Showing papers on "Space (mathematics) published in 1993"

The Geometry of Lagrange Spaces: Theory and Applications

Abstract: I. Fibre Bundles, General Theory. II. Connections in Fibre Bundles. III. Geometry of the Total Space of a Vector Bundle. IV. Geometrical Theory of Embeddings of Vector Bundles. V. Einstein Equations. VI. Generalized Einstein--Yang--Mills Equations. VII. Geometry of the Total Space of a Tangent Bundle. VIII. Finsler Spaces. IX. Lagrange Spaces. X. Generalized Lagrange Space. XI. Applications of the GLn Spaces with the Metric Tensor e2sigma(x,y)gammaij(x,y). XII. Relativistic Geometrical Optics. XIII. Geometry of Time Dependent Lagrangians. Bibliography. Index.

Approximations of continuous functionals by neural networks with application to dynamic systems

Abstract: The paper gives several strong results on neural network representation in an explicit form. Under very mild conditions a functional defined on a compact set in C(a, b) or L/sup p/(a, b), spaces of infinite dimensions, can be approximated arbitrarily well by a neural network with one hidden layer. The results are a significant development beyond earlier work, where theorems of approximating continuous functions defined on a finite-dimensional real space by neural networks with one hidden layer were given. All the results are shown to be applicable to the approximation of the output of dynamic systems at any particular time. >

Discrete Groups of Motions of Spaces of Constant Curvature

Abstract: Discrete groups of motions of spaces of constant curvature, as well as other groups that can be regarded as such (although they may be defined differently), arise naturally in different areas of mathematics and its applications. Examples are the symmetry groups of regular polyhedra, symmetry groups of ornaments and crystallographic structures, discrete groups of holomorphic transformations arising in the uniformization theory of Riemannian surfaces, fundamental groups of space forms, groups of integer Lorentz transformations etc. (see Chapter 1). Their study fills a brilliant page in the development of geometry.

Average Optimality in Dynamic Programming with General State Space

Abstract: A Markovian decision model with general state space, compact action space, and the average cost as criterion is considered. The existence of an optimal policy is shown via an optimality inequality in terms of the minimal average cost g and a relative value function w. The existence of some w is usually shown via relative compactness in a space of real-valued functions on the state space. Here it shall be shown that one can instead do with pointwise relative compactness in the set of real numbers if one makes use of a generalized lower limit of functions. An application to an inventory model is given.

Application of Hilbert-space coupled-cluster theory to simple (H 2 ) 2 model systems. II. Nonplanar models

Abstract: In this series, the recently developed explicit formalism of orthogonally spin-adapted Hibert space (or state universal), multireference (MR) coupled-cluster (CC) theory, exploiting the model space spanned by two closed-shell-type reference configurations, is applied to a simple minimum-basis-set four-electron model system consisting of two interacting hydrogen molecules in various geometrical arrangements. In this paper, we examine the nonplanar geometries of this system, generally referred to as the T4 models, and their special cases designated as P4 and V4 models. They correspond to different cross sections of the H[sub 4] potential-energy hypersurface, involving the dissociation or simultaneous stretching of two H---H bonds. They involve various quasidegeneracy types, including the orbital and configurational degeneracies, the twofold degeneracy of the ground electronic state and interesting cases of broken-symmetry solutions. We employ the CC with singles and doubles (SD) approximation, so that the cluster operators are approximated by their one- and two-body components. Comparing the resulting CC energies with exact values, which are easily obtained for these models by using the full configuration-interaction method, and performing a cluster analysis of the exact solutions, we assess the performance of various MRCC Hilbert-space approaches at both linear and nonlinear levels of approximation, while a continuous transition ismore » being made between the degenerate and nondegenerate or strongly correlated regimes. We elucidate the sources and the type of singular behavior in both linear and nonlinear versions of MRCC theory, examine the role played by various intruder states, and discuss the potential usefulness of broken-symmetry MRCCSD solutions.« less

