WorldWideScience

Sample records for riemannian geometry survey

  1. Topics in Riemannian geometry

    International Nuclear Information System (INIS)

    Ezin, J.P.

    1988-08-01

    The lectures given at the ''5th Symposium of Mathematics in Abidjan: Differential Geometry and Mechanics'' are presented. They are divided into four chapters: Riemannian metric on a differential manifold, curvature tensor fields on a Riemannian manifold, some classical functionals on Riemannian manifolds and questions. 11 refs

  2. Geometric control theory and sub-Riemannian geometry

    CERN Document Server

    Boscain, Ugo; Gauthier, Jean-Paul; Sarychev, Andrey; Sigalotti, Mario

    2014-01-01

    This volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as  sub-Riemannian, Finslerian  geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods  has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group  of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume.

  3. Riemannian geometry

    CERN Document Server

    Petersen, Peter

    2016-01-01

    Intended for a one year course, this text serves as a single source, introducing readers to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialize in Riemannian geometry. This is one of the few Works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory. The book will appeal to a readership that have a basic knowledge of standard manifold theory, including tensors, forms, and Lie groups. Important revisions to the third edition include: a substantial addition of unique and enriching exercises scattered throughout the text; inclusion of an increased number of coordinate calculations of connection and curvature; addition of general formulas for curvature on Lie Groups and submersions; integration of variational calculus into the text allowing for an early treatment of the Sphere theorem using a proof by Berger; incorporation of several recent results about manifolds with posit...

  4. Comparison theorems in Riemannian geometry

    CERN Document Server

    Cheeger, Jeff

    2008-01-01

    The central theme of this book is the interaction between the curvature of a complete Riemannian manifold and its topology and global geometry. The first five chapters are preparatory in nature. They begin with a very concise introduction to Riemannian geometry, followed by an exposition of Toponogov's theorem-the first such treatment in a book in English. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. A quick chapter of Morse theory is followed by one on the injectivity radius. Chapters 6-9 deal with many of the most re

  5. Riemannian geometry during the second half of the twentieth century

    CERN Document Server

    Berger, Marcel

    1999-01-01

    In the last fifty years of the twentieth century Riemannian geometry has exploded with activity. Berger marks the start of this period with Rauch's pioneering paper of 1951, which contains the first real pinching theorem and an amazing leap in the depth of the connection between geometry and topology. Since then, the field has become so rich that it is almost impossible for the uninitiated to find their way through it. Textbooks on the subject invariably must choose a particular approach, thus narrowing the path. In this book, Berger provides a truly remarkable survey of the main developments in Riemannian geometry in the last fifty years, focusing his main attention on the following five areas: Curvature and topology; the construction of and the classification of space forms; distinguished metrics, especially Einstein metrics; eigenvalues and eigenfunctions of the Laplacian; the study of periodic geodesics and the geodesic flow. Other topics are treated in less detail in a separate section. Berger's survey p...

  6. Needle decompositions in Riemannian geometry

    CERN Document Server

    Klartag, Bo'az

    2017-01-01

    The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.

  7. Riemannian geometry in an orthogonal frame

    CERN Document Server

    Cartan, Elie Joseph

    2001-01-01

    Foreword by S S Chern. In 1926-27, Cartan gave a series of lectures in which he introduced exterior forms at the very beginning and used extensively orthogonal frames throughout to investigate the geometry of Riemannian manifolds. In this course he solved a series of problems in Euclidean and non-Euclidean spaces, as well as a series of variational problems on geodesics. In 1960, Sergei P Finikov translated from French into Russian his notes of these Cartan's lectures and published them as a book entitled Riemannian Geometry in an Orthogonal Frame. This book has many innovations, such as the n

  8. Sub-Riemannian geometry and optimal transport

    CERN Document Server

    Rifford, Ludovic

    2014-01-01

    The book provides an introduction to sub-Riemannian geometry and optimal transport and presents some of the recent progress in these two fields. The text is completely self-contained: the linear discussion, containing all the proofs of the stated results, leads the reader step by step from the notion of distribution at the very beginning to the existence of optimal transport maps for Lipschitz sub-Riemannian structure. The combination of geometry presented from an analytic point of view and of optimal transport, makes the book interesting for a very large community. This set of notes grew from a series of lectures given by the author during a CIMPA school in Beirut, Lebanon.

  9. Dynamic graphs, community detection, and Riemannian geometry

    Energy Technology Data Exchange (ETDEWEB)

    Bakker, Craig; Halappanavar, Mahantesh; Visweswara Sathanur, Arun

    2018-03-29

    A community is a subset of a wider network where the members of that subset are more strongly connected to each other than they are to the rest of the network. In this paper, we consider the problem of identifying and tracking communities in graphs that change over time {dynamic community detection} and present a framework based on Riemannian geometry to aid in this task. Our framework currently supports several important operations such as interpolating between and averaging over graph snapshots. We compare these Riemannian methods with entry-wise linear interpolation and that the Riemannian methods are generally better suited to dynamic community detection. Next steps with the Riemannian framework include developing higher-order interpolation methods (e.g. the analogues of polynomial and spline interpolation) and a Riemannian least-squares regression method for working with noisy data.

  10. Conformal, Riemannian and Lagrangian geometry the 2000 Barrett lectures

    CERN Document Server

    Chang, Sun-Yung A; Grove, Karsten; Yang, Paul C; Freire, Alexandre

    2002-01-01

    Recent developments in topology and analysis have led to the creation of new lines of investigation in differential geometry. The 2000 Barrett Lectures present the background, context and main techniques of three such lines by means of surveys by leading researchers. The first chapter (by Alice Chang and Paul Yang) introduces new classes of conformal geometric invariants, and then applies powerful techniques in nonlinear differential equations to derive results on compactifications of manifolds and on Yamabe-type variational problems for these invariants. This is followed by Karsten Grove's lectures, which focus on the use of isometric group actions and metric geometry techniques to understand new examples and classification results in Riemannian geometry, especially in connection with positive curvature. The chapter written by Jon Wolfson introduces the emerging field of Lagrangian variational problems, which blends in novel ways the structures of symplectic geometry and the techniques of the modern calculus...

  11. Riemannian geometry and geometric analysis

    CERN Document Server

    Jost, Jürgen

    2017-01-01

    This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry. It offers insight into a wide range of topics, including fundamental concepts of Riemannian geometry, such as geodesics, connections and curvature; the basic models and tools of geometric analysis, such as harmonic functions, forms, mappings, eigenvalues, the Dirac operator and the heat flow method; as well as the most important variational principles of theoretical physics, such as Yang-Mills, Ginzburg-Landau or the nonlinear sigma model of quantum field theory. The present volume connects all these topics in a systematic geometric framework. At the same time, it equips the reader with the working tools of the field and enables her or him to delve into geometric research.  The 7th edition has been systematically reorganized and updated. Almost no page has been left unchanged. It also includes new material, for instance on symplectic geometry, as well as the B...

  12. The Riemannian geometry is not sufficient for the geometrization of the Maxwell's equations

    Science.gov (United States)

    Kulyabov, Dmitry S.; Korolkova, Anna V.; Velieva, Tatyana R.

    2018-04-01

    The transformation optics uses geometrized Maxwell's constitutive equations to solve the inverse problem of optics, namely to solve the problem of finding the parameters of the medium along the paths of propagation of the electromagnetic field. For the geometrization of Maxwell's constitutive equations, the quadratic Riemannian geometry is usually used. This is due to the use of the approaches of the general relativity. However, there arises the question of the insufficiency of the Riemannian structure for describing the constitutive tensor of the Maxwell's equations. The authors analyze the structure of the constitutive tensor and correlate it with the structure of the metric tensor of Riemannian geometry. It is concluded that the use of the quadratic metric for the geometrization of Maxwell's equations is insufficient, since the number of components of the metric tensor is less than the number of components of the constitutive tensor. A possible solution to this problem may be a transition to Finslerian geometry, in particular, the use of the Berwald-Moor metric to establish the structural correspondence between the field tensors of the electromagnetic field.

  13. 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

  14. 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.

  15. Quantum Riemannian geometry of phase space and nonassociativity

    Directory of Open Access Journals (Sweden)

    Beggs Edwin J.

    2017-04-01

    Full Text Available Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.

  16. 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.

  17. A Novel Riemannian Metric Based on Riemannian Structure and Scaling Information for Fixed Low-Rank Matrix Completion.

    Science.gov (United States)

    Mao, Shasha; Xiong, Lin; Jiao, Licheng; Feng, Tian; Yeung, Sai-Kit

    2017-05-01

    Riemannian optimization has been widely used to deal with the fixed low-rank matrix completion problem, and Riemannian metric is a crucial factor of obtaining the search direction in Riemannian optimization. This paper proposes a new Riemannian metric via simultaneously considering the Riemannian geometry structure and the scaling information, which is smoothly varying and invariant along the equivalence class. The proposed metric can make a tradeoff between the Riemannian geometry structure and the scaling information effectively. Essentially, it can be viewed as a generalization of some existing metrics. Based on the proposed Riemanian metric, we also design a Riemannian nonlinear conjugate gradient algorithm, which can efficiently solve the fixed low-rank matrix completion problem. By experimenting on the fixed low-rank matrix completion, collaborative filtering, and image and video recovery, it illustrates that the proposed method is superior to the state-of-the-art methods on the convergence efficiency and the numerical performance.

  18. 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)

  19. 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...

  20. Connections and curvatures on complex Riemannian manifolds

    International Nuclear Information System (INIS)

    Ganchev, G.; Ivanov, S.

    1991-05-01

    Characteristic connection and characteristic holomorphic sectional curvatures are introduced on a complex Riemannian manifold (not necessarily with holomorphic metric). For the class of complex Riemannian manifolds with holomorphic characteristic connection a classification of the manifolds with (pointwise) constant holomorphic characteristic curvature is given. It is shown that the conformal geometry of complex analytic Riemannian manifolds can be naturally developed on the class of locally conformal holomorphic Riemannian manifolds. Complex Riemannian manifolds locally conformal to the complex Euclidean space are characterized with zero conformal fundamental tensor and zero conformal characteristic tensor. (author). 12 refs

  1. Rigid supersymmetry on 5-dimensional Riemannian manifolds and contact geometry

    International Nuclear Information System (INIS)

    Pan, Yiwen

    2014-01-01

    In this note we generalize the methods of http://dx.doi.org/10.1007/JHEP08(2012)141, http://dx.doi.org/10.1007/JHEP01(2013)072 and http://dx.doi.org/10.1007/JHEP05(2013)017 to 5-dimensional Riemannian manifolds M. We study the relations between the geometry of M and the number of solutions to a generalized Killing spinor equation obtained from a 5-dimensional supergravity. The existence of 1 pair of solutions is related to almost contact metric structures. We also discuss special cases related to M=S 1 ×M 4 , which leads to M being foliated by submanifolds with special properties, such as Quaternion-Kähler. When there are 2 pairs of solutions, the closure of the isometry sub-algebra generated by the solutions requires M to be S 3 or T 3 -fibration over a Riemann surface. 4 pairs of solutions pin down the geometry of M to very few possibilities. Finally, we propose a new supersymmetric theory for N=1 vector multiplet on K-contact manifold admitting solutions to the Killing spinor equation

  2. 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...

  3. 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

  4. 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.

  5. 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.)

  6. 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.

  7. 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

  8. The three-body problem and equivariant Riemannian geometry

    Science.gov (United States)

    Alvarez-Ramírez, M.; García, A.; Meléndez, J.; Reyes-Victoria, J. G.

    2017-08-01

    We study the planar three-body problem with 1/r2 potential using the Jacobi-Maupertuis metric, making appropriate reductions by Riemannian submersions. We give a different proof of the Gaussian curvature's sign and the completeness of the space reported by Montgomery [Ergodic Theory Dyn. Syst. 25, 921-947 (2005)]. Moreover, we characterize the geodesics contained in great circles.

  9. Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics

    Energy Technology Data Exchange (ETDEWEB)

    Novello, Mario [Centro Brasileiro de Pesquisas Fisicas (ICRA/CBPF), Rio de Janeiro, RJ (Brazil). Instituto de Cosmologia Relatividade e Astrofisica; Bittencourt, Eduardo, E-mail: eduardo.bittencourt@icranet.org [Physics Department, La Sapienza University of Rome (Italy)

    2015-12-15

    We present an overview of recent developments concerning modifications of the geometry of space-time to describe various physical processes of interactions among classical and quantum configurations. We concentrate in two main lines of research: the Metric Relativity and the Dynamical Bridge. We describe the notion of equivalent (dragged) metric ĝ μ υ which is responsible to map the path of any accelerated body in Minkowski space-time onto a geodesic motion in such associatedĝ geometry. Only recently, the method introduced by Einstein in general relativity was used beyond the domain of gravitational forces to map arbitrary accelerated bodies submitted to non-Newtonian attractions onto geodesics of a modified geometry. This process has its roots in the very ancient idea to treat any dynamical problem in Classical Mechanics as nothing but a problem of static where all forces acting on a body annihilates themselves including the inertial ones. This general procedure, that concerns arbitrary forces - beyond the uses of General Relativity that is limited only to gravitational processes - is nothing but the relativistic version of the d'Alembert method in classical mechanics and consists in the principle of Metric Relativity. The main difference between gravitational interaction and all other forces concerns the universality of gravity which added to the interpretation of the equivalence principle allows all associated geometries-one for each different body in the case of non-gravitational forces-to be unified into a unique Riemannian space-time structure. The same geometrical description appears for electromagnetic waves in the optical limit within the context of nonlinear theories or material medium. Once it is largely discussed in the literature, the so-called analogue models of gravity, we will dedicate few sections on this emphasizing their relation with the new concepts introduced here. Then, we pass to the description of the Dynamical Bridge formalism

  10. 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.

  11. 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.

  12. 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.)

  13. 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

  14. 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.

  15. 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

  16. 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.

  17. 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 ...

  18. 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...

  19. 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.

  20. 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.

  1. 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.

  2. 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...

  3. 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...

  4. 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.

  5. 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 ...

  6. 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...

  7. 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.

  8. 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.

  9. 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.

  10. 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...

  11. 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.)

  12. 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 ...

  13. 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.

  14. 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.)

  15. 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...

  16. 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...

  17. 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.

  18. 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

  19. 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 ...

  20. 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...

  1. 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...

  2. 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...

  3. 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)

  4. 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.

  5. 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)

  6. 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...

  7. 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...

  8. 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

  9. 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.

  10. 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.

  11. 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.

  12. 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

  13. 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

  14. 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)

  15. 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)

  16. 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...

  17. 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

  18. 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.

  19. 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.

  20. 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

  1. 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

  2. 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.

  3. Special metrics and group actions in geometry

    CERN Document Server

    Fino, Anna; Musso, Emilio; Podestà, Fabio; Vezzoni, Luigi

    2017-01-01

    The volume is a follow-up to the INdAM meeting “Special metrics and quaternionic geometry” held in Rome in November 2015. It offers a panoramic view of a selection of cutting-edge topics in differential geometry, including 4-manifolds, quaternionic and octonionic geometry, twistor spaces, harmonic maps, spinors, complex and conformal geometry, homogeneous spaces and nilmanifolds, special geometries in dimensions 5–8, gauge theory, symplectic and toric manifolds, exceptional holonomy and integrable systems. The workshop was held in honor of Simon Salamon, a leading international scholar at the forefront of academic research who has made significant contributions to all these subjects. The articles published here represent a compelling testimony to Salamon’s profound and longstanding impact on the mathematical community. Target readership includes graduate students and researchers working in Riemannian and complex geometry, Lie theory and mathematical physics.

  4. 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)

  5. 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 ...

  6. 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

  7. 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.

  8. 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.

  9. 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...

  10. 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...

  11. 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

  12. 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.

  13. 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...

  14. 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.

  15. 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)

  16. 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...

  17. 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...

  18. 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...

  19. 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)

  20. 2d forward modelling of marine CSEM survey geometry for seabed logging

    International Nuclear Information System (INIS)

    Hussain, N.; Noh, M.; Yahya, N.B.

    2011-01-01

    Hydrocarbon reserve exploration in deep water is done by geophysical surveys. Previously seismic geophysical surveys were explicitly used but it has indistinct results for both water and hydrocarbon saturated reservoir. Recent development for the detection of hydrocarbon reservoir in deeper water is Marine Controlled Source Electromagnetic (MCSEM) geophysical survey. MCSEM is sensitive to electrical conductivity of rocks by which it can differentiate between hydrocarbon reservoir and water saturated reservoir. MCSEM survey geometry put vital role and may causes for anomalies in synthetic data. Consequentially MCSEM is sensitive to survey geometry (e.g. source dipping, rotation and speed, receivers' orientation etc) which causes anomalies. The interpretation for delineating subsurface structure from survey data need to well understand the effects of survey geometry anomalies. Forward modelling is an alternative rather real time survey to study the aforementioned anomalies. In this paper finite difference method (FDM) is implemented for 2D forward modelling in the sense of qualitative understanding to how induced Electromagnetic (EM) signal changes its overall pattern while interact with physical earth properties. A stratified earth structure is developed and modelled in MatLabTM software to study the behaviour of EM field with physical earth properties. Obtained results of 2D geological models are also discussed in this paper. (author)

  1. 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...

  2. 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

  3. 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

  4. 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.)

  5. Isometric C1-immersions for pairs of Riemannian metrics

    International Nuclear Information System (INIS)

    D'Ambra, Giuseppina; Datta, Mahuya

    2001-08-01

    Let h 1 , h 2 be two Euclidean metrics on R q , and let V be a C ∞ -manifold endowed with two Riemannian metrics g 1 and g 2 . We study the existence of C 1 -immersions f:(V,g 1 ,g 2 )→(R q ,h 1 ,h 2 ) such that f*(h i )=g i for i=1,2. (author)

  6. 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

  7. 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...

  8. 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 ...

  9. 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.

  10. 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

  11. 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

  12. 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

  13. 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.

  14. 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.

  15. 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.

  16. 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)

  17. 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)

  18. 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

  19. 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.

  20. A survey on the geometry of production models in economics

    Directory of Open Access Journals (Sweden)

    Alina-Daniela Vîlcu

    2017-01-01

    Full Text Available In this article we survey selected recent results on the geometry of production models, focussing on the main production functions that are usually analyzed in economics, namely homogeneous, homothetic, quasi-sum and quasi-product production functions.

  1. 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.

  2. 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.)

  3. 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...

  4. 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

  5. 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

  6. 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

  7. 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.

  8. 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.

  9. 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.)

  10. 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

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. 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).

  16. 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.

  17. 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)

  18. 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)

  19. 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.)

  20. 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.)

  1. 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....

  2. 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.

  3. 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 ...

  4. 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

  5. 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.

  6. On organizing principles of discrete differential geometry. Geometry of spheres

    International Nuclear Information System (INIS)

    Bobenko, Alexander I; Suris, Yury B

    2007-01-01

    Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. This survey contains a discussion of the following two fundamental discretization principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem treated here is discretization of curvature-line parametrized surfaces in Lie geometry. Systematic use of the discretization principles leads to a discretization of curvature-line parametrization which unifies circular and conical nets.

  7. 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...

  8. 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.

  9. 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.

  10. 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

  11. 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.

  12. 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.

  13. 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.

  14. Geometry

    CERN Document Server

    Pedoe, Dan

    1988-01-01

    ""A lucid and masterly survey."" - Mathematics Gazette Professor Pedoe is widely known as a fine teacher and a fine geometer. His abilities in both areas are clearly evident in this self-contained, well-written, and lucid introduction to the scope and methods of elementary geometry. It covers the geometry usually included in undergraduate courses in mathematics, except for the theory of convex sets. Based on a course given by the author for several years at the University of Minnesota, the main purpose of the book is to increase geometrical, and therefore mathematical, understanding and to he

  15. 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

  16. 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)

  17. Mathematical and conceptual foundations of 20th-century physics

    International Nuclear Information System (INIS)

    Emch, G.G.

    1984-01-01

    This volume presents a unified mathematical account of the conceptual foundations of 20th-century Physics. Part 1 provides a survey of classical physics divided in separate chapters on mechanics, thermodynamics and statistical mechanics, and electromagnetism. This study provides opportunities to place in perspective the successive advents of calculus, of probability and statistics, of differential and sympletic geometry, and of classical functional analysis. Relativity is presented in part 2 of this book and quantum theory in part 3. The motivation provided by physical problems in the development of mathematical disciplines such as, for instance, pseudo-Riemannian geometries, Hilbert spaces and operator algebras, are emphasized. (H.W.). refs.; figs.; schemes

  18. 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.

  19. 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)

  20. 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

  1. 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.

  2. 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.

  3. 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

  4. 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...

  5. 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.

  6. 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

  7. Local analytic geometry

    CERN Document Server

    Abhyankar, Shreeram Shankar

    1964-01-01

    This book provides, for use in a graduate course or for self-study by graduate students, a well-motivated treatment of several topics, especially the following: (1) algebraic treatment of several complex variables; (2) geometric approach to algebraic geometry via analytic sets; (3) survey of local algebra; (4) survey of sheaf theory. The book has been written in the spirit of Weierstrass. Power series play the dominant role. The treatment, being algebraic, is not restricted to complex numbers, but remains valid over any complete-valued field. This makes it applicable to situations arising from

  8. 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.)

  9. 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.

  10. Connections between algebra, combinatorics, and geometry

    CERN Document Server

    Sather-Wagstaff, Sean

    2014-01-01

    Commutative algebra, combinatorics, and algebraic geometry are thriving areas of mathematical research with a rich history of interaction. Connections Between Algebra, Combinatorics, and Geometry contains lecture notes, along with exercises and solutions, from the Workshop on Connections Between Algebra and Geometry held at the University of Regina from May 29-June 1, 2012. It also contains research and survey papers from academics invited to participate in the companion Special Session on Interactions Between Algebraic Geometry and Commutative Algebra, which was part of the CMS Summer Meeting at the University of Regina held June 2–3, 2012, and the meeting Further Connections Between Algebra and Geometry, which was held at the North Dakota State University, February 23, 2013. This volume highlights three mini-courses in the areas of commutative algebra and algebraic geometry: differential graded commutative algebra, secant varieties, and fat points and symbolic powers. It will serve as a useful resou...

  11. 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)

  12. 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...

  13. Advances in discrete differential geometry

    CERN Document Server

    2016-01-01

    This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. The authors take a closer look at discrete models in differential geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classical dynamical systems with continuous time can hardly be seen. This is the paradigm of structure-preserving discretizations. Current advances in this field are stimulated to a large extent by its relevance for computer graphics and mathematical physics. This book is written by specialists working together on a common research project. It is about differential geometry and dynamical systems, smooth and discrete theories, ...

  14. History of analytic geometry

    CERN Document Server

    Boyer, Carl B

    2012-01-01

    Designed as an integrated survey of the development of analytic geometry, this study presents the concepts and contributions from before the Alexandrian Age through the eras of the great French mathematicians Fermat and Descartes, and on through Newton and Euler to the "Golden Age," from 1789 to 1850.

  15. Geometry, structure and randomness in combinatorics

    CERN Document Server

    Nešetřil, Jaroslav; Pellegrini, Marco

    2014-01-01

    This book collects some surveys on current trends in discrete mathematics and discrete geometry. The areas covered include:  graph representations, structural graphs theory, extremal graph theory, Ramsey theory and constrained satisfaction problems.

  16. 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

  17. Assigned and unassigned distance geometry: applications to biological molecules and nanostructures

    Energy Technology Data Exchange (ETDEWEB)

    Billinge, Simon J. L. [Columbia Univ., New York, NY (United States). Applied Physics and Applied Mathematics; Brookhaven National Lab. (BNL), Upton, NY (United States). X-ray Scattering Group; Duxbury, Phillip M. [Michigan State Univ., East Lansing, MI (United States). Dept. of Physics and Astronomy; Gonçalves, Douglas S. [Univ. Federal de Santa Catarina,; Lavor, Carlile [Univ. of Campinas (UNICAMP), Sao Paulo (Brazil). Dept. of Applied Mathematics (IMECC-UNICAMP); Mucherino, Antonio [Univ. de Rennes, Rennes (France). Institut de Recherche en Informatique et Systemes Aleatoires

    2016-04-04

    Here, considering geometry based on the concept of distance, the results found by Menger and Blumenthal originated a body of knowledge called distance geometry. This survey covers some recent developments for assigned and unassigned distance geometry and focuses on two main applications: determination of three-dimensional conformations of biological molecules and nanostructures.

  18. 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.

  19. 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.

  20. Modular Theory, Non-Commutative Geometry and Quantum Gravity

    Directory of Open Access Journals (Sweden)

    Wicharn Lewkeeratiyutkul

    2010-08-01

    Full Text Available This paper contains the first written exposition of some ideas (announced in a previous survey on an approach to quantum gravity based on Tomita-Takesaki modular theory and A. Connes non-commutative geometry aiming at the reconstruction of spectral geometries from an operational formalism of states and categories of observables in a covariant theory. Care has been taken to provide a coverage of the relevant background on modular theory, its applications in non-commutative geometry and physics and to the detailed discussion of the main foundational issues raised by the proposal.

  1. 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)

  2. Architectural Geometry and Fabrication-Aware Design

    KAUST Repository

    Pottmann, Helmut

    2013-01-01

    . This is the source of numerous research problems many of which fall into the area of Geometric Computing and form part of a recently emerging research area, called "Architectural Geometry". The present paper provides a short survey of research in Architectural

  3. 3D Digital Surveying and Modelling of Cave Geometry: Application to Paleolithic Rock Art.

    Science.gov (United States)

    González-Aguilera, Diego; Muñoz-Nieto, Angel; Gómez-Lahoz, Javier; Herrero-Pascual, Jesus; Gutierrez-Alonso, Gabriel

    2009-01-01

    3D digital surveying and modelling of cave geometry represents a relevant approach for research, management and preservation of our cultural and geological legacy. In this paper, a multi-sensor approach based on a terrestrial laser scanner, a high-resolution digital camera and a total station is presented. Two emblematic caves of Paleolithic human occupation and situated in northern Spain, "Las Caldas" and "Peña de Candamo", have been chosen to put in practise this approach. As a result, an integral and multi-scalable 3D model is generated which may allow other scientists, pre-historians, geologists…, to work on two different levels, integrating different Paleolithic Art datasets: (1) a basic level based on the accurate and metric support provided by the laser scanner; and (2) a advanced level using the range and image-based modelling.

