Translation Invariant Set Research Articles

Critical point asymptotics for Gaussian random waves with densities of any Sobolev regularity

We consider Gaussian random monochromatic waves u on the plane depending on a real parameter s that is directly related to the regularity of its Fourier transform. Specifically, the Fourier transform of u is fdσ, where dσ is the Hausdorff measure on the unit circle and the density f is a function on the circle that, roughly speaking, has exactly s−12 derivatives in L2 almost surely. When s=0, one recovers the classical setting for random waves with a translation-invariant covariance kernel. The main thrust of this paper is to explore the connection between the regularity parameter s and the asymptotic behavior of the number N(∇u,R) of critical points that are contained in the disk of radius R≫1. More precisely, we show that the expectation EN(∇u,R) grows like the area of the disk when the regularity is low enough (s<32) and like the diameter when the regularity is high enough (s>52), and that the corresponding exponent changes according to a linear interpolation law in the intermediate regime. The transitions occurring at the endpoint cases involve the square root of the logarithm of the radius. Interestingly, the highest asymptotic growth rate occurs only in the classical translation-invariant setting, s=0. A key step of the proof of this result is the obtention of precise asymptotic expansions for certain Neumann series of Bessel functions. When the regularity parameter is s>5, we show that in fact N(∇u,R) grows like the diameter with probability 1, albeit the ratio is not a universal constant but a random variable.

Many-body topological and skin states without open boundaries

Robust boundary states have been the focus of much recent research, both as topologically protected states and as non-Hermitian skin states. In this paper, we show that many-body effects can also induce analogs of these robust states in place of actual physical boundaries. Particle statistics or suitably engineered interactions, i.e., in ultracold atomic lattices can restrict the accessible many-body Hilbert space, and introduce effective boundaries in a spatially periodic higher-dimensional configuration space. We demonstrate the emergence of topological chiral modes in a two-fermion hopping model without open boundaries, with fermion pairs confined and asymmetrically propagated by suitably chosen fluxes. Heterogeneous nonreciprocal hoppings across different particle species can also result in robust particle clumping in a translation invariant setting, reminiscent of skin mode accumulation at an open boundary. But unlike fixed open boundaries, effective boundaries correspond to the locations of impenetrable particles and are mobile, giving rise to fundamentally different many-body versus single-body spectra and corresponding dynamics. Since nonreciprocal accumulation is agnostic to the dimensionality of restricted Hilbert spaces, our many-body skin states generalize directly in the thermodynamic limit. The many-body topological states, however, are nontrivially dimension-dependent, and their detailed exploration will stimulate further studies in higher dimensional topological invariants.

Calculus of continuous matrix product states

We discuss various properties of the variational class of continuous matrix product states, a class of ansatz states for one-dimensional quantum fields that was recently introduced as the direct continuum limit of the highly successful class of matrix product states. We discuss both attributes of the physical states, e.g. by showing in detail how to compute expectation values, as well as properties intrinsic to the representation itself, such as the gauge freedom. We consider general translation non-invariant systems made of several particle species and derive certain regularity properties that need to be satisfied by the variational parameters. We also devote a section to the translation invariant setting in the thermodynamic limit and show how continuous matrix product states possess an intrinsic ultraviolet cutoff. Finally, we introduce a new set of states which are tangent to the original set of continuous matrix product states. For the case of matrix product states, this construction has recently proven relevant in the development of new algorithms for studying time evolution and elementary excitations of quantum spin chains. We thus lay the foundation for similar developments for one-dimensional quantum fields.

A Simple Proof of Vitali's Theorem for Signed Measures

Several theorems are named after the Italian mathematician Vitali. In this note, we provide a simple proof of an extension of Vitali's Theorem on the existence of nonmeasurable sets. Specifically we show, without using any decomposition theorems, that there does not exist a nontrivial, atom-less, σ-additive and translation invariant set function L from the power set of the real line to the extended real numbers with L([0, 1]) = 1. (Note that L is not assumed to be nonnegative.)

