In this paper we study the singularities of holomorphic functions of bicomplex variables introduced by G. B. Price ($An Introduction to Multicomplex Spaces and Functions$, Dekker, New York,1991). In particular, we use computational algebra techniques to show that even in the case of one bicomplex variable, there cannot be compact singularities. The same techniques allow us to prove a duality theorem for such functions.

We study minimal graphic functions on complete Riemannian manifolds $\Sigma$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Sigma$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Sigma$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

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.

The aim of this paper is to develop a mathematical framework for opto-elastography. In opto-elastography, a mechanical perturbation of the medium produces a decorrelation of optical speckle patterns due to the displacements of optical scatterers. To model this, we consider two optically random media, with the second medium obtained by shifting the first medium in some local region. We derive the radiative transfer equation for the cross-correlation of the wave fields in the media. Then we derive its diffusion approximation. In both the radiative transfer and the diffusion regimes, we relate the correlation of speckle patterns to the solutions of the radiative transfer and the diffusion equations. We present numerical simulations based on our model which are in agreement with recent experimental measurements.

Theorem. Let $\pi$ be a finite group of order $n$, $R$ be a Dedekind domain satisfying that (i) $\fn{char}R=0$, (ii) every prime divisor of $n$ is not invertible in $R$, and (iii) $p$ is unramified in $R$ for any prime divisor $p$ of $n$. Then all the flabby (resp.\ coflabby) $R\pi$-lattices are invertible if and only if all the Sylow subgroups of $\pi$ are cyclic. The above theorem was proved by Endo and Miyata when $R=\bm{Z}$ \cite[Theorem 1.5]{EM}. As applications of this theorem, we give a short proof and a partial generalization of a result of Torrecillas and Weigel \cite[Theorem A]{TW}, which was proved using cohomological Mackey functors.