Showing papers on "Space (mathematics) published in 2009"

Nonlinear Numerical Analysis in Reproducing Kernel Space

Abstract: Although the application of reproducing kernel has been explored in different fields in the past twenty to thirty years and the relevant researches are active in the recent five years, there is still not a book on the application of reproducing kernel. This book attempts to introduce to the readers engaged in mathematical application these solutions, especially the constructing theory of the reproducing kernel space that the authors originally created and gradually improved. Reproducing kernel space is a special Hilbert space. The authors have been engaged in the constructing theory research of the reproducing kernel space since 1980's, and worked out a series of specific structural methods for reproducing kernel space and reproducing kernel functions.

Weighted Morrey spaces and a singular integral operator

Abstract: In this paper, we shall introduce a weighted Morrey space and study the several properties of classical operatorsin harmonic analysis on this space (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Harmonic analysis on spaces of homogeneous type

Abstract: Calde?on-Zygmund Operator on Space of Homogeneous Type.- The Boundedness of Calderon-Zygmund Operators on Wavelet Spaces.- Wavelet Expansions on Spaces of Homogeneous Type.- Wavelets and Spaces of Functions and Distributions.- Littlewood-Paley Analysis on Non Homogeneous Spaces.

On Reducibility of Schrödinger Equations with Quasiperiodic in Time Potentials

Abstract: We prove that a linear d-dimensional Schrodinger equation with an x-periodic and t-quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector. The reduction is made by means of a non-autonomous linear transformation of the space of x-periodic functions. This transformation is a quasiperiodic function of t.

The S-Matrix in Twistor Space

Abstract: The simplicity and hidden symmetries of (Super) Yang-Mills and (Super)Gravity scattering amplitudes suggest the existence of a "weak-weak" dual formulation in which these structures are made manifest at the expense of manifest locality. We suggest that this dual description lives in (2,2) signature and is naturally formulated in twistor space. We recast the BCFW recursion relations in an on-shell form that begs to be transformed into twistor space. Our twistor transformation is inspired by Witten's, but differs in treating twistor and dual twistor variables more equally. In these variables the three and four-point amplitudes are amazingly simple; the BCFW relations are represented by diagrammatic rules that precisely define the "twistor diagrams" of Andrew Hodges. The "Hodges diagrams" for Yang-Mills theory are disks and not trees; they reveal striking connections between amplitudes and suggest a new form for them in momentum space. We also obtain a twistorial formulation of gravity. All tree amplitudes can be combined into an "S-Matrix" functional which is the natural holographic observable in asymptotically flat space; the BCFW formula turns into a quadratic equation for this "S-Matrix", providing a holographic description of N=4 SYM and N=8 Supergravity at tree level. We explore loop amplitudes in (2,2) signature and twistor space, beginning with a discussion of IR behavior. We find that the natural pole prescription renders the amplitudes well-defined and free of IR divergences. Loop amplitudes vanish for generic momenta, and in twistor space are even simpler than their tree-level counterparts! This further supports the idea that there exists a sharply defined object corresponding to the S-Matrix in (2,2) signature, computed by a dual theory naturally living in twistor space.

Band touching from real space topology in frustrated hopping models(Topological Aspects of Solid State Physics)

Abstract: We study ``frustrated'' hopping models, in which at least one energy band, at the maximum or minimum of the spectrum, is dispersionless. The states of the flat band(s) can be represented in a basis which is fully localized, having support on a vanishing fraction of the system in the thermodynamic limit. In the majority of examples, a dispersive band touches the flat band(s) at a number of discrete points in momentum space. We demonstrate that this band touching is related to states which exhibit non-trivial topology in real space. Specifically, these states have support on one-dimensional loops which wind around the entire system (with periodic boundary conditions). A counting argument is given that determines, in each case, whether there is band touching or not, in precise correspondence to the result of straightforward diagonalization. When they are present, the topological structure protects the band touchings in the sense that they can only be removed by perturbations which {\sl also} split the degeneracy of the flat band.

