Showing papers on "Space (mathematics) published in 2013"

A User’s Guide to Optimal Transport

Abstract: This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro. It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments, including the metric theory of gradient flows, geometric and functional inequalities related to optimal transportation, the first and second order differential calculus in the Wasserstein space and the synthetic theory of metric measure spaces with Ricci curvature bounded from below.

Planck 2013 results. XXV. Searches for cosmic strings and other topological defects

Abstract: Planck data have been used to provide stringent new constraints on cosmic strings and other defects. We describe forecasts of the CMB power spectrum induced by cosmic strings, calculating these from network models and simulations using line-of-sight Boltzmann solvers. We have studied Nambu-Goto cosmic strings, as well as field theory strings for which radiative effects are important, thus spanning the range of theoretical uncertainty in strings models. We have added the angular power spectrum from strings to that for a simple adiabatic model, with the extra fraction defined as $f_{10}$ at multipole $\ell=10$. This parameter has been added to the standard six parameter fit using COSMOMC with flat priors. For the Nambu-Goto string model, we have obtained a constraint on the string tension of $G\mu/c^2 < 1.5 x 10^{-7}$ and $f_{10} < 0.015$ at 95% confidence that can be improved to $G\mu/c^2 < 1.3 x 10^{-7}$ and $f_{10} < 0.010$ on inclusion of high-$\ell$ CMB data. For the abelian-Higgs field theory model we find, $G\mu_{AH}/c^2 < 3.2 x 10^{-7}$ and $f_{10} < 0.028$. The marginalized likelihoods for $f_{10}$ and in the $f_{10}$--$\Omega_b h^2$ plane are also presented. We have also obtained constraints on $f_{10}$ for models with semi-local strings and global textures for which $G\mu/c^2 < 1.1 x 10^{-6}$. We have made complementarity searches for the specific non-Gaussian signatures of cosmic strings, calibrating with all-sky Planck resolution CMB maps generated from networks of post-recombination strings. We have obtained upper limits on the string tension at 95% confidence of $G\mu/c^2 < 8.8 x 10^{-7}$ using modal bispectrum estimation and $G\mu/c^2 < 7.8 x 10^{-7}$ for real space searches with Minkowski functionals. These are conservative upper bounds because only post-recombination string contributions have been included in the non-Gaussian analysis.

Spaces of Neoliberal Experimentation: Soft Spaces, Postpolitics, and Neoliberal Governmentality

Abstract: This paper examines the proliferation of soft spaces of governance, focusing on planning. We move beyond more functional explanations to explore the politics of soft spaces, more specifi cally how soft space forms of governance operate as integral to processes of neoliberalisation, highlighting how such state forms facilitate neoliberalisation through their fl exibility and variability. Recent state restructuring of the planning sector and emerging trends for soft spaces in England under the Coalition government proposals are discussed.

Moduli of $p$-divisible groups

Abstract: We prove several results about moduli spaces of p-divisible groups such as Rapoport–Zink spaces. Our main goal is to prove that Rapoport–Zink spaces at infinite level carry a natural structure as a perfectoid space, and to give a description purely in terms of padic Hodge theory of these spaces. This allows us to formulate and prove duality isomorphisms between basic Rapoport–Zink spaces at infinite level in general. Moreover, we identify the image of the period morphism, reproving results of Faltings, [Fal10]. For this, we give a general classification of p-divisible groups over OC , where C is an algebraically closed complete extension of Qp, in the spirit of Riemann’s classification of complex abelian varieties. Another key ingredient is a full faithfulness result for the Dieudonne module functor for p-divisible groups over semiperfect rings (i.e. rings on which the Frobenius is surjective).

Local decay of waves on asymptotically flat stationary space-times

Abstract: Author(s): Tataru, Daniel | 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.