  4. The VSEPR model of molecular geometry

    CERN Document Server

    Gillespie, Ronald J

    2012-01-01

    Valence Shell Electron Pair Repulsion (VSEPR) theory is a simple technique for predicting the geometry of atomic centers in small molecules and molecular ions. This authoritative reference was written by Istvan Hartiggai and the developer of VSEPR theory, Ronald J. Gillespie. In addition to its value as a text for courses in molecular geometry and chemistry, it constitutes a classic reference for professionals.Starting with coverage of the broader aspects of VSEPR, this volume narrows its focus to a succinct survey of the methods of structural determination. Additional topics include the appli

  5. 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.

  6. 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.

  7. 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.

  8. 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”.

  9. 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

  10. Computational geometry lectures at the morningside center of mathematics

    CERN Document Server

    Wang, Ren-Hong

    2003-01-01

    Computational geometry is a borderline subject related to pure and applied mathematics, computer science, and engineering. The book contains articles on various topics in computational geometry, which are based on invited lectures and some contributed papers presented by researchers working during the program on Computational Geometry at the Morningside Center of Mathematics of the Chinese Academy of Science. The opening article by R.-H. Wang gives a nice survey of various aspects of computational geometry, many of which are discussed in more detail in other papers in the volume. The topics include problems of optimal triangulation, splines, data interpolation, problems of curve and surface design, problems of shape control, quantum teleportation, and others.

  11. 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.

  12. 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)

  13. 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...

  14. 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.

  15. 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.

  16. 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.

  17. Architectural Geometry and Fabrication-Aware Design

    KAUST Repository

    Pottmann, Helmut

    2013-04-27

    Freeform shapes and structures with a high geometric complexity play an increasingly important role in contemporary architecture. While digital models are easily created, the actual fabrication and construction remains a challenge. This is the source of numerous research problems many of which fall into the area of Geometric Computing and form part of a recently emerging research area, called "Architectural Geometry". The present paper provides a short survey of research in Architectural Geometry and shows how this field moves towards a new direction in Geometric Modeling which aims at combining shape design with important aspects of function and fabrication. © 2013 Kim Williams Books, Turin.

  18. 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.

  19. 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.)

  20. 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.

  1. 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

  2. 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

  3. 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.

  4. Conference on Strings, Duality, and Geometry

    CERN Document Server

    Phong, Duong; Yau, Shing-Tung; Mirror Symmetry IV

    2002-01-01

    This book presents contributions of participants of a workshop held at the Centre de Recherches Mathématiques (CRM), University of Montréal. It can be viewed as a sequel to Mirror Symmetry I (1998), Mirror Symmetry II (1996), and Mirror Symmetry III (1999), copublished by the AMS and International Press. The volume presents a broad survey of many of the noteworthy developments that have taken place in string theory, geometry, and duality since the mid 1990s. Some of the topics emphasized include the following: Integrable models and supersymmetric gauge theories; theory of M- and D-branes and noncommutative geometry; duality between strings and gauge theories; and elliptic genera and automorphic forms. Several introductory articles present an overview of the geometric and physical aspects of mirror symmetry and of corresponding developments in symplectic geometry. The book provides an efficient way for a very broad audience of mathematicians and physicists to explore the frontiers of research into this rapi...

  5. 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.

  6. 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

  7. A Short Description of Electromagnetism Using the Finsler Geometry

    Directory of Open Access Journals (Sweden)

    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.

  8. 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.

  9. 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...

  10. 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.

  11. From geometry to topology

    CERN Document Server

    Flegg, H Graham

    2001-01-01

    This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4-12 give a largely intuitive presentation of selected topics.

  12. 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

  13. 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.

  14. Lie groups, differential equations, and geometry advances and surveys

    CERN Document Server

    2017-01-01

    This book collects a series of contributions addressing the various contexts in which the theory of Lie groups is applied. A preliminary chapter serves the reader both as a basic reference source and as an ongoing thread that runs through the subsequent chapters. From representation theory and Gerstenhaber algebras to control theory, from differential equations to Finsler geometry and Lepage manifolds, the book introduces young researchers in Mathematics to a wealth of different topics, encouraging a multidisciplinary approach to research. As such, it is suitable for students in doctoral courses, and will also benefit researchers who want to expand their field of interest.

  15. 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...

  16. Foliation theory in algebraic geometry

    CERN Document Server

    McKernan, James; Pereira, Jorge

    2016-01-01

    Featuring a blend of original research papers and comprehensive surveys from an international team of leading researchers in the thriving fields of foliation theory, holomorphic foliations, and birational geometry, this book presents the proceedings of the conference "Foliation Theory in Algebraic Geometry," hosted by the Simons Foundation in New York City in September 2013.  Topics covered include: Fano and del Pezzo foliations; the cone theorem and rank one foliations; the structure of symmetric differentials on a smooth complex surface and a local structure theorem for closed symmetric differentials of rank two; an overview of lifting symmetric differentials from varieties with canonical singularities and the applications to the classification of AT bundles on singular varieties; an overview of the powerful theory of the variety of minimal rational tangents introduced by Hwang and Mok; recent examples of varieties which are hyperbolic and yet the Green-Griffiths locus is the whole of X; and a classificati...

  17. 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, ...

  18. 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)

  19. Digital Geometry Algorithms Theoretical Foundations and Applications to Computational Imaging

    CERN Document Server

    Barneva, Reneta

    2012-01-01

    Digital geometry emerged as an independent discipline in the second half of the last century. It deals with geometric properties of digital objects and is developed with the unambiguous goal to provide rigorous theoretical foundations for devising new advanced approaches and algorithms for various problems of visual computing. Different aspects of digital geometry have been addressed in the literature. This book is the first one that explicitly focuses on the presentation of the most important digital geometry algorithms. Each chapter provides a brief survey on a major research area related to the general volume theme, description and analysis of related fundamental algorithms, as well as new original contributions by the authors. Every chapter contains a section in which interesting open problems are addressed.

  20. 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.

  1. 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.

  2. 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)

  3. 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.

  4. 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 ...

  5. 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.

  6. Fourth International Meeting on Gravitation and Cosmology

    CERN Document Server

    Aguilar, José; Barrera, Luz; Accelerated Cosmic Expansion

    2014-01-01

    This volume provides both an update and a review of the state of alternative theories of gravity, in connection with the issue of the accelerated expansion of the universe. Different theoretical proposals explain the acceleration in cosmic expansion, generating the dark energy issue and opening the possibilities of alternative theories of gravity (besides general relativity). Related issues, such as the problem of dark matter, are also surveyed in order to give the readers profound insight on the subject from different points of view. Comprised of short talks and plenary lectures given by leading experts in the field, some of them with brilliant and historic contributions, this book allows the reader to find referenced surveys in topics like f(R) theories, the dark matter and dark energy issues, Modified Newtonian Dynamics (MOND) scenarios, f(T) theories, scalar-tensor theories derived from non-Riemannian geometries, emergent universes, the cosmological constant and other topics of current interest for physic...

  7. Tensorial spacetime geometries and background-independent quantum field theory

    International Nuclear Information System (INIS)

    Raetzel, Dennis

    2012-01-01

    Famously, Einstein read off the geometry of spacetime from Maxwell's equations. Today, we take this geometry that serious that our fundamental theory of matter, the standard model of particle physics, is based on it. However, it seems that there is a gap in our understanding if it comes to the physics outside of the solar system. Independent surveys show that we need concepts like dark matter and dark energy to make our models fit with the observations. But these concepts do not fit in the standard model of particle physics. To overcome this problem, at least, we have to be open to matter fields with kinematics and dynamics beyond the standard model. But these matter fields might then very well correspond to different spacetime geometries. This is the basis of this thesis: it studies the underlying spacetime geometries and ventures into the quantization of those matter fields independently of any background geometry. In the first part of this thesis, conditions are identified that a general tensorial geometry must fulfill to serve as a viable spacetime structure. Kinematics of massless and massive point particles on such geometries are introduced and the physical implications are investigated. Additionally, field equations for massive matter fields are constructed like for example a modified Dirac equation. In the second part, a background independent formulation of quantum field theory, the general boundary formulation, is reviewed. The general boundary formulation is then applied to the Unruh effect as a testing ground and first attempts are made to quantize massive matter fields on tensorial spacetimes.

  8. 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

  9. Experimental and Theoretical Methods in Algebra, Geometry and Topology

    CERN Document Server

    Veys, Willem; Bridging Algebra, Geometry, and Topology

    2014-01-01

    Algebra, geometry and topology cover a variety of different, but intimately related research fields in modern mathematics. This book focuses on specific aspects of this interaction. The present volume contains refereed papers which were presented at the International Conference “Experimental and Theoretical Methods in Algebra, Geometry and Topology”, held in Eforie Nord (near Constanta), Romania, during 20-25 June 2013. The conference was devoted to the 60th anniversary of the distinguished Romanian mathematicians Alexandru Dimca and Ştefan Papadima. The selected papers consist of original research work and a survey paper. They are intended for a large audience, including researchers and graduate students interested in algebraic geometry, combinatorics, topology, hyperplane arrangements and commutative algebra. The papers are written by well-known experts from different fields of mathematics, affiliated to universities from all over the word, they cover a broad range of topics and explore the research f...

  10. 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.

  11. 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 ...

  12. 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.

  13. 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.

  14. Conference on Complex Geometry and Mirror Symmetry

    CERN Document Server

    Vinet, Luc; Yau, Shing-Tung; Mirror Symmetry III

    1999-01-01

    This book presents surveys from a workshop held during the theme year in geometry and topology at the Centre de recherches mathématiques (CRM, University of Montréal). The volume is in some sense a sequel to Mirror Symmetry I (1998) and Mirror Symmetry II (1996), copublished by the AMS and International Press. Included are recent developments in the theory of mirror manifolds and the related areas of complex and symplectic geometry. The long introductory articles explain the key physical ideas and motivation, namely conformal field theory, supersymmetry, and string theory. Open problems are emphasized. Thus the book provides an efficient way for a very broad audience of mathematicians and physicists to reach the frontier of research in this fast expanding area. - See more at: http://bookstore.ams.org/amsip-10#sthash.DbxEFJDx.dpuf

  15. A study in derived algebraic geometry volume I : correspondences and duality

    CERN Document Server

    Gaitsgory, Dennis

    2017-01-01

    Derived algebraic geometry is a far-reaching generalization of algebraic geometry. It has found numerous applications in various parts of mathematics, most prominently in representation theory. This volume develops the theory of ind-coherent sheaves in the context of derived algebraic geometry. Ind-coherent sheaves are a "renormalization" of quasi-coherent sheaves and provide a natural setting for Grothendieck-Serre duality as well as geometric incarnations of numerous categories of interest in representation theory. This volume consists of three parts and an appendix. The first part is a survey of homotopical algebra in the setting of \\infty-categories and the basics of derived algebraic geometry. The second part builds the theory of ind-coherent sheaves as a functor out of the category of correspondences and studies the relationship between ind-coherent and quasi-coherent sheaves. The third part sets up the general machinery of the \\mathrm{(}\\infty, 2\\mathrm{)}-category of correspondences needed for the sec...

  16. Slab2 - Updated Subduction Zone Geometries and Modeling Tools

    Science.gov (United States)

    Moore, G.; Hayes, G. P.; Portner, D. E.; Furtney, M.; Flamme, H. E.; Hearne, M. G.

    2017-12-01

    The U.S. Geological Survey database of global subduction zone geometries (Slab1.0), is a highly utilized dataset that has been applied to a wide range of geophysical problems. In 2017, these models have been improved and expanded upon as part of the Slab2 modeling effort. With a new data driven approach that can be applied to a broader range of tectonic settings and geophysical data sets, we have generated a model set that will serve as a more comprehensive, reliable, and reproducible resource for three-dimensional slab geometries at all of the world's convergent margins. The newly developed framework of Slab2 is guided by: (1) a large integrated dataset, consisting of a variety of geophysical sources (e.g., earthquake hypocenters, moment tensors, active-source seismic survey images of the shallow slab, tomography models, receiver functions, bathymetry, trench ages, and sediment thickness information); (2) a dynamic filtering scheme aimed at constraining incorporated seismicity to only slab related events; (3) a 3-D data interpolation approach which captures both high resolution shallow geometries and instances of slab rollback and overlap at depth; and (4) an algorithm which incorporates uncertainties of contributing datasets to identify the most probable surface depth over the extent of each subduction zone. Further layers will also be added to the base geometry dataset, such as historic moment release, earthquake tectonic providence, and interface coupling. Along with access to several queryable data formats, all components have been wrapped into an open source library in Python, such that suites of updated models can be released as further data becomes available. This presentation will discuss the extent of Slab2 development, as well as the current availability of the model and modeling tools.

  17. On the geometry of Riemannian manifolds with a Lie structure at infinity

    Directory of Open Access Journals (Sweden)

    Bernd Ammann

    2004-01-01

    Full Text Available We study a generalization of the geodesic spray and give conditions for noncomapct manifolds with a Lie structure at infinity to have positive injectivity radius. We also prove that the geometric operators are generated by the given Lie algebra of vector fields. This is the first one in a series of papers devoted to the study of the analysis of geometric differential operators on manifolds with Lie structure at infinity.

  18. Arithmetic noncommutative geometry

    CERN Document Server

    Marcolli, Matilde

    2005-01-01

    Arithmetic noncommutative geometry denotes the use of ideas and tools from the field of noncommutative geometry, to address questions and reinterpret in a new perspective results and constructions from number theory and arithmetic algebraic geometry. This general philosophy is applied to the geometry and arithmetic of modular curves and to the fibers at archimedean places of arithmetic surfaces and varieties. The main reason why noncommutative geometry can be expected to say something about topics of arithmetic interest lies in the fact that it provides the right framework in which the tools of geometry continue to make sense on spaces that are very singular and apparently very far from the world of algebraic varieties. This provides a way of refining the boundary structure of certain classes of spaces that arise in the context of arithmetic geometry, such as moduli spaces (of which modular curves are the simplest case) or arithmetic varieties (completed by suitable "fibers at infinity"), by adding boundaries...

  19. Higher geometry an introduction to advanced methods in analytic geometry

    CERN Document Server

    Woods, Frederick S

    2005-01-01

    For students of mathematics with a sound background in analytic geometry and some knowledge of determinants, this volume has long been among the best available expositions of advanced work on projective and algebraic geometry. Developed from Professor Woods' lectures at the Massachusetts Institute of Technology, it bridges the gap between intermediate studies in the field and highly specialized works.With exceptional thoroughness, it presents the most important general concepts and methods of advanced algebraic geometry (as distinguished from differential geometry). It offers a thorough study

  20. CIMPA Summer School on Arithmetic and Geometry Around Hypergeometric Functions

    CERN Document Server

    Uludağ, A; Yoshida, Masaaki; Arithmetic and Geometry Around Hypergeometric Functions

    2007-01-01

    This volume comprises the Lecture Notes of the CIMPA Summer School "Arithmetic and Geometry around Hypergeometric Functions" held at Galatasaray University, Istanbul in 2005. It contains lecture notes, a survey article, research articles, and the results of a problem session. Key topics are moduli spaces of points on P1 and Picard-Terada-Deligne-Mostow theory, moduli spaces of K3 surfaces, complex hyperbolic geometry, ball quotients, GKZ hypergeometric structures, Hilbert and Picard modular surfaces, uniformizations of complex orbifolds, algebraicity of values of Schwartz triangle functions, and Thakur's hypergeometric function. The book provides a background, gives detailed expositions and indicates new research directions. It is directed to postgraduate students and researchers.

  1. Non-holonomic dynamics and Poisson geometry

    International Nuclear Information System (INIS)

    Borisov, A V; Mamaev, I S; Tsiganov, A V

    2014-01-01

    This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie-Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them. Bibliography: 95 titles

  2. Gravitation and relativity

    CERN Document Server

    Hoffmann, William F

    1964-01-01

    Remarks on the observational basis of general relativity ; Riemannian geometry ; gravitation as geometry ; gravitational waves ; Mach's principle and experiments on mass anisotropy ; the many faces of Mach ; the significance for the solar system of time-varying gravitation ; relativity principles and the role of coordinates in physics ; the superdense star and the critical nucleon number ; gravitation and light ; possible effects on the solar system of φ waves if they exist ; the Lyttleton-Bondi universe and charge equality ; quantization of general relativity ; Mach's principle as boundary condition for Einstein's equations.

  3. The Geometry Conference

    CERN Document Server

    Bárány, Imre; Vilcu, Costin

    2016-01-01

    This volume presents easy-to-understand yet surprising properties obtained using topological, geometric and graph theoretic tools in the areas covered by the Geometry Conference that took place in Mulhouse, France from September 7–11, 2014 in honour of Tudor Zamfirescu on the occasion of his 70th anniversary. The contributions address subjects in convexity and discrete geometry, in distance geometry or with geometrical flavor in combinatorics, graph theory or non-linear analysis. Written by top experts, these papers highlight the close connections between these fields, as well as ties to other domains of geometry and their reciprocal influence. They offer an overview on recent developments in geometry and its border with discrete mathematics, and provide answers to several open questions. The volume addresses a large audience in mathematics, including researchers and graduate students interested in geometry and geometrical problems.

  4. Hyperbolic geometry

    CERN Document Server

    Iversen, Birger

    1992-01-01

    Although it arose from purely theoretical considerations of the underlying axioms of geometry, the work of Einstein and Dirac has demonstrated that hyperbolic geometry is a fundamental aspect of modern physics

  5. Geometry of the Universe

    International Nuclear Information System (INIS)

    Gurevich, L.Eh.; Gliner, Eh.B.

    1978-01-01

    Problems of investigating the Universe space-time geometry are described on a popular level. Immediate space-time geometries, corresponding to three cosmologic models are considered. Space-time geometry of a closed model is the spherical Riemann geonetry, of an open model - is the Lobachevskij geometry; and of a plane model - is the Euclidean geometry. The Universe real geometry in the contemporary epoch of development is based on the data testifying to the fact that the Universe is infinitely expanding

  6. TT detector description and implementation of the survey measurements

    CERN Document Server

    Salzmann, C

    2008-01-01

    The TT geometry in the software has been updated to comply with the latest technical drawings. The main difference is in the description of the beam pipe insulation, where the amount of material has increased from $7.5\\%$ to $15.4\\%$ of $X_0$. Mother volumes are added to decrease the CPU consumption and finally several scans are made to compare the material budget between the DC06 geometry and the new 2008 geometry. In addition, the survey measurements of the TT detector have been analysed. These measurements can be subdivided into surveys of the detector box, photogrammetry of the balconies and metrology of the half-modules. The offsets with the nominal geometry are implemented in the alignment condition database.

  7. Sasakian manifolds and M-theory

    International Nuclear Information System (INIS)

    Figueroa-O’Farrill, José; Santi, Andrea

    2016-01-01

    We extend the link between Einstein Sasakian manifolds and Killing spinors to a class of η-Einstein Sasakian manifolds, both in Riemannian and Lorentzian settings, characterizing them in terms of generalized Killing spinors. We propose a definition of supersymmetric M-theory backgrounds on such a geometry and find a new class of such backgrounds, extending previous work of Haupt, Lukas and Stelle. (paper)

  8. Integrated approach to 3-D seismic acquisition geometry analysis : Emphasizing the influence of the inhomogeneous subsurface

    NARCIS (Netherlands)

    van Veldhuizen, E.J.

    2006-01-01

    The seismic reflection method for imaging of the earth's interior is an essential part of the exploration and exploitation of hydrocarbon resources. A seismic survey should be designed such that the acquired data leads to a sufficiently accurate subsurface image. The survey geometry analysis method

  9. Researches of real observation geometry in monitoring fuel-containing materials' subcriticality

    International Nuclear Information System (INIS)

    Vysotskij, E.D.; Shevchenko, V.G.; Shevchenko, M.V.

    2002-01-01

    The effectiveness of fuel-containing materials monitoring is discussed in the part related to the feasibilities of researches and realization of optimal geometry (detectors - source) of survey of neutron activity dynamics in nuclearly hazardous areas with clusters of fuel-containing materials concentrated in the premises 305/2

  10. Geometry and its applications

    CERN Document Server

    Meyer, Walter J

    2006-01-01

    Meyer''s Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters. The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry''s usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers.* Realistic applications integrated throughout the text, including (but not limited to): - Symmetries of artistic patterns- Physics- Robotics- Computer vision- Computer graphics- Stability of architectural structures- Molecular biology- Medicine- Pattern recognition* Historical notes included in many chapters...

  11. Beautiful geometry

    CERN Document Server

    Maor, Eli

    2014-01-01

    If you've ever thought that mathematics and art don't mix, this stunning visual history of geometry will change your mind. As much a work of art as a book about mathematics, Beautiful Geometry presents more than sixty exquisite color plates illustrating a wide range of geometric patterns and theorems, accompanied by brief accounts of the fascinating history and people behind each. With artwork by Swiss artist Eugen Jost and text by acclaimed math historian Eli Maor, this unique celebration of geometry covers numerous subjects, from straightedge-and-compass constructions to intriguing configur

  12. Revolutions of Geometry

    CERN Document Server

    O'Leary, Michael

    2010-01-01

    Guides readers through the development of geometry and basic proof writing using a historical approach to the topic. In an effort to fully appreciate the logic and structure of geometric proofs, Revolutions of Geometry places proofs into the context of geometry's history, helping readers to understand that proof writing is crucial to the job of a mathematician. Written for students and educators of mathematics alike, the book guides readers through the rich history and influential works, from ancient times to the present, behind the development of geometry. As a result, readers are successfull

  13. Analogy and Dynamic Geometry System Used to Introduce Three-Dimensional Geometry

    Science.gov (United States)

    Mammana, M. F.; Micale, B.; Pennisi, M.

    2012-01-01

    We present a sequence of classroom activities on Euclidean geometry, both plane and space geometry, used to make three dimensional geometry more catchy and simple. The activity consists of a guided research activity that leads the students to discover unexpected properties of two apparently distant geometrical entities, quadrilaterals and…

  14. Killing spinor equations in dimension 7 and geometry of integrable G2-manifolds

    International Nuclear Information System (INIS)

    Friedrich, Thomas; Ivanov, Stefan

    2001-12-01

    We compute the scalar curvature of 7-dimensional G 2 -manifolds admitting a connection with totally skew-symmetric torsion. We prove the formula for the general solution of the Killing spinor equation and express the Riemannian scalar curvature of the solution in terms of the dilation function and the NS 3-form field. In dimension n=7 the dilation function involved in the second fermionic string equation has an interpretation as a conformal change of the underlying integrable G 2 -structure into a cocalibrated one of pure type W 3 . (author)

  15. Ward identities for conformal models

    International Nuclear Information System (INIS)

    Lazzarini, S.; Stora, R.

    1988-01-01

    Ward identities which express the symmetry of conformal models are treated. Diffeomorphism invariance or locally holomorphic coordinate transformations are used. Diffeomorphism invariance is then understood in terms of Riemannian geometry. Two different sets of Ward identities expressing diffeomorphism invariance in a conformally invariant way are found for the free bosonic string. Using a geometrical argument, the correct invariance for a large class of conformal models is given

  16. Geometry essentials for dummies

    CERN Document Server

    Ryan, Mark

    2011-01-01

    Just the critical concepts you need to score high in geometry This practical, friendly guide focuses on critical concepts taught in a typical geometry course, from the properties of triangles, parallelograms, circles, and cylinders, to the skills and strategies you need to write geometry proofs. Geometry Essentials For Dummies is perfect for cramming or doing homework, or as a reference for parents helping kids study for exams. Get down to the basics - get a handle on the basics of geometry, from lines, segments, and angles, to vertices, altitudes, and diagonals Conque

  17. A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary

    Directory of Open Access Journals (Sweden)

    Stephen M. Paneitz

    2008-03-01

    Full Text Available This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key rôle in conformal differential geometry in dimension 4 and is now known as the Paneitz operator.

  18. Complex analysis and geometry

    CERN Document Server

    Silva, Alessandro

    1993-01-01

    The papers in this wide-ranging collection report on the results of investigations from a number of linked disciplines, including complex algebraic geometry, complex analytic geometry of manifolds and spaces, and complex differential geometry.

  19. 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.

  20. From classical to modern algebraic geometry Corrado Segre's mastership and legacy

    CERN Document Server

    Conte, Alberto; Gatto, Letterio; Giacardi, Livia; Marchisio, Marina; Verra, Alessandro

    2016-01-01

    This book commemorates the 150th birthday of Corrado Segre, one of the founders of the Italian School of Algebraic Geometry and a crucial figure in the history of Algebraic Geometry. It is the outcome of a conference held in Turin, Italy. One of the book's most unique features is the inclusion of a previously unpublished manuscript by Corrado Segre, together with a scientific commentary. Representing a prelude to Segre's seminal 1894 contribution on the theory of algebraic curves, this manuscript and other important archival sources included in the essays shed new light on the eminent role he played at the international level. Including both survey articles and original research papers, the book is divided into three parts: section one focuses on the implications of Segre's work in a historic light, while section two presents new results in his field, namely Algebraic Geometry. The third part features Segre's unpublished notebook: Sulla Geometria Sugli Enti Algebrici Semplicemente Infiniti (1890-1891). This v...

  1. Matrix Information Geometry

    CERN Document Server

    Bhatia, Rajendra

    2013-01-01

    This book is an outcome of the Indo-French Workshop on Matrix Information Geometries (MIG): Applications in Sensor and Cognitive Systems Engineering, which was held in Ecole Polytechnique and Thales Research and Technology Center, Palaiseau, France, in February 23-25, 2011. The workshop was generously funded by the Indo-French Centre for the Promotion of Advanced Research (IFCPAR).  During the event, 22 renowned invited french or indian speakers gave lectures on their areas of expertise within the field of matrix analysis or processing. From these talks, a total of 17 original contribution or state-of-the-art chapters have been assembled in this volume. All articles were thoroughly peer-reviewed and improved, according to the suggestions of the international referees. The 17 contributions presented  are organized in three parts: (1) State-of-the-art surveys & original matrix theory work, (2) Advanced matrix theory for radar processing, and (3) Matrix-based signal processing applications.  