A Multilayered Space-Event Model for Navigation in Indoor Spaces

Abstract: In this paper a new conceptual framework for indoor navigation is proposed. While route planning requires models which reflect the internal structure of a building, localization techniques require complementary models reflecting the characteristics of sensors and transmitters. Since the partitioning of building space differs in both cases, a conceptual separation of different space models into a multilayer representation is proposed. Concrete space models for topographic space and sensor space are introduced. Both are systematically subdivided into primal and dual space on the one hand and (Euclidean) geometry and topology on the other hand. While topographic space describes 3D models of buildings and their semantically subdivisions into storey’s and rooms, sensor space describes the positions and ranges of transmitters and sensors like Wi-Fi access points or RFID sensors. It is shown how the connection of the different layers of the space models describe a joint state of a moving subject or object and reduces uncertainty about its current position.

Q-learning and Pontryagin's Minimum Principle

Abstract: Q-learning is a technique used to compute an optimal policy for a controlled Markov chain based on observations of the system controlled using a non-optimal policy. It has proven to be effective for models with finite state and action space. This paper establishes connections between Q-learning and nonlinear control of continuous-time models with general state space and general action space. The main contributions are summarized as follows.

Local decay of waves on asymptotically flat stationary space-times

Abstract: In this article we study the pointwise decay properties of solutions to the wave equation on a class of stationary asymptotically flat backgrounds in three space dimensions. Under the assumption that uniform energy bounds and a weak form of local energy decay hold forward in time we establish a $t^{-3}$ local uniform decay rate for linear waves. This work was motivated by open problems concerning decay rates for linear waves on Schwarzschild and Kerr backgrounds, where such a decay rate has been conjectured by R. Price. Our results apply to both of these cases.

Well-Posedness for the Two-Dimensional Modified Zakharov–Kuznetsov Equation

Abstract: We prove that the initial value problem for the two-dimensional modified Zakharov–Kuznetsov equation is locally well-posed for data in $H^s(\mathbb{R}^2)$, $s>3/4$. Even though the critical space for this equation is $L^2(\mathbb{R}^2)$, we prove that well-posedness is not possible in such space. Global well-posedness and a sharp maximal function estimate are also established.

Bergman kernels and equilibrium measures for line bundles over projective manifolds

Abstract: Let L be a holomorphic line bundle over a compact complex projective Hermitian manifold X. Any fixed smooth hermitian metric h on L induces a Hilbert space structure on the space of global holomorphic sections with values in the k th tensor power of L. In this paper various convergence results are obtained for the corresponding Bergman kernels (i.e. orthogonal projection kernels). The convergence is studied in the large k limit and is expressed in terms of the equilibrium metric h_e associated to h, as well as in terms of the Monge-Ampere measure of h on a certain support set. It is also shown that the equilibrium metric h_e is in the class C^{1,1} on the complement of the augmented base locus of L. For L ample these results give generalizations of well-known results concerning the case when the curvature of h is globally positive (then h_e=h). In general, the results can be seen as local metrized versions of Fujita's approximation theorem for the volume of L.

Super-potentials of positive closed currents, intersection theory and dynamics

Abstract: We introduce a notion of super-potential for positive closed currents of bidegree (p, p) on projective spaces. This gives a calculus on positive closed currents of arbitrary bidegree. We define in particular the intersection of such currents and the pull-back operator by meromorphic maps. One of the main tools is the introduction of structural discs in the space of positive closed currents which gives a “geometry” on that space. We apply the theory of super-potentials to construct Green currents for rational maps and to study equidistribution problems for holomorphic endomorphisms and for polynomial automorphisms.