Geometry and integrability of topological-antitopological fusion

Abstract: Integrability of equations of topological-antitopological fusion (being proposed by Cecotti and Vafa) describing the ground state metric on a given 2D topological field theory (TFT) model, is proved. For massive TFT models these equations are reduced to a universal form (being independent on the given TFT model) by gauge transformations. For massive perturbations of topological conformal field theory models the separatrix solutions of the equations bounded at infinity are found by the isomonodromy deformations method. Also it is shown that the ground state metric together with some part of the underlined TFT structure can be parametrized by pluriharmonic maps of the coupling space to the symmetric space of real positive definite quadratic forms.

Long range scattering for non-linear Schrödinger and Hartree equations in space dimension n≥2

Abstract: We consider the scattering problem for the non-linear Schrodinger (NLS) equation with a power interaction with critical powerp=1+2/n in space dimensionsn=2 and 3 and for the Hartree equation with potential |x|−1 in space dimensionn≥2. We prove the existence of modified wave operators in theL2 sense on a dense set of small and sufficiently regular asymptotic states.

The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics

Abstract: The Riemannian metric structure of the shape space $\sum^k_m$ for $k$ labelled points in $\mathbb{R}^m$ was given by Kendall for the atypically simple situations in which $m = 1$ or 2 and $k \geq 2$. Here we deal with the general case $(m \geq 1, k \geq 2)$ by using the properties of Riemannian submersions and warped products as studied by O'Neill. The approach is via the associated size-and-shape space that is the warped product of the shape space and the half-line $\mathbb{R}_+$ (carrying size), the warping function being equal to the square of the size. When combined with parallel studies by Le of the corresponding global geodesic geometry, the results obtained here determine the environment in which shape-statistical calculations have to be acted out. Finally three different applications are discussed that illustrate the theory and its use in practice.

The singular value decomposition and long and short spaces of noisy matrices

Abstract: Using geometrical, algebraic, and statistical arguments, it is clarified why and when the singular value decomposition is successful in so-called subspace methods. First the concepts of long and short spaces are introduced, and a fundamental asymmetry in the consistency properties of the estimates is discussed. The model, which is associated with the short space, can be estimated consistently, but the estimates of the original data, which follow from the long space, are always inconsistent. An expression is found for the asymptotic bias in terms of canonical angles, which can be estimated from the data. This allows all equivalent reconstructions of the original signals to be described as a matrix ball, the center of which is the minimum variance estimate. Remarkably, the canonical angles also appear in the optimal weighting that is used in weighted subspace fitting approaches. The results are illustrated with a numerical simulation. A number of examples are discussed. >

The monomial-divisor mirror map

Abstract: For each family of Calabi-Yau hypersurfaces in toric varieties, Batyrev has proposed a possible mirror partner (which is also a family of Calabi-Yau hypersurfaces). We explain a natural construction of the isomorphism between certain Hodge groups of these hypersurfaces, as predicted by mirror symmetry, which we call the monomial-divisor mirror map. We indicate how this map can be interpreted as the differential of the expected mirror isomorphism between the moduli spaces of the two Calabi-Yau manifolds. We formulate a very precise conjecture about the form of that mirror isomorphism, which when combined with some earlier conjectures of the third author would completely specify it. We then conclude that the moduli spaces of the nonlinear sigma models whose targets are the different birational models of a Calabi-Yau space should be connected by analytic continuation, and that further analytic continuation should lead to moduli spaces of other kinds of conformal field theories. (This last conclusion was first drawn by Witten.)

