Showing papers on "Space (mathematics) published in 2011"

Reproducing Kernel Hilbert Spaces in Probability and Statistics

Abstract: 1 Theory.- 2 RKHS AND STOCHASTIC PROCESSES.- 3 Nonparametric Curve Estimation.- 4 Measures And Random Measures.- 5 Miscellaneous Applications.- 6 Computational Aspects.- 7 A Collection of Examples.- to Sobolev spaces.- A.l Schwartz-distributions or generalized functions.- A.1.1 Spaces and their topology.- A.1.2 Weak-derivative or derivative in the sense of distributions.- A.1.3 Facts about Fourier transforms.- A.2 Sobolev spaces.- A.2.1 Absolute continuity of functions of one variable.- A.2.2 Sobolev space with non negative integer exponent.- A.2.3 Sobolev space with real exponent.- A.2.4 Periodic Sobolev space.- A.3 Beppo-Levi spaces.

Weakly turbulent instability of anti-de Sitter spacetime.

Abstract: We study the nonlinear evolution of a weakly perturbed anti-de Sitter (AdS) space by solving numerically the four-dimensional spherically symmetric Einstein-massless-scalar field equations with negative cosmological constant. Our results suggest that AdS space is unstable under arbitrarily small generic perturbations. We conjecture that this instability is triggered by a resonant mode mixing which gives rise to diffusion of energy from low to high frequencies.

The principle of relative locality

Abstract: We propose a deepening of the relativity principle according to which the invariant arena for nonquantum physics is a phase space rather than spacetime. Descriptions of particles propagating and interacting in spacetimes are constructed by observers, but different observers, separated from each other by translations, construct different spacetime projections from the invariant phase space. Nonetheless, all observers agree that interactions are local in the spacetime coordinates constructed by observers local to them. This framework, in which absolute locality is replaced by relative locality, results from deforming energy-momentum space, just as the passage from absolute to relative simultaneity results from deforming the linear addition of velocities. Different aspects of energy-momentum space geometry, such as its curvature, torsion and nonmetricity, are reflected in different kinds of deformations of the energy-momentum conservation laws. These are in principle all measurable by appropriate experiments. We also discuss a natural set of physical hypotheses which singles out the cases of energy-momentum space with a metric compatible connection and constant curvature.

Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations

Abstract: In this paper we provide a well-posedness theory for weak measure solutions of the Cauchy problem for a family of nonlocal interaction equations. These equations are continuum models for interacting particle systems with attractive/repulsive pairwise interaction potentials. The main phenomenon of interest is that, even with smooth initial data, the solutions can concentrate mass in finite time. We develop an existence theory that enables one to go beyond the blow-up time in classical norms and allows for solutions to form atomic parts of the measure in finite time. The weak measure solutions are shown to be unique and exist globally in time. Moreover, in the case of sufficiently attractive potentials, we show the finite-time total collapse of the solution onto a single point for compactly supported initial measures. Our approach is based on the theory of gradient flows in the space of probability measures endowed with the Wasserstein metric. In addition to classical tools, we exploit the stability of the flow with respect to the transportation distance to greatly simplify many problems by reducing them to questions about particle approximations.

Hardy Spaces Associated to Non-Negative Self-Adjoint Operators Satisfying Davies-Gaffney Estimates

Abstract: Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

The generalized active space concept in multiconfigurational self-consistent field methods.

Abstract: A multiconfigurational self-consistent field method based on the concept of generalized active space (GAS) is presented. GAS wave functions are obtained by defining an arbitrary number of active spaces with arbitrary occupation constraints. By a suitable choice of the GAS spaces, numerous ineffective configurations present in a large complete active space (CAS) can be removed, while keeping the important ones in the CI space. As a consequence, the GAS self-consistent field approach retains the accuracy of the CAS self-consistent field (CASSCF) ansatz and, at the same time, can deal with larger active spaces, which would be unaffordable at the CASSCF level. Test calculations on the Gd atom, Gd2 molecule, and oxoMn(salen) complex are presented. They show that GAS wave functions achieve the same accuracy as CAS wave functions on systems that would be prohibitive at the CAS level.

