Related papers: Uniqueness of a constrained variational problem an…

We consider stochastic non-linear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational…

Physical understanding of how the interplay between symmetries and nonlinear effects can control the scaling and multiscaling properties in a coupled driven system, such as magnetohydrodynamic turbulence or turbulent binary fluid mixtures,…

Functional data analysis involves data described by regular functions rather than by a finite number of real valued variables. While some robust data analysis methods can be applied directly to the very high dimensional vectors obtained…

Using Carleman estimates, we give a lower bound for solutions to the discrete Schr\"odinger equation in both dynamic and stationary settings that allows us to prove uniqueness results, under some assumptions on the decay of the solutions.

We calculate universal finite-size scaling functions for systems with an n-component order parameter and algebraically decaying interactions. Just as previously has been found for short-range interactions, this leads to a singular…

Nonlinear filtering is a pivotal problem that has attracted significant attention from mathematicians, statisticians, engineers, and various other scientific disciplines. The solution to this problem is governed by the so-called filtering…

In this paper we prove the uniqueness and radial symmetry of minimizers for variational problems that model several phenomena. The uniqueness is a consequence of the convexity of the functional. The main technique is Fourier transform of…

The phenomenon of removable singularity is studied for overedetermined systems of differential equations. We show that the dimension of the characteristic variety plays a key role in the problem.

In this article the question on uniqueness of weak solution of the incompressible Navier-Stokes Equations in the 3-dimensional case is studied. Here the investigation is carried out with use of another approach. The uniqueness of velocity…

We study a free boundary problem which arises as the continuum version of a stochastic particles system in the context of Fourier law. Local existence and uniqueness of the classical solution are well known in the literature of free…

We introduce a bottleneck method for learning data representations based on information deficiency, rather than the more traditional information sufficiency. A variational upper bound allows us to implement this method efficiently. The…

In this paper we look at a class of random optimization problems. We discuss ways that can help determine typical behavior of their solutions. When the dimensions of the optimization problems are large such an information often can be…

Finite-size scaling is a key tool in statistical physics, used to infer critical behavior in finite systems. Here we use the analogous concept of finite-time scaling to describe the bifurcation diagram at finite times in discrete dynamical…

We present a theory of ensemble diversity, explaining the nature of diversity for a wide range of supervised learning scenarios. This challenge has been referred to as the holy grail of ensemble learning, an open research issue for over 30…

We examine an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters, where each cluster is assumed to be composed of identical units. In contrast to previous investigations into such…

Variational problems under uniform quasiconvex constraints on the gradient are studied. In particular, existence of solutions to such problems is proved as well as existence of lagrange multipliers associated to the uniform constraint. They…

A principled method to obtain approximate solutions of general constrained integer optimization problems is introduced. The approach is based on the calculation of a mean field probability distribution for the decision variables which is…

In this paper we give an explicit representation of the solutions of a characteristic Cauchy problem for a class of PDEs with singular coefficients. We give the explicit solutions in terms of the Gauss hypergeometric functions, which enable…

We investigate the Calder\'on problem for the fractional Schr\"odinger equation with drift, proving that the unknown drift and potential in a bounded domain can be determined simultaneously and uniquely by an infinite number of exterior…

We frame the problem of selecting an optimal audio encoding scheme as a supervised learning task. Through uniform convergence theory, we guarantee approximately optimal codec selection while controlling for selection bias. We present…