Algebraic Structures of Rough Sets

Abstract: This paper deals with some algebraic and set-theoretical properties of rough sets. Our considerations are based on the original conception of rough sets formulated by Pawlak [4, 5]. Let U be any fixed non-empty set traditionally called the universe and let R be an equivalence relation on U. The pair A = (U, R) is called the approximation space. We will call the equivalence classes of the relation R the elementary sets. We denote the family of elementary sets by U/R. We assume that the empty set is also an elementary set. Every union of elementary sets will be called a composed set. We denote the family of composed sets by ComR. We can characterize each set X ⊆ U using the composed sets [5].

Ricci flow, Einstein metrics and space forms

Abstract: The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) L 2 -pinched Riemannian manifolds can be deformed to space forms through Ricci flow

Quantum affine symmetry in vertex models

Abstract: We study the higher spin analogs of the six-vertex model on the basis of its symmetry under the quantum affine algebra . Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/ annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin 1/2, and that the n-particle space has an RSOS type structure rather than a simple tensor product of the one-particle space. This agrees with the picture proposed earlier by Reshetikhin.

Factorisation of generalised theta functions. I

Abstract: We prove a version of “factorisation”, relating the space of sections of theta bundles on the moduli spaces of (parabolic, rank 2) vector bundles on curves of genusg andg−1.

The dynamics of perturbations of the contracting Lorenz attractor

Abstract: We show here that by modifying the eigenvalues λ2 0 by acontracting condition λ3+λ1 < 0, we can obtain vector fields exhibiting transitive non-hyperbolic attractors which are persistent in the following measure theoretical sense: They correspond to a positive Lebesgue measure set in a twoparameter space Actually, there is a codimension-two submanifold in the space of all vector fields, whose elements are full density points for the set of vector fields that exhibit a contracting Lorenz-like attractor in generic two parameter families through them On the other hand, for an open and dense set of perturbations, the attractor breaks into one or at most two attracting periodic orbits, the singularity, a hyperbolic set and a set of wandering orbits linking these objects

String Propagation in Gravitational Wave Backgrounds

Abstract: The Conformal Field Theory of the current algebra of the centrally extended 2-d Euclidean group is analyzed. Its representations can be written in terms of four free fields (without background charge) with signature ($-$+++). We construct all irreducible representations of the current algebra with unitary base out of the free fields and their orbifolds. This is used to investigate the spectrum and scattering of strings moving in the background of a gravitational wave. We find that all the dynamics happens in the transverse space or the longitunal one but not both.

On Hibert's Metric for Simplices

Abstract: . For any bounded convex open subset C of a finite dimensional real vector space, we review the canonical Hilbert metric defined on C and we investigate the corresponding group of isometries. In case C is an open 2-simplex S , we show that the resulting space is isometric to ℝ 2 with a norm such that the unit ball is a regular hexagon, and that the central symmetry in this plane corresponds to the quadratic transformation associated to S . Finally, we discuss briefly Hilbert's metric for symmetric spaces and we state some open problems. Generalities on Hilbert metrics The first proposition below comes from a letter of D. Hilbert to F. Klein [Hil]. It is discussed in several other places, such as sections 28, 29 and 50 of [BuK], and chapter 18 of [Bui], and [Bea]. There are also nice applications of Hilbert metrics to the classical Perron-Frobenius Theorem [Sae], [KoP] and to various generalizations in functional analysis [Bir], [Bus]. Let V be a real affine space, assumed here to be finite dimensional (except in Remark 3.3), and let C be a non empty bounded convex open subset of V . We want to define a metric on C which, in the special case where C is the open unit disc of the complex plane, gives the projective model of the hyperbolic plane (sometimes called the “Klein model”). Let x,y ∈ C . If x = y , one sets obviously d ( x,y ) = 0. Otherwise, the well defined affine line l x,y ⊂ V containing x and y cuts the boundary of C in two points, say u on the side of x and v on the side of y ; see Figure 1.