High-precision predictions for the acoustic scale in the non-linear regime

Abstract: We measure shifts of the acoustic scale due to nonlinear growth and redshift distortions to a high precision using a very large volume of high-force-resolution simulations. We compare results from various sets of simulations that differ in their force, volume, and mass resolution. We find a consistency within 1.5-sigma for shift values from different simulations and derive shift alpha(z) -1 = (0.300\pm 0.015)% [D(z)/D(0)]^{2} using our fiducial set. We find a strong correlation with a non-unity slope between shifts in real space and in redshift space and a weak correlation between the initial redshift and low redshift. Density-field reconstruction not only removes the mean shifts and reduces errors on the mean, but also tightens the correlations: after reconstruction, we recover a slope of near unity for the correlation between the real and redshift space and restore a strong correlation between the low and the initial redshifts. We derive propagators and mode-coupling terms from our N-body simulations and compared with Zeldovich approximation and the shifts measured from the chi^2 fitting, respectively. We interpret the propagator and the mode-coupling term of a nonlinear density field in the context of an average and a dispersion of its complex Fourier coefficients relative to those of the linear density field; from these two terms, we derive a signal-to-noise ratio of the acoustic peak measurement. We attempt to improve our reconstruction method by implementing 2LPT and iterative operations: we obtain little improvement. The Fisher matrix estimates of uncertainty in the acoustic scale is tested using 5000 (Gpc/h)^3 of cosmological PM simulations from Takahashi et al. (2009). (abridged)

A note on the holography of Chern-Simons matter theories with flavour

Abstract: We study a three-dimensional = 3 U(N)k × U(N)−k Chern-Simons matter theory with flavour, corresponding to the = 6 Aharony-Bergman-Jafferis-Maldacena CSM theory coupled to 2Nf fundamental fields. The dual holographic description is given by the near-horizon geometry of N M2-branes at a particular hypertoric geometry 8. We explicitly construct the space 8 and match its isometries to the global symmetries of the field theory. We also discuss the model in the quenched approximation by embedding probe D6-branes in AdS4 × .

Quantum geometry from phase space reduction

Abstract: In this work, we give an explicit isomorphism between the usual spin network basis and the direct quantization of the reduced phase space of tetrahedra. The main outcome is a formula that describes the space of SU(2) invariant states by an integral over coherent states satisfying the closure constraint exactly or, equivalently, as an integral over the space of classical tetrahedra. This provides an explicit realization of theorems by Guillemin–Sternberg and Hall that describe the commutation of quantization and reduction. In the final part of the paper, we use our result to express the Freidel–Krasnov spin foam model as an integral over classical tetrahedra, and the asymptotics of the vertex amplitude is determined.

Correlation Functions in Non-Relativistic Holography

Abstract: Recently constructed gravity solutions with Schrodinger symmetry provide a new example of AdS/CFT-type dualities for the type of non-relativistic field theories relevant to certain cold atom systems. In this paper we use the gravity side to calculate n-point correlation functions of scalar fields by reducing the computation to that in ordinary AdS space via a particular Fourier transform. We evaluate the relevant integrals for 3- and 4-point functions and show that the results are consistent with the requirements of Schrodinger invariance, the implications of which we also work out for general n-point functions.

On the Eikonal approximation in AdS space

Abstract: We explore the eikonal approximation to graviton exchange in AdS5 space, as relevant to scattering in gauge theories. We restrict ourselves to the regime where conformal invariance of the dual gauge theory holds, and to large 't Hooft coupling where the computation involves pure gravity. We give a heuristic argument, a direct loop computation, and a shock wave derivation. The scalar propagator in AdS3 plays a key role, indicating that even at strong coupling, two-dimensional conformal invariance controls high-energy four-dimensional gauge-theory scattering.

Optimized general sparse grid approximation spaces for operator equations

Abstract: This paper is concerned with the construction of optimized sparse grid approximation spaces for elliptic pseudodifferential operators of arbitrary order. Based on the framework of tensor-product biorthogonal wavelet bases and stable subspace splittings, we construct operator-adapted subspaces with a dimension smaller than that of the standard full grid spaces but which have the same approximation order as the standard full grid spaces, provided that certain additional regularity assumptions on the solution are fulfilled. Specifically for operators of positive order, their dimension is O(2 J ) independent of the dimension n of the problem, compared to O(2 Jn ) for the full grid space. Also, for operators of negative order the overall cost is significantly in favor of the new approximation spaces. We give cost estimates for the case of continuous linear information. We show these results in a constructive manner by proposing a Galerkin method together with optimal preconditioning. The theory covers elliptic boundary value problems as well as boundary integral equations.

