symplectic, Poisson, and Noncommutative geometry

2014-01-01

Symplectic geometry originated in physics, but it has flourished as an independent subject in mathematics, together with its offspring, symplectic topology. Symplectic methods have even been applied back to mathematical physics. Noncommutative geometry has developed an alternative mathematical quantization scheme based on a geometric approach to operator algebras. Deformation quantization, a blend of symplectic methods and noncommutative geometry, approaches quantum mechanics from a more algebraic viewpoint, as it addresses quantization as a deformation of Poisson structures. This volume contains seven chapters based on lectures given by invited speakers at two May 2010 workshops held at the Mathematical Sciences Research Institute: Symplectic and Poisson Geometry in Interaction with Analysis, Algebra and Topology (honoring Alan Weinstein, one of the key figures in the field) and Symplectic Geometry, Noncommutative Geometry and Physics. The chapters include presentations of previously unpublished results and ...

Web metrics for library and information professionals

CERN Document Server

Stuart, David

2014-01-01

This is a practical guide to using web metrics to measure impact and demonstrate value. The web provides an opportunity to collect a host of different metrics, from those associated with social media accounts and websites to more traditional research outputs. This book is a clear guide for library and information professionals as to what web metrics are available and how to assess and use them to make informed decisions and demonstrate value. As individuals and organizations increasingly use the web in addition to traditional publishing avenues and formats, this book provides the tools to unlock web metrics and evaluate the impact of this content. The key topics covered include: bibliometrics, webometrics and web metrics; data collection tools; evaluating impact on the web; evaluating social media impact; investigating relationships between actors; exploring traditional publications in a new environment; web metrics and the web of data; the future of web metrics and the library and information professional.Th...

Partial rectangular metric spaces and fixed point theorems.

Science.gov (United States)

Shukla, Satish

2014-01-01

The purpose of this paper is to introduce the concept of partial rectangular metric spaces as a generalization of rectangular metric and partial metric spaces. Some properties of partial rectangular metric spaces and some fixed point results for quasitype contraction in partial rectangular metric spaces are proved. Some examples are given to illustrate the observed results.

Metric elasticity in a collapsing star: Gravitational radiation coupled to torsional motion

International Nuclear Information System (INIS)

Gerlach, U.H.; Scott, J.F.

1986-01-01

Torsional oscillatory matter motion as well as differential rotation couple via the linearized Einstein field equations to the gravitational degrees of freedom. For an arbitrary spherically symmetric background, such as that of a wildly pulsating or a catastrophically collapsing star, we exhibit (a) the strain tensor and (b) the corresponding stress-energy tensor. It is found that in the star there are two elasticity tensors. One expresses the familiar elasticity of matter, the other expresses the elasticity of the geometry. This metric elasticity is responsible for coupling the gravitational and matter degrees of freedom. The two coupled scalar wave equations for these degrees of freedom are exhibited. Also exhibited are their characteristics as well as the junction conditions for their solutions across any spherical surface of discontinuity

A Kerr-NUT metric

International Nuclear Information System (INIS)

Vaidya, P.C.; Patel, L.K.; Bhatt, P.V.

1976-01-01

Using Galilean time and retarded distance as coordinates the usual Kerr metric is expressed in form similar to the Newman-Unti-Tamburino (NUT) metric. The combined Kerr-NUT metric is then investigated. In addition to the Kerr and NUT solutions of Einstein's equations, three other types of solutions are derived. These are (i) the radiating Kerr solution, (ii) the radiating NUT solution satisfying Rsub(ik) = sigmaxisub(i)xisub(k), xisub(i)xisup(i) = 0, and (iii) the associated Kerr solution satisfying Rsub(ik) = 0. Solution (i) is distinct from and simpler than the one reported earlier by Vaidya and Patel (Phys. Rev.; D7:3590 (1973)). Solutions (ii) and (iii) gave line elements which have the axis of symmetry as a singular line. (author)

Exact Descriptions of General Relativity Derived from Newtonian Mechanics within Curved Geometries

Science.gov (United States)

Savickas, David

2015-04-01

General relativity and Newtonian mechanics are shown to be exactly related when Newton's second law is written in a curved geometry by using the physical components of a vector as is defined in tensor calculus. By replacing length within the momentum's velocity by the vector metric in a curved geometry the second law can then be shown to be exactly identical to the geodesic equation of motion occurring in general relativity. When time's vector direction is constant, as similarly occurs in Newtonian mechanics, this equation can be reduced to a curved three-dimensional equation of motion that yields the the Schwarzschild equations of motion for an isolated particle. They can be used to describe gravitational behavior for any array of masses for which the Newtonian gravitational potential is known, and is shown to describe a mass particle's behavior in the gravitational field of a thin mass-rod. This use of Newton's laws allows relativistic behavior to be described in a physically comprehensible manner. D. Savickas, Int. J. Mod. Phys. D 23 1430018, (2014).

