Showing papers on "Space (mathematics) published in 2012"

Nonlinear Potential Theory on Metric Spaces

Abstract: The p-Laplace equation is the main prototype for nonlinear elliptic problems and forms a basis for various applications, such as injection moulding of plastics, nonlinear elasticity theory and image processing Its solutions, called p-harmonic functions, have been studied in various contexts since the 1960s, first on Euclidean spaces and later on Riemannian manifolds, graphs and Heisenberg groups Nonlinear potential theory of p-harmonic functions on metric spaces has been developing since the 1990s and generalizes and unites these earlier theoriesThis monograph gives a unified treatment of the subject and covers most of the available results in the field, so far scattered over a large number of research papers The aim is to serve both as an introduction to the area for an interested reader and as a reference text for an active researcher The presentation is rather self-contained, but the reader is assumed to know measure theory and functional analysisThe first half of the book deals with Sobolev type spaces, so-called Newtonian spaces, based on upper gradients on general metric spaces In the second half, these spaces are used to study p-harmonic functions on metric spaces and a nonlinear potential theory is developed under some additional, but natural, assumptions on the underlying metric spaceEach chapter contains historical notes with relevant references and an extensive index is provided at the end of the book

and Model Choice Across a Large Model Space

Abstract: Abstract Since its introduction in 2001, MrBayes has grown in popularity as a software package for Bayesian phylogenetic inference using Markov chain Monte Carlo (MCMC) methods. With this note, we announce the release of version 3.2, a major upgrade to the latest official release presented in 2003. The new version provides convergence diagnostics and allows multiple analyses to be run in parallel with convergence progress monitored on the fly. The introduction of new proposals and automatic optimization of tuning parameters has improved convergence for many problems. The new version also sports significantly faster likelihood calculations through streaming single-instruction-multiple-data extensions (SSE) and support of the BEAGLE library, allowing likelihood calculations to be delegated to graphics processing units (GPUs) on compatible hardware. Speedup factors range from around 2 with SSE code to more than 50 with BEAGLE for codon problems. Checkpointing across all models allows long runs to be completed even when an analysis is prematurely terminated. New models include relaxed clocks, dating, model averaging across time-reversible substitution models, and support for hard, negative, and partial (backbone) tree constraints. Inference of species trees from gene trees is supported by full incorporation of the Bayesian estimation of species trees (BEST) algorithms. Marginal model likelihoods for Bayes factor tests can be estimated accurately across the entire model space using the stepping stone method. The new version provides more output options than previously, including samples of ancestral states, site rates, site dN/dS rations, branch rates, and node dates. A wide range of statistics on tree parameters can also be output for visualization in FigTree and compatible software.

Entanglement renormalization and holography

Abstract: We show how recent progress in real space renormalization group methods can be used to define a generalized notion of holography inspired by holographic dualities in quantum gravity. The generalization is based upon organizing information in a quantum state in terms of scale and defining a higher-dimensional geometry from this structure. While states with a finite correlation length typically give simple geometries, the state at a quantum critical point gives a discrete version of anti-de Sitter space. Some finite temperature quantum states include black hole-like objects. The gross features of equal time correlation functions are also reproduced.

Partition functions of N=(2,2) gauge theories on S^2 and vortices

Abstract: We apply localization techniques to compute the partition function of a two-dimensional N=(2,2) R-symmetric theory of vector and chiral multiplets on S^2. The path integral reduces to a sum over topological sectors of a matrix integral over the Cartan subalgebra of the gauge group. For gauge theories which would be completely Higgsed in the presence of a Fayet-Iliopoulos term in flat space, the path integral alternatively reduces to the product of a vortex times an antivortex partition functions, weighted by semiclassical factors and summed over isolated points on the Higgs branch. As applications we evaluate the partition function for some U(N) gauge theories, showing equality of the path integrals for theories conjectured to be dual by Hori and Tong and deriving new expressions for vortex partition functions.

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.

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.

Supersymmetry on Curved Spaces and Holography

Abstract: We study superconformal and supersymmetric theories on Euclidean four- and threemanifolds with a view toward holographic applications. Preserved supersymmetry for asymptotically locally AdS solutions implies the existence of a (charged) “conformal Killing spinor” on the boundary. We study the geometry behind the existence of such spinors. We show in particular that, in dimension four, they exist on any complex manifold. This implies that a superconformal theory has at least one supercharge on any such space, if we allow for a background field (in general complex) for the R-symmetry. We also show that this is actually true for any supersymmetric theory with an R-symmetry. We also analyze the three-dimensional case and provide examples of supersymmetric theories on Sasaki spaces.

Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

Abstract: We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Holder continuous way along the geodesic. We give examples that show that the Holder exponent, along with essentially all the other consequences that follow from this estimate, are sharp. Among the applications is that the regular set is convex for any non- collapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

Superstring Perturbation Theory Revisited

Abstract: Perturbative superstring theory is revisited, with the goal of giving a simpler and more direct demonstration that multi-loop amplitudes are gauge-invariant (apart from known anomalies), satisfy space-time supersymmetry when expected, and have the expected infrared behavior. The main technical tool is to make the whole analysis, including especially those arguments that involve integration by parts, on supermoduli space, rather than after descending to ordinary moduli space.

Membrane sigma-models and quantization of non-geometric flux backgrounds

Abstract: We develop quantization techniques for describing the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds M . Starting from a suitable Courant sigma-model on an open membrane with target space M , regarded as a topological sector of closed string dynamics in R-space, we derive a twisted Poisson sigma- model on the boundary of the membrane whose target space is the cotangent bundle T * M and whose quasi-Poisson structure coincides with those previously proposed. We argue that from the membrane perspective the path integral over multivalued closed string fields in Q-space is equivalent to integrating over open strings in R-space. The corresponding boundary correlation functions reproduce Kontsevich’s deformation quantization formula for the twisted Poisson manifolds. For constant R-flux, we derive closed formulas for the corresponding nonassociative star product and its associator, and compare them with previous proposals for a 3-product of fields on R-space. We develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones and add fluctuations to the R-flux background. We show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group associated to the T-dual doubled geometry, and hence clarify the relation to the twisted convolution products for topological nonassociative torus bundles. We further demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets.

New recursion relations and a flat space limit for AdS/CFT correlators

Abstract: We consider correlation functions of the stress-tensor or a conserved current in ${\mathrm{AdS}}_{d+1}/{\mathrm{CFT}}_{d}$ computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree-level. These relations have an advantage over the Britto-Cachazo-Feng-Witten (BCFW)-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including $d=3$. We also introduce a new method of extracting flat-space $S$-matrix elements from AdS/CFT correlators in momentum space. We show that the ($d+1$)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the $d$-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon, or graviton amplitude appears as its coefficient.

Morrey spaces in harmonic analysis

Abstract: Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

Bounded distortion mapping spaces for triangular meshes

Abstract: The problem of mapping triangular meshes into the plane is fundamental in geometric modeling, where planar deformations and surface parameterizations are two prominent examples. Current methods for triangular mesh mappings cannot, in general, control the worst case distortion of all triangles nor guarantee injectivity.This paper introduces a constructive definition of generic convex spaces of piecewise linear mappings with guarantees on the maximal conformal distortion, as-well as local and global injectivity of their maps. It is shown how common geometric processing objective functionals can be restricted to these new spaces, rather than to the entire space of piecewise linear mappings, to provide a bounded distortion version of popular algorithms.

Thermalization with a chemical potential from AdS spaces

Abstract: The time-scale of thermalization in holographic dual models with a chemical potential in diverse number of dimensions is systematically investigated using the gauge/gravity duality. We consider a model with a thin-shell of charged dust collapsing from the boundary toward the bulk interior of asymptotically anti-de Sitter (AdS) spaces. In the outer region there is a Reissner-Nordstrom-AdS black hole (RNAdS-BH), while in the inner region there is an anti-de Sitter space. We consider renormalized geodesic lengths and minimal area surfaces as probes of thermalization, which in the dual quantum field theory (QFT) correspond to two-point functions and expectation values of Wilson loops, respectively. We show how the behavior of these extensive probes changes for charged black holes in comparison with Schwarzschild-AdS black holes (AdS-BH), for different values of the black hole mass and charge. The full range of values of the chemical potential over temperature ratio in the dual QFT is investigated. In all cases, the structure of the thermalization curves shares similar features with those obtained from the AdS-BH. On the other hand, there is an important difference in comparison with the AdS-BH: the thermalization times obtained from the renormalized geodesic lengths and the minimal area surfaces are larger for the RNAdS-BH, and they increase as the black hole charge increases.

Composite Majorana fermion wave functions in nanowires

Abstract: We consider Majorana fermions (MFs) in quasi-one-dimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spin-orbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wave function in the weak and strong spin orbit interaction regimes. We find that the wave function for one single MF is a composite object formed by superpositions of different MF wave functions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wave functions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normal-superconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with numerical results obtained in previous studies.