Noncommutative Spacetimes: Symmetries in Noncommutative Geometry and Field Theory

Abstract: Deformed Field Theory: Physical Aspects.- Differential Calculus and Gauge Transformations on a Deformed Space.- Deformed Gauge Theories.- Einstein Gravity on Deformed Spaces.- Deformed Gauge Theory: Twist Versus Seiberg#x2013 Witten Approach.- Another Example of Noncommutative Spaces: #x03BA -Deformed Space.- Noncommutative Geometries: Foundations and Applications.- Noncommutative Spaces.- Quantum Groups, Quantum Lie Algebras, and Twists.- Noncommutative Symmetries and Gravity.- Twist Deformations of Quantum Integrable Spin Chains.- The Noncommutative Geometry of Julius Wess.

On an integral operator from the zygmund space to the bloch-type space on the unit ball

Abstract: In this paper, we introduce an integral operator on the unit ball . The boundedness and compactness of the operator from the Zygmund space to the Bloch-type space or the little Bloch-type space are investigated.

Local Heterotic Torsional Models

Abstract: We present a class of smooth supersymmetric heterotic solutions with a non-compact Eguchi-Hanson space. The non-compact geometry is embedded as the base of a six-dimensional non-Kahler manifold with a non-trivial torus fiber. We solve the non-linear anomaly equation in this background exactly. We also define a new charge that detects the non-Kahlerity of our solutions.

General conditions for the existence of non-standard Lagrangians for dissipative dynamical systems

Abstract: Equations of motion describing dissipative dynamical systems with coefficients varying either in time or in space are considered. To identify the equations that admit a Lagrangian description, two classes of non-standard Lagrangians are introduced and general conditions required for the existence of these Lagrangians are determined. The conditions are used to obtain some non-standard Lagrangians and derive equations of motion resulting from these Lagrangians.

A quantitative computational model for complete partial metric spaces via formal balls

Abstract: Given a partial metric space (X, p), we use (BX, ⊑dp) to denote the poset of formal balls of the associated quasi-metric space (X, dp). We obtain characterisations of complete partial metric spaces and sup-separable complete partial metric spaces in terms of domain-theoretic properties of (BX, ⊑dp). In particular, we prove that a partial metric space (X, p) is complete if and only if the poset (BX, ⊑dp) is a domain. Furthermore, for any complete partial metric space (X, p), we construct a Smyth complete quasi-metric q on BX that extends the quasi-metric dp such that both the Scott topology and the partial order ⊑dp are induced by q. This is done using the partial quasi-metric concept recently introduced and discussed by H. P. Kunzi, H. Pajoohesh and M. P. Schellekens (Kunzi et al. 2006). Our approach, which is inspired by methods due to A. Edalat and R. Heckmann (Edalat and Heckmann 1998), generalises to partial metric spaces the constructions given by R. Heckmann (Heckmann 1999) and J. J. M. M. Rutten (Rutten 1998) for metric spaces.

Gradient flows on Wasserstein spaces over compact Alexandrov spaces

Abstract: We establish the existence of Euclidean tangent cones on Wasserstein spaces over compact Alexandrov spaces of curvature bounded below. By using this Riemannian structure, we formulate and construct gradient flows of functions on such spaces. If the underlying space is a Riemannian manifold of nonnegative sectional curvature, then our gradient flow of the free energy produces a solution of the linear Fokker-Planck equation.

Canonical Wick Rotations in 3-Dimensional Gravity

Abstract: The authors develop a canonical Wick rotation-rescaling theory in 3-dimensional gravity. This includes: a simultaneous classification: this shows how maximal globally hyperbolic space times of arbitrary constant curvature, which admit a complete Cauchy surface and canonical cosmological time, as well as complex projective structures on arbitrary surfaces, are all different materializations of 'more fundamental' encoding structures; Canonical geometric correlations: this shows how space times of different curvature, that share a same encoding structure, are related to each other by canonical rescalings, and how they can be transformed by canonical Wick rotations in hyperbolic 3-manifolds, that carry the appropriate asymptotic projective structure. Both Wick rotations and rescalings act along the canonical cosmological time and have universal rescaling functions. These correlations are functorial with respect to isomorphisms of the respective geometric categories.