Showing papers on "Space (mathematics) published in 1999"

Higher Spin Gauge Theories: Star-Product and AdS Space

Abstract: We review the theory of higher spin gauge fields in 2+1 and 3+1 dimensional anti-de Sitter space and present some new results on the structure of higher spin currents and explicit solutions of the massless equations. A previously obtained d=3 integrating flow is generalized to d=4 and is shown to give rise to a perturbative solution of the d=4 nonlinear higher spin equations. A particular attention is paid to the relationship between the star-product origin of the higher spin symmetries, AdS geometry and the concept of space-time locality.

Little string theory in a double scaling limit

Abstract: A double scaling limit can be defined in string theory on a Calabi-Yau (CY) manifold by approaching a point in moduli space where the CY space develops an isolated singularity and at the same time taking the string coupling to zero, while keeping a particular combination of the two parameters fixed. This leads to a decoupled theory without gravity which has a weak coupling expansion, and can be studied using a holographically dual non-critical superstring description. The usual ``Little String Theory'' corresponds to the strong coupling limit of this theory. We use holography to compute two and three point functions in weakly coupled double scaled little string theory, and study the spectrum of the theory in various dimensions. We find a discrete spectrum of masses which exhibits Hagedorn growth.

String Theory and Noncommutative Geometry

Abstract: We extend earlier ideas about the appearance of noncommutative geometry in string theory with a nonzero B-field. We identify a limit in which the entire string dynamics is described by a minimally coupled (supersymmetric) gauge theory on a noncommutative space, and discuss the corrections away from this limit. Our analysis leads us to an equivalence between ordinary gauge fields and noncommutative gauge fields, which is realized by a change of variables that can be described explicitly. This change of variables is checked by comparing the ordinary Dirac-Born-Infeld theory with its noncommutative counterpart. We obtain a new perspective on noncommutative gauge theory on a torus, its T-duality, and Morita equivalence. We also discuss the D0/D4 system, the relation to M-theory in DLCQ, and a possible noncommutative version of the six-dimensional (2,0) theory.

On Representations and Integrability of Mathematical Structures in Energy-Conserving Physical Systems

Abstract: In the present paper we elaborate on the underlying Hamiltonian structure of interconnected energy-conserving physical systems. It is shown that a power-conserving interconnection of port-controlled generalized Hamiltonian systems leads to an implicit generalized Hamiltonian system, and a power-conserving partial interconnection to an implicit port-controlled Hamiltonian system. The crucial concept is the notion of a (generalized) Dirac structure, defined on the space of energy-variables or on the product of the space of energy-variables and the space of flow-variables in the port-controlled case. Three natural representations of generalized Dirac structures are treated. Necessary and sufficient conditions for closedness (or integrability) of Dirac structures in all three representations are obtained. The theory is applied to implicit port-controlled generalized Hamiltonian systems, and it is shown that the closedness condition for the Dirac structure leads to strong conditions on the input vector fields.

The Gravitational Action in Asymptotically AdS and Flat Spacetimes

Abstract: The divergences of the gravitational action are analyzed for spacetimes that are asymptotically anti-de Sitter and asymptotically flat. The gravitational action is rendered finite using a local counterterm prescription, and the relation of this method to the traditional reference spacetime is discussed. For AdS, an iterative procedure is devised that determines the counterterms efficiently. For asymptotically flat space, we use a different method to derive counterterms which are sufficient to remove divergences in most cases.

Three-dimensional quantum geometry and black holes

Abstract: We review some aspects of three-dimensional quantum gravity with emphasis in the `CFT -> Geometry' map that follows from the Brown-Henneaux conformal algebra. The general solution to the classical equations of motion with anti-de Sitter boundary conditions is displayed. This solution is parametrized by two functions which become Virasoro operators after quantisation. A map from the space of states to the space of classical solutions is exhibited. Some recent proposals to understand the Bekenstein-Hawking entropy are reviewed in this context. The origin of the boundary degrees of freedom arising in 2+1 gravity is analysed in detail using a Hamiltonian Chern-Simons formalism.

The local structure of length spaces with curvature bounded above

Abstract: We show that a number of different notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT(0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a flat in X.

