In order to measure the uncertainty of financial asset returns in the stock market, this paper presents a new model, called SV-dtC model, a stochastic volatility (SV) model assuming that the stock return has a doubly truncated Cauchy distribution, which takes into account the high peak and fat tail of the empirical distribution simultaneously. Under the Bayesian framework, a prior and posterior analysis for the parameters is made and Markov Chain Monte Carlo (MCMC) is used for computing the posterior estimates of the model parameters and forecasting in the empirical application of Shanghai Stock Exchange Composite Index (SSE-CI) with respect to the proposed SV-dtC model and two classic SV-N (SV model with Normal distribution) and SV-T (SV model with Student-t distribution) models. The empirical analysis shows that the proposed SV-dtC model has better performance by model checking, including independence test (Projection correlation test), Kolmogorov-Smirnov test(K-S test) and Q-Q plot. Additionally, deviance information criterion (DIC) also shows that the proposed model has a significant improvement in model fit over the others.

We investigate some relations between two kinds of semigroup regularities, namely the e-property and the eventual continuity, both of which contribute to the ergodicity for Markov processes on Polish spaces. More precisely, we prove that for Markov-Feller semigroup in discrete time and stochastically continuous Markov-Feller semigroup in continuous time, if there exists an ergodic measure whose support has a nonempty interior, then the e-property is satisfied on the interior of the support. In particular, it implies that, restricted on the support of each ergodic measure, the e-property and the eventual continuity are equivalent for the discrete-time and the stochastically continuous continuous-time Markov-Feller semigroups.

This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form $\dot{x}=-y+xf(x,y),\ \dot{y}=x+yf(x,y)$, where $f(x,y)=a_{1}x+a_{2}xy+a_{3}xy^{2}+a_{4}xy^{3}+a_{5}xy^4=x\sigma(y)$, and any zero of $1+a_{1}y+a_{2}y^2+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}$, $y=\overline{y}$ is an invariant straight line. At last, all global phase portraits are drawn on the Poincarédisk.

Let $\mathbf{G}=\{G_i: i\in[n]\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$, where $\mathbf{G}$ can be seen as an edge-colored (multi)graph and each $G_i$ is the set of edges with color $i$. A graph $F$ on $V$ is called rainbow if any two edges of $F$ come from different $G_i$s'. We say that $\mathbf{G}$ is rainbow pancyclic if there is a rainbow cycle $C_{\ell}$ of length $\ell$ in $\mathbf{G}$ for each integer $\ell\in [3,n]$. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem: If $\delta(G_i)\geq\frac{n}{2}$ for each $i\in[n]$, then there is a rainbow Hamiltonian cycle in $\mathbf{G}$. In this paper, under the same condition, we show that $\mathbf{G}$ is rainbow pancyclic except that $n$ is even and $\mathbf{G}$ consists of $n$ copies of $K_{\frac{n}{2},\frac{n}{2}}$. This result supports the famous meta-conjecture posed by Bondy.

The traditional simple random sampling (SRS) design method is inefficient in many cases. Statisticians proposed some new designs to increase efficiency. In this paper, as a variation of moving extremes ranked set sampling (MERSS), double MERSS (DMERSS) is proposed and its properties for estimating the population mean are considered. It turns out that, when the underlying distribution is symmetric, DMERSS gives unbiased estimators of the population mean. Also, it is found that DMERSS is more efficient than the SRS and MERSS methods for usual symmetric distributions (normal and uniform). For asymmetric distributions considered in this study, the DMERSS has a small bias and it is more efficient than SRS for usual asymmetric distribution (exponential) for small sample sizes.

Tensor data have been widely used in many fields, e.g., modern biomedical imaging, chemometrics, and economics, but often suffer from some common issues as in high dimensional statistics. How to find their low-dimensional latent structure has been of great interest for statisticians. To this end, we develop two efficient tensor sufficient dimension reduction methods based on the sliced average variance estimation (SAVE) to estimate the corresponding dimension reduction subspaces. The first one, entitled tensor sliced average variance estimation (TSAVE), works well when the response is discrete or takes finite values, but is not $\sqrt{n}$ consistent for continuous response; the second one, named bias-correction tensor sliced average variance estimation (CTSAVE), is a de-biased version of the TSAVE method. The asymptotic properties of both methods are derived under mild conditions. Simulations and real data examples are also provided to show the superiority of the efficiency of the developed methods.

