Related papers: Condensation and densification for sets of large d…
In solving Bayesian inverse problems, it is often desirable to use a common density parameterization to denote the prior and posterior. Typically we seek a density from the same family as the prior which closely approximates the true…
Many high dimensional integrals can be reduced to the problem of finding the relative measures of two sets. Often one set will be exponentially larger than the other, making it difficult to compare the sizes. A standard method of dealing…
We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $\abs{Q(x)-\xi}< \epsilon$ for a generic ternary form $Q$ is $\abs{x}\ll \epsilon^{-1}$. We also establish…
A finite relational structure A is called compact if for any infinite relational structure B of the same type, the existence of a homomorphism from B to A is equivalent to the existence of homomorphisms from all finite substructures of B to…
Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency,…
We study numerically the crystallization of a hard-sphere mixture with 8\% polydispersity. Although often used as a model glass former, for small system sizes we observe crystallization in molecular dynamics simulations. This opens the…
This paper deals with the dynamics of discretizations of isometries of $\mathbf R^n$, and more precisely the density of the successive images of $\mathbf Z^n$ by the discretizations of a generic sequence of isometries. We show that this…
We propose a novel numerical homogenization method based on the edge multiscale approach for solving indefinite time-harmonic Maxwell equations in heterogeneous media with large wavenumber. Numerical methods for these equations in…
The study of sums of finite sets of integers has mostly concentrated on sets with very small sumsets (Freiman's theorem and related work) and on sets with very large sumsets (Sidon sets and $B_h$-sets). This paper considers the full range…
We prove several results of the following type: given finite dimensional normed space V there exists another space X with log (dim X) = O(log (dim V)) and such that every subspace (or quotient) of X, whose dimension is not "too small,"…
The inverse problem of diffraction theory in essence amounts to the reconstruction of the atomic positions of a solid from its diffraction image. From a mathematical perspective, this is a notoriously difficult problem, even in the…
We develop a method for determining the density of squarefree values taken by certain multivariate integer polynomials that are invariants for the action of an algebraic group on a vector space. The method is shown to apply to the…
We show that there are sets of integers with asymptotic density arbitrarily close to 1 in which there is no solution to the equation ab=c, with a,b,c in the set. We also consider some natural generalizations, as well as a specific numerical…
We study the extent to which sets A in Z/NZ, N prime, resemble sets of integers from the additive point of view (``up to Freiman isomorphism''). We give a direct proof of a result of Freiman, namely that if |A + A| < K|A| and |A| < c(K)N…
A subset $A$ of a commutative semigroup $X$ is called a $B_h$ set in $X$ if the only solutions to $a_1+\dots+a_h = b_1 + \cdots +b_h$ (with $a_i,b_i \in A$) are the trivial solutions $\{a_1,\dots,a_h\} = \{b_1,\dots,b_h\}$ (as multisets).…
We consider density estimators based on the nearest neighbors method applied to discrete point distibutions in spaces of arbitrary dimensionality. If the density is constant, the volume of a hypersphere centered at a random location is…
A theorem of Glasner says that if $X$ is an infinite subset of the torus $\mathbb{T}$, then for any $\epsilon>0$, there exists an integer $n$ such that the dilation $nX=\{nx: x \in \mathbb{T} \}$ is $\epsilon$-dense (i.e, it intersects any…
This article is the first part of a two-fold study, the objective of which is the theoretical analysis and numerical investigation of new approximate corrector problems in the context of stochastic homogenization. We present here three new…
We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean…
We derive necessary density conditions for sampling and for interpolation in general reproducing kernel Hilbert spaces satisfying some natural conditions on the geometry of the space and the reproducing kernel. If the volume of shells is…