Towards an accurate model of the redshift space clustering of halos in the quasilinear regime

Abstract: Observations of redshift-space distortions in spectroscopic galaxy surveys offer an attractive method for measuring the build-up of cosmological structure, which depends both on the expansion rate of the Universe and our theory of gravity. Galaxies occupy dark matter halos, whose redshift space clustering has a complex dependence on bias that cannot be inferred from the behavior of matter. We identify two distinct corrections on quasilinear scales (~ 30-80 Mpc/h): the non-linear mapping between real and redshift space positions, and the non-linear suppression of power in the velocity divergence field. We model the first non-perturbatively using the scale-dependent Gaussian streaming model, which we show is accurate at the 10 (s>25) Mpc/h for the monopole (quadrupole) halo correlation functions. We use perturbation theory to predict the real space pairwise halo velocity statistics. Our fully analytic model is accurate at the 2 per cent level only on scales s > 40 Mpc/h. Recent models that neglect the corrections from the bispectrum and higher order terms from the non-linear real-to-redshift space mapping will not have the accuracy required for current and future observational analyses. Finally, we note that our simulation results confirm the essential but non-trivial assumption that on large scales, the bias inferred from real space clustering of halos is the same one that determines their pairwise infall velocity amplitude at the per cent level.

Decay of dissipative equations and negative Sobolev spaces

Abstract: We develop a general energy method for proving the optimal time decay rates of the solutions to the dissipative equations in the whole space. Our method is applied to classical examples such as the heat equation, the compressible Navier-Stokes equations and the Boltzmann equation. In particular, the optimal decay rates of the higher-order spatial derivatives of solutions are obtained. The negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates. We use a family of scaled energy estimates with minimum derivative counts and interpolations among them without linear decay analysis.

Stochastic Spectral Galerkin and Collocation Methods for PDEs with Random Coefficients: A Numerical Comparison

Abstract: Much attention has recently been devoted to the development of Stochastic Galerkin (SG) and Stochastic Collocation (SC) methods for uncertainty quantification. An open and relevant research topic is the comparison of these two methods. By introducing a suitable generalization of the classical sparse grid SC method, we are able to compare SG and SC on the same underlying multivariate polynomial space in terms of accuracy vs. computational work. The approximation spaces considered here include isotropic and anisotropic versions of Tensor Product (TP), Total Degree (TD), Hyperbolic Cross (HC) and Smolyak (SM) polynomials. Numerical results for linear elliptic SPDEs indicate a slight computational work advantage of isotropic SC over SG, with SC-SM and SG-TD being the best choices of approximation spaces for each method. Finally, numerical results corroborate the optimality of the theoretical estimate of anisotropy ratios introduced by the authors in a previous work for the construction of anisotropic approximation spaces.

Blow-up Theories for Semilinear Parabolic Equations

Abstract: 1 Introduction.- 2 A review of elliptic theories.- 3 A review of parabolic theories.- 4 A review of fixed point theorems.-5 Finite time Blow-up for evolution equations.- 6 Steady-State solutions.- 7 Blow-up rate.- 8 Asymptotically self-similar blow-up solutions.- 9 One space variable case

Wasserstein geometry of Gaussian measures

Abstract: This paper concerns the Riemannian/Alexandrov geometry of Gaussian measures, from the view point of the L 2 -Wasserstein geometry. The space of Gaussian measures is of finite dimension, which allows to write down the ex plicit Riemannian metric which in turn induces the L 2 -Wasserstein distance. Moreover, its completion as a metric space provides a complete picture of the singular behavior of the L 2 Wasserstein geometry. In particular, the singular set is st ratified according to the dimension of the support of the Gaussian measures, providing an explicit nontrivial example of Alexandrov space with extremal sets.