  2. Euclid Mission: Mapping the Geometry of the Dark Universe. Mission and Consortium Status

    Science.gov (United States)

    Rhodes, Jason

    2011-01-01

    Euclid concept: (1) High-precision survey mission to map the geometry of the Dark Universe (2) Optimized for two complementary cosmological probes: (2a) Weak Gravitational Lensing (2b) Baryonic Acoustic Oscillations (2c) Additional probes: clusters, redshift space distortions, ISW (3) Full extragalactic sky survey with 1.2m telescope at L2: (3a) Imaging: (3a-1) High precision imaging at visible wavelengths (3a-2) Photometry/Imaging in the near-infrared (3b) Near Infrared Spectroscopy (4) Synergy with ground based surveys (5) Legacy science for a wide range of in astronomy

  3. Fourth meeting entitled “Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data”

    CERN Document Server

    Vilanova, Anna; Burgeth, Bernhard; Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data

    2014-01-01

    Arising from the fourth Dagstuhl conference entitled Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data (2011), this book offers a broad and vivid view of current work in this emerging field. Topics covered range from applications of the analysis of tensor fields to research on their mathematical and analytical properties. Part I, Tensor Data Visualization, surveys techniques for visualization of tensors and tensor fields in engineering, discusses the current state of the art and challenges, and examines tensor invariants and glyph design, including an overview of common glyphs. The second Part, Representation and Processing of Higher-order Descriptors, describes a matrix representation of local phase, outlines mathematical morphological operations techniques, extended for use in vector images, and generalizes erosion to the space of diffusion weighted MRI. Part III, Higher Order Tensors and Riemannian-Finsler Geometry, offers powerful mathematical language to model and...

  4. Uniqueness of the electrostatic solution in Schwarzschild space

    International Nuclear Information System (INIS)

    Molnar, Pal G.; Elsaesser, Klaus

    2003-01-01

    In this Brief Report we give the proof that the solution of any static test charge distribution in Schwarzschild space is unique. In order to give the proof we derive the first Green's identity written with p-forms on (pseudo) Riemannian manifolds. Moreover, the proof of uniqueness can be shown for either any purely electric or purely magnetic field configuration. The spacetime geometry is not crucial for the proof

  5. Gauge theory and gravitation: an approach to a fiber bundle formalism

    International Nuclear Information System (INIS)

    Mello, L.A. de.

    1986-01-01

    The thesis is composed of two different parts. A formal complete and rigorous mathematical part-of topics of differential manilfolds, exterior calculus, riemannian geometry, principal fiber bundle (p.f.) with connections and linear connections and a second part of application of this mathematical formalism concerning physical theories, particularly the Maxwell eletromagnetism (EM), gauge theory of Yang-Mills (Y-M), the GRT, and the gravitation theory of Einstein-Cartan. (E.C.) [pt

  6. Complex algebraic geometry

    CERN Document Server

    Kollár, János

    1997-01-01

    This volume contains the lectures presented at the third Regional Geometry Institute at Park City in 1993. The lectures provide an introduction to the subject, complex algebraic geometry, making the book suitable as a text for second- and third-year graduate students. The book deals with topics in algebraic geometry where one can reach the level of current research while starting with the basics. Topics covered include the theory of surfaces from the viewpoint of recent higher-dimensional developments, providing an excellent introduction to more advanced topics such as the minimal model program. Also included is an introduction to Hodge theory and intersection homology based on the simple topological ideas of Lefschetz and an overview of the recent interactions between algebraic geometry and theoretical physics, which involve mirror symmetry and string theory.

  7. CMS geometry through 2020

    International Nuclear Information System (INIS)

    Osborne, I; Brownson, E; Eulisse, G; Jones, C D; Sexton-Kennedy, E; Lange, D J

    2014-01-01

    CMS faces real challenges with upgrade of the CMS detector through 2020 and beyond. One of the challenges, from the software point of view, is managing upgrade simulations with the same software release as the 2013 scenario. We present the CMS geometry description software model, its integration with the CMS event setup and core software. The CMS geometry configuration and selection is implemented in Python. The tools collect the Python configuration fragments into a script used in CMS workflow. This flexible and automated geometry configuration allows choosing either transient or persistent version of the same scenario and specific version of the same scenario. We describe how the geometries are integrated and validated, and how we define and handle different geometry scenarios in simulation and reconstruction. We discuss how to transparently manage multiple incompatible geometries in the same software release. Several examples are shown based on current implementation assuring consistent choice of scenario conditions. The consequences and implications for multiple/different code algorithms are discussed.

  8. Kinematic algebras, groups for elementary particles, and the geometry of momentum space

    International Nuclear Information System (INIS)

    Izmest'ev, A.A.

    1986-01-01

    It is shown that to each n-dimensional (n≥2) homogeneous isotropic Riemannian momentum (coordinate) space there corresponds a definite kinematic local algebra of operators N/sub a/, M/sub a//sub b/, P/sub a//sub ,/ ω(a,b = 1,2,...,n). In the three-dimensional case this gives the possibility of classifying particles in accordance with the algebras of the types of momentum space. The approach developed also makes it possible to obtain generalized equations describing particles of the different types. The operators under consideration satisfy not only the relevant algebra but also relations independent of the algebra that coincide in form with the Maxwell equations

  9. Analisis Keterampilan Geometri Siswa Dalam Memecahkan Masalah Geometri Berdasarkan Tingkat Berpikir Van Hiele

    OpenAIRE

    Muhassanah, Nuraini; Sujadi, Imam; Riyadi, Riyadi

    2014-01-01

    The objective of this research was to describe the VIII grade students geometry skills atSMP N 16 Surakarta in the level 0 (visualization), level 1 (analysis), and level 2 (informaldeduction) van Hiele level of thinking in solving the geometry problem. This research was aqualitative research in the form of case study analyzing deeply the students geometry skill insolving the geometry problem based on van Hiele level of thingking. The subject of this researchwas nine students of VIII grade at ...

  10. Algorithms in Algebraic Geometry

    CERN Document Server

    Dickenstein, Alicia; Sommese, Andrew J

    2008-01-01

    In the last decade, there has been a burgeoning of activity in the design and implementation of algorithms for algebraic geometric computation. Some of these algorithms were originally designed for abstract algebraic geometry, but now are of interest for use in applications and some of these algorithms were originally designed for applications, but now are of interest for use in abstract algebraic geometry. The workshop on Algorithms in Algebraic Geometry that was held in the framework of the IMA Annual Program Year in Applications of Algebraic Geometry by the Institute for Mathematics and Its

  11. Non-Euclidean geometry

    CERN Document Server

    Kulczycki, Stefan

    2008-01-01

    This accessible approach features two varieties of proofs: stereometric and planimetric, as well as elementary proofs that employ only the simplest properties of the plane. A short history of geometry precedes a systematic exposition of the principles of non-Euclidean geometry.Starting with fundamental assumptions, the author examines the theorems of Hjelmslev, mapping a plane into a circle, the angle of parallelism and area of a polygon, regular polygons, straight lines and planes in space, and the horosphere. Further development of the theory covers hyperbolic functions, the geometry of suff

  12. Efficient three-dimensional reconstruction of aquatic vegetation geometry: Estimating morphological parameters influencing hydrodynamic drag

    Science.gov (United States)

    Liénard, Jean; Lynn, Kendra; Strigul, Nikolay; Norris, Benjamin K.; Gatziolis, Demetrios; Mullarney, Julia C.; Bryan, Karin, R.; Henderson, Stephen M.

    2016-09-01

    Aquatic vegetation can shelter coastlines from energetic waves and tidal currents, sometimes enabling accretion of fine sediments. Simulation of flow and sediment transport within submerged canopies requires quantification of vegetation geometry. However, field surveys used to determine vegetation geometry can be limited by the time required to obtain conventional caliper and ruler measurements. Building on recent progress in photogrammetry and computer vision, we present a method for reconstructing three-dimensional canopy geometry. The method was used to survey a dense canopy of aerial mangrove roots, called pneumatophores, in Vietnam's Mekong River Delta. Photogrammetric estimation of geometry required 1) taking numerous photographs at low tide from multiple viewpoints around 1 m2 quadrats, 2) computing relative camera locations and orientations by triangulation of key features present in multiple images and reconstructing a dense 3D point cloud, and 3) extracting pneumatophore locations and diameters from the point cloud data. Step 3) was accomplished by a new 'sector-slice' algorithm, yielding geometric parameters every 5 mm along a vertical profile. Photogrammetric analysis was compared with manual caliper measurements. In all 5 quadrats considered, agreement was found between manual and photogrammetric estimates of stem number, and of number × mean diameter, which is a key parameter appearing in hydrodynamic models. In two quadrats, pneumatophores were encrusted with numerous barnacles, generating a complex geometry not resolved by hand measurements. In remaining cases, moderate agreement between manual and photogrammetric estimates of stem diameter and solid volume fraction was found. By substantially reducing measurement time in the field while capturing in greater detail the 3D structure, photogrammetry has potential to improve input to hydrodynamic models, particularly for simulations of flow through large-scale, heterogenous canopies.

  13. 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

  14. 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.

  15. 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.

  16. The limit space of a Cauchy sequence of globally hyperbolic spacetimes

    Energy Technology Data Exchange (ETDEWEB)

    Noldus, Johan [Universiteit Gent, Vakgroep Wiskundige analyse, Galglaan 2, 9000 Gent (Belgium)

    2004-02-21

    In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In section 2, I work gradually towards a construction of the limit space. I prove that the limit space is unique up to isometry. I also show that, in general, the limit space has quite complicated causal behaviour. This work prepares the final paper in which I shall study in more detail properties of the limit space and the moduli space of (compact) globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this paper a suitable definition of dimension of a Lorentz space in agreement with the one given by Gromov in the Riemannian case. The difference in philosophy between Lorentzian and Riemannian geometry is one of relativism versus absolutism. In the latter every point distinguishes itself while in the former in general two elements get distinguished by a third, different, one.

  17. The limit space of a Cauchy sequence of globally hyperbolic spacetimes

    International Nuclear Information System (INIS)

    Noldus, Johan

    2004-01-01

    In this second paper, I construct a limit space of a Cauchy sequence of globally hyperbolic spacetimes. In section 2, I work gradually towards a construction of the limit space. I prove that the limit space is unique up to isometry. I also show that, in general, the limit space has quite complicated causal behaviour. This work prepares the final paper in which I shall study in more detail properties of the limit space and the moduli space of (compact) globally hyperbolic spacetimes (cobordisms). As a fait divers, I give in this paper a suitable definition of dimension of a Lorentz space in agreement with the one given by Gromov in the Riemannian case. The difference in philosophy between Lorentzian and Riemannian geometry is one of relativism versus absolutism. In the latter every point distinguishes itself while in the former in general two elements get distinguished by a third, different, one

  18. System Geometries and Transit/Eclipse Probabilities

    Directory of Open Access Journals (Sweden)

    Howard A.

    2011-02-01

    Full Text Available Transiting exoplanets provide access to data to study the mass-radius relation and internal structure of extrasolar planets. Long-period transiting planets allow insight into planetary environments similar to the Solar System where, in contrast to hot Jupiters, planets are not constantly exposed to the intense radiation of their parent stars. Observations of secondary eclipses additionally permit studies of exoplanet temperatures and large-scale exo-atmospheric properties. We show how transit and eclipse probabilities are related to planet-star system geometries, particularly for long-period, eccentric orbits. The resulting target selection and observational strategies represent the principal ingredients of our photometric survey of known radial-velocity planets with the aim of detecting transit signatures (TERMS.

  19. Geometry and billiards

    CERN Document Server

    Tabachnikov, Serge

    2005-01-01

    Mathematical billiards describe the motion of a mass point in a domain with elastic reflections off the boundary or, equivalently, the behavior of rays of light in a domain with ideally reflecting boundary. From the point of view of differential geometry, the billiard flow is the geodesic flow on a manifold with boundary. This book is devoted to billiards in their relation with differential geometry, classical mechanics, and geometrical optics. The topics covered include variational principles of billiard motion, symplectic geometry of rays of light and integral geometry, existence and nonexistence of caustics, optical properties of conics and quadrics and completely integrable billiards, periodic billiard trajectories, polygonal billiards, mechanisms of chaos in billiard dynamics, and the lesser-known subject of dual (or outer) billiards. The book is based on an advanced undergraduate topics course (but contains more material than can be realistically taught in one semester). Although the minimum prerequisit...

  20. 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.

  1. Drawing Dynamic Geometry Figures Online with Natural Language for Junior High School Geometry

    Science.gov (United States)

    Wong, Wing-Kwong; Yin, Sheng-Kai; Yang, Chang-Zhe

    2012-01-01

    This paper presents a tool for drawing dynamic geometric figures by understanding the texts of geometry problems. With the tool, teachers and students can construct dynamic geometric figures on a web page by inputting a geometry problem in natural language. First we need to build the knowledge base for understanding geometry problems. With the…

  2. 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.)

  3. Hopf algebras in noncommutative geometry

    International Nuclear Information System (INIS)

    Varilly, Joseph C.

    2001-10-01

    We give an introductory survey to the use of Hopf algebras in several problems of non- commutative geometry. The main example, the Hopf algebra of rooted trees, is a graded, connected Hopf algebra arising from a universal construction. We show its relation to the algebra of transverse differential operators introduced by Connes and Moscovici in order to compute a local index formula in cyclic cohomology, and to the several Hopf algebras defined by Connes and Kreimer to simplify the combinatorics of perturbative renormalization. We explain how characteristic classes for a Hopf module algebra can be obtained from the cyclic cohomology of the Hopf algebra which acts on it. Finally, we discuss the theory of non- commutative spherical manifolds and show how they arise as homogeneous spaces of certain compact quantum groups. (author)

  4. KEMAJUAN BELAJAR SISWA PADA GEOMETRI TRANSFORMASI MENGGUNAKAN AKTIVITAS REFLEKSI GEOMETRI

    Directory of Open Access Journals (Sweden)

    Irkham Ulil Albab

    2014-10-01

    Full Text Available Abstrak: Penelitian ini bertujuan untuk mendeskripsikan kemajuan belajar siswa pada materi geometri transformasi yang didukung dengan serangkaian aktivitas belajar berdasarkan Pendidikan Matematika Realistik Indonesia. Penelitian didesain melalui tiga tahap, yaitu tahapan perancangan desain awal, pengujian desain melalui pembelajaran awal dan pembelajaran eksperimental, dan tahap analisis retrospektif. Dalam penelitian ini, Hypothetical Learning Trajectory, HLT (HLT berperan penting sebagai desain pembelajaran sekaligus instrumen penelitian. HLT diujikan terhadap 26 siswa kelas VII. Data dikumpulkan dengan teknik wawancara, pengamatan, dan catatan lapangan. Hasil penelitian menunjukkan bahwa desain pembelajaran ini mampu menstimulasi siswa untuk memberikan karakteristik refleksi dan transformasi geometri lainnya secara informal, mengklasifikasikannya dalam transformasi isometri pada level kedua, dan menemukan garis bantuan refleksi pada level yang lebih formal. Selain itu, garis bantuan refleksi digunakan oleh siswa untuk menggambar bayangan refleksi dan pola pencerminan serta memahami bentuk rotasi dan translasi sebagai kombinasi refleksi adalah level tertinggi. Keyword: transformasi geometri, kombinasi refleksi, rotasi, translasi, design research, HLT STUDENTS’ LEARNING PROGRESS ON TRANSFORMATION GEOMETRY USING THE GEOMETRY REFLECTION ACTIVITIES Abstract: This study was aimed at describing the students’ learning progress on transformation geometry supported by a set of learning activities based on Indonesian Realistic Mathematics Education. The study was designed into three stages, that is, the preliminary design stage, the design testing through initial instruction and experiment, and the restrospective analysis stage. In this study, Hypothetical Learning Trajectory (HLT played an important role as an instructional design and a research instrument. HLT was tested to 26 seventh grade students. The data were collected through interviews

  5. Software Geometry in Simulations

    Science.gov (United States)

    Alion, Tyler; Viren, Brett; Junk, Tom

    2015-04-01

    The Long Baseline Neutrino Experiment (LBNE) involves many detectors. The experiment's near detector (ND) facility, may ultimately involve several detectors. The far detector (FD) will be significantly larger than any other Liquid Argon (LAr) detector yet constructed; many prototype detectors are being constructed and studied to motivate a plethora of proposed FD designs. Whether it be a constructed prototype or a proposed ND/FD design, every design must be simulated and analyzed. This presents a considerable challenge to LBNE software experts; each detector geometry must be described to the simulation software in an efficient way which allows for multiple authors to easily collaborate. Furthermore, different geometry versions must be tracked throughout their use. We present a framework called General Geometry Description (GGD), written and developed by LBNE software collaborators for managing software to generate geometries. Though GGD is flexible enough to be used by any experiment working with detectors, we present it's first use in generating Geometry Description Markup Language (GDML) files to interface with LArSoft, a framework of detector simulations, event reconstruction, and data analyses written for all LAr technology users at Fermilab. Brett is the other of the framework discussed here, the General Geometry Description (GGD).

  6. Developments in special geometry

    International Nuclear Information System (INIS)

    Mohaupt, Thomas; Vaughan, Owen

    2012-01-01

    We review the special geometry of N = 2 supersymmetric vector and hypermultiplets with emphasis on recent developments and applications. A new formulation of the local c-map based on the Hesse potential and special real coordinates is presented. Other recent developments include the Euclidean version of special geometry, and generalizations of special geometry to non-supersymmetric theories. As applications we discuss the proof that the local r-map and c-map preserve geodesic completeness, and the construction of four- and five-dimensional static solutions through dimensional reduction over time. The shared features of the real, complex and quaternionic version of special geometry are stressed throughout.

  7. Introduction to General Relativity and Black Holes (5/5)

    CERN Multimedia

    CERN. Geneva

    2001-01-01

    Conceptual foundations of General Relativity (GR). Uniqueness of GR. Mathematical framework: tensor calculus, Riemannian geometry, connection, 'spin' connection, curvature, Cartan's form calculus. Hilbert-Einstein action, Einstein equations. Weak gravitational fields. Post Newtonian Approximation. Gravitanional Waves. Exact solutions. Killing vectors. Experimental tests. Black Holes: extensions of the Schwarzschild solution; Kerr-Newman holes; no-hair theorems; energtics of black holes; the membrane approach; quantum mechanics of black holes; Bekenstein entropy; Hawking temperature; black holes and string theory.

  8. Introduction to General Relativity and Black Holes (3/5)

    CERN Multimedia

    CERN. Geneva

    2001-01-01

    Conceptual foundations of General Relativity (GR). Uniqueness of GR. Mathematical framework: tensor calculus, Riemannian geometry, connection, 'spin' connection, curvature, Cartan's form calculus. Hilbert-Einstein action, Einstein equations. Weak gravitational fields. Post Newtonian Approximation. Gravitanional Waves. Exact solutions. Killing vectors. Experimental tests. Black Holes: extensions of the Schwarzschild solution; Kerr-Newman holes; no-hair theorems; energtics of black holes; the membrane approach; quantum mechanics of black holes; Bekenstein entropy; Hawking temperature; black holes and string theory.

  9. Introduction to General Relativity and Black Holes (1/5)

    CERN Multimedia

    CERN. Geneva

    2001-01-01

    Conceptual foundations of General Relativity (GR). Uniqueness of GR. Mathematical framework: tensor calculus, Riemannian geometry, connection, 'spin' connection, curvature, Cartan's form calculus. Hilbert-Einstein action, Einstein equations. Weak gravitational fields. Post Newtonian Approximation. Gravitanional Waves. Exact solutions. Killing vectors. Experimental tests. Black Holes: extensions of the Schwarzschild solution; Kerr-Newman holes; no-hair theorems; energtics of black holes; the membrane approach; quantum mechanics of black holes; Bekenstein entropy; Hawking temperature; black holes and string theory.

  10. Introduction to General Relativity and Black Holes (2/5)

    CERN Multimedia

    CERN. Geneva

    2001-01-01

    Conceptual foundations of General Relativity (GR). Uniqueness of GR. Mathematical framework: tensor calculus, Riemannian geometry, connection, 'spin' connection, curvature, Cartan's form calculus. Hilbert-Einstein action, Einstein equations. Weak gravitational fields. Post Newtonian Approximation. Gravitanional Waves. Exact solutions. Killing vectors. Experimental tests. Black Holes: extensions of the Schwarzschild solution; Kerr-Newman holes; no-hair theorems; energtics of black holes; the membrane approach; quantum mechanics of black holes; Bekenstein entropy; Hawking temperature; black holes and string theory.

  11. Introduction to General Relativity and Black Holes (4/5)

    CERN Multimedia

    CERN. Geneva

    2001-01-01

    Conceptual foundations of General Relativity (GR). Uniqueness of GR. Mathematical framework: tensor calculus, Riemannian geometry, connection, 'spin' connection, curvature, Cartan's form calculus. Hilbert-Einstein action, Einstein equations. Weak gravitational fields. Post Newtonian Approximation. Gravitanional Waves. Exact solutions. Killing vectors. Experimental tests. Black Holes: extensions of the Schwarzschild solution; Kerr-Newman holes; no-hair theorems; energtics of black holes; the membrane approach; quantum mechanics of black holes; Bekenstein entropy; Hawking temperature; black holes and string theory.

  12. The design of geometry teaching: learning from the geometry textbooks of Godfrey and Siddons

    OpenAIRE

    Fujita, Taro; Jones, Keith

    2002-01-01

    Deciding how to teach geometry remains a demanding task with one of major arguments being about how to combine the intuitive and deductive aspects of geometry into an effective teaching design. In order to try to obtain an insight into tackling this issue, this paper reports an analysis of innovative geometry textbooks which were published in the early part of the 20th Century, a time when significant efforts were being made to improve the teaching and learning of geometry. The analysis sugge...

  13. Sources of hyperbolic geometry

    CERN Document Server

    Stillwell, John

    1996-01-01

    This book presents, for the first time in English, the papers of Beltrami, Klein, and Poincaré that brought hyperbolic geometry into the mainstream of mathematics. A recognition of Beltrami comparable to that given the pioneering works of Bolyai and Lobachevsky seems long overdue-not only because Beltrami rescued hyperbolic geometry from oblivion by proving it to be logically consistent, but because he gave it a concrete meaning (a model) that made hyperbolic geometry part of ordinary mathematics. The models subsequently discovered by Klein and Poincaré brought hyperbolic geometry even further down to earth and paved the way for the current explosion of activity in low-dimensional geometry and topology. By placing the works of these three mathematicians side by side and providing commentaries, this book gives the student, historian, or professional geometer a bird's-eye view of one of the great episodes in mathematics. The unified setting and historical context reveal the insights of Beltrami, Klein, and Po...

  14. The geometry description markup language

    International Nuclear Information System (INIS)

    Chytracek, R.

    2001-01-01

    Currently, a lot of effort is being put on designing complex detectors. A number of simulation and reconstruction frameworks and applications have been developed with the aim to make this job easier. A very important role in this activity is played by the geometry description of the detector apparatus layout and its working environment. However, no real common approach to represent geometry data is available and such data can be found in various forms starting from custom semi-structured text files, source code (C/C++/FORTRAN), to XML and database solutions. The XML (Extensible Markup Language) has proven to provide an interesting approach for describing detector geometries, with several different but incompatible XML-based solutions existing. Therefore, interoperability and geometry data exchange among different frameworks is not possible at present. The author introduces a markup language for geometry descriptions. Its aim is to define a common approach for sharing and exchanging of geometry description data. Its requirements and design have been driven by experience and user feedback from existing projects which have their geometry description in XML

  15. Complex analysis and CR geometry

    CERN Document Server

    Zampieri, Giuseppe

    2008-01-01

    Cauchy-Riemann (CR) geometry is the study of manifolds equipped with a system of CR-type equations. Compared to the early days when the purpose of CR geometry was to supply tools for the analysis of the existence and regularity of solutions to the \\bar\\partial-Neumann problem, it has rapidly acquired a life of its own and has became an important topic in differential geometry and the study of non-linear partial differential equations. A full understanding of modern CR geometry requires knowledge of various topics such as real/complex differential and symplectic geometry, foliation theory, the geometric theory of PDE's, and microlocal analysis. Nowadays, the subject of CR geometry is very rich in results, and the amount of material required to reach competence is daunting to graduate students who wish to learn it. However, the present book does not aim at introducing all the topics of current interest in CR geometry. Instead, an attempt is made to be friendly to the novice by moving, in a fairly relaxed way, f...

  16. Global aspects of complex geometry

    CERN Document Server

    Catanese, Fabrizio; Huckleberry, Alan T

    2006-01-01

    Present an overview of developments in Complex Geometry. This book covers topics that range from curve and surface theory through special varieties in higher dimensions, moduli theory, Kahler geometry, and group actions to Hodge theory and characteristic p-geometry.

  17. Analytic geometry

    CERN Document Server

    Burdette, A C

    1971-01-01

    Analytic Geometry covers several fundamental aspects of analytic geometry needed for advanced subjects, including calculus.This book is composed of 12 chapters that review the principles, concepts, and analytic proofs of geometric theorems, families of lines, the normal equation of the line, and related matters. Other chapters highlight the application of graphing, foci, directrices, eccentricity, and conic-related topics. The remaining chapters deal with the concept polar and rectangular coordinates, surfaces and curves, and planes.This book will prove useful to undergraduate trigonometric st

  18. Vector geometry

    CERN Document Server

    Robinson, Gilbert de B

    2011-01-01

    This brief undergraduate-level text by a prominent Cambridge-educated mathematician explores the relationship between algebra and geometry. An elementary course in plane geometry is the sole requirement for Gilbert de B. Robinson's text, which is the result of several years of teaching and learning the most effective methods from discussions with students. Topics include lines and planes, determinants and linear equations, matrices, groups and linear transformations, and vectors and vector spaces. Additional subjects range from conics and quadrics to homogeneous coordinates and projective geom

  19. Physics- and engineering knowledge-based geometry repair system for robust parametric CAD geometries

    OpenAIRE

    Li, Dong

    2012-01-01

    In modern multi-objective design optimisation, an effective geometry engine is becoming an essential tool and its performance has a significant impact on the entire process. Building a parametric geometry requires difficult compromises between the conflicting goals of robustness and flexibility. The work presents a solution for improving the robustness of parametric geometry models by capturing and modelling relative engineering knowledge into a surrogate model, and deploying it automatically...

  20. The geometry of the active strike-slip El Tigre Fault, Precordillera of San Juan, Central-Western Argentina: integrating resistivity surveys with structural and geomorphological data

    Science.gov (United States)

    Fazzito, Sabrina Y.; Cortés, José M.; Rapalini, Augusto E.; Terrizzano, Carla M.