Fast interscale wavelet denoising of Poisson-corrupted images

We present a fast algorithm for image restoration in the presence of Poisson noise. Our approach is based on (1) the minimization of an unbiased estimate of the MSE for Poisson noise, (2) a linear parametrization of the denoising process and (3) the preservation of Poisson statistics across scales within the Haar DWT. The minimization of the MSE estimate is performed independently in each wavelet subband, but this is equivalent to a global image-domain MSE minimization, thanks to the orthogonality of Haar wavelets. This is an important difference with standard Poisson noise-removal methods, in particular those that rely on a non-linear preprocessing of the data to stabilize the variance. Our non-redundant interscale wavelet thresholding outperforms standard variance-stabilizing schemes, even when the latter are applied in a translation-invariant setting (cycle-spinning). It also achieves a quality similar to a state-of-the-art multiscale method that was specially developed for Poisson data. Considering that the computational complexity of our method is orders of magnitude lower, it is a very competitive alternative. The proposed approach is particularly promising in the context of low signal intensities and/or large data sets. This is illustrated experimentally with the denoising of low-count fluorescence micrographs of a biological sample.

CELLULAR NEURAL NETWORKS: SPACE-DEPENDENT TEMPLATE, MOSAIC PATTERNS AND SPATIAL CHAOS

We consider a Cellular Neural Network (CNN) with a bias term in the integer lattice tenpoint ℤ2 on the plane tenpoint ℤ2. We impose a space-dependent coupling (template) appropriate for CNN in the hexagonal lattice on tenpoint ℤ2. Stable mosaic patterns of such CNN are completely characterized. The spatial entropy of a tenpoint (p1, p2)-translation invariant set is proved to be well-defined and exists. Using such a theorem, we are also able to address the complexities of resulting mosaic patterns.

A Singular Field Method for Maxwell's Equations: Numerical Aspects for 2D Magnetostatics

The present paper deals with the solution of Maxwell-type problems by means of nodal H1 -conforming finite elements. In a nonconvex piecewise regular domain surrounded by a perfect conductor, such a discretization cannot in general approximate the singular behavior of the electromagnetic field near "reentrant" corners or edges. The singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior. The latter are deduced from the singular functions associated with the scalar Laplace operator. The theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting, but the study can be easily extended to numerous Maxwell-type problems. The numerical implementation of both variants is studied for a domain containing a single reentrant corner.

Local Decomposition of Refinable Spaces and Wavelets

A convenient setting for studyingmultiscale techniquesin various areas of applications is usually based on a sequence of nested closed subspaces of some function space script F which is often referred to asmultiresolution analysis.The concept ofwaveletsis a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of theorthogonaldifference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of open face Rs, say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding tobiorthogonal wavelets.The main objective of this paper is therefore to study possibly nonorthogonal but in a certain sensestableand evenlocaldecompositions of nested spaces and to develop tools which are not necessarily confined to the translation invariant setting. A convenient way ofparametrizingsuch decompositions is to reformulate them in terms of matrix relations. This allows one to characterizeallstable or local decompositions by identifying unique matrix transformations that carry one given decomposition into another one. It will be indicated how such a mechanism may help realizing several desirable features of multiscale decompositions and constructing stable multiscale bases with favorable properties. In particular, we apply these results to the identification of decompositions induced by local linear projectors. The importance of this particular application with regard to the construction of multiscale Riesz bases will be pointed out. Furthermore, we indicate possible specializations to orthogonal decompositions of spline spaces relative to nonuniform knot sequences, piecewise linear finite elements and principal shift invariant spaces. The common ground of all these examples as well as other situations of practical and theoretical interest is thatsome initialmultiscale basis is available. The techniques developed here can then be used to generate from such an initial decomposition other ones with desirable properties pertaining to moment conditions or stability properties.

Minimal Representations for Translation-Invariant Set Mappings by Mathematical Morphology

In his 1975 book, Matheron introduced a pair of dual representations, written in terms of erosions and dilations, for increasing translation invariant set mappings, using the concept of a kernel. Based on hit-miss topology, Maragos, in his 1985 Ph.D. thesis, has given sufficient conditions under which the increasing mappings have minimal representations. In this paper, a pair of dual representations for translation-invariant set mappings (not necessarily increasing) is presented. It is shown that under the same sufficient conditions such mappings have minimal representations. Actually, the representations of Matheron and Maragos are special cases of the proposed ones. Finally, some examples are given to illustrate the theory.