Nonassociative gravity in string theory

Abstract: In an on-shell conformal field theory approach, we find indications of a three-bracket structure for target space coordinates in general closed string backgrounds. This generalizes the appearance of noncommutative gauge theories for open strings in two-form backgrounds to a putative noncommutative/nonassociative gravity theory for closed strings probing curved backgrounds with non-vanishing three-form flux. Several aspects and consequences of the three-bracket structure are discussed and a new type of generalized uncertainty principle is proposed.

KKM mappings in cone b-metric spaces

Abstract: In this paper we establish some topological properties of the cone b-metric spaces and then improve some recent results about KKM mappings in the setting of a cone b-metric space. We also prove some fixed point existence results for multivalued mappings defined on such spaces.

Global mild solutions of Navier-Stokes equations

Abstract: We establish a global well-posedness of mild solutions to the three-dimensional, incompressible Navier-Stokes equations if the initial data are in the space X -1 defined by X -1 = {f E D'(R 3 ): ∫ ℝ 3 |ξ| -1 |f|dξ < ∞} and if the norms of the initial data in X -1 are bounded exactly by the viscosity coefficient μ.

Convenient categories of smooth spaces

Abstract: A 'Chen space' is a set X equipped with a collection of 'plots', i.e., maps from convex sets to X, satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's 'diffeological spaces' share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Penon and Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of 'concrete sheaves on a concrete site'. As a result, the categories of such spaces are locally Cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. © 2011 John C. Baez and Alexander E. Hoffnung.

U(N) coherent states for loop quantum gravity

Abstract: We investigate the geometry of the space of N-valent SU(2) intertwiners. We propose a new set of holomorphic operators acting on this space and a new set of coherent states which are covariant under U(N) transformations. These states are labeled by elements of the Grassmannian GrN, 2, they possess a direct geometrical interpretation in terms of framed polyhedra and are shown to be related to the well-known coherent intertwiners.

Addendum to Fast Scramblers

Abstract: This paper is an addendum to [arXiv:0808.2096] in which I point out that both de Sitter space and Rindler space are fast scramblers. This fact naturally suggests that the holographic description of a causal patch of de Sitter space may be a matrix quantum mechanics at finite temperature. The same can be said of Rindler space. Some qualitative features of these spaces can be understood from the matrix description.

Poisson process Fock space representation, chaos expansion and covariance inequalities

Abstract: We consider a Poisson process η on an arbitrary measurable space with an arbitrary sigma-finite intensity measure. We establish an explicit Fock space representation of square integrable functions of η. As a consequence we identify explicitly, in terms of iterated difference operators, the integrands in the Wiener–Ito chaos expansion. We apply these results to extend well-known variance inequalities for homogeneous Poisson processes on the line to the general Poisson case. The Poincare inequality is a special case. Further applications are covariance identities for Poisson processes on (strictly) ordered spaces and Harris–FKG-inequalities for monotone functions of η.

Higher-Spin Interactions: four-point functions and beyond

Abstract: In this work we construct an infinite class of four-point functions for massless higher-spin fields in flat space that are consistent with the gauge symmetry. In the Lagrangian picture, these reflect themselves in a peculiar non-local nature of the corresponding non-abelian higher-spin couplings implied by the Noether procedure that starts from the fourth order. We also comment on the nature of the colored spin-2 excitation present both in the open string spectrum and in the Vasiliev system, highlighting how some aspects of String Theory appear to reflect key properties of Field Theory that go beyond its low energy limit. A generalization of these results to n-point functions, fermions and mixed-symmetry fields is also addressed.