Wall-Crossing in Coupled 2d-4d Systems

Abstract: We introduce a new wall-crossing formula which combines and generalizes the Cecotti-Vafa and Kontsevich-Soibelman formulas for supersymmetric 2d and 4d systems respectively. This 2d-4d wall-crossing formula governs the wall-crossing of BPS states in an N = 2 supersymmetric 4d gauge theory coupled to a supersymmetric surface defect. When the theory and defect are compactied on a circle, we get a 3d theory with a su- persymmetric line operator, corresponding to a hyperholomorphic connection on a vector bundle over a hyperkahler space. The 2d-4d wall-crossing formula can be interpreted as a smoothness condition for this hyperholomorphic connection. We explain how the 2d-4d BPS spectrum can be determined for 4d theories of classS, that is, for those theories ob- tained by compactifying the six-dimensional (0; 2) theory with a partial topological twist on a punctured Riemann surface C. For such theories there are canonical surface defects. We illustrate with several examples in the case of A1 theories of classS. Finally, we indi- cate how our results can be used to produce solutions to the A1 Hitchin equations on the Riemann surface C.

A Lagrangian Approach for the Incompressible Navier-Stokes Equations with Variable Density

Abstract: We investigate the Cauchy problem for the inhomogeneous Navier-Stokes equations in the whole n-dimensional space. Under some smallness assumption on the data, we show the existence of global-in-time unique solutions in a critical functional framework. The initial density is required to belong to the multiplier space of \input amssym $\dot {B}^{n/p-1}_{p,1}({\Bbb R}^n)$. In particular, piecewise-constant initial densities are admissible data provided the jump at the interface is small enough and generate global unique solutions with piecewise constant densities. Using Lagrangian coordinates is the key to our results, as it enables us to solve the system by means of the basic contraction mapping theorem. As a consequence, conditions for uniqueness are the same as for existence. © 2012 Wiley Periodicals, Inc.

A covariant approach to general field space metric in multi-field inflation

Abstract: We present a covariant formalism for general multi-field system which enables us to obtain higher order action of cosmological perturbations easily and systematically. The effects of the field space geometry, described by the Riemann curvature tensor of the field space, are naturally incorporated. We explicitly calculate up to the cubic order action which is necessary to estimate non-Gaussianity and present those geometric terms which have not yet been known before.

Geometry of fractional spaces

Abstract: We introduce fractional flat space, described by a continuous geometry with constant non-integer Hausdorff and spectral dimensions. This is the analogue of Euclidean space, but with anomalous scaling and diffusion properties. The basic tool is fractional calculus, which is cast in a way convenient for the definition of the differential structure, distances, volumes, and symmetries. By an extensive use of concepts and techniques of fractal geometry, we clarify the relation between fractional calculus and fractals, showing that fractional spaces can be regarded as fractals when the ratio of their Hausdorff and spectral dimension is greater than one. All the results are analytic and constitute the foundation for field theories living on multi-fractal spacetimes, which are presented in a companion paper.

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.

The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators

Abstract: We investigate mapping properties for the Bargmann transform on an extended family of modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are valid for modulation spaces with moderate weights. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators on these modulation spaces, and on Gelfand–Shilov spaces.

Smooth compactness of self-shrinkers

Abstract: We prove a smooth compactness theorem for the space of embedded self-shrinkers in $\RR^3$. Since self-shrinkers model singularities in mean curvature flow, this theorem can be thought of as a compactness result for the space of all singularities and it plays an important role in studying generic mean curvature flow.

Greedy Algorithms for Reduced Bases in Banach Spaces

Abstract: Given a Banach space X and one of its compact sets F, we consider the problem of finding a good n dimensional space X_n \subset X which can be used to approximate the elements of F. The best possible error we can achieve for such an approximation is given by the Kolmogorov width d_n(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 A. Buffa, Y. Maday, A.T. Patera, C. Prud'homme, and G. Turinici, "A Priori convergence of the greedy algorithm for the parameterized reduced basis", M2AN Math. Model. Numer. Anal., 46(2012), 595-603 in the case X = H is a Hilbert space. The results there were significantly improved on in P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, "Convergence rates for greedy algorithms in reduced bases Methods", SIAM J. Math. Anal., 43 (2011), 1457-1472. 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.

Profiles of bounded radial solutions of the focusing, energy-critical wave equation

Abstract: In this paper we consider global and non-global bounded radial solutions of the focusing energy-critical wave equation in space dimension 3. We show that any of these solutions decouples, along a sequence of times that goes to the maximal time of existence, as a sum of modulated stationary solutions, a free radiation term and a term going to 0 in the energy space. In the case where there is only one stationary profile, we show that this expansion holds asymptotically without restriction to a subsequence.