This article is devoted to the exploration of finite element methods for magneto-heat coupling model, where the eddy current problem and the heat equation are coupled together with the heat convection and the radiation effects. Firstly, the decoupled scheme is established by applying backward Euler discretization in time and $N/acute{e}d/acute{e}lec$-Lagrange finite element in magnetic-temperature field, respectively. Secondly, the existence and uniqueness of the discretized scheme are proved by applying the theory of monotone operators. Then, under some regularity assumptions and time-step restriction

Voronoi formulas are Poisson-style summation formulas for automorphic forms (on GL(n)). They have been powerful tools in analytic number theory for a long time (with applications to the divisor problem, the circle problem, subconvexity of L-functions, etc). Firstly, we present a Voronoi formula for coefficients of a large class of L-functions, in the style of the classical converse theorem of Andre Weil. Our formula applies to full-level cusp forms, Rankin-Selberg convolutions, and isobaric sums. Secondly, we present the balanced Voronoi formula, with its both sides twisted by hyper-Kloosterman sums. Lastly, we will discuss how the formula will be generlized when the automorphic form has ramifications.

We will present the problem of bilinear controllability for Schrödinger equation in a bounded domain, i.e the possibility of controlling the equation by an action on a potential, with the difficulties and the few available results and methods up to now. We will then present a new result of local bilinear controllability in dimension 3 in a neighborhood of an eigenfunction corresponding to a simple eigenvalue and with a non-degeneracy condition of the normal derivative on a part of the boundary satisfying the geometric control condition.

Recently, we generalize the known DDVV-type inequalities for real (skew-)symmetric and complex (skew-)Hermitian matrices into arbitrary real, complex and quaternionic matrices. Inspired by the Erd/H{o}s-Mordell inequality, we establish DDVV-type inequalities for matrices in the subspaces spanned by a Clifford system or a Clifford algebra. We also generalize the B/"{o}ttcher-Wenzel inequality to quaternionic matrices. In this talk, we will briefly introduce the background, the new progress above, and some open questions.

Einstein metrics are natural Riemannian metrics on differentiable manifolds. In dimensions 2 and 3, they must have constant sectional curvature, while in dimension 4, they are much more complicated. For the complex setting, in 1990 Tian classified Kahler-Einstein four-manifolds with positive scalar curvature, and in 2012 LeBrun classified Hermitian, Einstein four-manifolds with positive scalar curvature. For the real setting, however less is known, even assuming a (strong) condition of positive sectional curvature.

Let ，let be a sequence of independent copies of a real-valued random variable , and set , . Motivated by a Theorem of Mikosch (1984), this talk is devoted to establishing a strong law of large numbers for the sequence . More specifically, necessary and sufficient conditions are given for where

The coverage control problem of a network of heterogeneous mobile sensors is addressed in this talk, where the goal is to minimize a coverage cost function which is defined to be the largest arrival time from the mobile sensor network to the points on a cricle. A necessary and sufficient condition for the global minimization of the coverage cost function is firstly derived. Then, distributed coverage control schemes with input saturation are developed to drive the sensors to the optimal configuration such that the necessary and sufficient condition is satisfied. Under the distributed coverage control schemes, the mobile sensors’ spatial order on the circle is preserved throughout the network’s evolution and thus collision between mobile sensors is avoided. Finally, simulation results are presented to illustrate the effectiveness of the proposed distributed control schemes.

In this paper, we pose two kinds of Minkowski problems involving the $p$-Laplacian operator. The Hadamard variational formulas for some $p$-Laplacian functionals are obtained. A good application is to prove symmetry results for solutions to some overdetermined problems of $p$-Laplacian equations.