We study the well-posedness of a unified system of coupled forward-backward stochastic differential equations (FB-SDEs) with Levy jumps and double completely-S skew reflections. Owing to the reflections, the solution to an embedded Skorohod problem may be not unique, i.e., bifurcations may occur at reflection boundaries, the well-known contraction mapping approach can not be extended directly to solve our problem. Thus, we develop a weak convergence method to prove the well-posedness of an adapted 6-tuple weak solution in the sense of distribution to the unified system. The proof heavily depends on newly established Malliavin calculus for vector-valued Levy processes together with a generalized linear growth and Lipschitz condition that guarantees the well-posedness of the unified system even under a random environment. Nevertheless, if a more strict boundary condition is imposed, i.e., the spectral radii in certain sense for the reflections are strictly less than the unity, a unique adapted 6-tuple strong solution in the sense of sample pathwise is concerned. In addition, as applications and economical studies of our unified system, we also develop new techniques including deriving a generalized mutual information formula for signal processing over possible non-Gaussian channels with multi-input multi-output (MIMO) antennas and dynamics driven by Levy processes.

We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard's theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems.

In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the standard sphere if we fix its geometric center in the ambient space. Finally, we discuss a Shi-Tam type of inequality for the Schwarzschild manifold as an application of our findings.

We consider the reconstruction of the Robin impedance coefficient of a heat conduction system in a two-dimensional spatial domain from the time-average measurement specified on the boundary. By applying the potential representation of a solution, this nonlinear inverse problem is transformed into an ill-posed integral system coupling the density function for potential and the unknown boundary impedance. The uniqueness as well as the conditional stability of this inverse problem is established from the integral system. Then we propose to find the boundary impedance by solving a non-convex regularizing optimization problem. The well-posedness of this optimization problem together with the convergence property of the minimizer is analyzed. Finally, based on the singularity decomposition of the potential representation of the solution, two iteration schemes with their numerical realizations are proposed to solve this optimization problem

We consider an inverse problem of recovering a time-dependent factor of an unknown source on some subboundary for a diffusion equation with time fractional derivative by nonlocal measurement data. Such fractional-order equations describe anomalous diffusion of some contaminants in heterogeneous media such as soil and model the contamination process from an unknown source located on a part of the boundary of the concerned domain. For this inverse problem, we firstly establish the well-posedness in some Sobolev space. Then we propose two regularizing schemes in order to reconstruct an unknown boundary source stably in terms of the noisy measurement data. The first regularizing scheme is based on an integral equation of the second kind which an unknown boundary source solves, and we prove a convergence rate of regularized solutions with a suitable choice strategy of the regularizing parameter. The second regularizing scheme relies directly on discretization by the radial basis function for the initial-boundary value problem for fractional diffusion equation, and we carry out numerical tests, which show the validity of our proposed regularizing scheme.