Showing papers on "Space (mathematics) published in 2010"

Applications of Hyperstructure Theory

Abstract: Introduction. Basic notions and results on Hyperstructure Theory. 1: Some topics of Geometry. 1. Descriptive geometries and join spaces. 2. Spherical geometries and join spaces. 3. Projective geometries and join spaces. 4. Multivalued loops and projective geometries. 2: Graphs and Hypergraphs. 1. Generalized graphs and hypergroups. 2. Chromatic quasi-canonical hypergroups. 3. Hypergroups induced by paths of a direct graph. 4. Hypergraphs and hypergroups. 5. On the hypergroup HGamma associated with a hypergraph Gamma. 6. Other hyperstructures associated with hypergraphs. 3: Binary Relations. 1. Quasi-order hypergroups. 2. Hypergroups associated with binary relations. 3. Hypergroups associated with union, intersection, direct product, direct limit of relations. 4. Relation beta in semihypergroups. 4: Lattices. 1. Distributive lattices and join spaces. 2. Lattice ordered join space. 3. Modular lattices and join spaces. 4. Direct limit and inverse limit of join spaces associated with lattices. 5. Hyperlattices and join spaces. 5: Fuzzy sets and rough sets. 2. Direct limit and inverse limit of join spaces associated with fuzzy subsets. 3. Rough sets, fuzzy subsets and join spaces. 4. Direct limits and inverse limits of join spaces associated with rough sets. 5. Hyperstructures and Factor Spaces. 6. Hypergroups induced by a fuzzy subset. Fuzzy hypergroups. 7. Fuzzy subhypermodules over fuzzy hyperrings. 8. On Chinese hyperstructures. 6: Automata. 1. Language theory and hyperstructures. 2. Automata and hyperstructures. 3. Automata and quasi-order hypergroups. 7: Cryptography. 1. Algebraic cryptography and hypergroupoids. 2. Cryptographic interpretation of some hyperstructures. 8: Codes. 1. Steiner hypergroupoids and Steiner systems. 2. Some basic notions about codes. 3. Steiner hypergroups and codes. 9: Median algebras, Relation algebras, C-algebras. 1. Median algebras and join spaces. 2. Relation algebras and quasi-canonical hypergroups. 3. C-algebras and quasi-canonical hypergroups. 10: Artificial Intelligence. 1. Generalized intervals. Connections with quasi-canonical hypergroups. 2. Weak representations of interval algebras. 11: Probabilities. Bibliography.

Analytic Continuation Of Chern-Simons Theory

Abstract: The title of this article refers to analytic continuation of three-dimensional Chern-Simons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S^3). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the 'physical Hilbert space' of a twisted version of N=4 super Yang-Mills theory in four dimensions.

Twisted geometries: A geometric parametrization of SU(2) phase space

Abstract: A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this paper we show how to parametrize this phase space in terms of quantities describing the intrinsic and extrinsic geometry of the triangulation dual to the graph. These are defined by the assignment to each face of its area, the two unit normals as seen from the two polyhedra sharing it, and an additional angle related to the extrinsic curvature. These quantities do not define a Regge geometry, since they include extrinsic data, but a looser notion of discrete geometry which is twisted in the sense that it is locally well-defined, but the local patches lack a consistent gluing among each other. We give the Poisson brackets among the new variables, and exhibit a symplectomorphism which maps them into the Poisson brackets of loop gravity. The new parametrization has the advantage of a simple description of the gauge-invariant reduced phase space, which is given by a product of phase spaces associated to edges and vertices, and it also provides an Abelianization of the SU(2) connection. The results are relevant for the construction of coherent states and, as a byproduct, contribute to clarify the connection between loop gravity and its subset corresponding to Regge geometries.