Multiconfigurational Second-Order Perturbation Theory Restricted Active Space (RASPT2) Studies on Mononuclear First-Row Transition-Metal Systems

Abstract: A series of model transition-metal complexes, CrF6, ferrocene, Cr(CO)6, ferrous porphin, cobalt corrole, and FeO/FeO(-), have been studied using second-order perturbation theory based on a restricted active space self-consistent field reference wave function (RASPT2). Several important properties (structures, relative energies of different structural minima, binding energies, spin state energetics, and electronic excitation energies) were investigated. A systematic investigation was performed on the effect of: (a) the size and composition of the global RAS space, (b) different (RAS1/RAS2/RAS3) subpartitions of the global RAS space, and (c) different excitation levels (out of RAS1/into RAS3) within the RAS space. Calculations with active spaces, including up to 35 orbitals, are presented. The results obtained with smaller active spaces (up to 16 orbitals) were compared to previous and current results obtained with a complete active space self-consistent field reference wave function (CASPT2). Higly accurate RASPT2 results were obtained for the heterolytic binding energy of ferrocene and for the electronic spectrum of Cr(CO)6, with errors within chemical accuracy. For ferrous porphyrin the intermediate spin (3)A2g ground state is (for the first time with a wave function-based method) correctly predicted, while its high magnetic moment (4.4 μB) is attributed to spin-orbit coupling with very close-lying (5)A1g and (3)Eg states. The toughest case met in this work is cobalt corrole, for which we studied the relative energy of several low-lying Co(II)-corrole π radical states with respect to the Co(III) ground state. Very large RAS spaces (25-33 orbitals) are required for this system, making compromises on the size of RAS2 and/or the excitation level unavoidable, thus increasing the uncertainty of the RASPT2 results by 0.1-0.2 eV. Still, also for this system, the RASPT2 method is shown to provide distinct improvements over CASPT2, by overcoming the strict limitations in the size of the active space inherent to the latter method.

Smooth metric measure spaces with non-negative curvature

Abstract: We study both function theoretic and spectral properties on complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ with nonnegative Bakry-\'{E}mery Ricci curvature. Among other things, we derive a gradient estimate for positive $f$-harmonic functions and obtain as a consequence the strong Liouville property under the optimal sublinear growth assumption on $f.$ We also establish a sharp upper bound of the bottom spectrum of the $f$-Laplacian in terms of the linear growth rate of $f.$ Moreover, we show that if equality holds and $M$ is not connected at infinity, then $M$ must be a cylinder. As an application, we conclude steady Ricci solitons must be connected at infinity.

On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT

Abstract: We study the relation between two sets of correlators in interacting quantum eld theory on de Sitter space. The rst are correlators computed using in-in perturbation theory in the expanding cosmological patch of de Sitter space (also known as the conformal patch, or the Poincar e patch), and for which the free propagators are taken to be those of the free Euclidean vacuum. The second are correlators obtained by analytic continuation from Euclidean de Sitter; i.e., they are correlators in the fully interacting Hartle-Hawking state. We give an analytic argument that these correlators coincide for interacting massive scalar elds with any m 2 > 0. We also verify this result via direct calculation in simple examples. The correspondence holds diagram by diagram, and at any nite value of an appropriate Pauli-Villars regulator mass M. Along the way, we note interesting connections between various prescriptions for perturbation theory in general static spacetimes with bifurcate Killing horizons.

The space of stability conditions on the local projective plane

Abstract: We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply-connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to 1(3). Finally, we show that there is a submanifold isomorphic to the univer- sal covering of a moduli space of elliptic curves with 1(3)-level struc- ture. The morphism is 1(3)-equivariant, and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of - stability and by mirror symmetry.

New properties of Besov and Triebel-Lizorkin spaces on RD-spaces

Abstract: An RD-space \({\mathcal X}\) is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in \({\mathcal X}\). In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on \({\mathcal X}\) are independent of the choice of the regularity \({\epsilon\in (0,1)}\); as a result of this, the Besov and Triebel-Lizorkin spaces on \({\mathcal X}\) are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.

On the isoperimetric deficit in gauss space

Abstract: We prove a sharp quantitative version of the isoperimetric inequality in the space ${\Bbb R}^n$ endowed with the Gaussian measure.