In this paper, we present a minimum residual based gradient iterative method for solving a class of matrix equations including Sylvester matrix equations and general coupled matrix equations. The iterative method uses a negative gradient as steepest direction and seeks for an optimal step size to minimize the residual norm of next iterate. It is shown that the iterative sequence converges unconditionally to the exact solution for any initial guess and that the norm of the residual matrix and error matrix decrease monotonically. Numerical tests are presented to show the efficiency of the proposed method and confirm the theoretical results.

In this paper we mainly deal with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data. We first establish the global well-posedness of solution in the Besov space, then we obtain the optimal decay rates of solutions by virtue of the frequency decomposition method. Specifically, for the low frequency part, we use the Fourier splitting method of Schonbek and the spectrum analysis method, and for the high frequency part, we use the global energy estimate and the behavior of exponentially decay operator.

In this paper, we are concerned with the the Schrödinger-Newton system with $L^2$-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at $k$ different critical points of $V(x)$ under certain assumptions on asymptotic behavior of $V(x)$ and its first derivatives near these points. Especially, the critical points of $V(x)$ in this paper must be degenerate.
The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term.

We investigate the global structures of the non-selfsimilar solutions for $n$-dimensional ($n$-D) non-homogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a $({n-1})$-dimensional sphere. We first obtain the expressions of $n$-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves, we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the $n$-D shock waves. The asymptotic behaviors with geometric structures are also proved.

The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper. We first show that there exist both continuous and discontinuous stationary solutions. Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition. In addition, we demonstrate the influences of the diffusion coefficient on stationary solutions. The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem. Finally, some numerical simulations are given to illustrate the theoretical results.

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any two-coloring of the edges of $K_{N}$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the number of vertices in a minimum color class among all proper $\chi(G)$-colorings of $G$. Burr showed that $R(G,H)\geq (v(G)-1)(\chi(H)-1)+s(H)$ if $G$ is connected and $v(G)\geq s(H)$. A connected graph $G$ is $H$-good if $R(G,H)=(v(G)-1)(\chi(H)-1)+s(H)$. %Ramsey goodness is a special property of graph. Let $tH$ denote the disjoint union of $t$ copies of graph $H$, and let $G\vee H$ denote the join of $G$ and $H$. Denote a complete graph on $n$ vertices by $K_n$, and a tree on $n$ vertices by $T_n$. Denote a book with $n$ pages by $B_n$, i.e., the join $K_2\vee \overline{K_n}$. Erdös, Faudree, Rousseau and Schelp proved that $T_n$ is $B_m$-good if $n\geq 3m-3$. In this paper, we obtain the exact Ramsey number of $T_n$ versus $2B_2$. Our result implies that $T_n$ is $2B_2$-good if $n\geq5$.

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $\sum\limits_{A} x^{|A|}$, where the sum is over all independent sets $A$ of $G$. In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal. Although this unimodality conjecture has attracted many researchers' attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal. In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.

A coloring of graph $G$ is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number $\chi_i(G)$ of $G$ is the least integer $k$ such that $G$ has an injective $k$-coloring. In this paper, we prove that (1) if $G$ is a planar graph with girth $g\geq 6$ and maximum degree $\Delta \geq 7$, then $\chi_i(G)\leq \Delta +2$; (2) if $G$ is a planar graph with $\Delta \geq24$ and without 3,4,7-cycles, then $\chi_i(G)\leq \Delta +2$.