Solving POMDPs by Searching the Space of Finite Policies

Abstract: Solving partially observable Markov decision processes (POMDPs) is highly intractable in general, at least in part because the optimal policy may be infinitely large. In this paper, we explore the problem of finding the optimal policy from a restricted set of policies, represented as finite state automata of a given size. This problem is also intractable, but we show that the complexity can be greatly reduced when the POMDP and/or policy are further constrained. We demonstrate good empirical results with a branch-and-bound method for finding globally optimal deterministic policies, and a gradient-ascent method for finding locally optimal stochastic policies.

Spin-3 Gravity in Three-Dimensional Flat Space

Abstract: We present the first example of a nontrivial higher spin theory in three-dimensional flat space. We propose flat-space boundary conditions and prove their consistency for this theory. We find that the asymptotic symmetry algebra is a (centrally extended) higher spin generalization of the Bondi-Metzner-Sachs algebra, which we describe in detail. We also address higher spin analogues of flat space cosmology solutions and possible generalizations.

The splitting theorem in non-smooth context

Abstract: We prove that an infinitesimally Hilbertian CD(0,N) space containing a line splits as the product of $R$ and an infinitesimally Hilbertian CD(0,N-1) space. By `infinitesimally Hilbertian' we mean that the Sobolev space $W^{1,2}(X,d,m)$, which in general is a Banach space, is an Hilbert space. When coupled with a curvature-dimension bound, this condition is known to be stable with respect to measured Gromov-Hausdorff convergence.

Metric structures for Riemannian and non-Riemannian space

Abstract: Metric structures for Riemannian and non-Riemannian space , Metric structures for Riemannian and non-Riemannian space , کتابخانه دیجیتال جندی شاپور اهواز

Unidirectional acoustic transmission through a prism with near-zero refractive index

Abstract: We propose a two-dimensional model of acoustic prism comprising metamaterials with near-zero refractive index to yield high efficiency unidirectional transmission and demonstrate an implementation by coiling up space. Due to the acoustic tunneling effect, the waveform is kept consistent between input and output waves, and the transmitted angle can be controlled by reshaping the prism, even in the presence of hard scatterer. A directional waveguide is also designed whose transmission property can be freely switched between all possible states. Our design may have potential for practical applications of acoustic one-way devices in various fields such as ultrasound imaging and treatment.

On an Isoperimetric Problem with a Competing Nonlocal Term II: The General Case

Abstract: This paper is the continuation of a previous paper (H. Knupfer and C. B. Muratov, Comm. Pure Appl. Math. 66 (2013), 1129‐1162). We investigate the classical isoperimetric problem modified by an addition of a nonlocal repulsive term generated by a kernel given by an inverse power of the distance. In this work, we treat the case of a general space dimension. We obtain basic existence results for minimizers with sufficiently small masses. For certain ranges of the exponent in the kernel, we also obtain nonexistence results for sufficiently large masses, as well as a characterization of minimizers as balls for sufficiently small masses and low spatial dimensionality. The physically important special case of three space dimensions and Coulombic repulsion is included in all the results mentioned above. In particular, our work yields a negative answer to the question if stable atomic nuclei at arbitrarily high atomic numbers can exist in the framework of the classical liquid drop model of nuclear matter. In all cases the minimal energy scales linearly with mass for large masses, even if the infimum of energy cannot be attained. © 2014 Wiley Periodicals, Inc.

Local discontinuous Galerkin methods for fractional diffusion equations

Abstract: We consider the development and analysis of local discontinuous Galerkin methods for fractional diffusion problems in one space dimension, characterized by having fractional derivatives, parameterized by beta in [1,2]. After demonstrating that a classic approach fails to deliver optimal order of convergence, we introduce a modified local numerical flux which exhibits optimal order of convergence O(h^(k+1)) uniformly across the continuous range between pure advection (beta=1) and pure diffusion (beta=2). In the two classic limits, known schemes are recovered. We discuss stability and present an error analysis for the space semi-discretized scheme, which is supported through a few examples.

The Fast Convergence of Incremental PCA

Abstract: We consider a situation in which we see samples Xn ∈ ℝd drawn i.i.d. from some distribution with mean zero and unknown covariance A. We wish to compute the top eigenvector of A in an incremental fashion - with an algorithm that maintains an estimate of the top eigenvector in O(d) space, and incrementally adjusts the estimate with each new data point that arrives. Two classical such schemes are due to Krasulina (1969) and Oja (1983). We give finite-sample convergence rates for both.