Potential Theory and Dynamics on the Berkovich Projective Line

Abstract: The purpose of this book is to develop the foundations of potential theory and rational dynamics on the Berkovich projective line over an arbitrary complete, algebraically closed non-Archimedean field. In addition to providing a concrete and 'elementary' introduction to Berkovich analytic spaces and to potential theory and rational iteration on the Berkovich line, the book contains applications to arithmetic geometry and arithmetic dynamics. A number of results in the book are new, and most have not previously appeared in book form. Three appendices - on analysis, R-trees, and Berkovich's general theory of analytic spaces - are included to make the book as self-contained as possible. The authors first give a detailed description of the topological structure of the Berkovich projective line and then introduce the Hsia kernel, the fundamental kernel for potential theory. Using the theory of metrized graphs, they define a Laplacian operator on the Berkovich line and construct theories of capacities, harmonic and subharmonic functions, and Green's functions, all of which are strikingly similar to their classical complex counterparts. After developing a theory of multiplicities for rational functions, they give applications to non-Archimedean dynamics, including local and global equidistribution theorems, fixed point theorems, and Berkovich space analogues of many fundamental results from the classical Fatou-Julia theory of rational iteration. They illustrate the theory with concrete examples and exposit Rivera-Letelier's results concerning rational dynamics over the field of p-adic complex numbers. They also establish Berkovich space versions of arithmetic results such as the Fekete-Szego theorem and Bilu's equidistribution theorem.

Domain Decomposition Preconditioners for Multiscale Flows in High Contrast Media: Reduced Dimension Coarse Spaces

Abstract: In this paper, robust preconditioners for multiscale flow problems are investigated. We consider elliptic equations with highly varying coefficients. We design and analyze two-level domain decomposition preconditioners that converge independent of the contrast in the media properties. The coarse spaces are constructed using selected eigenvectors of a local spectral problem. Our new construction enriches any given initial coarse space to make it suitable for high-contrast problems. Using the initial coarse space we construct local mass matrices for the local eigenvalue problems. We show that there is a gap in the spectrum of the eigenvalue problem when high-conductivity regions are disconnected. The eigenvectors corresponding to small, asymptotically vanishing eigenvalues are chosen to construct an enrichment of the initial coarse space. Only via a judicious choice of the initial space do we reduce the dimension of the resulting coarse space. Classical coarse basis functions such as multiscale or energy mi...

Transfer Learning on Heterogenous Feature Spaces via Spectral Transformation

Abstract: Labeled examples are often expensive and time-consuming to obtain. One practically important problem is: can the labeled data from other related sources help predict the target task, even if they have (a) different feature spaces (e.g., image vs. text data), (b) different data distributions, and (c) different output spaces? This paper proposes a solution and discusses the conditions where this is possible and highly likely to produce better results. It works by first using spectral embedding to unify the different feature spaces of the target and source data sets, even when they have completely different feature spaces. The principle is to cast into an optimization objective that preserves the original structure of the data, while at the same time, maximizes the similarity between the two. Second, a judicious sample selection strategy is applied to select only those related source examples. At last, a Bayesian-based approach is applied to model the relationship between different output spaces. The three steps can bridge related heterogeneous sources in order to learn the target task. Among the 12 experiment data sets, for example, the images with wavelet-transformed-based features are used to predict another set of images whose features are constructed from color-histogram space. By using these extracted examples from heterogeneous sources, the models can reduce the error rate by as much as~50\%, compared with the methods using only the examples from the target task.

Global aspects of the space of 6D N = 1 supergravities

Abstract: We perform a global analysis of the space of consistent 6D quantum gravity theories with N = 1 supersymmetry, including models with multiple tensor multiplets. We prove that for theories with fewer than T = 9 tensor multiplets, a finite number of distinct gauge groups and matter content are possible. We find infinite families of field combinations satisfying anomaly cancellation and admitting physical gauge kinetic terms for T > 8. We find an integral lattice associated with each apparently-consistent supergravity theory; this lattice is determined by the form of the anomaly polynomial. For models which can be realized in F-theory, this anomaly lattice is related to the intersection form on the base of the F-theory elliptic fibration. The condition that a supergravity model have an F-theory realization imposes constraints which can be expressed in terms of this lattice. The analysis of models which satisfy known low-energy consistency conditions and yet violate F-theory constraints suggests possible novel constraints on low-energy supergravity theories.

Light and heavy mesons in a soft-wall holographic approach

Abstract: We study the mass spectrum and decay constants of light and heavy mesons in a soft-wall holographic approach, using the correspondence of string theory in Anti-de Sitter space and conformal field theory in physical space-time.