    2013-07-01

    The geometry and related geomorphological features of the right-lateral strike-slip El Tigre Fault, one of the main morphostructural discontinuities in the Central-Western Precordillera of Argentina, were investigated. Achievements of this survey include: recognition of structural and geometrical discontinuities along the fault trace, identification and classification of landforms associated with local transpressional and transtensional sectors, observation of significant changes in the fault strike and detection of right and left bends of different wavelength. In the Central Segment of the El Tigre Fault, 2D electrical resistivity tomography surveys were carried out across the fault zone. The resistivity imaging permitted to infer the orientation of the main fault surface, the presence of blind fault branches along the fault zone, tectonic tilting of the Quaternary sedimentary cover, subsurface structure of pressure ridges and depth to the water table. Based on this information, it is possible to characterize the El Tigre Fault also as an important hydro-geological barrier. Our survey shows that the main fault surface changes along different segments from a high-angle to a subvertical setting whilst the vertical-slip component is either reverse or normal, depending on the local transpressive or transtensive regime induced by major bends along the trace. These local variations are expressed as sections of a few kilometres in length with relatively homogeneous behaviour and frequently separated by oblique or transversal structures.

  1. Applied geometry and discrete mathematics

    CERN Document Server

    Sturm; Gritzmann, Peter; Sturmfels, Bernd

    1991-01-01

    This volume, published jointly with the Association for Computing Machinery, comprises a collection of research articles celebrating the occasion of Victor Klee's sixty-fifth birthday in September 1990. During his long career, Klee has made contributions to a wide variety of areas, such as discrete and computational geometry, convexity, combinatorics, graph theory, functional analysis, mathematical programming and optimization, and theoretical computer science. In addition, Klee made important contributions to mathematics education, mathematical methods in economics and the decision sciences, applications of discrete mathematics in the biological and social sciences, and the transfer of knowledge from applied mathematics to industry. In honor of Klee's achievements, this volume presents more than forty papers on topics related to Klee's research. While the majority of the papers are research articles, a number of survey articles are also included. Mirroring the breadth of Klee's mathematical contributions, th...

  2. Geomorphological and hydrological implications of a given hydraulic geometry relationship, beyond the power-law

    Science.gov (United States)

    Kim, JongChun; Paik, Kyungrock

    2015-04-01

    implications. U. S. Geological Survey Professional Paper, 252. Paik, K., & Kumar, P. (2004). Hydraulic geometry and the nonlinearity of the network instantaneous response, Water Resource Research, 40, W03602. Stall, J. B., & Fok, Y. S. (1968). Hydraulic geometry of Illinois streams. University of Illinois Water Resources Center Research Report, 15. Stall, J. B., & Yang, C. T. (1970). Hydraulic geometry of 12 selected stream systems of the United States. University of Illinois Water Resources Center Research Report, 32.

  3. Noncommutative geometry

    CERN Document Server

    Connes, Alain

    1994-01-01

    This English version of the path-breaking French book on this subject gives the definitive treatment of the revolutionary approach to measure theory, geometry, and mathematical physics developed by Alain Connes. Profusely illustrated and invitingly written, this book is ideal for anyone who wants to know what noncommutative geometry is, what it can do, or how it can be used in various areas of mathematics, quantization, and elementary particles and fields.Key Features* First full treatment of the subject and its applications* Written by the pioneer of this field* Broad applications in mathemat

  4. Geometry Revealed

    CERN Document Server

    Berger, Marcel

    2010-01-01

    Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces,

  5. Discrete differential geometry. Consistency as integrability

    OpenAIRE

    Bobenko, Alexander I.; Suris, Yuri B.

    2005-01-01

    A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete differential geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not ...

  6. 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.

  7. Spinorial Geometry and Branes

    International Nuclear Information System (INIS)

    Sloane, Peter

    2007-01-01

    We adapt the spinorial geometry method introduced in [J. Gillard, U. Gran and G. Papadopoulos, 'The spinorial geometry of supersymmetric backgrounds,' Class. Quant. Grav. 22 (2005) 1033 [ (arXiv:hep-th/0410155)

  8. An introduction to incidence geometry

    CERN Document Server

    De Bruyn, Bart

    2016-01-01

    This book gives an introduction to the field of Incidence Geometry by discussing the basic families of point-line geometries and introducing some of the mathematical techniques that are essential for their study. The families of geometries covered in this book include among others the generalized polygons, near polygons, polar spaces, dual polar spaces and designs. Also the various relationships between these geometries are investigated. Ovals and ovoids of projective spaces are studied and some applications to particular geometries will be given. A separate chapter introduces the necessary mathematical tools and techniques from graph theory. This chapter itself can be regarded as a self-contained introduction to strongly regular and distance-regular graphs. This book is essentially self-contained, only assuming the knowledge of basic notions from (linear) algebra and projective and affine geometry. Almost all theorems are accompanied with proofs and a list of exercises with full solutions is given at the end...

  9. Spinorial Geometry and Branes

    Energy Technology Data Exchange (ETDEWEB)

    Sloane, Peter [Department of Mathematics, King' s College, University of London, Strand, London WC2R 2LS (United Kingdom)

    2007-09-15

    We adapt the spinorial geometry method introduced in [J. Gillard, U. Gran and G. Papadopoulos, 'The spinorial geometry of supersymmetric backgrounds,' Class. Quant. Grav. 22 (2005) 1033 [ (arXiv:hep-th/0410155)

  10. Introduction to non-Euclidean geometry

    CERN Document Server

    Wolfe, Harold E

    2012-01-01

    One of the first college-level texts for elementary courses in non-Euclidean geometry, this concise, readable volume is geared toward students familiar with calculus. A full treatment of the historical background explores the centuries-long efforts to prove Euclid's parallel postulate and their triumphant conclusion. Numerous original exercises form an integral part of the book.Topics include hyperbolic plane geometry and hyperbolic plane trigonometry, applications of calculus to the solutions of some problems in hyperbolic geometry, elliptic plane geometry and trigonometry, and the consistenc

  11. Optical geometry across the horizon

    International Nuclear Information System (INIS)

    Jonsson, Rickard

    2006-01-01

    In a recent paper (Jonsson and Westman 2006 Class. Quantum Grav. 23 61), a generalization of optical geometry, assuming a non-shearing reference congruence, is discussed. Here we illustrate that this formalism can be applied to (a finite four-volume) of any spherically symmetric spacetime. In particular we apply the formalism, using a non-static reference congruence, to do optical geometry across the horizon of a static black hole. While the resulting geometry in principle is time dependent, we can choose the reference congruence in such a manner that an embedding of the geometry always looks the same. Relative to the embedded geometry the reference points are then moving. We discuss the motion of photons, inertial forces and gyroscope precession in this framework

  12. Pseudo-random data acquisition geometry in 3D seismic survey; Sanjigen jishin tansa ni okeru giji random data shutoku reiauto ni tsuite

    Energy Technology Data Exchange (ETDEWEB)

    Minegishi, M; Tsuburaya, Y [Japan National Oil Corp., Tokyo (Japan). Technology Research Center

    1996-10-01

    Influence of pseudo-random geometry on the imaging for 3D seismic exploration data acquisition has been investigate using a simple model by comparing with the regular geometry. When constituting wave front by the interference of elemental waves, pseudo-random geometry data did not always provide good results. In the case of a point diffractor, the imaging operation, where the constituted wave front was returned to the point diffractor by the interference of elemental waves for the spatial alias records, did not always give clear images. In the case of multi point diffractor, good images were obtained with less noise generation in spite of alias records. There are a lot of diffractors in the actual geological structures, which corresponds to the case of multi point diffractors. Finally, better images could be obtained by inputting records acquired using the pseudo-random geometry rather than by inputting spatial alias records acquired using the regular geometry. 7 refs., 6 figs.

  13. 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.

  14. Complex numbers, quantum mechanics and the beginning of time

    International Nuclear Information System (INIS)

    Gibbons, G.W.; Pohle, H.J.

    1993-01-01

    A basic problem in quantizing a field in curved space is the decomposition of the classical modes in positive and negative frequency. The decomposition is equivalent to a choice of a complex structure in the space of classical solutions. In our construction the real tunneling geometries provide the link between this complex structure and analytic properties of the classical solutions in a riemannian section of space. This is related to the Osterwalder-Schrader approach to euclidean field theory. (orig.)

  15. On pseudoparticle solutions in Yang's theory of gravity

    International Nuclear Information System (INIS)

    Mielke, E.W.

    1980-03-01

    Within the framework of differential geometry, Yang's parallel-displacement gauge theory is considered with respect to ''pure'' gravitational fields. In a four-dimensional Riemannian manifold it is shown that the double self-dual solutions obey Einstein's vacuum equations with cosmological term, whereas the double anti-self-dual configurations satisfy the Rainich conditions of Wheeler's geometrodynamics. Conformal methods reveal that the gravitational analogue of the ''instanton'' or pseudoparticle solution of Yang-Mills theory was already known to Riemann. (author)

  16. A Continuum Mechanical Approach to Geodesics in Shape Space

    Science.gov (United States)

    2010-01-01

    mean curvature flow equation. Calc. Var., 3:253–271, 1995. [30] Siddharth Manay, Daniel Cremers , Byung-Woo Hong, Anthony J. Yezzi, and Stefano Soatto...P. W. Michor and D. Mumford. Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc., 8:1–48, 2006. 37 [33] Peter W. Michor, David ... Cremers . Shape matching by variational computation of geodesics on a manifold. In Pattern Recognition, LNCS 4174, pages 142–151, 2006. [38] P

  17. Color-weak compensation using local affine isometry based on discrimination threshold matching

    OpenAIRE

    Mochizuki, Rika; Kojima, Takanori; Lenz, Reiner; Chao, Jinhui

    2015-01-01

    We develop algorithms for color-weak compensation and color-weak simulation based on Riemannian geometry models of color spaces. The objective function introduced measures the match of color discrimination thresholds of average normal observers and a color-weak observer. The developed matching process makes use of local affine maps between color spaces of color-normal and color-weak observers. The method can be used to generate displays of images that provide color-normal and color-weak obser...

  18. Convection in Slab and Spheroidal Geometries

    Science.gov (United States)

    Porter, David H.; Woodward, Paul R.; Jacobs, Michael L.

    2000-01-01

    Three-dimensional numerical simulations of compressible turbulent thermally driven convection, in both slab and spheroidal geometries, are reviewed and analyzed in terms of velocity spectra and mixing-length theory. The same ideal gas model is used in both geometries, and resulting flows are compared. The piecewise-parabolic method (PPM), with either thermal conductivity or photospheric boundary conditions, is used to solve the fluid equations of motion. Fluid motions in both geometries exhibit a Kolmogorov-like k(sup -5/3) range in their velocity spectra. The longest wavelength modes are energetically dominant in both geometries, typically leading to one convection cell dominating the flow. In spheroidal geometry, a dipolar flow dominates the largest scale convective motions. Downflows are intensely turbulent and up drafts are relatively laminar in both geometries. In slab geometry, correlations between temperature and velocity fluctuations, which lead to the enthalpy flux, are fairly independent of depth. In spheroidal geometry this same correlation increases linearly with radius over the inner 70 percent by radius, in which the local pressure scale heights are a sizable fraction of the radius. The effects from the impenetrable boundary conditions in the slab geometry models are confused with the effects from non-local convection. In spheroidal geometry nonlocal effects, due to coherent plumes, are seen as far as several pressure scale heights from the lower boundary and are clearly distinguishable from boundary effects.

  19. Complex and symplectic geometry

    CERN Document Server

    Medori, Costantino; Tomassini, Adriano

    2017-01-01

    This book arises from the INdAM Meeting "Complex and Symplectic Geometry", which was held in Cortona in June 2016. Several leading specialists, including young researchers, in the field of complex and symplectic geometry, present the state of the art of their research on topics such as the cohomology of complex manifolds; analytic techniques in Kähler and non-Kähler geometry; almost-complex and symplectic structures; special structures on complex manifolds; and deformations of complex objects. The work is intended for researchers in these areas.

  20. Initiation to global Finslerian geometry

    CERN Document Server

    Akbar-Zadeh, Hassan

    2006-01-01

    After a brief description of the evolution of thinking on Finslerian geometry starting from Riemann, Finsler, Berwald and Elie Cartan, the book gives a clear and precise treatment of this geometry. The first three chapters develop the basic notions and methods, introduced by the author, to reach the global problems in Finslerian Geometry. The next five chapters are independent of each other, and deal with among others the geometry of generalized Einstein manifolds, the classification of Finslerian manifolds of constant sectional curvatures. They also give a treatment of isometric, affine, p

  1. Algebraic geometry in India

    Indian Academy of Sciences (India)

    algebraic geometry but also in related fields like number theory. ... every vector bundle on the affine space is trivial. (equivalently ... les on a compact Riemann surface to unitary rep- ... tial geometry and topology and was generalised in.

  2. Generalizing optical geometry

    International Nuclear Information System (INIS)

    Jonsson, Rickard; Westman, Hans

    2006-01-01

    We show that by employing the standard projected curvature as a measure of spatial curvature, we can make a certain generalization of optical geometry (Abramowicz M A and Lasota J-P 1997 Class. Quantum Grav. A 14 23-30). This generalization applies to any spacetime that admits a hypersurface orthogonal shearfree congruence of worldlines. This is a somewhat larger class of spacetimes than the conformally static spacetimes assumed in standard optical geometry. In the generalized optical geometry, which in the generic case is time dependent, photons move with unit speed along spatial geodesics and the sideways force experienced by a particle following a spatially straight line is independent of the velocity. Also gyroscopes moving along spatial geodesics do not precess (relative to the forward direction). Gyroscopes that follow a curved spatial trajectory precess according to a very simple law of three-rotation. We also present an inertial force formalism in coordinate representation for this generalization. Furthermore, we show that by employing a new sense of spatial curvature (Jonsson R 2006 Class. Quantum Grav. 23 1)) closely connected to Fermat's principle, we can make a more extensive generalization of optical geometry that applies to arbitrary spacetimes. In general this optical geometry will be time dependent, but still geodesic photons move with unit speed and follow lines that are spatially straight in the new sense. Also, the sideways experienced (comoving) force on a test particle following a line that is straight in the new sense will be independent of the velocity

  3. Geometry of Quantum States

    International Nuclear Information System (INIS)

    Hook, D W

    2008-01-01

    A geometric framework for quantum mechanics arose during the mid 1970s when authors such as Cantoni explored the notion of generalized transition probabilities, and Kibble promoted the idea that the space of pure quantum states provides a natural quantum mechanical analogue for classical phase space. This central idea can be seen easily since the projection of Schroedinger's equation from a Hilbert space into the space of pure spaces is a set of Hamilton's equations. Over the intervening years considerable work has been carried out by a variety of authors and a mature description of quantum mechanics in geometric terms has emerged with many applications. This current offering would seem ideally placed to review the last thirty years of progress and relate this to the most recent work in quantum entanglement. Bengtsson and Zyczkowski's beautifully illustrated volume, Geometry of Quantum States (referred to as GQS from now on) attempts to cover considerable ground in its 466 pages. Its topics range from colour theory in Chapter 1 to quantum entanglement in Chapter 15-to say that this is a whirlwind tour is, perhaps, no understatement. The use of the work 'introduction' in the subtitle of GQS, might suggest to the reader that this work be viewed as a textbook and I think that this interpretation would be incorrect. The authors have chosen to present a survey of different topics with the specific aim to introduce entanglement in geometric terms-the book is not intended as a pedagogical introduction to the geometric approach to quantum mechanics. Each of the fifteen chapters is a short, and mostly self-contained, essay on a particular aspect or application of geometry in the context of quantum mechanics with entanglement being addressed specifically in the final chapter. The chapters fall into three classifications: those concerned with the mathematical background, those which discuss quantum theory and the foundational aspects of the geometric framework, and

  4. Introduction to combinatorial geometry

    International Nuclear Information System (INIS)

    Gabriel, T.A.; Emmett, M.B.

    1985-01-01

    The combinatorial geometry package as used in many three-dimensional multimedia Monte Carlo radiation transport codes, such as HETC, MORSE, and EGS, is becoming the preferred way to describe simple and complicated systems. Just about any system can be modeled using the package with relatively few input statements. This can be contrasted against the older style geometry packages in which the required input statements could be large even for relatively simple systems. However, with advancements come some difficulties. The users of combinatorial geometry must be able to visualize more, and, in some instances, all of the system at a time. Errors can be introduced into the modeling which, though slight, and at times hard to detect, can have devastating effects on the calculated results. As with all modeling packages, the best way to learn the combinatorial geometry is to use it, first on a simple system then on more complicated systems. The basic technique for the description of the geometry consists of defining the location and shape of the various zones in terms of the intersections and unions of geometric bodies. The geometric bodies which are generally included in most combinatorial geometry packages are: (1) box, (2) right parallelepiped, (3) sphere, (4) right circular cylinder, (5) right elliptic cylinder, (6) ellipsoid, (7) truncated right cone, (8) right angle wedge, and (9) arbitrary polyhedron. The data necessary to describe each of these bodies are given. As can be easily noted, there are some subsets included for simplicity

  5. Graded geometry and Poisson reduction

    OpenAIRE

    Cattaneo, A S; Zambon, M

    2009-01-01

    The main result of [2] extends the Marsden-Ratiu reduction theorem [4] in Poisson geometry, and is proven by means of graded geometry. In this note we provide the background material about graded geometry necessary for the proof in [2]. Further, we provide an alternative algebraic proof for the main result. ©2009 American Institute of Physics

  6. Geometry of multihadron production

    Energy Technology Data Exchange (ETDEWEB)

    Bjorken, J.D.

    1994-10-01

    This summary talk only reviews a small sample of topics featured at this symposium: Introduction; The Geometry and Geography of Phase space; Space-Time Geometry and HBT; Multiplicities, Intermittency, Correlations; Disoriented Chiral Condensate; Deep Inelastic Scattering at HERA; and Other Contributions.

  7. Geometry of multihadron production

    International Nuclear Information System (INIS)

    Bjorken, J.D.

    1994-10-01

    This summary talk only reviews a small sample of topics featured at this symposium: Introduction; The Geometry and Geography of Phase space; Space-Time Geometry and HBT; Multiplicities, Intermittency, Correlations; Disoriented Chiral Condensate; Deep Inelastic Scattering at HERA; and Other Contributions

  8. Geometry of higher-dimensional black hole thermodynamics

    International Nuclear Information System (INIS)

    Aaman, Jan E.; Pidokrajt, Narit

    2006-01-01

    We investigate thermodynamic curvatures of the Kerr and Reissner-Nordstroem (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four-dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for a d=5 Kerr black hole is curved and divergent in the extremal limit. For a d≥6 Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For the RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In d≥5 the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in d=5 with double angular momenta

  9. Lectures on Symplectic Geometry

    CERN Document Server

    Silva, Ana Cannas

    2001-01-01

    The goal of these notes is to provide a fast introduction to symplectic geometry for graduate students with some knowledge of differential geometry, de Rham theory and classical Lie groups. This text addresses symplectomorphisms, local forms, contact manifolds, compatible almost complex structures, Kaehler manifolds, hamiltonian mechanics, moment maps, symplectic reduction and symplectic toric manifolds. It contains guided problems, called homework, designed to complement the exposition or extend the reader's understanding. There are by now excellent references on symplectic geometry, a subset of which is in the bibliography of this book. However, the most efficient introduction to a subject is often a short elementary treatment, and these notes attempt to serve that purpose. This text provides a taste of areas of current research and will prepare the reader to explore recent papers and extensive books on symplectic geometry where the pace is much faster. For this reprint numerous corrections and cl...

  10. Computational synthetic geometry

    CERN Document Server

    Bokowski, Jürgen

    1989-01-01

    Computational synthetic geometry deals with methods for realizing abstract geometric objects in concrete vector spaces. This research monograph considers a large class of problems from convexity and discrete geometry including constructing convex polytopes from simplicial complexes, vector geometries from incidence structures and hyperplane arrangements from oriented matroids. It turns out that algorithms for these constructions exist if and only if arbitrary polynomial equations are decidable with respect to the underlying field. Besides such complexity theorems a variety of symbolic algorithms are discussed, and the methods are applied to obtain new mathematical results on convex polytopes, projective configurations and the combinatorics of Grassmann varieties. Finally algebraic varieties characterizing matroids and oriented matroids are introduced providing a new basis for applying computer algebra methods in this field. The necessary background knowledge is reviewed briefly. The text is accessible to stud...

  11. Designs and finite geometries

    CERN Document Server

    1996-01-01

    Designs and Finite Geometries brings together in one place important contributions and up-to-date research results in this important area of mathematics. Designs and Finite Geometries serves as an excellent reference, providing insight into some of the most important research issues in the field.

  12. d-geometries revisited

    CERN Document Server

    Ceresole, Anna; Gnecchi, Alessandra; Marrani, Alessio

    2013-01-01

    We analyze some properties of the four dimensional supergravity theories which originate from five dimensions upon reduction. They generalize to N>2 extended supersymmetries the d-geometries with cubic prepotentials, familiar from N=2 special K\\"ahler geometry. We emphasize the role of a suitable parametrization of the scalar fields and the corresponding triangular symplectic basis. We also consider applications to the first order flow equations for non-BPS extremal black holes.

  13. Geometry success in 20 minutes a day

    CERN Document Server

    LLC, LearningExpress

    2014-01-01

    Whether you're new to geometry or just looking for a refresher, Geometry Success in 20 Minutes a Day offers a 20-step lesson plan that provides quick and thorough instruction in practical, critical skills. Stripped of unnecessary math jargon but bursting with geometry essentials, Geometry Success in 20 Minutes a Day: Covers all vital geometry skills, from the basic building blocks of geometry to ratio, proportion, and similarity to trigonometry and beyond Provides hundreds of practice exercises in test format Applies geometr

  14. The application of *-products to noncommutative geometry and gauge theory

    International Nuclear Information System (INIS)

    Sykora, A.

    2004-06-01

    Due to the singularities arising in quantum field theory and the difficulties in quantizing gravity it is often believed that the description of spacetime by a smooth manifold should be given up at small length scales or high energies. In this work we will replace spacetime by noncommutative structures arising within the framework of deformation quantization. The ordinary product between functions will be replaced by a *-product, an associative product for the space of functions on a manifold. We develop a formalism to realize algebras defined by relations on function spaces. For this purpose we construct the Weyl-ordered *-product and present a method how to calculate *-products with the help of commuting vector fields. Concepts developed in noncommutative differential geometry will be applied to this type of algebras and we construct actions for noncommutative field theories. In the classical limit these noncommutative theories become field theories on manifolds with nonvanishing curvature. It becomes clear that the application of *-products is very fruitful to the solution of noncommutative problems. In the semiclassical limit every *-product is related to a Poisson structure, every derivation of the algebra to a vector field on the manifold. Since in this limit many problems are reduced to a couple of differential equations the *-product representation makes it possible to construct noncommutative spaces corresponding to interesting Riemannian manifolds. Derivations of *-products makes it further possible to extend noncommutative gauge theory in the Seiberg-Witten formalism with covariant derivatives. The resulting noncommutative gauge fields may be interpreted as one forms of a generalization of the exterior algebra of a manifold. For the Formality *-product we prove the existence of the abelian Seiberg-Witten map for derivations of these *-products. We calculate the enveloping algebra valued non abelian Seiberg-Witten map pertubatively up to second order for

  15. 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...

  16. Introduction to tropical geometry

    CERN Document Server

    Maclagan, Diane

    2015-01-01

    Tropical geometry is a combinatorial shadow of algebraic geometry, offering new polyhedral tools to compute invariants of algebraic varieties. It is based on tropical algebra, where the sum of two numbers is their minimum and the product is their sum. This turns polynomials into piecewise-linear functions, and their zero sets into polyhedral complexes. These tropical varieties retain a surprising amount of information about their classical counterparts. Tropical geometry is a young subject that has undergone a rapid development since the beginning of the 21st century. While establishing itself as an area in its own right, deep connections have been made to many branches of pure and applied mathematics. This book offers a self-contained introduction to tropical geometry, suitable as a course text for beginning graduate students. Proofs are provided for the main results, such as the Fundamental Theorem and the Structure Theorem. Numerous examples and explicit computations illustrate the main concepts. Each of t...

  17. Geometry Euclid and beyond

    CERN Document Server

    Hartshorne, Robin

    2000-01-01

    In recent years, I have been teaching a junior-senior-level course on the classi­ cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa­ rately. The remainder of the book is an exploration of questions that arise natu­ rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And ...

  18. Basic algebraic geometry, v.2

    CERN Document Server

    Shafarevich, Igor Rostislavovich

    1994-01-01

    Shafarevich Basic Algebraic Geometry 2 The second edition of Shafarevich's introduction to algebraic geometry is in two volumes. The second volume covers schemes and complex manifolds, generalisations in two different directions of the affine and projective varieties that form the material of the first volume. Two notable additions in this second edition are the section on moduli spaces and representable functors, motivated by a discussion of the Hilbert scheme, and the section on Kähler geometry. The book ends with a historical sketch discussing the origins of algebraic geometry. From the Zentralblatt review of this volume: "... one can only respectfully repeat what has been said about the first part of the book (...): a great textbook, written by one of the leading algebraic geometers and teachers himself, has been reworked and updated. As a result the author's standard textbook on algebraic geometry has become even more important and valuable. Students, teachers, and active researchers using methods of al...

  19. 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

  20. Tensor calculus, relativity, and cosmology a first course

    CERN Document Server

    Dalarsson, M

    2005-01-01

    This book combines relativity, astrophysics, and cosmology in a single volume, providing an introduction to each subject that enables students to understand more detailed treatises as well as the current literature. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The next section on relativistic astrophysics discusses

  1. 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...

  2. The Beauty of Geometry

    Science.gov (United States)

    Morris, Barbara H.

    2004-01-01

    This article describes a geometry project that used the beauty of stained-glass-window designs to teach middle school students about geometric figures and concepts. Three honors prealgebra teachers and a middle school mathematics gifted intervention specialist created a geometry project that covered the curriculum and also assessed students'…

  3. Fractal Geometry and Stochastics V

    CERN Document Server

    Falconer, Kenneth; Zähle, Martina

    2015-01-01

    This book brings together leading contributions from the fifth conference on Fractal Geometry and Stochastics held in Tabarz, Germany, in March 2014. The book is divided into five sections covering different facets of this fast developing area: geometric measure theory, self-similar fractals and recurrent structures, analysis and algebra on fractals, multifractal theory, and random constructions. There are state-of-the-art surveys as well as papers highlighting more specific recent advances. The authors are world-experts who present their topics comprehensibly and attractively. The book provides an accessible gateway to the subject for newcomers as well as a reference for recent developments for specialists. Authors include: Krzysztof Barański, Julien Barral, Kenneth Falconer, De-Jun Feng, Peter J. Grabner, Rostislav Grigorchuk, Michael Hinz, Stéphane Jaffard, Maarit Järvenpää, Antti Käenmäki, Marc Kesseböhmer, Michel Lapidus, Klaus Mecke, Mark Pollicott,  Michał Rams, Pablo Shmerkin, and András Te...