Let $\Omega$ be a bounded smooth domain in ${\mathbb{R}}^N \ (N\geq3)$. Assuming that 0<s<1, 1p,q)≠(N+2s/N-2s,N+2s/N-2s), and $a,b>0$ are constants, we consider the existence results for positive solutions of a class of fractional elliptic system below, \begin{align*} \left\{\begin{array}{ll} (a+b[u]^2_s)(-\Delta)^su=v^p+h_1(x,u,v,\nabla u,\nabla v), &\quad x\in\Omega,\\ (-\Delta)^sv=u^q+h_2(x,u,v,\nabla u,\nabla v), &\quad x\in\Omega,\\ u,v>0, &\quad x\in\Omega,\\ u=v=0, &\quad x\in \mathbb{R}^N\backslash\Omega. \end{array}\right. \end{align*} Under some assumptions of $h_i(x,u,v,\nabla u,\nabla v)(i=1,2)$, we get a priori bounds of the positive solutions to the problem (1.1) by the blow-up methods and rescaling argument. Based on these estimates and degree theory, we establish the existence of positive solutions to problem (1.1).

In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PDEs, determines four major physical unknowns: the pressure, the concentrations of brine and radionuclide, and the temperature. The pressure is solved by a conservative mixed finite volume element method, and the computational accuracy is improved for Darcy velocity. Other unknowns are computed by a composite scheme of upwind approximation and mixed finite volume element. Numerical dispersion and nonphysical oscillation are eliminated, and the convection-dominated diffusion problems are solved well with high order computational accuracy. The mixed finite volume element is conservative locally, and get the objective functions and their adjoint vector functions simultaneously. The conservation nature is an important character in numerical simulation of underground fluid. Fractional step difference is introduced to solve the concentrations of radionuclide factors, and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems. By the theory and technique of a priori estimates of differential equations, we derive an optimal order estimates in $L^2$ norm. Finally, numerical examples show the effectiveness and practicability for some actual problems.

In this paper, we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject. The $k$-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data, and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered. In addition, we also consider two other clustering methods, $k$-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis. Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index. The approaches are further illustrated through empirical analysis of human mortality data.

Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\cdots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 2(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\delta(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.

The spatial and spatiotemporal autoregressive conditional heteroscedasticity (STARCH) models receive increasing attention. In this paper, we introduce a spatiotemporal autoregressive (STAR) model with STARCH errors, which can capture the spatiotemporal dependence in mean and variance simultaneously. The Bayesian estimation and model selection are considered for our model. By Monte Carlo simulations, it is shown that the Bayesian estimator performs better than the corresponding maximum-likelihood estimator, and the Bayesian model selection can select out the true model in most times. Finally, two empirical examples are given to illustrate the superiority of our models in fitting those data.

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that there are no bichromatic cycles in $G$. The acyclic chromatic index $\chi'_a(G)$ of $G$ is the smallest $k$ such that $G$ has an acyclic edge coloring using $k$ colors. It was conjectured that every simple graph $G$ with maximum degree $\Delta$ has $\chi'_a(G)\le \Delta+2$. A 1-planar graph is a graph that can be drawn in the plane so that each edge is crossed by at most one other edge. In this paper, we show that every 1-planar graph $G$ without $4$-cycles has $\chi'_a(G)\le \Delta+22$.

This paper is concerned with the large time behavior of the Cauchy problem for Navier-Stokes/Allen-Cahn system describing the interface motion of immiscible two-phase flow in 3-D. The existence and uniqueness of global solutions and the stability of the phase separation state are proved under the small initial perturbations. Moreover, the optimal time decay rates are obtained for higher-order spatial derivatives of density, velocity and phase. Our results imply that if the immiscible two-phase flow is initially located near the phase separation state, then under small perturbation conditions, the solution exists globally and decays algebraically to the complete separation state of the two-phase flow, that is, there will be no interface fracture, vacuum, shock wave, mass concentration at any time, and the interface thickness tends to zero as the time $t\rightarrow+\infty$.