Euclidean quantum gravity

Abstract: In these lectures I am going to describe an approach to Quantum Gravity using path integrals in the Euclidean regime i.e. over positive definite metrics. (Strictly speaking, Riemannian would be more appropriate but it has the wrong connotations). The motivation for this is the belief that the topological properties of the gravitational fields play an essential role in Quantum Theory. Attempts to quantize gravity ignoring the topological possibilities and simply drawing Feynman diagrams corresponding to perturbations around flat space have not been very successful: there seem to be an infinite sequence of undetermined renormalization parameters. The situation is slightly better with supergravity theories; the undetermined renormalization parameters seem to come in only at the third and higher loops around flat space but perturbations around metrics that are topologically non-trivial introduce undetermined parameters even at the one loop level [1] [27] as I shall show later on.

Real hypersurfaces of indefinite Kaehler manifolds

Abstract: We show the existence of ( ϵ )-almost contact metric structures and give examples of ( ϵ )-Sasakian manifolds. Then we get a classification theorem for real hypersurfaces of indefinite complex space-forms with parallel structure vector field. We prove that ( ϵ )-Sasakian real hypersurfaces of a semi-Euclidean space are either open sets of the pseudosphere S2S2n

Examples of singular reduction

Abstract: The construction of the reduced space for a symplectic manifold with symmetry, as formalized by Marsden and Weinstein [13], has proved to be very useful in many areas of mathematics ranging from classical mechanics to algebraic geometry. In the ideal situation, which requires the value of the moment map to be weakly regular, the reduced space is again a symplectic manifold. A lot of work has been done in the last ten years in the hope of finding a ‘correct’ reduction procedure in the case of singular values. For example, Arms, Gotay and Jennings describe several approaches to reduction in [4]. At some point it has also been observed by workers in the field that in all examples the level set of a moment map modulo the appropriate group action is a union of symplectic manifolds. Recently Otto has proved that something similar does indeed hold, namely that such a quotient is a union of symplectic orbifolds [16]. Independently two of us, R. Sjamaar and E. Lerman, have proved a stronger result [21]. We proved that in the case of proper actions the reduced space, which we simply took to be the level set modulo the action, is a stratified symplectic space. Thereby we obtained a global description of the possible dynamics, a procedure for lifting the dynamics to the original space and a local characterization of the singularities of the reduced space. (The precise definitions will be given below.) The goal of this paper is twofold. First of all, we would like to present a number of examples that illustrate the general theory. Secondly, in computing the examples we have noticed that many familiar methods for computing reduced spaces work nicely in the singular situations. For instance, in the case of a lifted action on a cotangent bundle the reduced space at the zero level is the ‘cotangent bundle’ of the orbit space. And in some cases the reduced space can be identified with the closure of a coadjoint orbit.

A radius sphere theorem.

Abstract: The purpose of this paper is to present an optimal sphere theorem for metric spaces analogous to the celebrated Rauch-Berger-Klingenberg Sphere Theorem and the Diameter Sphere Theorem in riemannian geometry. There has lately been considerable interest in studying spaces which are more singular than riemannian manifolds. A natural reason for doing this is because Gromov-Hausdorff limits of riemannian manifolds are almost never riemannian manifolds, but usually only inner metric spaces with various nice properties. The kind of spaces we wish to study here are the so-called Alexandrov spaces. Alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the distance comparison sense. This definition might seem a little ambiguous since there are many ways in which one can define finite dimensionality and lower curvature bounds. The foundational work by Plaut in [PI], however, shows that these different possibilities for definitions are equivalent. The structure of Alexandrov spaces was studied in [BGP], [PI] and [P]. In particular if X is an Alexandrov space and p e X then the space of directions Ep at p is an Alexandrov space of one less dimension and with curvature > 1. Furthermore a neighborhood of p in X is homeomorphic to the linear cone over £„ . One of the important implications of this is that the local structure of n-dimensional Alexandrov spaces is determined by the structure of (n-l)-dimensional Alexandrov spaces with curvature > 1. Sphere theorems in this context seem to be particularly interesting. For if one can give geometric characterizations of