In this paper, we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimizers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo and Shu about the explicit characterization of minimizers, and present a new proof, which has the advantage of being extendable to higher dimensions. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.

Each metric graph has canonically associated to it a polarized real torus called its tropical Jacobian. A fundamental real-valued invariant associated to each polarized real torus is its tropical moment. We give an explicit and efficiently computable formula for the tropical moment of a tropical Jacobian in terms of potential theory on the underlying metric graph. We show that there exists a universal linear relation between the tropical moment, a certain capacity called the tau invariant, and the total length of a metric graph. To put our formula in a broader context, we relate our work to the computation of heights attached to principally polarized abelian varieties.

This paper presents a theoremon universality on orthogonal polynomials/randommatrices under a weak local condition on the weight function $w$ . With a new inequality for polynomials and with the use of fast decreasing polynomials, it is shown that an approach of D. S. Lubinsky is applicable. The proof works at all points that are Lebesgue-points for both the weight function $w$ and $\log \,w$ .

We describe our implementation of a parallel fast multipole method for evaluating potentials for discrete and continuous source distributions. The first requires summation over the source points and the second requiring integration over a continuous source density. Both problems require (N2) complexity when computed directly; however, can be accelerated to (N) time using FMM. In our PVFMM software library, we use kernel independent FMM and this allows us to compute potentials for a wide range of elliptic kernels. Our method is high order, adaptive and scalable. In this paper, we discuss several algorithmic improvements and performance optimizations including cache locality, vectorization, shared memory parallelism and use of coprocessors. Our distributed memory implementation uses space-filling curve for partitioning data and a hypercube communication scheme. We present convergence results for Laplace, Stokes and Helmholtz (low wavenumber) kernels for both particle and volume FMM. We measure efficiency of our method in terms of CPU cycles per unknown for different accuracies and different kernels. We also demonstrate scalability of our implementation up to several thousand processor cores on the Stampede platform at the Texas Advanced Computing Center.

In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

An occupation measure describes the expected amount of time a stochastic process spends in different parts of its state space prior to a given random time. It is shown that a basic identity involving occupation measures provides a unified approach to a variety of moment identities for Markov chains, and some connections with potential theory are made.