Let $p\geqslant 2$ be a prime and $\mathbb{Z}_p$ be the ring of $p$-adic integers. For any $\alpha,\beta,z\in \mathbb{Z}_p$, define $f_{\alpha,\beta}(z)=\alpha z+\beta$. The first part of this paper studies all minimal subsystems of semigroup dynamical systems $(\mathbb{Z}_p,G)$ when $f_{\alpha_1,\beta_1}$ and $f_{\alpha_2,\beta_2}$ are commutative, where the semigroup $G=\{f_{\alpha_1,\beta_1}^n \circ f_{\alpha_2,\beta_2}^m: m,n \in \mathbb{N}\}$. In particular, we find the semigroup dynamical system $(\mathbb{Z}_p,G)\ (p\geqslant 3)$ is minimal if and only if $(\mathbb{Z}_p,f_{\alpha_1,\beta_1})$ or $(\mathbb{Z}_p,f_{\alpha_2,\beta_2})$ is minimal and we determine all the cases that $(\mathbb{Z}_2,G)$ is minimal. In the second part, we study weakly essentially minimal affine semigroup dynamical systems on $\mathbb{Z}_p$, which is a kind of minimal semigroup systems without any minimal single action. It is shown that such semigroup is non-commutative when $p\geqslant 3$. Moreover, for a fixed prime $p$, we find the least number of generators of a weakly essentially minimal affine semigroup on $\mathbb{Z}_p$. We show that such number is $2$ for $p=2$ and $3$ for $p=3$. Also, we show that such number is not greater than $p$.

Iteration is a simple repetition of the same operation. However, it may be complex in some simple mappings such as polynomial mappings. In this paper, we discuss the iteration of a special class of nonmonotonic mappings called Markov mappings, and give the concrete expressions of iteration of the mappings which have either one, or two, or finitely many nonmonotonic points respectively.

In this paper, we prove the Lyapunov behavior and dynamical localization for the quasi-periodic CMV matrices with most frequencies and Verblunsky coefficients defined by the skew-shift, in the regime of positive Lyapunov exponents.

This paper mainly studies the global inviscid limit of the Ginzburg-Landau system while the initial data is taken in $L^2(\mathbb{R}^n){\times}L^2(\mathbb{R}^n)\operatorname{or}H^1(\mathbb{R}^n){\times}H^1(\mathbb{R}^n).$ Specifically, we investigate the difference between the solution of the Ginzburg-Landau system and the nonlinear Schrödinger system, and use energy estimate to deal with the difference．It is obtained that the inviscid limit of the solution of the Ginzburg-Landau system is the solution of the nonlinear Schrödinger system．

This paper studies the existence and uniqueness of transonic shock solutions to the steady compressible Euler equations with gravity in a three-dimensional spherically symmetric divergent nozzle. Assuming that the influence of gravity on the fluid is sufficiently small and the supersonic initial conditions are given at the entrance, it can be proved that when the pressure $p$ at the exit falls in certain range, there exists a unique transonic shock solution within the nozzle by demonstrating that the pressure at the outlet is a strictly monotone function of the shock location.

This paper studies the cavitation and concentration phenomena of the Riemann solutions for a reduced two-phase mixtures model with non-isentropic dusty gas state as the pressure vanishes. Firstly, we construct the Riemann entropy solutions by characteristic analysis method in $ (p, u, s) $ coordinate system. Secondly, we conclude that, the pressureless limit of Riemann solutions for the reduced two-phase mixtures model is just the Riemann solutions for the reduced 2-dimensional pressureless gas dynamics model. Finally, we present numerical simulations which are consistent with our theoretical analysis.

In this paper, the indirect stabilization of strongly coupled wave equations with variable coefficients and boundary damping is studied. It is important to note that only one equation in the system is directly affected by boundary damping. By using Riemannian geometry method and higher order energy method, it is proved that the decay rate of the globally coupled system is affected by the type of boundary conditions. The results show that when the undamped equations have Dirichlet boundary conditions, the system exhibits exponential stability, while when the undamped equations have Neumann boundary conditions, the system has only polynomial stability. Finally, the exponential stability of the locally coupled system is established under Dirichlet and Neumann boundary conditions.