Banach spaces with the daugavet property

Abstract: A Banach space X is said to have the Daugavet property if every operator T: X -* X of rank 1 satisfies 11 Id +Tl= 1 + flTIl. We show that then every weakly compact operator satisfies this equation as well and that X contains a copy of t1. However, X need not contain a copy of L1. We also study pairs of spaces X C Y and operators T: X -* Y satisfying I I J + T I I -_ 1-4- 1f T I I, where J: X -* Y is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with f Id +Tll 1 + Il T l I is as small as possible and give characterisations in terms of a smoothness condition.

The holographic electron density theorem and quantum similarity measures

Abstract: How much information about the complete molecule is present in a part of the molecule? Quantum similarity measures provide comparisons between molecular electron densities based on integration over the whole space. Such integration involves boundaryless electron densities, whereas an early application of the Hohenberg—Kohn theorem to local subsystems of molecules requires these molecules to be confined to bounded, finite regions of the space. However, actual molecules have no boundaries, they are not confined to any finite region of the space. In order to find deterministic relations between local and global, boundaryless electron densities, and to classify the link between quantum similarity measures involving the full space and local subsystems, the unique extension property called the holographic property of subsystems of complete, boundaryless electron densities is established. Any nonzero volume piece of the ground state electron density completely determines the electron density of the complete, bou...

Boundary observability for the space semi-discretizations of the 1 - D wave equation

Abstract: We consider space semi-discretizations of the 1 − d wave equation in a bounded interval with homogeneous Dirichlet boundary conditions. We analyze the problem of boundary observability,i.e., the problem of whether the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the net-spacing h ! 0. We prove that, due to the spurious modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. We prove however a uniform bound in a subspace of solutions generated by the low frequencies of the discrete system. When h ! 0 this nite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We consider both nite-dierence and nite-element semi-discretizations.

Superconformal Hypermultiplets

Abstract: We present theories of N=2 hypermultiplets in four spacetime dimensions that are invariant under rigid or local superconformal symmetries The target spaces of theories with rigid superconformal invariance are (4n)-dimensional {\it special} hyper-Kahler manifolds Such manifolds can be described as cones over tri-Sasakian metrics and are locally the product of a flat four-dimensional space and a quaternionic manifold The latter manifolds appear in the coupling of hypermultiplets to N=2 supergravity We employ local sections of an Sp$(n)\times{\rm Sp}(1)$ bundle in the formulation of the Lagrangian and transformation rules, thus allowing for arbitrary coordinatizations of the hyper-Kahler and quaternionic manifolds

Higher spin gauge theories: Star product and AdS space

Abstract: We review the theory of higher spin gauge fields in 2+1 and 3+1 dimensional anti-de Sitter space and present some new results on the structure of higher spin currents and explicit solutions of the massless equations. A previously obtained d=3 integrating flow is generalized to d=4 and is shown to give rise to a perturbative solution of the d=4 nonlinear higher spin equations. A particular attention is paid to the relationship between the star-product origin of the higher spin symmetries, AdS geometry and the concept of space-time locality.

A Multiplicative Ergodic Theorem and Nonpositively Curved Spaces

Abstract: We study integrable cocycles u.n; x/ over an ergodic measure preserving transformation that take values in a semigroup of nonexpanding maps of a nonpositively curved space Y , e.g. a Cartan-Hadamard space or a uniformly convex Banach space. It is proved that for any y 2 Y and almost all x; there exist A 0 and a unique geodesic ray .t; x/in Y starting at y such that lim n!1 1 n d. .An; x/; u.n; x/y/ D 0: In the case where Y is the symmetric space GLN.R/=ON.R/ and the cocycles take values in GLN.R/; this is equivalent to the multiplicative ergodic theorem of Oseledec. Two applications are also described. The first concerns the determination of Poisson boundaries and the second concerns Hilbert-Schmidt operators.

The geometry of domains in space

Abstract: This treatment of domains in space emphasizes the growing interaction between analysis and geometry. Geometric analysis is an important area of study for both pure and applied mathematicians and engineers.