Systems of dyadic cubes in a doubling metric space

Abstract: A number of recent results in Euclidean Harmonic Analysis have exploited several adjacent systems of dyadic cubes, instead of just one fixed system. In this paper, we extend such constructions to general spaces of homogeneous type, making these tools available for Analysis on metric spaces. The results include a new (non-random) construction of boundedly many adjacent dyadic systems with useful covering properties, and a streamlined version of the random construction recently devised by H. Martikainen and the first author. We illustrate the usefulness of these constructions with applications to weighted inequalities and the BMO space; further applications will appear in forthcoming work.

Holographic renormalization as a canonical transformation

Abstract: The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in asymptotically AdS spaces these problems are systematically addressed by the method of holographic renormalization. We argue that this class of a priori ill defined variational problems extends far beyond the realm of holographic dualities. As we show, exactly the same issues arise in gravity in non asymptotically AdS spaces, in point particles with certain unbounded from below potentials, and even fundamental strings in flat or AdS backgrounds. We show that the variational problem in all such cases can be made well defined by the following procedure, which is intrinsic to the system in question and does not rely on the existence of a holographically dual theory: (i) The first step is the construction of the space of the most general asymptotic solutions of the classical equations of motion that inherits a well defined symplectic form from that on phase space. The requirement of a well defined symplectic form is essential and often leads to a necessary repackaging of the degrees of freedom. (ii) Once the space of asymptotic solutions has been constructed in terms of the correct degrees of freedom, then there exists a boundary term that is obtained as a certain solution of the Hamilton-Jacobi equation which simultaneously makes the variational problem well defined and preserves the symplectic form. This procedure is identical to holographic renormalization in the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian system.

Twistors to twisted geometries

Abstract: In a previous paper we showed that the phase space of loop quantum gravity on a fixed graph can be parametrized in terms of twisted geometries, quantities describing the intrinsic and extrinsic discrete geometry of a cellular decomposition dual to the graph. Here we unravel the origin of the phase space from a geometric interpretation of twistors.

Geometric Group Theory

Abstract: The aim of this talk is to define the space of ends of a f.g. (finitely generated) group. First, define the space of ends Ends(X) for a metric space X. Define what quasi-isometries are, briefly sketch the concept of proper and geodesic spaces and show that quasi-isometries between such spaces induce homeomorphisms between their spaces of ends. This can be found in [3], p.144/45. Define the Cayley graph for a group presentation and give the main ideas of Theorem 1.5 in [10]. Conclude.

A framework for non-homogeneous analysis on metric spaces, and the RBMO space of Tolsa

Abstract: A new class of metric measure spaces is introduced and studied. This class generalises the well-established doubling metric measure spaces as well as the spaces (Rn, μ) with μ(B(α, r)) ≤ Crd, in which non-doubling harmonic analysis has recently been developed. It seems to be a promising framework for an abstract extension of this theory. Tolsa’s space of regularised BMO functions is defined in this new setting, and the John-Nirenberg inequality is proven.

On the Origin of Gravity and the Laws of Newton

Abstract: Starting from first principles and general assumptions Newton's law of gravitation is shown to arise naturally and unavoidably in a theory in which space is emergent through a holographic scenario. Gravity is explained as an entropic force caused by changes in the information associated with the positions of material bodies. A relativistic generalization of the presented arguments directly leads to the Einstein equations. When space is emergent even Newton's law of inertia needs to be explained. The equivalence principle leads us to conclude that it is actually this law of inertia whose origin is entropic.

Analytic Continuation Of Chern-Simons Theory

Abstract: The title of this article refers to analytic continuation of three-dimensional Chern-Simons gauge theory away from integer values of the usual coupling parameter k, to explore questions such as the volume conjecture, or analytic continuation of three-dimensional quantum gravity (to the extent that it can be described by gauge theory) from Lorentzian to Euclidean signature. Such analytic continuation can be carried out by rotating the integration cycle of the Feynman path integral. Morse theory or Picard-Lefschetz theory gives a natural framework for describing the appropriate integration cycles. An important part of the analysis involves flow equations that turn out to have a surprising four-dimensional symmetry. After developing a general framework, we describe some specific examples (involving the trefoil and figure-eight knots in S^3). We also find that the space of possible integration cycles for Chern-Simons theory can be interpreted as the "physical Hilbert space" of a twisted version of N=4 super Yang-Mills theory in four dimensions.