Inspired by the transmission characteristics of the coronavirus disease 2019 (COVID-19), an epidemic model with quarantine and standard incidence rate is first developed, then a novel analysis approach is proposed for finding the ultimate lower bound of the number of infected individuals, which means that the epidemic is uniformly persistent if the control reproduction number $\mathcal{R}_{c}>1$. This approach can be applied to the related biomathematical models, and some existing works can be improved by using that. In addition, the infection-free equilibrium $V^0$ of the model is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$ and linearly stable if $\mathcal{R}_{c}=1$; while $V^0$ is unstable if $\mathcal{R}_{c}>1$.

In this paper, the insurance company considers venture capital and risk-free investment in a constant proportion. The surplus process is perturbed by diffusion. At first, the integro-differential equations satisfied by the expected discounted dividend payments and the Gerber-Shiu function are derived. Then, the approximate solutions of the integro-differential equations are obtained through the sinc method. Finally, the numerical examples are given when the claim sizes follow different distributions. Furthermore, the errors between the explicit solution and the numerical solution are discussed in a special case.

Let $G$ be a simple graph and $G^{\sigma}$ be the oriented graph with $G$ as its underlying graph and orientation $\sigma$. The rank of the adjacency matrix of $G$ is called the rank of $G$ and is denoted by $r(G)$. The rank of the skew-adjacency matrix of $G^{\sigma}$ is called the skew-rank of $G^{\sigma}$ and is denoted by $sr(G^{\sigma})$. Let $V(G)$ be the vertex set and $E(G)$ be the edge set of $G$. The cyclomatic number of $G$, denoted by $c(G)$, is equal to $|E(G)|-|V(G)|+\omega(G)$, where $\omega(G)$ is the number of the components of $G$. It is proved for any oriented graph $G^{\sigma}$ that $-2c(G)\leqslant sr(G^{\sigma})-r(G)\leqslant2c(G)$. In this paper, we prove that there is no oriented graph $G^{\sigma}$ with $sr(G^{\sigma})-r(G)=2c(G)-1$, and in addition, there are infinitely many oriented graphs $G^{\sigma}$ with connected underlying graphs such that $c(G)=k$ and $sr(G^{\sigma})-r(G)=2c(G)-\ell$ for every integers $k, \ell$ satisfying $0\leqslant\ell\leqslant4k$ and $\ell\neq1$.

The classical Archimedean approximation of $\pi$ uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in $\mathbb{R}^2 $ and it is well-known that by using linear combinations of these basic estimates, modern extrapolation techniques can greatly speed up the approximation process. % reduce the associated approximation errors. Similarly, when $n$ vertices are randomly selected on the circle, the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to $\pi$ almost surely as $ n \to \infty $, and by further applying extrapolation processes, faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons. In this paper, we further develop nonlinear extrapolation methods for approximating $\pi$ through certain nonlinear functions of the semiperimeter and area of such polygons. We focus on two types of extrapolation estimates of the forms $ \mathcal{X}_n = \mathcal{S}_n^{\alpha} \mathcal{A}_n^{\beta} $ and $ \mathcal{Y}_n (p) = \left( \alpha \mathcal{S}_n^p + \beta \mathcal{A}_n^p \right)^{1/p} $ where $ \alpha + \beta = 1 $, $ p \neq 0 $, and $ \mathcal{S}_n $ and $ \mathcal{A}_n $ respectively represents the semiperimeter and area of a random $n$-gon inscribed in the unit circle in $ \mathbb{R}^2 $, and $ \mathcal{X}_n $ may be viewed as the limit of $ \mathcal{Y}_n (p) $ when $ p \to 0 $. By deriving probabilistic asymptotic expansions with carefully controlled error estimates for $ \mathcal{X}_n $ and $ \mathcal{Y}_n (p) $, we show that the choice $ \alpha = 4/3 $, $ \beta = -1/3 $ minimizes the approximation error in both cases, and their distributions are also asymptotically normal.

This paper considers a multi-state repairable system that is composed of two classes of components, one of which has a priority for repair. First, we investigate the well-posedenss of the system by applying the operator semigroup theory. Then, using Greiner's idea and the spectral properties of the corresponding operator, we obtain that the time-dependent solution of the system converges strongly to its steady-state solution.