Time-Resolved Observation of Ultrahigh Intensity Laser-Produced Electron Jets Propagating through Transparent Solid Targets

Abstract: We report on shadowgraphic measurements showing the first space- and time-resolved snapshots of ultraintense laser pulse-generated fast electrons propagating through a solid target. A remarkable result is the formation of highly collimated jets ( $l20\ensuremath{-}\ensuremath{\mu}\mathrm{m}$) traveling at the velocity of light and extending up to 1 mm. This feature clearly indicates a magnetically assisted regime of electron transport, of critical interest for the fast ignitor scheme. Along with these jets, we detect a slower ( $\ensuremath{\approx}c/2$) and broader (up to 1 mm) ionization front consistent with collisional hot electron energy transport.

Algebraic Holography

Abstract: A rigorous (and simple) proof is given that there is a one-to-one correspondence between causal anti-deSitter covariant quantum field theories on anti-deSitter space and causal conformally covariant quantum field theories on its conformal boundary The correspondence is given by the explicit identification of observables localized in wedge regions in anti-deSitter space and observables localized in double-cone regions in its boundary It takes vacuum states into vacuum states, and positive-energy representations into positive-energy representations

Optimizing Shake-and-Bake for proteins.

Abstract: Shake-and-Bake is a direct-methods procedure which has provided ab initio solutions for protein structures containing as many as 1000 independent non-H atoms. This algorithm extends the range of conventional direct methods by repetitively, unconditionally and automatically alternating reciprocal-space phase refinement with filtering in real space to impose constraints. The application of SnB to protein-sized molecules is significantly affected by the choice made for certain critical parameters, including the number of peaks used for density modification, the choice of phase-refinement method and the number of refinement cycles. The effects of parameter variation have been studied for six protein structures, all of which are solvable by Shake-and-Bake using data at 1.1 A or higher resolution. Solvability in the resolution range 1.2–1.4 A appears to be enhanced by the presence of heavier atoms (S, Cl). Furthermore, it appears that in this range the ratio of refinement cycles and triplet phase invariants to atoms in the structure must be increased. Large structures lacking atoms of any element heavier than oxygen also require non-traditional parameter values.

Renormalizability of the Supersymmetric Yang-Mills Theories on the Noncommutative Torus

Abstract: We argue that Yang-Mills theory on noncommutative torus, expressed in the Fourrier modes, is described by a gauge theory in a usual commutative space, the gauge group being a generalization of the area-preserving diffeomorphisms to the noncommutative case. In this way, performing the loop calculation in this gauge theory in the continuum limit we show that this theory is {\it one loop renormalizable}, and discuss the UV and IR limits. The moduli space of the vacua of the noncommutative super Yang-Mills theories in (2+1) dimensions is discussed.

Algebraic analysis of solvable lattice models

Abstract: Background of the problem The spin $1/2$ XXZ model for $\Delta <-1$ The six-vertex model in the anti-ferroelectric regime Solvability and symmetry Correlation functions-physical derivation Level one modules and bosonization Vertex operators Space of states-mathematical picture Traces of vertex operators Correlation functions and form factors The $XXX$ limit $q\rightarrow-1$ Discussions List of formulas

The resolution of the Navier-Stokes equations in anisotropic spaces

Abstract: In this paper we prove global existence and uniqueness for solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which are Hdi in the i-th direction, d1 + d2 + d3 = 1/2, -1/2 < di < 1/2 and in a space which is L2 in the first two directions and B2,11/2 in the third direction, where H and B denote the usual homogeneous Sobolev and Besov spaces.

A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem

Abstract: A fully polynomial time approximation scheme (FPTAS) is presented for the classical 0-1 knapsack problem. The new approach considerably improves the necessary space requirements. The two best previously known approaches need O(n + 1/e3) and O(n · 1/e) space, respectively. Our new approximation scheme requires only O(n + 1/e2) space while also reducing the running time.

Hopf solitons on S**3 and R**3

Abstract: The Skyrme-Faddeev system, a modified O(3) sigma model in three space dimensions, admits topological solitons with nonzero Hopf number. One may learn something about these solitons by considering the system on the 3-sphere of radius R. In particular, the Hopf map is a solution which is unstable if .