Topological anti-topological fusion in four-dimensional superconformal field theories

Abstract: We present some new exact results for general four-dimensional superconformal field theories. We derive differential equations governing the coupling constant dependence of chiral primary correlators. For \( \mathcal{N} = 2 \) theories we show that the Zamolodchikov metric on the moduli space and the operator mixing of chiral primaries are quasi-topological quantities and constrained by holomorphy. The equations that we find are the four-dimensional analogue of the tt* equations in two-dimensions, discovered by the method of “topological anti-topological fusion” by Cecotti and Vafa. Our analysis relies on conformal perturbation theory and the superconformal Ward identities and does not use a topological twist.

Elliptic Equations in Divergence Form with Partially BMO Coefficients

Abstract: The solvability in Sobolev spaces is proved for divergence form second order elliptic equations in the whole space, a half space, and a bounded Lipschitz domain. For equations in the whole space or a half space, the leading coefficients aij are assumed to be only measurable in one direction and have locally small BMO semi-norms in the other directions. For equations in a bounded domain, additionally we assume that aij have small BMO semi-norms in a neighborhood of the boundary of the domain. We give a unified approach of both the Dirichlet boundary problem and the conormal derivative problem. We also investigate elliptic equations in Sobolev spaces with mixed norms under the same assumptions on the coefficients.

Quasi-compactness of transfer operators for contact Anosov flows

Abstract: For any Cr contact Anosov flow with r ≥ 3, we construct a scale of Hilbert spaces, which are embedded in the space of distributions on the phase space and contain all the Cr functions, such that the one-parameter family of transfer operators for the flow extend to them boundedly and that the extensions are quasi-compact. We also give explicit bounds on the essential spectral radii of those extensions in terms of differentiability r and the hyperbolicity exponents of the flow.

Interpolation properties of Besov spaces defined on metric spaces

Abstract: Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi-norms When q = ∞ the usual change from integral to supremum is made in the definition. The Besov space Bp, qα (X) is the set of those functions f in Llocp(X) for which the semi-norm ‖f ‖ is finite. We will show that if a doubling metric measure space (X, d, μ) supports a (1, p)-Poincare inequality, then the Besov space Bp, qα (X) coincides with the real interpolation space (Lp (X), KS1, p(X))α, q, where KS1, p(X) is the Sobolev space defined by Korevaar and Schoen [15]. This results in (sharp) imbedding theorems. We further show that our definition of a Besov space is equivalent with the definition given by Bourdon and Pajot [3], and establish a trace theorem (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Global solutions of the cauchy problem for quasi‐linear first‐order equations in several space variables

Abstract: In recent years, there has been some interest in quasi-linear differential equations of conservation type, and much progress has been made in the case of a single space variable (see [ l ] for a survey of the literature). The results obtained here extend part of this theory to several space variables. To be more precise, we are concerned with solutions, which are defined in the region t 2 0, of the Cauchy problem

On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients

Abstract: We prove the solvability in Sobolev spaces for both divergence and non-divergence form higher order parabolic and elliptic systems in the whole space, on a half space, and on a bounded domain. The leading coefficients are assumed to be merely measurable in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders. For the proof, we develop a set of new techniques to produce mean oscillation estimates for systems on a half space.

Compactness by the Hausdorff measure of noncompactness

Abstract: In the present paper, we establish some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators given by infinite matrices that map an arbitrary B K -space into the sequence spaces c 0 , c , l ∞ and l 1 , and into the matrix domains of triangles in these spaces. Furthermore, by using the Hausdorff measure of noncompactness, we apply our results to characterize some classes of compact operators on the B K -spaces.

The wave equation and general plane wave solutions in fractional space

Abstract: This work presents the analytical solution of vector wave equation in fractional space. General plane wave solution to the wave equation for flelds in source-free and lossless media is obtained in fractional space. The obtained solution is a generalization of wave equation from integer dimensional space to a non-integer dimensional space. The classical results are recovered when integer-dimensional space is considered.

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces

Abstract: In this paper we develop a general, systematic, microlocal frame- work for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This frame- work, described in Section 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis. Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose's b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactication, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics. The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new con- struction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. These results are also available in a follow-up paper which is more expository in nature, (54). The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.