  4. Teaching Spatial Geometry in a Virtual World

    DEFF Research Database (Denmark)

    Förster, Klaus-Tycho

    2017-01-01

    Spatial geometry is one of the fundamental mathematical building blocks of any engineering education. However, it is overshadowed by planar geometry in the curriculum between playful early primary education and later analytical geometry, leaving a multi-year gap where spatial geometry is absent...

  5. Trends and developments in computational geometry

    NARCIS (Netherlands)

    Berg, de M.

    1997-01-01

    This paper discusses some trends and achievements in computational geometry during the past five years, with emphasis on problems related to computer graphics. Furthermore, a direction of research in computational geometry is discussed that could help in bringing the fields of computational geometry

  6. An approach for management of geometry data

    Science.gov (United States)

    Dube, R. P.; Herron, G. J.; Schweitzer, J. E.; Warkentine, E. R.

    1980-01-01

    The strategies for managing Integrated Programs for Aerospace Design (IPAD) computer-based geometry are described. The computer model of geometry is the basis for communication, manipulation, and analysis of shape information. IPAD's data base system makes this information available to all authorized departments in a company. A discussion of the data structures and algorithms required to support geometry in IPIP (IPAD's data base management system) is presented. Through the use of IPIP's data definition language, the structure of the geometry components is defined. The data manipulation language is the vehicle by which a user defines an instance of the geometry. The manipulation language also allows a user to edit, query, and manage the geometry. The selection of canonical forms is a very important part of the IPAD geometry. IPAD has a canonical form for each entity and provides transformations to alternate forms; in particular, IPAD will provide a transformation to the ANSI standard. The DBMS schemas required to support IPAD geometry are explained.

  7. Surveys on surgery theory

    CERN Document Server

    Cappell, Sylvain; Rosenberg, Jonathan

    2014-01-01

    Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. The sixtieth birthday (on December 14, 1996) of C.T.C. Wall, a leading member of the subject''s founding generation, led the editors of this volume to reflect on the extraordinary accomplishments of surgery theory as well as its current enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source surveying surgery theory and its applications. Because no one person could write such a survey, the editors ask

  8. Surveys on surgery theory

    CERN Document Server

    Cappell, Sylvain; Rosenberg, Jonathan

    2014-01-01

    Surgery theory, the basis for the classification theory of manifolds, is now about forty years old. There have been some extraordinary accomplishments in that time, which have led to enormously varied interactions with algebra, analysis, and geometry. Workers in many of these areas have often lamented the lack of a single source that surveys surgery theory and its applications. Indeed, no one person could write such a survey. The sixtieth birthday of C. T. C. Wall, one of the leaders of the founding generation of surgery theory, provided an opportunity to rectify the situation and produce a

  9. "WGL," a Web Laboratory for Geometry

    Science.gov (United States)

    Quaresma, Pedro; Santos, Vanda; Maric, Milena

    2018-01-01

    The role of information and communication technologies (ICT) in education is nowadays well recognised. The "Web Geometry Laboratory," is an e-learning, collaborative and adaptive, Web environment for geometry, integrating a well known dynamic geometry system. In a collaborative session, teachers and students, engaged in solving…

  10. Analytische Geometrie

    Science.gov (United States)

    Kemnitz, Arnfried

    Der Grundgedanke der Analytischen Geometrie besteht darin, dass geometrische Untersuchungen mit rechnerischen Mitteln geführt werden. Geometrische Objekte werden dabei durch Gleichungen beschrieben und mit algebraischen Methoden untersucht.

  11. Algebraic Geometry and Number Theory Summer School

    CERN Document Server

    Sarıoğlu, Celal; Soulé, Christophe; Zeytin, Ayberk

    2017-01-01

    This lecture notes volume presents significant contributions from the “Algebraic Geometry and Number Theory” Summer School, held at Galatasaray University, Istanbul, June 2-13, 2014. It addresses subjects ranging from Arakelov geometry and Iwasawa theory to classical projective geometry, birational geometry and equivariant cohomology. Its main aim is to introduce these contemporary research topics to graduate students who plan to specialize in the area of algebraic geometry and/or number theory. All contributions combine main concepts and techniques with motivating examples and illustrative problems for the covered subjects. Naturally, the book will also be of interest to researchers working in algebraic geometry, number theory and related fields.

  12. The Idea of Order at Geometry Class.

    Science.gov (United States)

    Rishel, Thomas

    The idea of order in geometry is explored using the experience of assignments given to undergraduates in a college geometry course "From Space to Geometry." Discussed are the definition of geometry, and earth measurement using architecture, art, and common experience. This discussion concludes with a consideration of the question of whether…

  13. Special geometry

    International Nuclear Information System (INIS)

    Strominger, A.

    1990-01-01

    A special manifold is an allowed target manifold for the vector multiplets of D=4, N=2 supergravity. These manifolds are of interest for string theory because the moduli spaces of Calabi-Yau threefolds and c=9, (2,2) conformal field theories are special. Previous work has given a local, coordinate-dependent characterization of special geometry. A global description of special geometries is given herein, and their properties are studied. A special manifold M of complex dimension n is characterized by the existence of a holomorphic Sp(2n+2,R)xGL(1,C) vector bundle over M with a nowhere-vanishing holomorphic section Ω. The Kaehler potential on M is the logarithm of the Sp(2n+2,R) invariant norm of Ω. (orig.)

  14. Using Dynamic Geometry Software to Improve Eight Grade Students' Understanding of Transformation Geometry

    Science.gov (United States)

    Guven, Bulent

    2012-01-01

    This study examines the effect of dynamic geometry software (DGS) on students' learning of transformation geometry. A pre- and post-test quasi-experimental design was used. Participants in the study were 68 eighth grade students (36 in the experimental group and 32 in the control group). While the experimental group students were studying the…

  15. 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.

  16. Disformal transformation in Newton-Cartan geometry

    Energy Technology Data Exchange (ETDEWEB)

    Huang, Peng [Zhejiang Chinese Medical University, Department of Information, Hangzhou (China); Sun Yat-Sen University, School of Physics and Astronomy, Guangzhou (China); Yuan, Fang-Fang [Nankai University, School of Physics, Tianjin (China)

    2016-08-15

    Newton-Cartan geometry has played a central role in recent discussions of the non-relativistic holography and condensed matter systems. Although the conformal transformation in non-relativistic holography can easily be rephrased in terms of Newton-Cartan geometry, we show that it requires a nontrivial procedure to arrive at the consistent form of anisotropic disformal transformation in this geometry. Furthermore, as an application of the newly obtained transformation, we use it to induce a geometric structure which may be seen as a particular non-relativistic version of the Weyl integrable geometry. (orig.)

  17. Geometry and symmetry

    CERN Document Server

    Yale, Paul B

    2012-01-01

    This book is an introduction to the geometry of Euclidean, affine, and projective spaces with special emphasis on the important groups of symmetries of these spaces. The two major objectives of the text are to introduce the main ideas of affine and projective spaces and to develop facility in handling transformations and groups of transformations. Since there are many good texts on affine and projective planes, the author has concentrated on the n-dimensional cases.Designed to be used in advanced undergraduate mathematics or physics courses, the book focuses on ""practical geometry,"" emphasi

  18. Optimizing solar-cell grid geometry

    Science.gov (United States)

    Crossley, A. P.

    1969-01-01

    Trade-off analysis and mathematical expressions calculate optimum grid geometry in terms of various cell parameters. Determination of the grid geometry provides proper balance between grid resistance and cell output to optimize the energy conversion process.

  19. Geometry and Cloaking Devices

    Science.gov (United States)

    Ochiai, T.; Nacher, J. C.

    2011-09-01

    Recently, the application of geometry and conformal mappings to artificial materials (metamaterials) has attracted the attention in various research communities. These materials, characterized by a unique man-made structure, have unusual optical properties, which materials found in nature do not exhibit. By applying the geometry and conformal mappings theory to metamaterial science, it may be possible to realize so-called "Harry Potter cloaking device". Although such a device is still in the science fiction realm, several works have shown that by using such metamaterials it may be possible to control the direction of the electromagnetic field at will. We could then make an object hidden inside of a cloaking device. Here, we will explain how to design invisibility device using differential geometry and conformal mappings.

  20. A first course in geometry

    CERN Document Server

    Walsh, Edward T

    2014-01-01

    This introductory text is designed to help undergraduate students develop a solid foundation in geometry. Early chapters progress slowly, cultivating the necessary understanding and self-confidence for the more rapid development that follows. The extensive treatment can be easily adapted to accommodate shorter courses. Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter incl

  1. Global affine differential geometry of hypersurfaces

    CERN Document Server

    Li, An-Min; Zhao, Guosong; Hu, Zejun

    2015-01-01

    This book draws a colorful and widespread picture of global affine hypersurface theory up to the most recent state. Moreover, the recent development revealed that affine differential geometry- as differential geometry in general- has an exciting intersection area with other fields of interest, like partial differential equations, global analysis, convex geometry and Riemann surfaces.

  2. Spectral dimension of quantum geometries

    International Nuclear Information System (INIS)

    Calcagni, Gianluca; Oriti, Daniele; Thürigen, Johannes

    2014-01-01

    The spectral dimension is an indicator of geometry and topology of spacetime and a tool to compare the description of quantum geometry in various approaches to quantum gravity. This is possible because it can be defined not only on smooth geometries but also on discrete (e.g., simplicial) ones. In this paper, we consider the spectral dimension of quantum states of spatial geometry defined on combinatorial complexes endowed with additional algebraic data: the kinematical quantum states of loop quantum gravity (LQG). Preliminarily, the effects of topology and discreteness of classical discrete geometries are studied in a systematic manner. We look for states reproducing the spectral dimension of a classical space in the appropriate regime. We also test the hypothesis that in LQG, as in other approaches, there is a scale dependence of the spectral dimension, which runs from the topological dimension at large scales to a smaller one at short distances. While our results do not give any strong support to this hypothesis, we can however pinpoint when the topological dimension is reproduced by LQG quantum states. Overall, by exploring the interplay of combinatorial, topological and geometrical effects, and by considering various kinds of quantum states such as coherent states and their superpositions, we find that the spectral dimension of discrete quantum geometries is more sensitive to the underlying combinatorial structures than to the details of the additional data associated with them. (paper)

  3. Algebraic geometry

    CERN Document Server

    Lefschetz, Solomon

    2005-01-01

    An introduction to algebraic geometry and a bridge between its analytical-topological and algebraical aspects, this text for advanced undergraduate students is particularly relevant to those more familiar with analysis than algebra. 1953 edition.

  4. General relativity and gravitational waves

    CERN Document Server

    Weber, Johanna

    1961-01-01

    An internationally famous physicist and electrical engineer, the author of this text was a pioneer in the investigation of gravitational waves. Joseph Weber's General Relativity and Gravitational Waves offers a classic treatment of the subject. Appropriate for upper-level undergraduates and graduate students, this text remains ever relevant. Brief but thorough in its introduction to the foundations of general relativity, it also examines the elements of Riemannian geometry and tensor calculus applicable to this field.Approximately a quarter of the contents explores theoretical and experimenta

  5. A geometry calibration method for rotation translation trajectory

    International Nuclear Information System (INIS)

    Zhang Jun; Yan Bin; Li Lei; Lu Lizhong; Zhang Feng

    2013-01-01

    In cone-beam CT imaging system, it is difficult to directly measure the geometry parameters. In this paper, a geometry calibration method for rotation translation trajectory is proposed. Intrinsic parameters are solved from the relationship built on geometry parameter of the system and projection trajectory of calibration object. Parameters of rotation axis are extrapolated from the unified intrinsic parameter, and geometry parameters of the idle trajectory are acquired too. The calibration geometry can be analytically determined using explicit formulae, it can avoid getting into local optimum in iterative way. Simulation experiments are carried out on misaligned geometry, experiment results indicate that geometry artifacts due to misaligned geometry are effectively depressed by the proposed method, and the image quality is enhanced. (authors)

  6. Construction of hexahedral elements mesh capturing realistic geometries of Bayou Choctaw SPR site

    Energy Technology Data Exchange (ETDEWEB)

    Park, Byoung Yoon [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Roberts, Barry L. [Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

    2015-09-01

    The three-dimensional finite element mesh capturing realistic geometries of Bayou Choctaw site has been constructed using the sonar and seismic survey data obtained from the field. The mesh is consisting of hexahedral elements because the salt constitutive model is coded using hexahedral elements. Various ideas and techniques to construct finite element mesh capturing artificially and naturally formed geometries are provided. The techniques to reduce the number of elements as much as possible to save on computer run time with maintaining the computational accuracy is also introduced. The steps and methodologies could be applied to construct the meshes of Big Hill, Bryan Mound, and West Hackberry strategic petroleum reserve sites. The methodology could be applied to the complicated shape masses for not only various civil and geological structures but also biological applications such as artificial limbs.

  7. Geometry modeling for SAM-CE Monte Carlo calculations

    International Nuclear Information System (INIS)

    Steinberg, H.A.; Troubetzkoy, E.S.

    1980-01-01

    Three geometry packages have been developed and incorporated into SAM-CE, for representing in three dimensions the transport medium. These are combinatorial geometry - a general (non-lattice) system, complex combinatorial geometry - a very general system with lattice capability, and special reactor geometry - a special purpose system for light water reactor geometries. Their different attributes are described

  8. 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).

  9. Proof of the holographic formula for entanglement entropy

    International Nuclear Information System (INIS)

    Fursaev, Dmitri V.

    2006-01-01

    Entanglement entropy for a spatial partition of a quantum system is studied in theories which admit a dual description in terms of the anti-de Sitter (AdS) gravity one dimension higher. A general proof of the holographic formula which relates the entropy to the area of a codimension 2 minimal hypersurface embedded in the bulk AdS space is given. The entanglement entropy is determined by a partition function which is defined as a path integral over Riemannian AdS geometries with non-trivial boundary conditions. The topology of the Riemannian spaces puts restrictions on the choice of the minimal hypersurface for a given boundary conditions. The entanglement entropy is also considered in Randall-Sundrum braneworld models where its asymptotic expansion is derived when the curvature radius of the brane is much larger than the AdS radius. Special attention is paid to the geometrical structure of anomalous terms in the entropy in four dimensions. Modification of the holographic formula by the higher curvature terms in the bulk is briefly discussed

  10. Molecular motion in restricted geometries

    Indian Academy of Sciences (India)

    Molecular dynamics in restricted geometries is known to exhibit anomalous behaviour. Diffusion, translational or rotational, of molecules is altered significantly on confinement in restricted geometries. Quasielastic neutron scattering (QENS) offers a unique possibility of studying molecular motion in such systems. Both time ...

  11. 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

  12. Ideal geometries in the military architecture of F. di Giorgio Martini. The defence system fortress-tower of Cagli

    Directory of Open Access Journals (Sweden)

    Paolo Clini

    2015-09-01

    Full Text Available  The subject of the military architecture was widely studied and analyzed by the architect Francesco di Giorgio Martini (1439-1501, who created important defensive works in the duchy of Montefeltro. A defensive line connects a series of medieval villages of the Marches: b Between the Martini’s fortifications, Cagli has still a fortified tower perfectly preserved and some traces of the fortress. The analysis conducted on them involves the information inferable by the essay, by historical surveys and the recent laser scanner survey. The aim of the research was to identify the generating geometry of the architecture: taking the precepts of Vitruvius, in the Renaissance period geometry was synonymous with proportion, order and symmetry. Two are subjects: the first one is the in-depth knowledge of the tower through the digital data obtained from the point cloud, the other one is the reconstruction of a global image of the complex tower-fortress.

  13. An Investigation into Conversion from Non-Uniform Rational B-Spline Boundary Representation Geometry to Constructive Solid Geometry

    Science.gov (United States)

    2015-12-01

    ARL-SR-0347 ● DEC 2015 US Army Research Laboratory An Investigation into Conversion from Non-Uniform Rational B-Spline Boundary...US Army Research Laboratory An Investigation into Conversion from Non-Uniform Rational B-Spline Boundary Representation Geometry to...from Non-Uniform Rational B-Spline Boundary Representation Geometry to Constructive Solid Geometry 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c

  14. Head First 2D Geometry

    CERN Document Server

    Fallow), Stray

    2009-01-01

    Having trouble with geometry? Do Pi, The Pythagorean Theorem, and angle calculations just make your head spin? Relax. With Head First 2D Geometry, you'll master everything from triangles, quads and polygons to the time-saving secrets of similar and congruent angles -- and it'll be quick, painless, and fun. Through entertaining stories and practical examples from the world around you, this book takes you beyond boring problems. You'll actually use what you learn to make real-life decisions, like using angles and parallel lines to crack a mysterious CSI case. Put geometry to work for you, and

  15. Numerically robust geometry engine for compound solid geometries

    International Nuclear Information System (INIS)

    Vlachoudis, V.; Sinuela-Pastor, D.

    2013-01-01

    Monte Carlo programs heavily rely on a fast and numerically robust solid geometry engines. However the success of solid modeling, depends on facilities for specifying and editing parameterized models through a user-friendly graphical front-end. Such a user interface has to be fast enough in order to be interactive for 2D and/or 3D displays, but at the same time numerically robust in order to display possible modeling errors at real time that could be critical for the simulation. The graphical user interface Flair for FLUKA currently employs such an engine where special emphasis has been given on being fast and numerically robust. The numerically robustness is achieved by a novel method of estimating the floating precision of the operations, which dynamically adapts all the decision operations accordingly. Moreover a predictive caching mechanism is ensuring that logical errors in the geometry description are found online, without compromising the processing time by checking all regions. (authors)

  16. Quantification of Porcine Vocal Fold Geometry.

    Science.gov (United States)

    Stevens, Kimberly A; Thomson, Scott L; Jetté, Marie E; Thibeault, Susan L

    2016-07-01

    The aim of this study was to quantify porcine vocal fold medial surface geometry and three-dimensional geometric distortion induced by freezing the larynx, especially in the region of the vocal folds. The medial surface geometries of five excised porcine larynges were quantified and reported. Five porcine larynges were imaged in a micro-CT scanner, frozen, and rescanned. Segmentations and three-dimensional reconstructions were used to quantify and characterize geometric features. Comparisons were made with geometry data previously obtained using canine and human vocal folds as well as geometries of selected synthetic vocal fold models. Freezing induced an overall expansion of approximately 5% in the transverse plane and comparable levels of nonuniform distortion in sagittal and coronal planes. The medial surface of the porcine vocal folds was found to compare reasonably well with other geometries, although the compared geometries exhibited a notable discrepancy with one set of published human female vocal fold geometry. Porcine vocal folds are qualitatively geometrically similar to data available for canine and human vocal folds, as well as commonly used models. Freezing of tissue in the larynx causes distortion of around 5%. The data can provide direction in estimating uncertainty due to bulk distortion of tissue caused by freezing, as well as quantitative geometric data that can be directly used in developing vocal fold models. Copyright © 2016 The Voice Foundation. Published by Elsevier Inc. All rights reserved.

  17. Non-Euclidean stress-free configuration of arteries accounting for curl of axial strips sectioned from vessels.

    Science.gov (United States)

    Takamizawa, Keiichi; Nakayama, Yasuhide

    2013-11-01

    It is well known that arteries are subject to residual stress. In earlier studies, the residual stress in the arterial ring relieved by a radial cut was considered in stress analysis. However, it has been found that axial strips sectioned from arteries also curled into arcs, showing that the axial residual stresses were relieved from the arterial walls. The combined relief of circumferential and axial residual stresses must be considered to accurately analyze stress and strain distributions under physiological loading conditions. In the present study, a mathematical model of a stress-free configuration of artery was proposed using Riemannian geometry. Stress analysis for arterial walls under unloaded and physiologically loaded conditions was performed using exponential strain energy functions for porcine and human common carotid arteries. In the porcine artery, the circumferential stress distribution under physiological loading became uniform compared with that without axial residual strain, whereas a gradient of axial stress distribution increased through the wall thickness. This behavior showed almost the same pattern that was observed in a recent study in which approximate analysis accounting for circumferential and axial residual strains was performed, whereas the circumferential and axial stresses increased from the inner surface to the outer surface under a physiological condition in the human common carotid artery of a two-layer model based on data of other recent studies. In both analyses, Riemannian geometry was appropriate to define the stress-free configurations of the arterial walls with both circumferential and axial residual strains.

  18. 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.

  19. Discrete and computational geometry

    CERN Document Server

    Devadoss, Satyan L

    2011-01-01

    Discrete geometry is a relatively new development in pure mathematics, while computational geometry is an emerging area in applications-driven computer science. Their intermingling has yielded exciting advances in recent years, yet what has been lacking until now is an undergraduate textbook that bridges the gap between the two. Discrete and Computational Geometry offers a comprehensive yet accessible introduction to this cutting-edge frontier of mathematics and computer science. This book covers traditional topics such as convex hulls, triangulations, and Voronoi diagrams, as well as more recent subjects like pseudotriangulations, curve reconstruction, and locked chains. It also touches on more advanced material, including Dehn invariants, associahedra, quasigeodesics, Morse theory, and the recent resolution of the Poincaré conjecture. Connections to real-world applications are made throughout, and algorithms are presented independently of any programming language. This richly illustrated textbook also fe...

  20. A prediction for bubbling geometries

    OpenAIRE

    Okuda, Takuya

    2007-01-01

    We study the supersymmetric circular Wilson loops in N=4 Yang-Mills theory. Their vacuum expectation values are computed in the parameter region that admits smooth bubbling geometry duals. The results are a prediction for the supergravity action evaluated on the bubbling geometries for Wilson loops.

  1. Geometry -----------~--------------RESONANCE

    Indian Academy of Sciences (India)

    Parallel: A pair of lines in a plane is said to be parallel if they do not meet. Mathematicians were at war ... Subsequently, Poincare, Klein, Beltrami and others refined non-. Euclidean geometry. ... plane divides the plane into two half planes and.

  2. Intermediate algebra & analytic geometry

    CERN Document Server

    Gondin, William R

    1967-01-01

    Intermediate Algebra & Analytic Geometry Made Simple focuses on the principles, processes, calculations, and methodologies involved in intermediate algebra and analytic geometry. The publication first offers information on linear equations in two unknowns and variables, functions, and graphs. Discussions focus on graphic interpretations, explicit and implicit functions, first quadrant graphs, variables and functions, determinate and indeterminate systems, independent and dependent equations, and defective and redundant systems. The text then examines quadratic equations in one variable, system

  3. 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).

  4. A survey of TRIPOLI-4

    International Nuclear Information System (INIS)

    Both, J.P.; Derriennic, H.; Morillon, B.; Nimal, J.C.

    1994-01-01

    A survey of the new version of the TRIPOLI code used in shielding calculations, is presented. The main new features introduced in this version are: combinatorial geometry, a multigroup automatic weighting scheme, and complete treatment of nuclear evaluation files; a simulation implementation including strategy for parallelization of the code is presented. Some benchmark calculations are also presented. 7 figs., 5 tabs., 4 refs

  5. Geometry The Language of Space and Form (Revised Edition)

    CERN Document Server

    Tabak, John

    2011-01-01

    Geometry, Revised Edition describes geometry in antiquity. Beginning with a brief description of some of the geometry that preceded the geometry of the Greeks, it takes up the story of geometry during the European Renaissance as well as the significant mathematical progress in other areas of the world. It also discusses the analytic geometry of Ren Descartes and Pierre Fermat, the alternative coordinate systems invented by Isaac Newton, and the solid geometry of Leonhard Euler. Also included is an overview of the geometry of one of the most successful mathematicians of the 19th century, Bernha

  6. From geometry to algebra and vice versa: Realistic mathematics education principles for analyzing geometry tasks

    Science.gov (United States)

    Jupri, Al

    2017-04-01

    In this article we address how Realistic Mathematics Education (RME) principles, including the intertwinement and the reality principles, are used to analyze geometry tasks. To do so, we carried out three phases of a small-scale study. First we analyzed four geometry problems - considered as tasks inviting the use of problem solving and reasoning skills - theoretically in the light of the RME principles. Second, we tested two problems to 31 undergraduate students of mathematics education program and other two problems to 16 master students of primary mathematics education program. Finally, we analyzed student written work and compared these empirical to the theoretical results. We found that there are discrepancies between what we expected theoretically and what occurred empirically in terms of mathematization and of intertwinement of mathematical concepts from geometry to algebra and vice versa. We conclude that the RME principles provide a fruitful framework for analyzing geometry tasks that, for instance, are intended for assessing student problem solving and reasoning skills.

  7. Differential geometry curves, surfaces, manifolds

    CERN Document Server

    Kohnel, Wolfgang

    2002-01-01

    This carefully written book is an introduction to the beautiful ideas and results of differential geometry. The first half covers the geometry of curves and surfaces, which provide much of the motivation and intuition for the general theory. Special topics that are explored include Frenet frames, ruled surfaces, minimal surfaces and the Gauss-Bonnet theorem. The second part is an introduction to the geometry of general manifolds, with particular emphasis on connections and curvature. The final two chapters are insightful examinations of the special cases of spaces of constant curvature and Einstein manifolds. The text is illustrated with many figures and examples. The prerequisites are undergraduate analysis and linear algebra.

  8. Projective Geometry

    Indian Academy of Sciences (India)

    mathematicians are trained to use very precise language, and so find it hard to simplify and state .... thing. If you take a plane on which there are two such triangles which enjoy the above ... within this geometry to simplify things if needed.

  9. 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

  10. Second International workshop Geometry and Symbolic Computation

    CERN Document Server

    Walczak, Paweł; Geometry and its Applications

    2014-01-01

    This volume has been divided into two parts: Geometry and Applications. The geometry portion of the book relates primarily to geometric flows, laminations, integral formulae, geometry of vector fields on Lie groups, and osculation; the articles in the applications portion concern some particular problems of the theory of dynamical systems, including mathematical problems of liquid flows and a study of cycles for non-dynamical systems. This Work is based on the second international workshop entitled "Geometry and Symbolic Computations," held on May 15-18, 2013 at the University of Haifa and is dedicated to modeling (using symbolic calculations) in differential geometry and its applications in fields such as computer science, tomography, and mechanics. It is intended to create a forum for students and researchers in pure and applied geometry to promote discussion of modern state-of-the-art in geometric modeling using symbolic programs such as Maple™ and Mathematica®, as well as presentation of new results. ...

