In algebraic combinatorics, the first step of the classification of interesting objects is usually to find all their feasible parameters. The feasible parameters are often integral solutions of some complicated Diophantine equations, which cannot be solved by known methods. In this paper, we develop a method to solve such Diophantine equations in $3$ variables. We demonstrate it by giving a classification of finite subsets that are spherical $2$-distance sets and spherical $\{4,2,1\}$-designs at the same time.

Let $\mathbf{A} := \mathbb{F}_q[t]$ be a polynomial ring over a finite field $\mathbb{F}_q$ of odd characteristic and let $D \in \mathbf{A}$ be a square-free polynomial. Denote by $\mathbf{N}_{D}(n,q)$ the number of polynomials $f$ in $\mathbf{A}$ of degree $n$ which may be represented in the form $u \cdot f = A^2-DB^2$ for some $A,B \in \mathbf{A}$ and $u \in \mathbb{F}_q^{\times}$, and by $\mathbf{B}_{\mathcal{D}}(n,q)$ the number of polynomials in $\mathbf{A}$ of degree $n$ which can be represented by a primitive quadratic form of a given discriminant $\mathcal{D} \in \mathbf{A}$, not necessary square-free. If the class number of the maximal order of $\mathbb{F}_q(t,\sqrt{D})$ is one, then we give very precise asymptotic formulas for $\mathbf{N}_{D}(n,q)$. Moreover, we also give very precise asymptotic formulas for $\mathbf{B}_{\mathcal{D}}(n,q)$.

In this paper, we deduce several types of generating functions for $q$-2D Hermite polynomial by the method of homogeneous $q$-difference equations. Besides, we deduce a multilinear generating function for $q$-2D Hermite polynomials as a generalization of Andrew's result. Moreover, we build a transformation identity involving the generalized $q$-2D Hermite polynomials by the method of homogeneous $q$-difference equations. As an application, we give a transformation identity involving $D_{q}(m,n)$ and $D_{q}^{*}(m,n)$.

Suppose that $R$ is a prime ring of characteristic different from $2$ with Utumi quotient ring $U$, $C = Z(U)$ the extended centroid of $R$, and $f(x_1,\ldots,x_n)$ a noncentral multilinear polynomial over $C$. If $F$, $G$ and $H$ are three nonzero generalized derivations of $R$ such that \[ F\big( G(f(X)) f(X) \big) = f(X) H(f(X)) \] for all $X = (x_{1},\ldots,x_{n}) \in R^n$, then we describe the nature of the maps $F$, $G$ and $H$.

This paper deals with the Cauchy problem and Neumann initial boundary value problem for a quasilinear diffusion equation with nonlocal Fisher-KPP type reaction terms. We establish the uniform boundedness and global existence of solutions to the problems by using multipliers technique and modified Moser's iteration argument for some ranges of parameters. Moreover, the ranges of parameters have similar structure to that of the classical critical Fujita exponent.

We studied the existence of a positive solution to Schrödinger-type semipositone problems with mixed nonlinear boundary conditions. By considering the cases when the reaction term with a parameter satisfies a superlinear and a sublinear growth condition at infinity, we established the existence of a positive solution for the large and small values of the parameter, respectively. The proofs are mainly based on the sub- and supersolution method for the sublinear case and the mountain pass lemma with $C^{1,\alpha}(\overline{\Omega})$-regularity for the superlinear case.

In this study we propose a construction framework utilizing stepwise integrations and the Sherman-Morrison-Woodbury formula to seek pseudospectral integration preconditioning matrices for differential operators. We illustrate this framework through formulating an inverse matrix for the Helmholtz differential operator of the form $\frac{d}{dx} a(x) \frac{d}{dx} + c(x)$. Numerical experiments were conducted to examine the performance of the derived operator. The results show that the inverse matrix is an effective solution operator to numerically solve general second order differential equations.

This paper investigates the time optimal control problems to a new class of fractional non-instantaneous impulsive stochastic partial differential inclusions with Clarke subdifferential in Hilbert spaces. Firstly, using the fractional calculus, properties of fractional resolvent operators and a fixed-point theorem, the existence of mild solutions for these systems is presented. Secondly, the existence of time optimal control of system governed by fractional stochastic control inclusions with Clarke subdifferential and non-instantaneous impulses is also obtained. Finally, an example is given to illustrate our main results.

In this paper, a nonsmooth vector optimization problem with cone and equality constraints is considered. We establish some relations between the notions of vector critical points in the sense of Fritz John and in the sense of Karush-Kuhn-Tucker and weakly $K$-efficient and $K$-efficient solutions for the constrained vector optimization problem in which every component of the involved functions is locally Lipschitz. These relationships are stated under cone-$FJ$-pseudo-invexity and cone-$KT$-pseudo-invexity hypotheses defined for the considered vector optimization problem with cone inequality and also equality constraints and via the Clarke generalized gradient for vector-valued functions.