Biharmonic mappings are an important class of geometric mappings, but the partial differential equations that are satisfied are very complex, making their regularity study difficult. In order to study this class of problems, in this note we consider a class of biharmonic map-type fourth order elliptic partial differential equation system

where $B_1=\{x\in\mathbb{R}^{n}:|x|<1\}$ with $n\ge4$, and $Q_{1},{\rm Q_{2}}$ satisfy critical growth conditions with respect to $\nabla u$ and $\nabla^2 u$. Then, under suitable smallness assumption, this note proves that the solutions of this system of equations all have Hölder regularity, thus generalising related results in the literature. This result helps to deepen the understanding of the structure of biharmonic mappings and the research on the regularity theory.

The 3D generalized incompressible MHD-Boussinesq equations with temperature-dependent thermal diffusivity and electrical resistivity are considered in this paper. We prove that there is a unique global strong solution of the 3D generalized incompressible MHD-Boussinesq equations with temperature-dependent thermal diffusivity and electrical resistivity in the Sobolev spaces $H^{s}$ for any $s>2$.

In this paper, we are interested in the existence of normalized solutions for some fractional Kirchhoff equations with $p$-Laplacian operator. For the existence and nonexistence of normalized solutions, using the method of energy estimates, we give a complete classification with respect to nonlinear term exponent $q$ and an explicit threshold value of $c$ (with $\int_{\mathbb{R}^N}|u|^p{\rm d}x=c^p$) in the range $q\in (p,p+\frac{2sp^2}{N})$. We also derive some existence of mountain pass type normalized solutions on the $L^2$ manifold in the range $q\geq p+\frac{2sp^2}{N}$. Furthermore, some asymptotic behaviors with respect to $c$ were also given.

In this paper, we focus on dealing with a class of critical Kirchhoff type equation

where $ a,b > 0$ are constants and $ \lambda > 0 $. The nonlinear growth of $ |u|^{2}u $ reaches the Sobolev critical exponent since $ 2^{*}= 4 $ in dimension 4. Assume that $ V $ is the nonnegative continuous potential, which represents a potential well with the bottom $ V^{-1}(0) $ and $ f \in C(\mathbb{R},\mathbb{R}) $ satisfies suitable conditions. By the variational methods, the existence of at least a ground state solution is obtained. Moreover, we study the concentration behavior of the ground state solutions as $ \lambda \rightarrow\infty $ and their asymptotic behavior as $ b\rightarrow 0 $ and $ \lambda \rightarrow\infty $, respectively.

The paper study the initial-boundary value problem for a class of fractional $p$-Laplace diffusion equation with logarithmic nonlinearity. Using the Galerkin approximation, potential well theory and Nehari manifold methods, the global existence of solutions in subcritical and critical states is proven. Then, by constructing auxiliary functions and applying differential inequality techniques, the existence of blow-up solutions in finite time is established.

This paper is concerned with the existence of periodic forced waves on a two-dimensional lattice for Lotka-Volterra cooperative systems with a shifting habitat. First, we prove the existence and uniqueness of solutions to the initial value problem for the Lotka-Volterra system and establish a comparison principle. Second, we construct a pair of upper and lower solutions of the Lotka-Volterra cooperative system and prove the existence of periodic forced waves consistent with the habitat movement velocity by monotone iteration technique and combining the upper and lower solution methods.

In this paper, the Hessian quotient equations with prescribed contact angle boundary value or oblique derivative boundary value problem are studied. Finally a priori global gradient estimate for the $k$-admissible solutions is derived.

As a simplified version of a three-component chemotaxis system introduced by Jones et al to model the spatio-temporal behavior of criminal activities under the effects of police deployment, a two-component non-local system with singular sensitivity is considered over a bounded domain $\Omega\subset \mathbb{R}^n$ with $n\geq3$. For all reasonably regular initial data, the existence of classical solution of the corresponding initial-boundary value problem is established globally in time. In particular, we enlarge the range of chemotactic sensitivity $\chi_1$, compared to known results on the Short et al model which describes the spatio-temporal behavior of criminal activities without the effects of police deployment, which reveals that the police deployment has a regularization effect on solution. This is also a theoretical supplement to the previously numerical result unveiled by Jones, Brantingham and Chayes (Math Models MethodsAppl Sci, 2010) that the police deployment is beneficial for suppressing the formation of criminal hotspots.