  11. Geometry intuitive, discrete, and convex : a tribute to László Fejes Tóth

    CERN Document Server

    Böröczky, Károly; Tóth, Gábor; Pach, János

    2013-01-01

    The present volume is a collection of a dozen survey articles, dedicated to the memory of the famous Hungarian geometer, László Fejes Tóth, on the 99th anniversary of his birth. Each article reviews recent progress in an important field in intuitive, discrete, and convex geometry. The mathematical work and perspectives of all editors and most contributors of this volume were deeply influenced by László Fejes Tóth.

  12. Network geometry with flavor: From complexity to quantum geometry

    Science.gov (United States)

    Bianconi, Ginestra; Rahmede, Christoph

    2016-03-01

    Network geometry is attracting increasing attention because it has a wide range of applications, ranging from data mining to routing protocols in the Internet. At the same time advances in the understanding of the geometrical properties of networks are essential for further progress in quantum gravity. In network geometry, simplicial complexes describing the interaction between two or more nodes play a special role. In fact these structures can be used to discretize a geometrical d -dimensional space, and for this reason they have already been widely used in quantum gravity. Here we introduce the network geometry with flavor s =-1 ,0 ,1 (NGF) describing simplicial complexes defined in arbitrary dimension d and evolving by a nonequilibrium dynamics. The NGF can generate discrete geometries of different natures, ranging from chains and higher-dimensional manifolds to scale-free networks with small-world properties, scale-free degree distribution, and nontrivial community structure. The NGF admits as limiting cases both the Bianconi-Barabási models for complex networks, the stochastic Apollonian network, and the recently introduced model for complex quantum network manifolds. The thermodynamic properties of NGF reveal that NGF obeys a generalized area law opening a new scenario for formulating its coarse-grained limit. The structure of NGF is strongly dependent on the dimensionality d . In d =1 NGFs grow complex networks for which the preferential attachment mechanism is necessary in order to obtain a scale-free degree distribution. Instead, for NGF with dimension d >1 it is not necessary to have an explicit preferential attachment rule to generate scale-free topologies. We also show that NGF admits a quantum mechanical description in terms of associated quantum network states. Quantum network states evolve by a Markovian dynamics and a quantum network state at time t encodes all possible NGF evolutions up to time t . Interestingly the NGF remains fully classical but

  13. The Persistification of the ATLAS Geometry

    CERN Document Server

    AUTHOR|(INSPIRE)INSPIRE-00068562; The ATLAS collaboration; Bianchi, Riccardo-Maria

    2016-01-01

    The complex geometry of the whole detector of the ATLAS experiment at LHC is currently stored only in custom online databases, from which it is built on-the- y on request. Accessing the online geometry guarantees accessing the latest version of the detector description, but requires the setup of the full ATLAS so ware framework “Athena”, which provides the online services and the tools to retrieve the data from the database. is operation is cumbersome and slows down the applications that need to access the geometry. Moreover, all applications that need to access the detector geom- etry need to be built and run on the same platform as the ATLAS framework, preventing the usage of the actual detector geometry in stand-alone applications. Here we propose a new mechanism to persistify and serve the geometry of HEP experiments. e new mechanism is composed by a new le format and a REST API. e new le format allows to store the whole detector description locally in a at le, and it is especially optimized to descri...

  14. Implosions and hypertoric geometry

    DEFF Research Database (Denmark)

    Dancer, A.; Kirwan, F.; Swann, A.

    2013-01-01

    The geometry of the universal hyperkahler implosion for SU (n) is explored. In particular, we show that the universal hyperkahler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperkahler implosion.......The geometry of the universal hyperkahler implosion for SU (n) is explored. In particular, we show that the universal hyperkahler implosion naturally contains a hypertoric variety described in terms of quivers. Furthermore, we discuss a gauge theoretic approach to hyperkahler implosion....

  15. Granular flows in constrained geometries

    Science.gov (United States)

    Murthy, Tejas; Viswanathan, Koushik

    Confined geometries are widespread in granular processing applications. The deformation and flow fields in such a geometry, with non-trivial boundary conditions, determine the resultant mechanical properties of the material (local porosity, density, residual stresses etc.). We present experimental studies of deformation and plastic flow of a prototypical granular medium in different nontrivial geometries- flat-punch compression, Couette-shear flow and a rigid body sliding past a granular half-space. These geometries represent simplified scaled-down versions of common industrial configurations such as compaction and dredging. The corresponding granular flows show a rich variety of flow features, representing the entire gamut of material types, from elastic solids (beam buckling) to fluids (vortex-formation, boundary layers) and even plastically deforming metals (dead material zone, pile-up). The effect of changing particle-level properties (e.g., shape, size, density) on the observed flows is also explicitly demonstrated. Non-smooth contact dynamics particle simulations are shown to reproduce some of the observed flow features quantitatively. These results showcase some central challenges facing continuum-scale constitutive theories for dynamic granular flows.

  16. A vector space approach to geometry

    CERN Document Server

    Hausner, Melvin

    2010-01-01

    The effects of geometry and linear algebra on each other receive close attention in this examination of geometry's correlation with other branches of math and science. In-depth discussions include a review of systematic geometric motivations in vector space theory and matrix theory; the use of the center of mass in geometry, with an introduction to barycentric coordinates; axiomatic development of determinants in a chapter dealing with area and volume; and a careful consideration of the particle problem. 1965 edition.

  17. Integral geometry and valuations

    CERN Document Server

    Solanes, Gil

    2014-01-01

    Valuations are finitely additive functionals on the space of convex bodies. Their study has become a central subject in convexity theory, with fundamental applications to integral geometry. In the last years there has been significant progress in the theory of valuations, which in turn has led to important achievements in integral geometry. This book originated from two courses delivered by the authors at the CRM and provides a self-contained introduction to these topics, covering most of the recent advances. The first part, by Semyon Alesker, is devoted to the theory of convex valuations, with emphasis on the latest developments. A special focus is put on the new fundamental structures of the space of valuations discovered after Alesker's irreducibility theorem. Moreover, the author describes the newly developed theory of valuations on manifolds. In the second part, Joseph H. G. Fu gives a modern introduction to integral geometry in the sense of Blaschke and Santaló, based on the notions and tools presented...

  18. CBM RICH geometry optimization

    Energy Technology Data Exchange (ETDEWEB)

    Mahmoud, Tariq; Hoehne, Claudia [II. Physikalisches Institut, Giessen Univ. (Germany); Collaboration: CBM-Collaboration

    2016-07-01

    The Compressed Baryonic Matter (CBM) experiment at the future FAIR complex will investigate the phase diagram of strongly interacting matter at high baryon density and moderate temperatures in A+A collisions from 2-11 AGeV (SIS100) beam energy. The main electron identification detector in the CBM experiment will be a RICH detector with a CO{sub 2} gaseous-radiator, focusing spherical glass mirrors, and MAPMT photo-detectors being placed on a PMT-plane. The RICH detector is located directly behind the CBM dipole magnet. As the final magnet geometry is now available, some changes in the RICH geometry become necessary. In order to guarantee a magnetic field of 1 mT at maximum in the PMT plane for effective operation of the MAPMTs, two measures have to be taken: The PMT plane is moved outwards of the stray field by tilting the mirrors by 10 degrees and shielding boxes have been designed. In this contribution the results of the geometry optimization procedure are presented.

  19. Kaehler geometry and SUSY mechanics

    International Nuclear Information System (INIS)

    Bellucci, Stefano; Nersessian, Armen

    2001-01-01

    We present two examples of SUSY mechanics related with Kaehler geometry. The first system is the N = 4 supersymmetric one-dimensional sigma-model proposed in hep-th/0101065. Another system is the N = 2 SUSY mechanics whose phase space is the external algebra of an arbitrary Kaehler manifold. The relation of these models with antisymplectic geometry is discussed

  20. GPS: Geometry, Probability, and Statistics

    Science.gov (United States)

    Field, Mike

    2012-01-01

    It might be said that for most occupations there is now less of a need for mathematics than there was say fifty years ago. But, the author argues, geometry, probability, and statistics constitute essential knowledge for everyone. Maybe not the geometry of Euclid, but certainly geometrical ways of thinking that might enable us to describe the world…

  1. FINAL REPORT: GEOMETRY AND ELEMENTARY PARTICLE PHYSICS

    Energy Technology Data Exchange (ETDEWEB)

    Singer, Isadore M.

    2008-03-04

    The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists’ quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energy for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.

  2. Final Report: Geometry And Elementary Particle Physics

    International Nuclear Information System (INIS)

    Singer, Isadore M.

    2008-01-01

    The effect on mathematics of collaborations between high-energy theoretical physics and modern mathematics has been remarkable. Mirror symmetry has revolutionized enumerative geometry, and Seiberg-Witten invariants have greatly simplified the study of four manifolds. And because of their application to string theory, physicists now need to know cohomology theory, characteristic classes, index theory, K-theory, algebraic geometry, differential geometry, and non-commutative geometry. Much more is coming. We are experiencing a deeper contact between the two sciences, which will stimulate new mathematics essential to the physicists quest for the unification of quantum mechanics and relativity. Our grant, supported by the Department of Energy for twelve years, has been instrumental in promoting an effective interaction between geometry and string theory, by supporting the Mathematical Physics seminar, postdoc research, collaborations, graduate students and several research papers.

  3. Discrete quantum geometries and their effective dimension

    International Nuclear Information System (INIS)

    Thuerigen, Johannes

    2015-01-01

    In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension d on low energy scales to a real number between 0 and d on high energy scales. In the particular case of 1 these results allow to understand the quantum geometry as effectively fractal.

  4. An introduction to the Kähler-Ricci flow

    CERN Document Server

    Eyssidieux, Philippe; Guedj, Vincent

    2013-01-01

    This volume collects lecture notes from courses offered at several conferences and workshops, and provides the first exposition in book form of the basic theory of the Kähler-Ricci flow and its current state-of-the-art. While several excellent books on Kähler-Einstein geometry are available, there have been no such works on the Kähler-Ricci flow. The book will serve as a valuable resource for graduate students and researchers in complex differential geometry, complex algebraic geometry and Riemannian geometry, and will hopefully foster further developments in this fascinating area of research.   The Ricci flow was first introduced by R. Hamilton in the early 1980s, and is central in G. Perelman’s celebrated proof of the Poincaré conjecture. When specialized for Kähler manifolds, it becomes the Kähler-Ricci flow, and reduces to a scalar PDE (parabolic complex Monge-Ampère equation). As a spin-off of his breakthrough, G. Perelman proved the convergence of the Kähler-Ricci flow on Kähler-Einstein man...

  5. Introduction into integral geometry and stereology

    DEFF Research Database (Denmark)

    Kiderlen, Markus

    Statistics and Random Fields and is a self-containing introduction into integral geometry and its applications in stereology. The most important integral geometric tools for stereological applications are kinematic formulas and results of Blaschke-Petkantschin type. Therefore, Crofton's formula......This text is the extended version of two talks held at the Summer Academy Stochastic Geometry, Spatial Statistics and Random Fields in the Soellerhaus, Germany, in September 2009. It forms (with slight modifications) a chapter of the Springer lecture notes Lectures on Stochastic Geometry, Spatial...

  6. Proceedings of the Third International Workshop on Mathematical Foundations of Computational Anatomy

    DEFF Research Database (Denmark)

    the mathematical community around shapes and the MICCAI community in view of computational anatomy applications. It targets more particularly researchers investigating the combination of statistical and geometrical aspects in the modeling of the variability of biological shapes. The workshop is a forum...... of the workshop: statistics on manifolds and diff eomorphisms for surface or longitudinal registration. One session gathers papers exploring new mathematical structures beyond Riemannian geometry while the last oral session deals with the emerging theme of statistics on graphs and trees. Finally, a poster session......Computational anatomy is an emerging discipline at the interface of geometry, statistics and image analysis which aims at modeling and analyzing the biological shape of tissues and organs. The goal is to estimate representative organ anatomies across diseases, populations, species or ages, to model...

  7. String theory compactifications

    CERN Document Server

    Graña, Mariana

    2017-01-01

    The lectures in this book provide graduate students and non-specialist researchers with a concise introduction to the concepts and formalism required to reduce the ten-dimensional string theories to the observable four-dimensional space-time - a procedure called string compactification. The text starts with a very brief introduction to string theory, first working out its massless spectrum and showing how the condition on the number of dimensions arises. It then dwells on the different possible internal manifolds, from the simplest to the most relevant phenomenologically, thereby showing that the most elegant description is through an extension of ordinary Riemannian geometry termed generalized geometry, which was first introduced by Hitchin. Last but not least, the authors review open problems in string phenomenology, such as the embedding of the Standard Model and obtaining de Sitter solutions.

  8. Université de Genève

    CERN Multimedia

    2008-01-01

    École de physique: Département de physique nucléaire et corspusculaire 24, Quai Ernest-Ansermet 1211 GENEVE 4 Tél: (022) 379 62 73 - Fax: (022) 379 69 92 Friday 18 January 2008 Colloquium at 17:00 – Stückelberg Auditorium The Fabric of Spacetime Prof. Demetrios Christodoulou / Laureate of he Tomalla Prize for gravity research The first part of the lecture is a review of the geometric and physical basis of general relativity theory. It begins with Gauss‘s intrinsic geometry of surfaces, proceeding to Riemannian geometry, special relativity and the concept of spacetime with its geometry, the equivalence principle, the identification of tidal gravitational force with spacetime curvature, concluding with the formulation of general relativity and Einstein’s equations. The second part of the lecture begins with the incompleteness theorem of Penrose and its geometric antecedents. Demetrios Christodoulou then proceeds to describe his work on the stabilityof the Minkowski ...

  9. Surrogate Modeling for Geometry Optimization

    DEFF Research Database (Denmark)

    Rojas Larrazabal, Marielba de la Caridad; Abraham, Yonas; Holzwarth, Natalie

    2009-01-01

    A new approach for optimizing the nuclear geometry of an atomic system is described. Instead of the original expensive objective function (energy functional), a small number of simpler surrogates is used.......A new approach for optimizing the nuclear geometry of an atomic system is described. Instead of the original expensive objective function (energy functional), a small number of simpler surrogates is used....

  10. 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...

  11. A Whirlwind Tour of Computational Geometry.

    Science.gov (United States)

    Graham, Ron; Yao, Frances

    1990-01-01

    Described is computational geometry which used concepts and results from classical geometry, topology, combinatorics, as well as standard algorithmic techniques such as sorting and searching, graph manipulations, and linear programing. Also included are special techniques and paradigms. (KR)

  12. Monte Carlo simulation of fully Markovian stochastic geometries

    International Nuclear Information System (INIS)

    Lepage, Thibaut; Delaby, Lucie; Malvagi, Fausto; Mazzolo, Alain

    2010-01-01

    The interest in resolving the equation of transport in stochastic media has continued to increase these last years. For binary stochastic media it is often assumed that the geometry is Markovian, which is never the case in usual environments. In the present paper, based on rigorous mathematical theorems, we construct fully two-dimensional Markovian stochastic geometries and we study their main properties. In particular, we determine a percolation threshold p c , equal to 0.586 ± 0.0015 for such geometries. Finally, Monte Carlo simulations are performed through these geometries and the results compared to homogeneous geometries. (author)

  13. Tidal stresses and energy gaps in microstate geometries

    Science.gov (United States)

    Tyukov, Alexander; Walker, Robert; Warner, Nicholas P.

    2018-02-01

    We compute energy gaps and study infalling massive geodesic probes in the new families of scaling, microstate geometries that have been constructed recently and for which the holographic duals are known. We find that in the deepest geometries, which have the lowest energy gaps, the geodesic deviation shows that the stress reaches the Planck scale long before the probe reaches the cap of the geometry. Such probes must therefore undergo a stringy transition as they fall into microstate geometry. We discuss the scales associated with this transition and comment on the implications for scrambling in microstate geometries.

  14. Transformational plane geometry

    CERN Document Server

    Umble, Ronald N

    2014-01-01

    Axioms of Euclidean Plane Geometry The Existence and Incidence Postulates The Distance and Ruler Postulates The Plane Separation Postulate The Protractor Postulate The Side-Angle-Side Postulate and the Euclidean Parallel Postulate Theorems of Euclidean Plane Geometry The Exterior Angle Theorem Triangle Congruence Theorems The Alternate Interior Angles Theorem and the Angle Sum Theorem Similar Triangles Introduction to Transformations, Isometries, and Similarities Transformations Isometries and SimilaritiesAppendix: Proof of Surjectivity Translations, Rotations, and Reflections Translations Rotations Reflections Appendix: Geometer's Sketchpad Commands Required by Exploratory Activities Compositions of Translations, Rotations, and Reflections The Three Points Theorem Rotations as Compositions of Two Reflections Translations as Compositions of Two Halfturns or Two Reflections The Angle Addition Theorem Glide Reflections Classification of Isometries The Fundamental Theorem and Congruence Classification of Isometr...

  15. Multilevel geometry optimization

    Science.gov (United States)

    Rodgers, Jocelyn M.; Fast, Patton L.; Truhlar, Donald G.

    2000-02-01

    Geometry optimization has been carried out for three test molecules using six multilevel electronic structure methods, in particular Gaussian-2, Gaussian-3, multicoefficient G2, multicoefficient G3, and two multicoefficient correlation methods based on correlation-consistent basis sets. In the Gaussian-2 and Gaussian-3 methods, various levels are added and subtracted with unit coefficients, whereas the multicoefficient Gaussian-x methods involve noninteger parameters as coefficients. The multilevel optimizations drop the average error in the geometry (averaged over the 18 cases) by a factor of about two when compared to the single most expensive component of a given multilevel calculation, and in all 18 cases the accuracy of the atomization energy for the three test molecules improves; with an average improvement of 16.7 kcal/mol.

  16. Fractal geometry mathematical foundations and applications

    CERN Document Server

    Falconer, Kenneth

    2013-01-01

    The seminal text on fractal geometry for students and researchers: extensively revised and updated with new material, notes and references that reflect recent directions. Interest in fractal geometry continues to grow rapidly, both as a subject that is fascinating in its own right and as a concept that is central to many areas of mathematics, science and scientific research. Since its initial publication in 1990 Fractal Geometry: Mathematical Foundations and Applications has become a seminal text on the mathematics of fractals.  The book introduces and develops the general theory and applica

  17. 3D seismic surveys for shallow targets

    Energy Technology Data Exchange (ETDEWEB)

    Lawton, D.C.; Stewart, R.R.; Bertram, M.B. [Calgary Univ., AB (Canada). Dept. of Geoscience, Consortium for Research in Elastic Wave Exploration Seismology

    2008-07-01

    Although 3D seismic surveys are generally used to map deep hydrocarbon plays, this study demonstrated that they can be useful for characterizing shallow targets, such as oilsands deposits. A high-resolution 3D seismic survey was undertaken to map shallow stratigraphy near Calgary, Alberta. The project demonstrated the efficacy of reflection seismic surveys for shallow targets ranging from 100 to 500 metres. The purpose of the program was to map shallow stratigraphy and structure to depths of up to 500m, and to investigate shallow aquifers in the study area. The results of the survey illustrated the opportunity that 3D seismic surveys provide for mapping shallow reflectors and the acquisition geometry needed to image them. Applications include mapping the distribution of shallow aquifers, delineating shallow coals and investigating oilsands deposits. 2 refs., 5 figs.

  18. Planning for Evolution in a Production Environment: Migration from a Legacy Geometry Code to an Abstract Geometry Modeling Language in STAR

    Science.gov (United States)

    Webb, Jason C.; Lauret, Jerome; Perevoztchikov, Victor

    2012-12-01

    Increasingly detailed descriptions of complex detector geometries are required for the simulation and analysis of today's high-energy and nuclear physics experiments. As new tools for the representation of geometry models become available during the course of an experiment, a fundamental challenge arises: how best to migrate from legacy geometry codes developed over many runs to the new technologies, such as the ROOT/TGeo [1] framework, without losing touch with years of development, tuning and validation. One approach, which has been discussed within the community for a number of years, is to represent the geometry model in a higher-level language independent of the concrete implementation of the geometry. The STAR experiment has used this approach to successfully migrate its legacy GEANT 3-era geometry to an Abstract geometry Modelling Language (AgML), which allows us to create both native GEANT 3 and ROOT/TGeo implementations. The language is supported by parsers and a C++ class library which enables the automated conversion of the original source code to AgML, supports export back to the original AgSTAR[5] representation, and creates the concrete ROOT/TGeo geometry implementation used by our track reconstruction software. In this paper we present our approach, design and experience and will demonstrate physical consistency between the original AgSTAR and new AgML geometry representations.

  19. MIFT: GIFT Combinatorial Geometry Input to VCS Code

    Science.gov (United States)

    1977-03-01

    r-w w-^ H ^ß0318is CQ BRL °RCUMr REPORT NO. 1967 —-S: ... MIFT: GIFT COMBINATORIAL GEOMETRY INPUT TO VCS CODE Albert E...TITLE (and Subtitle) MIFT: GIFT Combinatorial Geometry Input to VCS Code S. TYPE OF REPORT & PERIOD COVERED FINAL 6. PERFORMING ORG. REPORT NUMBER...Vehicle Code System (VCS) called MORSE was modified to accept the GIFT combinatorial geometry package. GIFT , as opposed to the geometry package

  20. 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

  1. 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

  2. Transformasi Geometri Rotasi Berbantuan Software Geogebra

    Directory of Open Access Journals (Sweden)

    Muhamad Hanafi

    2018-02-01

    Full Text Available Penelitian  ini bertujuan untuk membantu visualisasi dan menemukan konsep pada Transformasi geometri Rotasi di titik Pusat  dengan menggunakan software GeoGebra. Penelitian ini mengulas tentang Koordinat Kartesius dan Polar, dan selanjutntya Transformasi geometri Rotasi di titik Pusat .

  3. A consistent data set of Antarctic ice sheet topography, cavity geometry, and global bathymetry

    Directory of Open Access Journals (Sweden)

    R. Timmermann

    2010-12-01

    Full Text Available Sub-ice shelf circulation and freezing/melting rates in ocean general circulation models depend critically on an accurate and consistent representation of cavity geometry. Existing global or pan-Antarctic topography data sets have turned out to contain various inconsistencies and inaccuracies. The goal of this work is to compile independent regional surveys and maps into a global data set. We use the S-2004 global 1-min bathymetry as the backbone and add an improved version of the BEDMAP topography (ALBMAP bedrock topography for an area that roughly coincides with the Antarctic continental shelf. The position of the merging line is individually chosen in different sectors in order to capture the best of both data sets. High-resolution gridded data for ice shelf topography and cavity geometry of the Amery, Fimbul, Filchner-Ronne, Larsen C and George VI Ice Shelves, and for Pine Island Glacier are carefully merged into the ambient ice and ocean topographies. Multibeam survey data for bathymetry in the former Larsen B cavity and the southeastern Bellingshausen Sea have been obtained from the data centers of Alfred Wegener Institute (AWI, British Antarctic Survey (BAS and Lamont-Doherty Earth Observatory (LDEO, gridded, and blended into the existing bathymetry map. The resulting global 1-min Refined Topography data set (RTopo-1 contains self-consistent maps for upper and lower ice surface heights, bedrock topography, and surface type (open ocean, grounded ice, floating ice, bare land surface. The data set is available in NetCDF format from the PANGAEA database at doi:10.1594/pangaea.741917.

  4. Algebra, Geometry and Mathematical Physics Conference

    CERN Document Server

    Paal, Eugen; Silvestrov, Sergei; Stolin, Alexander

    2014-01-01

    This book collects the proceedings of the Algebra, Geometry and Mathematical Physics Conference, held at the University of Haute Alsace, France, October 2011. Organized in the four areas of algebra, geometry, dynamical symmetries and conservation laws and mathematical physics and applications, the book covers deformation theory and quantization; Hom-algebras and n-ary algebraic structures; Hopf algebra, integrable systems and related math structures; jet theory and Weil bundles; Lie theory and applications; non-commutative and Lie algebra and more. The papers explore the interplay between research in contemporary mathematics and physics concerned with generalizations of the main structures of Lie theory aimed at quantization, and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non-commutative geometry and applications in physics and beyond. The book benefits a broad audience of researchers a...

  5. Homological mirror symmetry and tropical geometry

    CERN Document Server

    Catanese, Fabrizio; Kontsevich, Maxim; Pantev, Tony; Soibelman, Yan; Zharkov, Ilia

    2014-01-01

    The relationship between Tropical Geometry and Mirror Symmetry goes back to the work of Kontsevich and Y. Soibelman (2000), who applied methods of non-archimedean geometry (in particular, tropical curves) to Homological Mirror Symmetry. In combination with the subsequent work of Mikhalkin on the “tropical” approach to Gromov-Witten theory, and the work of Gross and Siebert, Tropical Geometry has now become a powerful tool. Homological Mirror Symmetry is the area of mathematics concentrated around several categorical equivalences connecting symplectic and holomorphic (or algebraic) geometry. The central ideas first appeared in the work of Maxim Kontsevich (1993). Roughly speaking, the subject can be approached in two ways: either one uses Lagrangian torus fibrations of Calabi-Yau manifolds (the so-called Strominger-Yau-Zaslow picture, further developed by Kontsevich and Soibelman) or one uses Lefschetz fibrations of symplectic manifolds (suggested by Kontsevich and further developed by Seidel). Tropical Ge...

  6. Use of information technologies in teaching course "Analytical geometry" in higher schools on example of software "ANALYTICAL GEOMETRY"

    OpenAIRE

    V. B. Grigorieva

    2009-01-01

    In article are considered the methodical questions of using of computer technologies, for example, the software "Analytical geometry", in process of teaching course of analytical geometry in the higher school.

  7. Variable geometry Darrieus wind machine

    Science.gov (United States)

    Pytlinski, J. T.; Serrano, D.