Supermoduli Space Is Not Projected

Abstract: We prove that for genus greater than or equal to 5, the moduli space of super Riemann surfaces is not projected (and in particular is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.

Uniqueness in Calderón’s problem with Lipschitz conductivities

Abstract: We use Xs,b-inspired spaces to prove a uniqueness result for Calderon’s problem in a Lipschitz domain Ω under the assumption that the conductivity lies in the space W1,∞(Ω‾). For Lipschitz conductivities, we obtain uniqueness for conductivities close to the identity in a suitable sense. We also prove uniqueness for arbitrary C1 conductivities.

One-particle-irreducible consistency relations for cosmological perturbations

Abstract: We derive consistency relations for correlators of scalar cosmological perturbations that hold in the ``squeezed limit'' in which one or more of the external momenta become soft. Our results are formulated as relations between suitably defined one-particle-irreducible $N$-point and ($N\ensuremath{-}1$)-point functions that follow from residual spatial conformal diffeomorphisms of the unitary gauge Lagrangian. As such, some of these relations are exact to all orders in perturbation theory and do not rely on approximate de Sitter invariance or other dynamical assumptions (e.g., properties of the operator product expansion or the behavior of modes at the horizon crossing). The consistency relations apply model-independently to cosmological scenarios in which the time evolution is driven by a single scalar field. Besides reproducing the known results for single-field inflation in the slow-roll limit, we verify that our consistency relations hold more generally, for instance, in ghost condensate models in flat space. We comment on possible extensions of our results to multifield models.

Greedy Algorithms for Reduced Bases in Banach Spaces

Abstract: Given a Banach space X and one of its compact sets $\mathcal{F}$ , we consider the problem of finding a good n-dimensional space X n ⊂X which can be used to approximate the elements of $\mathcal{F}$ . The best possible error we can achieve for such an approximation is given by the Kolmogorov width $d_{n}(\mathcal{F})_{X}$ . However, finding the space which gives this performance is typically numerically intractable. Recently, a new greedy strategy for obtaining good spaces was given in the context of the reduced basis method for solving a parametric family of PDEs. The performance of this greedy algorithm was initially analyzed in Buffa et al. (Model. Math. Anal. Numer. 46:595–603, 2012) in the case $X=\mathcal{H}$ is a Hilbert space. The results of Buffa et al. (Model. Math. Anal. Numer. 46:595–603, 2012) were significantly improved upon in Binev et al. (SIAM J. Math. Anal. 43:1457–1472, 2011). The purpose of the present paper is to give a new analysis of the performance of such greedy algorithms. Our analysis not only gives improved results for the Hilbert space case but can also be applied to the same greedy procedure in general Banach spaces.

The backreaction of anti-D3 branes on the Klebanov-Strassler geometry

Abstract: We present the full numerical solution for the 15-dimensional space of linearized deformations of the Klebanov-Strassler background which preserve the SU(2) × SU(2) × $ {{\mathbb{Z}}_2} $ symmetries. We identify within this space the solution corresponding to anti-D3 branes, (modulo the presence of a certain “subleading” singularity in the infrared). All the 15 integration constants of this solution are fixed in terms of the number of anti-D3 branes, and the solution differs in the UV from the supersymmetric solution into which it is supposed to decay by a mode corresponding to a rescaling of the field theory coordinates. Deciding whether two solutions that differ in the UV by a rescaling mode are dual to the same theory is involved even for supersymmetric Klebanov-Strassler solutions, and we explain in detail some of the subtleties associated to this.

Space dependent Fermi velocity in strained graphene

Abstract: We investigate some apparent discrepancies between two different models for curved graphene: the one based on tight-binding and elasticity theory, and the covariant approach based on quantum field theory in curved space. We demonstrate that strained or corrugated samples will have a space-dependent Fermi velocity in either approach that can affect the interpretation of local probe experiments in graphene. We also generalize the tight-binding approach to general inhomogeneous strain and find a gauge field proportional to the derivative of the strain tensor that has the same form as the one obtained in the covariant approach.