Quaternion algebra is an algebraic structure that satisfies the associative law but not the commutative law. It has important theoretical significance and application value for studying equations and operators in high-dimensional spaces. By first proving the Privalov theorem with parameter variables in quaternion analysis, then proving the commutation formula of non principal value products, and finally using mathematical analysis methods to take limits on both sides, the Poincaré-Bertrand formula on smooth surfaces is proved.

In this paper, we consider cubic polynomial normal forms on $\mathbb{R}^2$. By employing the theory of polynomial algebra to find the minimal irreducible decomposition of the corresponding varieties, we obtain smooth classifications of a kind of planar cubic polynomials via global conjugations. Our results are also applied to study the problems of iterative roots and embedding flows on $\mathbb{R}^2$.

In this note, we research the Abelian Higgs model subject to the Born-Infeld theory for which the BPS equations can be reduced to a quasi-linear differential equation. We show that the equation exists a unique solution under two interesting boundary conditions which realize the corresponding phase transition. The solution is constructed using a dynamical shooting method for which the correct shooting slope is unique. We also obtain the sharp asymptotic estimate for the solution at infinity.

A singularly perturbed Volterra integro-differential equation is considered. The problem is discretized by using a simple first-order finite difference scheme on a Vulanović-Bakhvalov mesh, the accuracy of which is first-order uniformly convergent with respect to the perturbation parameter $\varepsilon$. Furthermore, based on the Richardson extrapolation technique, the $\varepsilon$-uniform accuracy of the presented approximation scheme can be improved from $O(N^{-1})$ to $O(N^{-2})$, where $N$ is the number of mesh intervals. Finally, the theoretical finds are illustrated by two numerical experiments.

Orthogonal arrays with repeated rows have been widely used, which can reduce the complexity and cost of experiments and improve the reliability of experimental results. The optimal orthogonal arrays with repeated rows has better statistical properties and combinatorial structure, but little is known about this kind of optimal orthogonal arrays. In this paper, we mainly study the methods of constructing the optimal orthogonal arrays with repeated rows by various permutation of levels and permutation of columns. Firstly, a method of constructing saturated orthogonal arrays by using independent columns is introduced. Then, the construction methods of optimal orthogonal arrays with repeated rows and $m$-optimal orthogonal arrays are proposed by using various permutation of levels or columns. Finally, several infinite class of optimal orthogonal arrays with repeated rows are obtained and provided for users.

This article primarily focuses on the study of the limit cycle bifurcation problem of a class of near-Hamiltonian polynomial systems using Abel integral. First, by utilizing analytical techniques, approximate expansions of Abel integral are derived around the central singularity and in the vicinity of the heteroclinic loop, along with the calculated expressions for the coefficients. These results can be utilized to analyze the Hopf bifurcation or heteroclinic bifurcation of the perturbed system. Specifically, it is shown that the discussed near-Hamiltonian polynomial system can produce $[\frac{n+1}{4}]+[\frac{n-1}{4}]+1$ limit cycles near the central singularity and branch out $2[\frac{n+1}{4}]+[\frac{n-1}{4}]$ limit cycles near the heteroclinic loop.

Based on the relationship between the average curvature integrals of the convex and flat convex bodies in ${{\mathbb{R}}^{n}}$, the geometric probability that $h$ linear subspaces intersecting the convex body $K$ within $K$ is given. On this basis, the probability of the existence of a sphere with a radius of $r$ and a common point with all random planes in the convex body $K$ is given. Furthermore, the convergence probabilities of the points in the one-dimensional and two-dimensional cases are discussed.

Fractional optimization with sum of squares convex-concave polynomials is often involved in the uncertain data processing. This paper is concerned with its robust optimal solutions. We first give optimality conditions of robust optimal solutions for the uncertain fractional polynomial optimization problem in terms of robust optimization and a normal cone constraint qualification condition. Then, we give a robust dual problem to this uncertain fractional polynomial optimization problem and establish robust weak and strong duality properties between them. Moreover, we obtain exact sum of squares relaxation results for this uncertain fractional polynomial optimization problem.

By using the properties of the approximate subdifferentials of involved functions and generators of quasiconvex functions, we introduce a new constraint qualification. Under this constraint qualification, characterizations of the quasi $(\alpha,\varepsilon)$-optimal solution, the approximate saddle point theorems and the approximate mixed type duality theorems for quasiconvex programming are established.