    1983-08-01

    A variable geometry Darrieus wind machine is proposed. The lower attachment of the blades to the rotor can move freely up and down the axle allowing the blades of change shape during rotation. Experimental data for a 17 m. diameter Darrieus rotor and a theoretical model for multiple streamtube performance prediction were used to develop a computer simulation program for studying parameters that affect the machine's performance. This new variable geometry concept is described and interrelated with multiple streamtube theory through aerodynamic parameters. The computer simulation study shows that governor behavior of a Darrieus turbine can not be attained by a standard turbine operating within normally occurring rotational velocity limits. A second generation variable geometry Darrieus wind turbine which uses a telescopic blade is proposed as a potential improvement on the studied concept.

  8. Flux compactifications and generalized geometries

    International Nuclear Information System (INIS)

    Grana, Mariana

    2006-01-01

    Following the lectures given at CERN Winter School 2006, we present a pedagogical overview of flux compactifications and generalized geometries, concentrating on closed string fluxes in type II theories. We start by reviewing the supersymmetric flux configurations with maximally symmetric four-dimensional spaces. We then discuss the no-go theorems (and their evasion) for compactifications with fluxes. We analyse the resulting four-dimensional effective theories for Calabi-Yau and Calabi-Yau orientifold compactifications, concentrating on the flux-induced superpotentials. We discuss the generic mechanism of moduli stabilization and illustrate with two examples: the conifold in IIB and a T 6 /(Z 3 x Z 3 ) torus in IIA. We finish by studying the effective action and flux vacua for generalized geometries in the context of generalized complex geometry

  9. Flux compactifications and generalized geometries

    Energy Technology Data Exchange (ETDEWEB)

    Grana, Mariana [Service de Physique Theorique, CEA/Saclay, 91191 Gif-sur-Yvette Cedex (France)

    2006-11-07

    Following the lectures given at CERN Winter School 2006, we present a pedagogical overview of flux compactifications and generalized geometries, concentrating on closed string fluxes in type II theories. We start by reviewing the supersymmetric flux configurations with maximally symmetric four-dimensional spaces. We then discuss the no-go theorems (and their evasion) for compactifications with fluxes. We analyse the resulting four-dimensional effective theories for Calabi-Yau and Calabi-Yau orientifold compactifications, concentrating on the flux-induced superpotentials. We discuss the generic mechanism of moduli stabilization and illustrate with two examples: the conifold in IIB and a T{sup 6} /(Z{sub 3} x Z{sub 3}) torus in IIA. We finish by studying the effective action and flux vacua for generalized geometries in the context of generalized complex geometry.

  10. Curvature tensor copies in affine geometry

    International Nuclear Information System (INIS)

    Srivastava, P.P.

    1981-01-01

    The sets of space-time and spin-connections which give rise to the same curvature tensor are constructed. The corresponding geometries are compared. Results are illustrated by an explicit calculation and comment on the copies in Einstein-Cartan and Weyl-Cartan geometries. (Author) [pt

  11. Poisson geometry from a Dirac perspective

    Science.gov (United States)

    Meinrenken, Eckhard

    2018-03-01

    We present proofs of classical results in Poisson geometry using techniques from Dirac geometry. This article is based on mini-courses at the Poisson summer school in Geneva, June 2016, and at the workshop Quantum Groups and Gravity at the University of Waterloo, April 2016.

  12. Geometry, analysis and probability in honor of Jean-Michel Bismut

    CERN Document Server

    Hofer, Helmut; Labourie, François; Jan, Yves; Ma, Xiaonan; Zhang, Weiping

    2017-01-01

    This volume presents original research articles and extended surveys related to the mathematical interest and work of Jean-Michel Bismut. His outstanding contributions to probability theory and global analysis on manifolds have had a profound impact on several branches of mathematics in the areas of control theory, mathematical physics and arithmetic geometry. Contributions by: K. Behrend N. Bergeron S. K. Donaldson J. Dubédat B. Duplantier G. Faltings E. Getzler G. Kings R. Mazzeo J. Millson C. Moeglin W. Müller R. Rhodes D. Rössler S. Sheffield A. Teleman G. Tian K-I. Yoshikawa H. Weiss W. Werner The collection is a valuable resource for graduate students and researchers in these fields.

  13. King George Island ice cap geometry updated with airborne GPR measurements

    Directory of Open Access Journals (Sweden)

    M. Rückamp

    2012-07-01

    Full Text Available Ice geometry is a mandatory requirement for numerical modelling purposes. In this paper we present a consistent data set for the ice thickness, the bedrock topography and the ice surface topography of the King George Island ice cap (Arctowski icefield and the adjacent central part. The new data set is composed of ground based and airborne ground penetrating radar (GPR and differential GPS (DGPS measurements, obtained during several field campaigns. Blindow et al. (2010 already provided a comprehensive overview of the ground based measurements carried out in the safely accessible area of the ice cap. The updated data set incorporates airborne measurements in the heavily crevassed coastal areas. Therefore, in this paper special attention is paid to the airborne measurements by addressing the instrument used, survey procedure, and data processing in more detail. In particular, the inclusion of airborne GPR measurements with the 30 MHz BGR-P30-System developed at the Institute of Geophysics (University of Münster completes the picture of the ice geometry substantially. The compiled digital elevation model of the bedrock shows a rough, highly variable topography with pronounced valleys, ridges, and troughs. Mean ice thickness is 240 ± 6 m, with a maximum value of 422 ± 10 m in the surveyed area. Noticeable are bounded areas in the bedrock topography below sea level where marine based ice exists. The provided data set is required as a basis for future monitoring attempts or as input for numerical modelling experiments. The data set is available from the PANGAEA database at http://dx.doi.org/10.1594/PANGAEA.770567.

  14. Development of the geometry database for the CBM experiment

    Science.gov (United States)

    Akishina, E. P.; Alexandrov, E. I.; Alexandrov, I. N.; Filozova, I. A.; Friese, V.; Ivanov, V. V.

    2018-01-01

    The paper describes the current state of the Geometry Database (Geometry DB) for the CBM experiment. The main purpose of this database is to provide convenient tools for: (1) managing the geometry modules; (2) assembling various versions of the CBM setup as a combination of geometry modules and additional files. The CBM users of the Geometry DB may use both GUI (Graphical User Interface) and API (Application Programming Interface) tools for working with it.

  15. SABRINA, Geometry Plot Program for MCNP

    International Nuclear Information System (INIS)

    SEIDL, Marcus

    2003-01-01

    1 - Description of program or function: SABRINA is an interactive, three-dimensional, geometry-modeling code system, primarily for use with CCC-200/MCNP. SABRINA's capabilities include creation, visualization, and verification of three-dimensional geometries specified by either surface- or body-base combinatorial geometry; display of particle tracks are calculated by MCNP; and volume fraction generation. 2 - Method of solution: Rendering is performed by ray tracing or an edge and intersection algorithm. Volume fraction calculations are made by ray tracing. 3 - Restrictions on the complexity of the problem: A graphics display with X Window capability is required

  16. Differential geometry and topology of curves

    CERN Document Server

    Animov, Yu

    2001-01-01

    Differential geometry is an actively developing area of modern mathematics. This volume presents a classical approach to the general topics of the geometry of curves, including the theory of curves in n-dimensional Euclidean space. The author investigates problems for special classes of curves and gives the working method used to obtain the conditions for closed polygonal curves. The proof of the Bakel-Werner theorem in conditions of boundedness for curves with periodic curvature and torsion is also presented. This volume also highlights the contributions made by great geometers. past and present, to differential geometry and the topology of curves.

  17. Geometry of surfaces a practical guide for mechanical engineers

    CERN Document Server

    Radzevich, Stephen P

    2012-01-01

    Presents an in-depth analysis of geometry of part surfaces and provides the tools for solving complex engineering problems Geometry of Surfaces: A Practical Guide for Mechanical Engineers is a comprehensive guide to applied geometry of surfaces with focus on practical applications in various areas of mechanical engineering. The book is divided into three parts on Part Surfaces, Geometry of Contact of Part Surfaces and Mapping of the Contacting Part Surfaces. Geometry of Surfaces: A Practical Guide for Mechanical Engineers combines differential geometry and gearing theory and presents new developments in the elementary theory of enveloping surfaces. Written by a leading expert of the field, this book also provides the reader with the tools for solving complex engineering problems in the field of mechanical engineering. Presents an in-depth analysis of geometry of part surfaces Provides tools for solving complex engineering problems in the field of mechanical engineering Combines differential geometry an...

  18. Multilevel geometry optimization

    Energy Technology Data Exchange (ETDEWEB)

    Rodgers, Jocelyn M. [Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431 (United States); Fast, Patton L. [Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431 (United States); Truhlar, Donald G. [Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431 (United States)

    2000-02-15

    Geometry optimization has been carried out for three test molecules using six multilevel electronic structure methods, in particular Gaussian-2, Gaussian-3, multicoefficient G2, multicoefficient G3, and two multicoefficient correlation methods based on correlation-consistent basis sets. In the Gaussian-2 and Gaussian-3 methods, various levels are added and subtracted with unit coefficients, whereas the multicoefficient Gaussian-x methods involve noninteger parameters as coefficients. The multilevel optimizations drop the average error in the geometry (averaged over the 18 cases) by a factor of about two when compared to the single most expensive component of a given multilevel calculation, and in all 18 cases the accuracy of the atomization energy for the three test molecules improves; with an average improvement of 16.7 kcal/mol. (c) 2000 American Institute of Physics.

  19. Machine learning spatial geometry from entanglement features

    Science.gov (United States)

    You, Yi-Zhuang; Yang, Zhao; Qi, Xiao-Liang

    2018-02-01

    Motivated by the close relations of the renormalization group with both the holography duality and the deep learning, we propose that the holographic geometry can emerge from deep learning the entanglement feature of a quantum many-body state. We develop a concrete algorithm, call the entanglement feature learning (EFL), based on the random tensor network (RTN) model for the tensor network holography. We show that each RTN can be mapped to a Boltzmann machine, trained by the entanglement entropies over all subregions of a given quantum many-body state. The goal is to construct the optimal RTN that best reproduce the entanglement feature. The RTN geometry can then be interpreted as the emergent holographic geometry. We demonstrate the EFL algorithm on a 1D free fermion system and observe the emergence of the hyperbolic geometry (AdS3 spatial geometry) as we tune the fermion system towards the gapless critical point (CFT2 point).

  20. GEOMETRY – AN IMPORTANT MEANS OF EDUCATION IN THE CIVILISATION SCOPE

    OpenAIRE

    Liliana TOCARIU, PhD

    2017-01-01

    Geometry (from the Greek: γεωμετρία; geo = earth, metria = measure) is a genuine science, rooted in mathematics, which studies the plane and spatial forms of bodies from the objective or conceptual reality and the nature of the relationship that exists between them. Due to its complexity, geometry is divided into: Euclidian geometry, analytical geometry, descriptive geometry, projective geometry, kinematic geometry, surface and curve differential geometry, axiomatic geometry,...

  1. Fractal geometry in an expanding, one-dimensional, Newtonian universe.

    Science.gov (United States)

    Miller, Bruce N; Rouet, Jean-Louis; Le Guirriec, Emmanuel

    2007-09-01

    Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with new, larger, sample sizes from recent surveys, it is difficult to extract information concerning fractal properties with confidence. Similarly, three-dimensional N-body simulations with a billion particles only provide a thousand particles per dimension, far too small for accurate conclusions. With one-dimensional models these limitations can be overcome by carrying out simulations with on the order of a quarter of a million particles without compromising the computation of the gravitational force. Here the multifractal properties of two of these models that incorporate different features of the dynamical equations governing the evolution of a matter dominated universe are compared. For each model at least two scaling regions are identified. By employing criteria from dynamical systems theory it is shown that only one of them can be geometrically significant. The results share important similarities with galaxy observations, such as hierarchical clustering and apparent bifractal geometry. They also provide insights concerning possible constraints on length and time scales for fractal structure. They clearly demonstrate that fractal geometry evolves in the mu (position, velocity) space. The observed patterns are simply a shadow (projection) of higher-dimensional structure.

  2. Random geometry and Yang-Mills theory

    International Nuclear Information System (INIS)

    Froehlich, J.

    1981-01-01

    The author states various problems and discusses a very few preliminary rigorous results in a branch of mathematics and mathematical physics which one might call random (or stochastic) geometry. Furthermore, he points out why random geometry is important in the quantization of Yang-Mills theory. (Auth.)

  3. Quantification of variability in bedform geometry

    NARCIS (Netherlands)

    van der Mark, C.F.; Blom, Astrid; Hulscher, Suzanne J.M.H.

    2008-01-01

    We analyze the variability in bedform geometry in laboratory and field studies. Even under controlled steady flow conditions in laboratory flumes, bedforms are irregular in size, shape, and spacing, also in case of well-sorted sediment. Our purpose is to quantify the variability in bedform geometry.

  4. Geometry optimization of supersymmetrical molecules in quantum chemical ab-initio calculations

    International Nuclear Information System (INIS)

    Gruenbichler, H.

    1985-01-01

    One-dimensional geometry optimizations in ab-initio SCF-calculations are investigated. It is shown, that the well known standard algorithms are sometimes too expensive and can be replaced or accompanied by more recent algorithms. Two alternatives were realized in the molecule calculating program GAUSSIAN 80, basing on the Fibonacci algorithm and Kryachco potential adjustment. The algorithms were compared in terms of accuracy of results, CPU-time used and reliability of the method. The results are presented in various tables, showing the efficiency of the various methods. A survey of the usual model potentials is given and the compatibility with ab-initio data is evaluated. (Author, shortened and translated by A.N.)

  5. Conference on Number Theory and Arithmetic Geometry

    CERN Document Server

    Silverman, Joseph; Stevens, Glenn; Modular forms and Fermat’s last theorem

    1997-01-01

    This volume contains expanded versions of lectures given at an instructional conference on number theory and arithmetic geometry held August 9 through 18, 1995 at Boston University. Contributor's includeThe purpose of the conference, and of this book, is to introduce and explain the many ideas and techniques used by Wiles in his proof that every (semi-stable) elliptic curve over Q is modular, and to explain how Wiles' result can be combined with Ribet's theorem and ideas of Frey and Serre to show, at long last, that Fermat's Last Theorem is true. The book begins with an overview of the complete proof, followed by several introductory chapters surveying the basic theory of elliptic curves, modular functions, modular curves, Galois cohomology, and finite group schemes. Representation theory, which lies at the core of Wiles' proof, is dealt with in a chapter on automorphic representations and the Langlands-Tunnell theorem, and this is followed by in-depth discussions of Serre's conjectures, Galois deformations, ...

  6. DOGBONE GEOMETRY FOR RECIRCULATING ACCELERATORS

    International Nuclear Information System (INIS)

    BERG, J.S.; JOHNSTONE, C.; SUMMERS, D.

    2001-01-01

    Most scenarios for accelerating muons require recirculating acceleration. A racetrack shape for the accelerator requires particles with lower energy in early passes to traverse almost the same length of arc as particles with the highest energy. This extra arc length may lead to excess decays and excess cost. Changing the geometry to a dogbone shape, where there is a single linac and the beam turns completely around at the end of the linac, returning to the same end of the linac from which it exited, addresses this problem. In this design, the arc lengths can be proportional to the particle's momentum. This paper proposes an approximate cost model for a recirculating accelerator, attempts to make cost-optimized designs for both racetrack and dogbone geometries, and demonstrates that the dogbone geometry does appear to be more cost effective

  7. Increasing insightful thinking in analytic geometry

    NARCIS (Netherlands)

    Timmer, Mark; Verhoef, Neeltje Cornelia

    Elsewhere in this issue Ferdinand Verhulst described the discussion of the interaction of analysis and geometry in the 19th century. In modern times such discussions come up again and again. As of 2014, synthetic geometry will not be part of the Dutch 'vwo - mathematics B' programme anymore.

  8. Development and application of CATIA-GDML geometry builder

    International Nuclear Information System (INIS)

    Belogurov, S; Chernogorov, A; Ovcharenko, E; Schetinin, V; Berchun, Yu; Malzacher, P

    2014-01-01

    Due to conceptual difference between geometry descriptions in Computer-Aided Design (CAD) systems and particle transport Monte Carlo (MC) codes direct conversion of detector geometry in either direction is not feasible. The paper presents an update on functionality and application practice of the CATIA-GDML geometry builder first introduced at CHEP2010. This set of CATIAv5 tools has been developed for building a MC optimized GEANT4/ROOT compatible geometry based on the existing CAD model. The model can be exported via Geometry Description Markup Language (GDML). The builder allows also import and visualization of GEANT4/ROOT geometries in CATIA. The structure of a GDML file, including replicated volumes, volume assemblies and variables, is mapped into a part specification tree. A dedicated file template, a wide range of primitives, tools for measurement and implicit calculation of parameters, different types of multiple volume instantiation, mirroring, positioning and quality check have been implemented. Several use cases are discussed.

  9. Remarks on Hamiltonian structures in G2-geometry

    International Nuclear Information System (INIS)

    Cho, Hyunjoo; Salur, Sema; Todd, A. J.

    2013-01-01

    In this article, we treat G 2 -geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G 2 -structure; in particular, we discuss existence and make a number of identifications of the spaces of Hamiltonian structures associated to the two multisymplectic structures associated to an integrable G 2 -structure. Along the way, we prove some results in multisymplectic geometry that are generalizations of results from symplectic geometry

  10. Classical geometry Euclidean, transformational, inversive, and projective

    CERN Document Server

    Leonard, I E; Liu, A C F; Tokarsky, G W

    2014-01-01

    Features the classical themes of geometry with plentiful applications in mathematics, education, engineering, and science Accessible and reader-friendly, Classical Geometry: Euclidean, Transformational, Inversive, and Projective introduces readers to a valuable discipline that is crucial to understanding bothspatial relationships and logical reasoning. Focusing on the development of geometric intuitionwhile avoiding the axiomatic method, a problem solving approach is encouraged throughout. The book is strategically divided into three sections: Part One focuses on Euclidean geometry, which p

  11. MOCUM: A two-dimensional method of characteristics code based on constructive solid geometry and unstructured meshing for general geometries

    International Nuclear Information System (INIS)

    Yang Xue; Satvat, Nader

    2012-01-01

    Highlight: ► A two-dimensional numerical code based on the method of characteristics is developed. ► The complex arbitrary geometries are represented by constructive solid geometry and decomposed by unstructured meshing. ► Excellent agreement between Monte Carlo and the developed code is observed. ► High efficiency is achieved by parallel computing. - Abstract: A transport theory code MOCUM based on the method of characteristics as the flux solver with an advanced general geometry processor has been developed for two-dimensional rectangular and hexagonal lattice and full core neutronics modeling. In the code, the core structure is represented by the constructive solid geometry that uses regularized Boolean operations to build complex geometries from simple polygons. Arbitrary-precision arithmetic is also used in the process of building geometry objects to eliminate the round-off error from the commonly used double precision numbers. Then, the constructed core frame will be decomposed and refined into a Conforming Delaunay Triangulation to ensure the quality of the meshes. The code is fully parallelized using OpenMP and is verified and validated by various benchmarks representing rectangular, hexagonal, plate type and CANDU reactor geometries. Compared with Monte Carlo and deterministic reference solution, MOCUM results are highly accurate. The mentioned characteristics of the MOCUM make it a perfect tool for high fidelity full core calculation for current and GenIV reactor core designs. The detailed representation of reactor physics parameters can enhance the safety margins with acceptable confidence levels, which lead to more economically optimized designs.

  12. Redshift Survey Strategies

    Science.gov (United States)

    Jones, A. W.; Bland-Hawthorn, J.; Kaiser, N.

    1994-12-01

    In the first half of 1995, the Anglo-Australian Observatory is due to commission a wide field (2.1(deg) ), 400-fiber, double spectrograph system (2dF) at the f/3.3 prime focus of the AAT 3.9m bi-national facility. The instrument should be able to measure ~ 4000 galaxy redshifts (assuming a magnitude limit of b_J ~\\ 20) in a single dark night and is therefore ideally suited to studies of large-scale structure. We have carried out simple 3D numerical simulations to judge the relative merits of sparse surveys and contiguous surveys. We generate a survey volume and fill it randomly with particles according to a selection function which mimics a magnitude-limited survey at b_J = 19.7. Each of the particles is perturbed by a gaussian random field according to the dimensionless power spectrum k(3) P(k) / 2pi (2) determined by Feldman, Kaiser & Peacock (1994) from the IRAS QDOT survey. We introduce some redshift-space distortion as described by Kaiser (1987), a `thermal' component measured from pairwise velocities (Davis & Peebles 1983), and `fingers of god' due to rich clusters at random density enhancements. Our particular concern is to understand how the window function W(2(k)) of the survey geometry compromises the accuracy of statistical measures [e.g., P(k), xi (r), xi (r_sigma ,r_pi )] commonly used in the study of large-scale structure. We also examine the reliability of various tools (e.g. genus) for describing the topological structure within a contiguous region of the survey.

  13. 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.

  14. Dayside merging and cusp geometry

    International Nuclear Information System (INIS)

    Crooker, N.U.

    1979-01-01

    Geometrical considerations are presented to show that dayside magnetic merging when constrained to act only where the fields are antiparallel results in lines of merging that converge at the polar cusps. An important consequence of this geometry is that no accelerated flows are predicted across the dayside magnetopause. Acceleration owing to merging acts in opposition to the magnetosheath flow at the merging point and produces the variably directed, slower-than-magnetosheath flows observed in the entry layer. Another consequence of the merging geometry is that much of the time closed field lines constitute the subsolar region of the magnetopause. The manner in which the polar cap convection patterns predicted by the proposed geometry change as the interplanetary field is rotated through 360 0 provides a unifying description of how the observed single circular vortex and the crescent-shaped double vortex patterns mutually evolve under the influence of a single operating principle

  15. The harmonic oscillator in the forceless mechanics of Hertz and in the Riemannian space-time geometry

    International Nuclear Information System (INIS)

    Fueloep, L.

    1987-10-01

    The forceless mechanics of Hertz is a reformulation of the classical mechanics in a curved configuration space. The relationship between the forceless mechanics and the general relativity theory which uses curved Riemann spaces as well is investigated on the simple example of the harmonic oscillator. The mathematical similarities and differences and the different interpretations of similar formulas are discussed. Some formal constants of the Hertz mechanics have got concrete physical meanings in the general relativity. (D.Gy.)

  16. 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.)

  17. 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

  18. 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....

  19. Digital and discrete geometry theory and algorithms

    CERN Document Server

    Chen, Li

    2014-01-01

    This book provides comprehensive coverage of the modern methods for geometric problems in the computing sciences. It also covers concurrent topics in data sciences including geometric processing, manifold learning, Google search, cloud data, and R-tree for wireless networks and BigData.The author investigates digital geometry and its related constructive methods in discrete geometry, offering detailed methods and algorithms. The book is divided into five sections: basic geometry; digital curves, surfaces and manifolds; discretely represented objects; geometric computation and processing; and a

  20. Attitudes of High School Students towards Geometry

    Directory of Open Access Journals (Sweden)

    Esat Avcı

    2014-12-01

    Full Text Available In this research, attitudes of high school students towards geometry were investigated in terms of gender, grade, types of the field and school. Population of research includes students who were studying at high school in five distincs of Mersin in 2013-2014 academical year. Sample of research includes 935 students from twelve high schools. Attitude scale which was developed by Su-Özenir (2008 was used for data collection. For data analysis, mean, standart deviation, t test and ANOVA were used. A meaningful difference between students’ attitudes towards geometry and variance of gender and grade level wasn’t observed, on the other hand a meaningful difference according to field and school type is observed.Key Words:    Attitudes towards geometry, high school geometry lesson, attitude scale

  1. Commutative and Non-commutative Parallelogram Geometry: an Experimental Approach

    OpenAIRE

    Bertram, Wolfgang

    2013-01-01

    By "parallelogram geometry" we mean the elementary, "commutative", geometry corresponding to vector addition, and by "trapezoid geometry" a certain "non-commutative deformation" of the former. This text presents an elementary approach via exercises using dynamical software (such as geogebra), hopefully accessible to a wide mathematical audience, from undergraduate students and high school teachers to researchers, proceeding in three steps: (1) experimental geometry, (2) algebra (linear algebr...

  2. Some applications of mathematics in theoretical physics - A review

    International Nuclear Information System (INIS)

    Bora, Kalpana

    2016-01-01

    Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

  3. Some applications of mathematics in theoretical physics - A review

    Energy Technology Data Exchange (ETDEWEB)

    Bora, Kalpana [Physics Department, Gauhati University, Guwahati-781014, Assam (India)

    2016-06-21

    Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

  4. Some applications of mathematics in theoretical physics - A review

    Science.gov (United States)

    Bora, Kalpana

    2016-06-01

    Mathematics is a very beautiful subject-very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like-differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

  5. 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...

  6. A Study of Geometry Content Knowledge of Elementary Preservice Teachers

    Directory of Open Access Journals (Sweden)

    Fatma ASLAN-TUTAK

    2015-06-01

    Full Text Available The purpose of this research is to examine preservice elementary school teachers’ geometry learning as investigated by both qualitative and quantitative methods. For the qualitative investigation, narrative analysis and thematic analysis methods were used. The findings of narrative analysis indicated two main kinds of stories: as a learner and as a beginning teacher. The thematic analysis findings yield to three themes: history of learning geometry, perceptions about geometry, effective geometry instructional practices. The findings informed the quantitative investigation on geometry content knowledge for the case of quadrilaterals. During the second phase of the study, 102 participants who enrolled in the methods course completed pre and post test of teachers’ geometry content knowledge. Treatment group participants (n=54 received series of activities (geometry activities and student work analysis focusing on quadrilaterals, and control group participants (n=48 received traditional instruction. Repeated measures ANOVA results showed a significant change in treatment group participants’ geometry content knowledge. The mixed ANOVA results indicated a significant main effect of knowledge but no significant interaction between geometry content knowledge and grouping. Even though treatment group participants’ geometry content knowledge growth was significant, the difference between treatment group and control group participants’ growth in geometry content knowledge was not significant. This study informs mathematics teacher education in three important areas; limited knowledge of preservice teachers’ geometry content knowledge, integrating mathematics content and the context of teaching into methods course, and use of student work with preservice teachers.