Generalised functions of bounded deformation

Abstract: We introduce the space GBD of generalized functions of bounded defor- mation and study the structure properties of these functions: the rectiability and the slicing properties of their jump sets, and the existence of their approximate symmetric gradients. We conclude by proving a compactness results for GBD , which leads to a compactness result for the space GSBD of generalized special functions of bounded de- formation. The latter is connected to the existence of solutions to a weak formulation of some variational problems arising in fracture mechanics in the framework of linearized elasticity.

A fully variational spin-orbit coupled complete active space self-consistent field approach: Application to electron paramagnetic resonance g-tensors

Abstract: In this work, a relativistic version of the state-averaged complete active space self-consistent field method is developed (spin-orbit coupled state-averaged complete active space self-consistent field; CAS-SOC). The program follows a “one-step strategy” and treats the spin-orbit interaction (SOC) on the same footing as the electron-electron interaction. As opposed to other existing approaches, the program employs an intermediate coupling scheme in which spin and space symmetry adapted configuration space functions are allowed to interact via SOC. This adds to the transparency and computational efficiency of the procedure. The approach requires the utilization of complex-valued configuration interaction coefficients, but the molecular orbital coefficients can be kept real-valued without loss of generality. Hence, expensive arithmetic associated with evaluation of complex-valued transformed molecular integrals is completely avoided. In order to investigate the quality of the calculated wave function, we extended the method to the calculation of electronic g-tensors. As the SOC is already treated to all orders in the SA-CASSCF process, first order perturbation theory with the Zeeman operator is sufficient to accomplish this task. As a test-set, we calculated g-tensors of a set of diatomics, a set of d1 transition metal complexes MOX4n−, and a set of 5f1 actinide complexes AnX6n−. These calculations reveal that the effect of the wavefunction relaxation due to variation inclusion of SOC is of the same order of magnitude as the effect of inclusion of dynamic correlation and hence cannot be neglected for the accurate prediction of electronic g-tensors.

Nonlocal theories of gravity: the flat space propagator

Abstract: It was recently found that there are classes of nonlocal gravity theories that are free of ghosts and singularities in their Newtonian limit [PRL, 108 (2012), 031101]. In these proceedings, a detailed and pedagogical derivation of a main result, the flat space propagator for an arbitrary covariant metric theory of gravitation, is presented. The result is applied to analyse f(R) models, Gauss-Bonnet theory, Weyl-squared gravity and the potentially asymptotically free nonlocal theories.

Tunable transport with broken space-time symmetries

Abstract: Transport properties of particles and waves in spatially periodic structures that are driven by external time-dependent forces manifestly depend on the space-time symmetries of the corresponding equations of motion. A systematic analysis of these symmetries uncovers the conditions necessary for obtaining directed transport. In this work we give a unified introduction into the symmetry analysis and demonstrate its action on the motion in one-dimensional periodic, both in time and space, potentials. We further generalize the analysis to quasi-periodic drivings, higher space dimensions, and quantum dynamics. Recent experimental results on the transport of cold and ultracold atomic ensembles in ac-driven optical potentials are reviewed as illustrations of theoretical considerations.

Transport Equation with Nonlocal Velocity in Wasserstein Spaces: Convergence of Numerical Schemes

Abstract: Motivated by pedestrian modelling, we study evolution of measures in the Wasserstein space. In particular, we consider the Cauchy problem for a transport equation, where the velocity field depends on the measure itself. We deal with numerical schemes for this problem and prove convergence of a Lagrangian scheme to the solution, when the discretization parameters approach zero. We also prove convergence of an Eulerian scheme, under more strict hypotheses. Both schemes are discretizations of the push-forward formula defined by the transport equation. As a by-product, we obtain existence and uniqueness of the solution. All the results of convergence are proved with respect to the Wasserstein distance. We also show that L 1 spaces are not natural for such equations, since we lose uniqueness of the solution.