Stabilization of the Wave Equation with Boundary Time-Varying Delay

Directory of Open Access Journals (Sweden)

Hao Li

2014-01-01

Full Text Available We study the stabilization of the wave equation with variable coefficients in a bounded domain and a time-varying delay term in the time-varying, weakly nonlinear boundary feedbacks. By the Riemannian geometry methods and a suitable assumption of nonlinearity, we obtain the uniform decay of the energy of the closed loop system.

Background metric in supergravity theories

International Nuclear Information System (INIS)

Yoneya, T.

1978-01-01

In supergravity theories, we investigate the conformal anomaly of the path-integral determinant and the problem of fermion zero modes in the presence of a nontrivial background metric. Except in SO(3) -invariant supergravity, there are nonvanishing conformal anomalies. As a consequence, amplitudes around the nontrivial background metric contain unpredictable arbitrariness. The fermion zero modes which are explicitly constructed for the Euclidean Schwarzschild metric are interpreted as an indication of the supersymmetric multiplet structure of a black hole. The degree of degeneracy of a black hole is 2/sup 4n/ in SO(n) supergravity

Daylight metrics and energy savings

Energy Technology Data Exchange (ETDEWEB)

Mardaljevic, John; Heschong, Lisa; Lee, Eleanor

2009-12-31

The drive towards sustainable, low-energy buildings has increased the need for simple, yet accurate methods to evaluate whether a daylit building meets minimum standards for energy and human comfort performance. Current metrics do not account for the temporal and spatial aspects of daylight, nor of occupants comfort or interventions. This paper reviews the historical basis of current compliance methods for achieving daylit buildings, proposes a technical basis for development of better metrics, and provides two case study examples to stimulate dialogue on how metrics can be applied in a practical, real-world context.

Information Geometry of Non-Equilibrium Processes in a Bistable System with a Cubic Damping

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Rainer Hollerbach

2017-06-01

Full Text Available A probabilistic description is essential for understanding the dynamics of stochastic systems far from equilibrium, given uncertainty inherent in the systems. To compare different Probability Density Functions (PDFs, it is extremely useful to quantify the difference among different PDFs by assigning an appropriate metric to probability such that the distance increases with the difference between the two PDFs. This metric structure then provides a key link between stochastic systems and information geometry. For a non-equilibrium process, we define an infinitesimal distance at any time by comparing two PDFs at times infinitesimally apart and sum these distances in time. The total distance along the trajectory of the system quantifies the total number of different states that the system undergoes in time and is called the information length. By using this concept, we investigate the information geometry of non-equilibrium processes involved in disorder-order transitions between the critical and subcritical states in a bistable system. Specifically, we compute time-dependent PDFs, information length, the rate of change in information length, entropy change and Fisher information in disorder-to-order and order-to-disorder transitions and discuss similarities and disparities between the two transitions. In particular, we show that the total information length in order-to-disorder transition is much larger than that in disorder-to-order transition and elucidate the link to the drastically different evolution of entropy in both transitions. We also provide the comparison of the results with those in the case of the transition between the subcritical and supercritical states and discuss implications for fitness.

Detecting Anisotropic Inclusions Through EIT

Science.gov (United States)

Cristina, Jan; Päivärinta, Lassi

2017-12-01

We study the evolution equation {partialtu=-Λtu} where {Λt} is the Dirichlet-Neumann operator of a decreasing family of Riemannian manifolds with boundary {Σt}. We derive a lower bound for the solution of such an equation, and apply it to a quantitative density estimate for the restriction of harmonic functions on M}=Σ_{0 to the boundaries of {partialΣt}. Consequently we are able to derive a lower bound for the difference of the Dirichlet-Neumann maps in terms of the difference of a background metrics g and an inclusion metric {g+χ_{Σ}(h-g)} on a manifold M.

Metrics for energy resilience

International Nuclear Information System (INIS)

Roege, Paul E.; Collier, Zachary A.; Mancillas, James; McDonagh, John A.; Linkov, Igor

2014-01-01

Energy lies at the backbone of any advanced society and constitutes an essential prerequisite for economic growth, social order and national defense. However there is an Achilles heel to today's energy and technology relationship; namely a precarious intimacy between energy and the fiscal, social, and technical systems it supports. Recently, widespread and persistent disruptions in energy systems have highlighted the extent of this dependence and the vulnerability of increasingly optimized systems to changing conditions. Resilience is an emerging concept that offers to reconcile considerations of performance under dynamic environments and across multiple time frames by supplementing traditionally static system performance measures to consider behaviors under changing conditions and complex interactions among physical, information and human domains. This paper identifies metrics useful to implement guidance for energy-related planning, design, investment, and operation. Recommendations are presented using a matrix format to provide a structured and comprehensive framework of metrics relevant to a system's energy resilience. The study synthesizes previously proposed metrics and emergent resilience literature to provide a multi-dimensional model intended for use by leaders and practitioners as they transform our energy posture from one of stasis and reaction to one that is proactive and which fosters sustainable growth. - Highlights: • Resilience is the ability of a system to recover from adversity. • There is a need for methods to quantify and measure system resilience. • We developed a matrix-based approach to generate energy resilience metrics. • These metrics can be used in energy planning, system design, and operations

Balanced metrics for vector bundles and polarised manifolds

DEFF Research Database (Denmark)