Communications in Contemporary Mathematics

This paper deals with a nonlocal diffusion elliptic eigenvalue problem. Specifically, the diffusion of the unknown variable at a point of the domain depends on its value in a neighborhood of the point. We apply bifurcation arguments and appropriate approximation to obtain our results. Some applications to the population dynamics will be given.

We determine the largest rate of exponential decay at infinity for non-trivial solutions to the Dirac equation

Let $(M,𝜃)$ be a compact, connected, strictly pseudo-convex CR manifold. In this paper, we give some properties of the CR Yamabe Operator $L_{𝜃}$. We present an upper bound for the Second CR Yamabe Invariant, when the First CR Yamabe Invariant is negative, and the existence of a minimizer for the Second CR Yamabe Invariant, under some conditions.

Assume that $\mathrm{\Phi }:{𝕄}_n(ℂ)\to {𝕄}_n(ℂ)$ is a superoperator which preserves hermiticity. We give an algorithm determining whether $\mathrm{\Phi }$ preserves semipositivity (we call $\mathrm{\Phi }$ positive in this case). Our approach to the problem has a model-theoretic nature, namely, we apply techniques of quantifier elimination theory for real numbers. An approach based on these techniques seems to be the only one that allows to decide whether an arbitrary hermiticity-preserving $\mathrm{\Phi }$ is positive. Before we go to detailed analysis of the problem, we argue that quantifier elimination for real numbers (and also for complex numbers) can play a significant role in quantum information theory and other areas as well.

In this paper, we prove the local null controllability property for a nonlinear coupled system of two Korteweg–de Vries equations posed on a bounded interval and with a source term decaying exponentially on $t=T$. The system was introduced by Gear and Grimshaw to model the interactions of two-dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman, we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in $L^2$ have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

The topology of the moduli space for Lamé functions of degree $m$ is determined: this is a Riemann surface which consists of two connected components when $m\ge 2$; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn’s polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle $2\pi (2m+1)$ on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves.

We show that direct limit completions of vertex tensor categories inherit vertex and braided tensor category structures, under conditions that hold for example for all known Virasoro and affine Lie algebra tensor categories. A consequence is that the theory of vertex operator (super)algebra extensions also applies to infinite-order extensions. As an application, we relate rigid and non-degenerate vertex tensor categories of certain modules for both the affine vertex superalgebra of $𝔬𝔰𝔭(1|2)$ and the $N=1$ super Virasoro algebra to categories of Virasoro algebra modules via certain cosets.