Numerical multilinear algebra (or called tensor computation), in which instead of matrices and vectors the higher-order tensors are considered in numerical viewpoint, is a new branch of computational mathematics. Although it is an extension of numerical linear algebra, it has many essential differences from numerical linear algebra and more difficulties than it. In this paper, we present a survey on the state of the art knowledge on this topic, which is incomplete, and indicate some new trends for further research. Our survey also contains a detailed bibliography as its important part. We hope that this new area will be receiving more attention of more scholars.

This paper deals with the problem of maximizing the expected utility of the terminal wealth when the stock price satisfies a stochastic differential equation with instantaneous rates of return modelled as an Ornstein-Uhlenbeck process. Here, only the stock price and interest rate can be observable for an investor. It is reduced to a partially observed stochastic control problem. Combining the filtering theory with the dynamic programming approach, explicit representations of the optimal value functions and corresponding optimal strategies are derived. Moreover, closed-form solutions are provided in two cases of exponential utility and logarithmic utility. In particular, logarithmic utility is considered under the restriction of short-selling and borrowing.

This paper is concerned with the optimal control of jump-type stochastic differential equations associated with polar-decomposed Lévy measures with the feature of explicit construction on the jump term. The concrete construction is then utilized for analysis of two portfolio optimization problems for financial market models driven by stable-like processes.

We extend the results for 2-D Boussinesq equations from ℝ² to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U_t + A(t, U) = 0. In stead of using the methods of fundamental solutions in the case of entire ℝ², we study the qualities of F(u, υ) = (u · ▽)υ to get some useful estimates for A(t, U), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control $\left\| \theta \right\|_{H^s (\Omega )} $$ and $\left\| u \right\|_{H^s (\Omega )} $$ by $\left\| {\nabla \theta } \right\|_{L^\infty (\Omega )} $$ and $\left\| {\nabla u} \right\|_{L^\infty (\Omega )} $$. Furthermore, $\left\| {\nabla \theta } \right\|_{L^\infty (\Omega )} $$ can control $\left\| {\nabla u} \right\|_{L^\infty (\Omega )} $$ by using vorticity transportation equations. At last, $\left\| {\nabla \theta } \right\|_{M_\phi (\Omega )} $$ can control $\left\| {\nabla \theta } \right\|_{L^\infty (\Omega )} $$. Thus, we can find a blow-up criterion in the form of $\lim _{t \to T^ * } \int_0^t {\left\| {\nabla \theta ( \cdot ,\tau )} \right\|_{M_\phi (\Omega )} d\tau = \infty } $$.

Explicit sufficient and necessary conditions are presented for reaction-diffusion type Dirichlet forms on Polish spaces studied by Röckner and Wang [Potential Anal., 2006, 24: 223–243] to be (quasi-)regular. As preparations, the (quasi-)regularity of the sum of two or countably many Dirichlet forms is investigated.

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that $\int_{\mathbb{R}^n } {x^\gamma \varphi (x)dx = 0} $ for all multiindices γ. Let (ℐ(ℝⁿ))′ and (ℐ_∞(ℝⁿ))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝⁿ))′ and (ℐ_∞(ℝⁿ))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley function of f belongs to L^p(ℝⁿ) for some p ∊ (0,1], the condition f ∊ (ℐ_∞(ℝⁿ))′ is equivalent to that f ∊ (ℐ(ℝⁿ))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝⁿ) and ℐ_∞(ℝⁿ), also including their corresponding Hardy spaces.

On the basis of an extension for an early result of Khintchine, the class of asymptotic distributions of the standardized Ψ-sums for a class of distributions is obtained in this paper.

In this paper, we study the irreducible decompositions of determinantal varieties of matrices given by rank conditions on upper left submatrices. Using the concept of essential rank function and the Ehresmann partial order on the set of all simple matrices, we design an algorithm to write a determinantal variety as a union of its irreducible components. This solves a problem raised by B. Sturmfels.