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We give a survey of classical and recent results on sharp constants and symmetry/asymmetry of extremal functions in $1$-dimensional functional inequalities.

The problem of finding an appropriate geometrical/physical index for measuring a degree of inhomogeneity for a given space-time manifold is posed. Interrelations with the problem of understanding the gravitational/informational entropy are…

We establish spectral convergence results of approximations of unbounded non-selfadjoint linear operators with compact resolvents by operators that converge in generalized strong resolvent sense. The aim is to establish general assumptions…

We synthesize and unify notions of regularity, both of individual sets and of collections of sets, as they appear in the convergence theory of projection methods for consistent feasibility problems. Several new characterizations of…

In this work, we investigate the convergence of numerical approximations to coercivity constants of variational problems. These constants are essential components of rigorous error bounds for reduced-order modeling; extension of these…

The problem of reconstructing functions from their asymptotic expansions in powers of a small variable is addressed by deriving a novel type of approximants. The derivation is based on the self-similar approximation theory, which presents…

The design of fixed point algorithms is at the heart of monotone operator theory, convex analysis, and of many modern optimization problems arising in machine learning and control. This tutorial reviews recent advances in understanding the…

We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate…

We study the approximation of non-negative multi-variate couplings in the uniform norm while matching given single-variable marginal constraints.

We define and axiomatize approximate exclusion atoms in the team semantic setting. A team is a set of assignments, which can be seen as a mathematical model of a uni-relational database, and we say that an approximate exclusion atom is…

We consider high-dimensional estimation problems where the number of parameters diverges with the sample size. General conditions are established for consistency, uniqueness, and asymptotic normality in both unpenalized and penalized…

We propose a novel foundation for calculus that focuses on the notion of approximations while avoiding the use of limits altogether. Continuity is defined as approximation at a point, while differentiability is defined as approximation with…

Molecular dynamics simulations are used to study different definitions of contact at the atomic scale. The roles of temperature, adhesive interactions and atomic structure are studied for simple geometries. An elastic, crystalline substrate…

We introduce a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod…

We obtain new results on the existence and multiplicity of fixed points of Hammerstein equations in very general cones. In order to achieve this, we combine a new formulation of cones in terms of continuous functionals with fixed point…

We consider the geometry relaxation of an isolated point defect embedded in a homogeneous crystalline solid, within an atomistic description. We prove a sharp convergence rate for a periodic supercell approximation with respect to uniform…

Assuming string theorists will not soon provide a compelling case for the primary theory underlying particle physics, the field will proceed as it has historically: with data stimulating and testing ideas. Ideally the soft supersymmetry…

We define a general notion of entropy in elementary, algebraic terms. Based on that, weak forms of a scalar product and a distance measure are derived. We give basic properties of these quantities, generalize the Cauchy-Schwarz inequality,…

We consider the hardness of approximation of optimization problems from the point of view of definability. For many NP-hard optimization problems it is known that, unless P = NP, no polynomial-time algorithm can give an approximate solution…

This paper analyzes general spatially-coupled (SC) systems with multi-dimensional coupling. A continuum approximation is used to derive potential functions that characterize the performance of the SC systems. For any dimension of coupling,…