  7. A study of geometry content knowledge of elementary preservice teachers

    Directory of Open Access Journals (Sweden)

    Fatma Aslan Tutak

    2015-06-01

    Full Text Available The purpose of this research is to examine preservice elementary school teachers’ geometry learning as investigated by both qualitative and quantitative methods. For the qualitative investigation, narrative analysis and thematic analysis methods were used. The findings of narrative analysis indicated two main kinds of stories: as a learner and as a beginning teacher. The thematic analysis findings yield to three themes: history of learning geometry, perceptions about geometry, effective geometry instructional practices. The findings informed the quantitative investigation on geometry content knowledge for the case of quadrilaterals. During the second phase of the study, 102 participants who enrolled in the methods course completed pre and post test of teachers’ geometry content knowledge. Treatment group participants (n=54 received series of activities (geometry activities and student work analysis focusing on quadrilaterals, and control group participants (n=48 received traditional instruction. Repeated measures ANOVA results showed a significant change in treatment group participants’ geometry content knowledge. The mixed ANOVA results indicated a significant main effect of knowledge but no significant interaction between geometry content knowledge and grouping. Even though treatment group participants’ geometry content knowledge growth was significant, the difference between treatment group and control group participants’ growth in geometry content knowledge was not significant. This study informs mathematics teacher education in three important areas; limited knowledge of preservice teachers’ geometry content knowledge, integrating mathematics content and the context of teaching into methods course, and use of student work with preservice teachers.

  8. Combinatorial geometry in the plane

    CERN Document Server

    Hadwiger, Hugo; Klee, Victor

    2014-01-01

    Geared toward advanced undergraduates familiar with analysis and college geometry, this concise book discusses theorems on topics restricted to the plane such as convexity, coverings, and graphs. In addition to helping students cultivate rigorous thought, the text encourages the development of mathematical intuition and clarifies the nature of mathematical research.The two-part treatment begins with specific topics including integral distances, covering problems, point set geometry and convexity, simple paradoxes involving point sets, and pure combinatorics, among other subjects. The second pa

  9. Topology and geometry for physicists

    CERN Document Server

    Nash, Charles

    1983-01-01

    Differential geometry and topology are essential tools for many theoretical physicists, particularly in the study of condensed matter physics, gravity, and particle physics. Written by physicists for physics students, this text introduces geometrical and topological methods in theoretical physics and applied mathematics. It assumes no detailed background in topology or geometry, and it emphasizes physical motivations, enabling students to apply the techniques to their physics formulas and research. ""Thoroughly recommended"" by The Physics Bulletin, this volume's physics applications range fr

  10. Fundamental concepts of geometry

    CERN Document Server

    Meserve, Bruce E

    1983-01-01

    Demonstrates relationships between different types of geometry. Provides excellent overview of the foundations and historical evolution of geometrical concepts. Exercises (no solutions). Includes 98 illustrations.

  11. 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

  12. 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...

  13. Number theory III Diophantine geometry

    CERN Document Server

    1991-01-01

    From the reviews of the first printing of this book, published as Volume 60 of the Encyclopaedia of Mathematical Sciences: "Between number theory and geometry there have been several stimulating influences, and this book records of these enterprises. This author, who has been at the centre of such research for many years, is one of the best guides a reader can hope for. The book is full of beautiful results, open questions, stimulating conjectures and suggestions where to look for future developments. This volume bears witness of the broad scope of knowledge of the author, and the influence of several people who have commented on the manuscript before publication ... Although in the series of number theory, this volume is on diophantine geometry, and the reader will notice that algebraic geometry is present in every chapter. ... The style of the book is clear. Ideas are well explained, and the author helps the reader to pass by several technicalities. Reading and rereading this book I noticed that the topics ...

  14. Graphical debugging of combinational geometry

    International Nuclear Information System (INIS)

    Burns, T.J.; Smith, M.S.

    1992-01-01

    A graphical debugger for combinatorial geometry being developed at Oak Ridge National Laboratory is described. The prototype debugger consists of two parts: a FORTRAN-based ''view'' generator and a Microsoft Windows application for displaying the geometry. Options and features of both modules are discussed. Examples illustrating the various options available are presented. The potential for utilizing the images produced using the debugger as a visualization tool for the output of the radiation transport codes is discussed as is the future direction of the development

  15. Introduction to topology and geometry

    CERN Document Server

    Stahl, Saul

    2014-01-01

    An easily accessible introduction to over three centuries of innovations in geometry Praise for the First Edition ". . . a welcome alternative to compartmentalized treatments bound to the old thinking. This clearly written, well-illustrated book supplies sufficient background to be self-contained." -CHOICE This fully revised new edition offers the most comprehensive coverage of modern geometry currently available at an introductory level. The book strikes a welcome balance between academic rigor and accessibility, providing a complete and cohesive picture of the science with an unparallele

  16. Description of SSG Geometry - phase 1

    DEFF Research Database (Denmark)

    Margheritini, Lucia; Kofoed, Jens Peter

    The purpose of the study is to define the optimized geometry for the SSG in Svaheia, Norway and to provide the responsible for the turbines with useful information to their work.......The purpose of the study is to define the optimized geometry for the SSG in Svaheia, Norway and to provide the responsible for the turbines with useful information to their work....

  17. Fractal geometry of high temperature superconductors

    International Nuclear Information System (INIS)

    Mosolov, A.B.

    1989-01-01

    Microstructural geometry of superconducting structural composites of Ag-Yba 2 Cu 3 O x system with a volumetric shave of silver from 0 to 60% is investigated by light and electron microscopy methods. It is ascertained that the structure of cermets investigated is characterized by fractal geometry which is sufficient for describing the electrical and mechanical properties of these materials

  18. Morphing the feature-based multi-blocks of normative/healthy vertebral geometries to scoliosis vertebral geometries: development of personalized finite element models.

    Science.gov (United States)

    Hadagali, Prasannaah; Peters, James R; Balasubramanian, Sriram

    2018-03-12

    Personalized Finite Element (FE) models and hexahedral elements are preferred for biomechanical investigations. Feature-based multi-block methods are used to develop anatomically accurate personalized FE models with hexahedral mesh. It is tedious to manually construct multi-blocks for large number of geometries on an individual basis to develop personalized FE models. Mesh-morphing method mitigates the aforementioned tediousness in meshing personalized geometries every time, but leads to element warping and loss of geometrical data. Such issues increase in magnitude when normative spine FE model is morphed to scoliosis-affected spinal geometry. The only way to bypass the issue of hex-mesh distortion or loss of geometry as a result of morphing is to rely on manually constructing the multi-blocks for scoliosis-affected spine geometry of each individual, which is time intensive. A method to semi-automate the construction of multi-blocks on the geometry of scoliosis vertebrae from the existing multi-blocks of normative vertebrae is demonstrated in this paper. High-quality hexahedral elements were generated on the scoliosis vertebrae from the morphed multi-blocks of normative vertebrae. Time taken was 3 months to construct the multi-blocks for normative spine and less than a day for scoliosis. Efforts taken to construct multi-blocks on personalized scoliosis spinal geometries are significantly reduced by morphing existing multi-blocks.

  19. Distinguished figures in descriptive geometry and its applications for mechanism science from the middle ages to the 17th century

    CERN Document Server

    2016-01-01

    This book consists of chapters that focus specifically on single figures that worked on Descriptive Geometry and also in Mechanisms Sciences and contain biographical notes, a survey of their work and their achievements, together with a modern interpretation of their legacy. Since Vitruvius in ancient times, and with Brunelleschi in the Renaissance, the two disciplines began to share a common direction which, over the centuries, took shape through less well-known figures until the more recent times in which Gaspard Monge worked. Over the years, a gap has been created between Descriptive Geometry and Mechanism Science, which now appear to belong to different worlds. In reality, however, there is a very close relationship between the two disciplines, with a link based on extremely solid foundations. Without the theoretical foundations of Geometry it would not be possible to draw and design mechanical parts such as gears, while in Kinematics it would be less easy to design and predict the reciprocal movements of ...

  20. Vectorising the detector geometry to optimize particle transport

    CERN Document Server

    Apostolakis, John; Carminati, Federico; Gheata, Andrei; Wenzel, Sandro

    2014-01-01

    Among the components contributing to particle transport, geometry navigation is an important consumer of CPU cycles. The tasks performed to get answers to "basic" queries such as locating a point within a geometry hierarchy or computing accurately the distance to the next boundary can become very computing intensive for complex detector setups. So far, the existing geometry algorithms employ mainly scalar optimisation strategies (voxelization, caching) to reduce their CPU consumption. In this paper, we would like to take a different approach and investigate how geometry navigation can benefit from the vector instruction set extensions that are one of the primary source of performance enhancements on current and future hardware. While on paper, this form of microparallelism promises increasing performance opportunities, applying this technology to the highly hierarchical and multiply branched geometry code is a difficult challenge. We refer to the current work done to vectorise an important part of the critica...

  1. Students’ Errors in Geometry Viewed from Spatial Intelligence

    Science.gov (United States)

    Riastuti, N.; Mardiyana, M.; Pramudya, I.

    2017-09-01

    Geometry is one of the difficult materials because students must have ability to visualize, describe images, draw shapes, and know the kind of shapes. This study aim is to describe student error based on Newmans’ Error Analysis in solving geometry problems viewed from spatial intelligence. This research uses descriptive qualitative method by using purposive sampling technique. The datas in this research are the result of geometri material test and interview by the 8th graders of Junior High School in Indonesia. The results of this study show that in each category of spatial intelligence has a different type of error in solving the problem on the material geometry. Errors are mostly made by students with low spatial intelligence because they have deficiencies in visual abilities. Analysis of student error viewed from spatial intelligence is expected to help students do reflection in solving the problem of geometry.

  2. Equations for estimating bankfull channel geometry and discharge for streams in Massachusetts

    Science.gov (United States)

    Bent, Gardner C.; Waite, Andrew M.

    2013-01-01

    Regression equations were developed for estimating bankfull geometry—width, mean depth, cross-sectional area—and discharge for streams in Massachusetts. The equations provide water-resource and conservation managers with methods for estimating bankfull characteristics at specific stream sites in Massachusetts. This information can be used for the adminstration of the Commonwealth of Massachusetts Rivers Protection Act of 1996, which establishes a protected riverfront area extending from the mean annual high-water line corresponding to the elevation of bankfull discharge along each side of a perennial stream. Additionally, information on bankfull channel geometry and discharge are important to Federal, State, and local government agencies and private organizations involved in stream assessment and restoration projects. Regression equations are based on data from stream surveys at 33 sites (32 streamgages and 1 crest-stage gage operated by the U.S. Geological Survey) in and near Massachusetts. Drainage areas of the 33 sites ranged from 0.60 to 329 square miles (mi2). At 27 of the 33 sites, field data were collected and analyses were done to determine bankfull channel geometry and discharge as part of the present study. For 6 of the 33 sites, data on bankfull channel geometry and discharge were compiled from other studies done by the U.S. Geological Survey, Natural Resources Conservation Service of the U.S. Department of Agriculture, and the Vermont Department of Environmental Conservation. Similar techniques were used for field data collection and analysis for bankfull channel geometry and discharge at all 33 sites. Recurrence intervals of the bankfull discharge, which represent the frequency with which a stream fills its channel, averaged 1.53 years (median value 1.34 years) at the 33 sites. Simple regression equations were developed for bankfull width, mean depth, cross-sectional area, and discharge using drainage area, which is the most significant explanatory

  3. Non-euclidean geometry

    CERN Document Server

    Coxeter, HSM

    1965-01-01

    This textbook introduces non-Euclidean geometry, and the third edition adds a new chapter, including a description of the two families of 'mid-lines' between two given lines and an elementary derivation of the basic formulae of spherical trigonometry and hyperbolic trigonometry, and other new material.

  4. Multiple-view, Multiple-selection Visualization of Simulation Geometry in CMS

    International Nuclear Information System (INIS)

    Bauerdick, L A T; Eulisse, G; Jones, C; McCauley, T; Osborne, I; Kovalskyi, D; Mrak Tadel, A; Tadel, M; Yagil, A

    2012-01-01

    Fireworks, the event-display program of CMS, was extended with an advanced geometry visualization package. ROOT's TGeo geometry is used as internal representation, shared among several geometry views. Each view is represented by a GUI list-tree widget, implemented as a flat vector to allow for fast searching, selection, and filtering by material type, node name, and shape type. Display of logical and physical volumes is supported. Color, transparency, and visibility flags can be modified for each node or for a selection of nodes. Further operations, like opening of a new view or changing of the root node, can be performed via a context menu. Node selection and graphical properties determined by the list-tree view can be visualized in any 3D graphics view of Fireworks. As each 3D view can display any number of geometry views, a user is free to combine different geometry-view selections within the same 3D view. Node-selection by proximity to a given point is possible. A visual clipping box can be set for each geometry view to limit geometry drawing into a specified region. Visualization of geometric overlaps, as detected by TGeo, is also supported. The geometry visualization package is used for detailed inspection and display of simulation geometry with or without the event data. It also serves as a tool for geometry debugging and inspection, facilitating development of geometries for CMS detector upgrades and for SLHC.

  5. Regional Curves of Bankfull Channel Geometry for Non-Urban Streams in the Piedmont Physiographic Province, Virginia

    Science.gov (United States)

    Lotspeich, R. Russell

    2009-01-01

    Natural-channel design involves constructing a stream channel with the dimensions, slope, and plan-view pattern that would be expected to transport water and sediment and yet maintain habitat and aesthetics consistent with unimpaired stream segments, or reaches. Regression relations for bankfull stream characteristics based on drainage area, referred to as 'regional curves,' are used in natural stream channel design to verify field determinations of bankfull discharge and stream channel characteristics. One-variable, ordinary least-squares regressions relating bankfull discharge, bankfull cross-sectional area, bankfull width, bankfull mean depth, and bankfull slope to drainage area were developed on the basis of data collected at 17 streamflow-gaging stations in rural areas with less than 20 percent urban land cover within the basin area (non-urban areas) of the Piedmont Physiographic Province in Virginia. These regional curves can be used to estimate the bankfull discharge and bankfull channel geometry when the drainage area of a watershed is known. Data collected included bankfull cross-sectional geometry, flood-plain geometry, and longitudinal profile data. In addition, particle-size distributions of streambed material were determined, and data on basin characteristics were compiled for each reach. Field data were analyzed to determine bankfull cross-sectional area, bankfull width, bankfull mean depth, bankfull discharge, bankfull channel slope, and D50 and D84 particle sizes at each site. The bankfull geometry from the 17 sites surveyed during this study represents the average of two riffle cross sections for each site. Regional curves developed for the 17 sites had coefficient of determination (R2) values of 0.950 for bankfull cross-sectional area, 0.913 for bankfull width, 0.915 for bankfull mean depth, 0.949 for bankfull discharge, and 0.497 for bankfull channel slope. The regional curves represent conditions for streams with defined channels and bankfull

  6. Understanding Students’ Ideas about the Geometry of the Universe

    Science.gov (United States)

    Coble, Kimberly A.; Conlon, Mallory; Bailey, Janelle M.

    2017-06-01

    As astronomers further develop an understanding of the geometry of the Universe, it is essential to study students’ ideas so that instructors can communicate the field’s current status more effectively to their students. In this study, we examine undergraduate students’ pre- instruction ideas in general education astronomy courses (ASTRO 101) at three institutions through pre-course surveys given during the first week of instruction [N ~ 265]. We also examine students’ post-instruction ideas at a single institution through exam questions [N ~ 75] and interviews. Responses are analyzed through an iterative process of identifying self-emergent themes. We examine not only what students think the curvature of the universe is, but also "how we know." We find that many students think the Universe is “round” or that we cannot measure its curvature. Additionally, popular visualizations may enforce incorrect ideas.

  7. Geometry, topology, and string theory

    Energy Technology Data Exchange (ETDEWEB)

    Varadarajan, Uday [Univ. of California, Berkeley, CA (United States)

    2003-01-01

    A variety of scenarios are considered which shed light upon the uses and limitations of classical geometric and topological notions in string theory. The primary focus is on situations in which D-brane or string probes of a given classical space-time see the geometry quite differently than one might naively expect. In particular, situations in which extra dimensions, non-commutative geometries as well as other non-local structures emerge are explored in detail. Further, a preliminary exploration of such issues in Lorentzian space-times with non-trivial causal structures within string theory is initiated.

  8. Stochastic geometry and its applications

    CERN Document Server

    Chiu, Sung Nok; Kendall, Wilfrid S; Mecke, Joseph

    2013-01-01

    An extensive update to a classic text Stochastic geometry and spatial statistics play a fundamental role in many modern branches of physics, materials sciences, engineering, biology and environmental sciences. They offer successful models for the description of random two- and three-dimensional micro and macro structures and statistical methods for their analysis. The previous edition of this book has served as the key reference in its field for over 18 years and is regarded as the best treatment of the subject of stochastic geometry, both as a subject with vital a

  9. Spatial geometry and special relativity

    DEFF Research Database (Denmark)

    Kneubil, Fabiana Botelho

    2016-01-01

    In this work, it is shown the interplay of relative and absolute entities, which are present in both spatial geometry and special relativity. In order to strengthen the understanding of special relativity, we discuss firstly an instance of geometry and the existence of both frame......-dependent and frame-independent entities. We depart from a subject well known by students, which is the three-dimensional geometric space in order to compare, afterwards, with the treatment of four-dimensional space in the special relativity. The differences and similarities between these two subjects are also...

  10. Geometry, topology, and string theory

    International Nuclear Information System (INIS)

    Varadarajan, Uday

    2003-01-01

    A variety of scenarios are considered which shed light upon the uses and limitations of classical geometric and topological notions in string theory. The primary focus is on situations in which D-brane or string probes of a given classical space-time see the geometry quite differently than one might naively expect. In particular, situations in which extra dimensions, non-commutative geometries as well as other non-local structures emerge are explored in detail. Further, a preliminary exploration of such issues in Lorentzian space-times with non-trivial causal structures within string theory is initiated

  11. Simultaneous calibration phantom commission and geometry calibration in cone beam CT

    Science.gov (United States)

    Xu, Yuan; Yang, Shuai; Ma, Jianhui; Li, Bin; Wu, Shuyu; Qi, Hongliang; Zhou, Linghong

    2017-09-01

    Geometry calibration is a vital step for describing the geometry of a cone beam computed tomography (CBCT) system and is a prerequisite for CBCT reconstruction. In current methods, calibration phantom commission and geometry calibration are divided into two independent tasks. Small errors in ball-bearing (BB) positioning in the phantom-making step will severely degrade the quality of phantom calibration. To solve this problem, we propose an integrated method to simultaneously realize geometry phantom commission and geometry calibration. Instead of assuming the accuracy of the geometry phantom, the integrated method considers BB centers in the phantom as an optimized parameter in the workflow. Specifically, an evaluation phantom and the corresponding evaluation contrast index are used to evaluate geometry artifacts for optimizing the BB coordinates in the geometry phantom. After utilizing particle swarm optimization, the CBCT geometry and BB coordinates in the geometry phantom are calibrated accurately and are then directly used for the next geometry calibration task in other CBCT systems. To evaluate the proposed method, both qualitative and quantitative studies were performed on simulated and realistic CBCT data. The spatial resolution of reconstructed images using dental CBCT can reach up to 15 line pair cm-1. The proposed method is also superior to the Wiesent method in experiments. This paper shows that the proposed method is attractive for simultaneous and accurate geometry phantom commission and geometry calibration.

  12. Gravity is Geometry.

    Science.gov (United States)

    MacKeown, P. K.

    1984-01-01

    Clarifies two concepts of gravity--those of a fictitious force and those of how space and time may have geometry. Reviews the position of Newton's theory of gravity in the context of special relativity and considers why gravity (as distinct from electromagnetics) lends itself to Einstein's revolutionary interpretation. (JN)

  13. Stochastic analysis for Poisson point processes Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry

    CERN Document Server

    Peccati, Giovanni

    2016-01-01

    Stochastic geometry is the branch of mathematics that studies geometric structures associated with random configurations, such as random graphs, tilings and mosaics. Due to its close ties with stereology and spatial statistics, the results in this area are relevant for a large number of important applications, e.g. to the mathematical modeling and statistical analysis of telecommunication networks, geostatistics and image analysis. In recent years – due mainly to the impetus of the authors and their collaborators – a powerful connection has been established between stochastic geometry and the Malliavin calculus of variations, which is a collection of probabilistic techniques based on the properties of infinite-dimensional differential operators. This has led in particular to the discovery of a large number of new quantitative limit theorems for high-dimensional geometric objects. This unique book presents an organic collection of authoritative surveys written by the principal actors in this rapidly evolvi...

  14. Statistics and geometry of cosmic voids

    International Nuclear Information System (INIS)

    Gaite, José

    2009-01-01

    We introduce new statistical methods for the study of cosmic voids, focusing on the statistics of largest size voids. We distinguish three different types of distributions of voids, namely, Poisson-like, lognormal-like and Pareto-like distributions. The last two distributions are connected with two types of fractal geometry of the matter distribution. Scaling voids with Pareto distribution appear in fractal distributions with box-counting dimension smaller than three (its maximum value), whereas the lognormal void distribution corresponds to multifractals with box-counting dimension equal to three. Moreover, voids of the former type persist in the continuum limit, namely, as the number density of observable objects grows, giving rise to lacunar fractals, whereas voids of the latter type disappear in the continuum limit, giving rise to non-lacunar (multi)fractals. We propose both lacunar and non-lacunar multifractal models of the cosmic web structure of the Universe. A non-lacunar multifractal model is supported by current galaxy surveys as well as cosmological N-body simulations. This model suggests, in particular, that small dark matter halos and, arguably, faint galaxies are present in cosmic voids

  15. Recent applications of harmonic analysis to function spaces, differential equations, and data science novel methods in harmonic analysis

    CERN Document Server

    Gia, Quoc; Mayeli, Azita; Mhaskar, Hrushikesh; Zhou, Ding-Xuan

    2017-01-01

    The second of a two volume set on novel methods in harmonic analysis, this book draws on a number of original research and survey papers from well-known specialists detailing the latest innovations and recently discovered links between various fields. Along with many deep theoretical results, these volumes contain numerous applications to problems in signal processing, medical imaging, geodesy, statistics, and data science. The chapters within cover an impressive range of ideas from both traditional and modern harmonic analysis, such as: the Fourier transform, Shannon sampling, frames, wavelets, functions on Euclidean spaces, analysis on function spaces of Riemannian and sub-Riemannian manifolds, Fourier analysis on manifolds and Lie groups, analysis on combinatorial graphs, sheaves, co-sheaves, and persistent homologies on topological spaces. Volume II is organized around the theme of recent applications of harmonic analysis to function spaces, differential equations, and data science, covering topics such a...

  16. Frames and other bases in abstract and function spaces novel methods in harmonic analysis

    CERN Document Server

    Gia, Quoc; Mayeli, Azita; Mhaskar, Hrushikesh; Zhou, Ding-Xuan

    2017-01-01

    The first of a two volume set on novel methods in harmonic analysis, this book draws on a number of original research and survey papers from well-known specialists detailing the latest innovations and recently discovered links between various fields. Along with many deep theoretical results, these volumes contain numerous applications to problems in signal processing, medical imaging, geodesy, statistics, and data science. The chapters within cover an impressive range of ideas from both traditional and modern harmonic analysis, such as: the Fourier transform, Shannon sampling, frames, wavelets, functions on Euclidean spaces, analysis on function spaces of Riemannian and sub-Riemannian manifolds, Fourier analysis on manifolds and Lie groups, analysis on combinatorial graphs, sheaves, co-sheaves, and persistent homologies on topological spaces. Volume I is organized around the theme of frames and other bases in abstract and function spaces, covering topics such as: The advanced development of frames, including ...

  17. Considering Variable Road Geometry in Adaptive Vehicle Speed Control

    Directory of Open Access Journals (Sweden)

    Xinping Yan

    2013-01-01

    Full Text Available Adaptive vehicle speed control is critical for developing Advanced Driver Assistance Systems (ADAS. Vehicle speed control considering variable road geometry has become a hotspot in ADAS research. In this paper, first, an exploration of intrinsic relationship between vehicle operation and road geometry is made. Secondly, a collaborative vehicle coupling model, a road geometry model, and an AVSC, which can respond to variable road geometry in advance, are developed. Then, based on H∞ control method and the minimum energy principle, a performance index is specified by a cost function for the proposed AVSC, which can explicitly consider variable road geometry in its optimization process. The proposed AVSC is designed by the Hamilton-Jacobi Inequality (HJI. Finally, simulations are carried out by combining the vehicle model with the road geometry model, in an aim of minimizing the performance index of the AVSC. Analyses of the simulation results indicate that the proposed AVSC can automatically and effectively regulate speed according to variable road geometry. It is believed that the proposed AVSC can be used to improve the economy, comfort, and safety effects of current ADAS.

  18. A Gyrovector Space Approach to Hyperbolic Geometry

    CERN Document Server

    Ungar, Abraham

    2009-01-01

    The mere mention of hyperbolic geometry is enough to strike fear in the heart of the undergraduate mathematics and physics student. Some regard themselves as excluded from the profound insights of hyperbolic geometry so that this enormous portion of human achievement is a closed door to them. The mission of this book is to open that door by making the hyperbolic geometry of Bolyai and Lobachevsky, as well as the special relativity theory of Einstein that it regulates, accessible to a wider audience in terms of novel analogies that the modern and unknown share with the classical and familiar. T

  19. Computational commutative and non-commutative algebraic geometry

    CERN Document Server

    Cojocaru, S; Ufnarovski, V

    2005-01-01

    This publication gives a good insight in the interplay between commutative and non-commutative algebraic geometry. The theoretical and computational aspects are the central theme in this study. The topic is looked at from different perspectives in over 20 lecture reports. It emphasizes the current trends in commutative and non-commutative algebraic geometry and algebra. The contributors to this publication present the most recent and state-of-the-art progresses which reflect the topic discussed in this publication. Both researchers and graduate students will find this book a good source of information on commutative and non-commutative algebraic geometry.

  20. Supersymmetric geometries of IIA supergravity III

    International Nuclear Information System (INIS)

    Gran, Ulf; Papadopoulos, George; Schultz, Christian von

    2016-01-01

    We find that (massive) IIA backgrounds that admit a G 2 ⋉ℝ 8 invariant Killing spinor must exhibit a null Killing vector field which leaves the Killing spinor invariant and that the rotation of the Killing vector field satisfies a certain g 2 instanton condition. This result together with those in http://dx.doi.org/10.1007/JHEP05(2014)024 and http://dx.doi.org/10.1007/JHEP12(2015)113 complete the classification of geometries of all (massive) IIA backgrounds that preserve one supersymmetry. We also explore the geometry of a class of backgrounds which admit a G 2 ⋉ℝ 8 invariant Killing spinor and where in addition an appropriate 1-form bilinear vanishes. In all cases, we express the fluxes of the theory in terms of the geometry.