In this paper, we introduce the application of random matrices in mathematical physics including Riemann-Hilbert problem, nuclear physics, big data, image processing, compressed sensing and so on. We start with the Riemann-Hilbert problem and state the relation between the probability distribution of nontrivial zeros and the eigenvalues of the random matrices. Through the random matrices theory, we derive the distribution of Neutron width and probability density between energy levels. In addition, the application of random matrices in quantum chromo dynamics and two dimensional Einstein gravity equations is also presented in this paper.

In this paper, we define Sumudu transform with convergence conditions in bicomplex space. Also, we derive some of its basic properties and its inverse. Applications of bicomplex Sumudu transform are illustrated to find the solution of differential equation of bicomplex-valued functions and find the solution for Cartesian transverse electric magnetic (TEM) waves in homogeneous space.

An $n×n$ ray pattern $A$ is said to be spectrally arbitrary if for every monic $n$th degree polynomial $f(x)$ with coefficients from $\mathbb{C},$ there is a complex matrix in the ray pattern class of $A$ such that its characteristic polynomial is $f(x).$ In this paper, a family ray patterns is proved to be spectrally arbitrary by using Nilpotent-Jacobian method.

We design a family of 2D $H^m$-nonconforming finite elements using the full $P_{2m−3}$ degree polynomial space, for solving $2m$th elliptic partial differential equations. The consistent error is estimated and the optimal order of convergence is proved. Numerical tests on the new elements for solving tri-harmonic, 4-harmonic, 5-harmonic and 6-harmonic equations are presented, to verify the theory.

A predator-prey system with a constant proportion of prey refuge and stage-structure for prey species is proposed and studied in this paper. A set of conditions for the permanence of the system is obtained. The local stability of the system is discussed by the sign of eigenvalues. Furthermore, by using the iterative method, some suitable sufficient conditions for the global attractivity of the interior equilibrium is obtained. Our study shows that the constant proportion of prey refuge could lead to more complicated dynamic behaviors. Numerical simulations are also presented to illustrate the feasibility of the main results.

This paper deals with the class of $Q$-tensors, that is, a $Q$-tensor is a real tensor $\mathcal{A}$ such that the tensor complementarity problem $(q, \mathcal{A}):$ finding an $x ∈\mathbb{R}^n$ such that $x ≥ 0,$ $q+\mathcal{A}x^{m−1} ≥ 0,$ and $x^⊤(q+\mathcal{A}x^{m−1}) = 0,$ has a solution for each vector $q ∈ \mathbb{R}^n.$ Several subclasses of $Q$-tensors are given: $P$-tensors, $R$-tensors, strictly semi-positive tensors and semi-positive $R_0$-tensors. We prove that a nonnegative tensor is a $Q$-tensor if and only if all of its principal diagonal entries are positive, and so the equivalence of $Q$-tensor, $R$-tensors, strictly semi-positive tensors was showed if they are nonnegative tensors. We also show that a tensor is an $R_0$-tensor if and only if the tensor complementarity problem $(0, \mathcal{A})$ has no non-zero vector solution, and a tensor is a $R$-tensor if and only if it is an $R_0$-tensor and the tensor complementarity problem $(e, A)$ has no non-zero vector solution, where $e = (1, 1 · · · , 1)^⊤.$

We study the norm retrieval by projections on an infinite-dimensional Hilbert space $H.$ Let $\{e_i\}_{i∈I}$ be an orthonormal basis in $H$ and $W_i = \{e_i\}^⊥$ for all $i ∈ I.$ We show that $\{W_i\}_{i∈I}$ does norm retrieval if and only if $I$ is an infinite subset of $N.$ We also give some properties of norm retrieval by projections.