In this article, we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. We establish general strong law and complete convergence theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. Our results of strong limit theorems are more general than some related results previously obtained by Thrum (1987), Li et al. (1995) and Wu (2010) in classical probability space.

In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed, $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, \\ c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, \ \ t>0, \\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, \ \ t>0, \\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array} \right. (KSS) $$ subject to the boundary conditions $(\nabla n-nS(x,n,c)\nabla c)\cdot\nu=\nabla c\cdot\nu=0$ and $u=0$, and suitably regular initial data $(n_0 (x),c_0 (x),u_0 (x))$, where $\Omega\subset \mathbb{R}^3$ is a bounded domain with smooth boundary $\partial\Omega$. Here $S$ is a chemotactic sensitivity satisfying $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$ with some $C_S> 0$ and $\alpha> 0$. The greatest contribution of this paper is to consider the large time behavior of solutions for the system (KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system (KSS) decays to $(\frac{1}{|\Omega|}\int_{\Omega}n_0,\frac{1}{|\Omega|}\int_{\Omega}n_0,0)$ exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient $C_S$ of chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.

In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the $\vartheta(\lambda_{i}) \ (i=0, 1)$ would like to be imaginary, which results in the $\delta_{\lambda_{i}}^{0}$ contains an increasing $t^{\frac{\pm Im\vartheta(\lambda_{i})}{2}}$, and then the asymptotic behavior for nonlocal Hirota equation becomes differently.

In this paper, we study a global zero-relaxation limit problem of the electro-diffusion model arising in electro-hydrodynamics which is the coupled Planck-Nernst-Poisson and Navier-Stokes equations. That is, the paper deals with a singular limit problem of \begin{eqnarray*}\label{1.2} \left\{ \begin{array}{ll} u_t^{\epsilon}+u^{\epsilon}\cdot\nabla u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}=\Delta \phi^{\epsilon}\nabla\phi^{\epsilon}, \ \ \ &{\rm in}\ \mathbb{R}^{3}\times(0, \infty), \\[5pt] \nabla\cdot u^{\epsilon}=0, \ \ \ &{\rm in}\ \mathbb{R}^{3}\times(0, \infty), \\[5pt] n_t^{\epsilon}+u^{\epsilon}\cdot\nabla n^{\epsilon}-\Delta n^{\epsilon}=-\nabla\cdot(n^{\epsilon}\nabla \phi^{\epsilon}), &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] c_t^{\epsilon}+u^{\epsilon}\cdot\nabla c^{\epsilon}-\Delta c^{\epsilon}=\nabla\cdot(c^{\epsilon}\nabla\phi^{\epsilon}), &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] \epsilon^{-1} \phi^{\epsilon}_t= \Delta \phi^{\epsilon}- n^{\epsilon}+ c^{\epsilon}, &{\rm in}\ \mathbb{R}^{3}\times(0, \infty),\\[5pt] (u^{\epsilon}, n^{\epsilon}, c^{\epsilon},\phi^{\epsilon})|_{t=0}= (u_{0}, n_{0}, c_{0},\phi_{0}), &{\rm in}\ \mathbb{R}^{3} \end{array} \right. \end{eqnarray*} involving with a positive, large parameter $\epsilon$. The present work show a case that $(u^{\epsilon}, n^{\epsilon}, c^{\epsilon})$ stabilizes to $(u^{\infty}, n^{\infty}, c^{\infty}):=(u, n, c)$ uniformly with respect to the time variable as $\epsilon\rightarrow+\infty$ with respect to the strong topology in a certain Fourier-Herz space.

This paper considers the random coefficient autoregressive model with time-functional variance noises, hereafter the RCA-TFV model. We first establish the consistency and asymptotic normality of the conditional least squares estimator for the constant coefficient. The semiparametric least squares estimator for the variance of the random coefficient and the nonparametric estimator for the variance function are constructed, and their asymptotic results are reported. A simulation study is presented along with an analysis of real data to assess the performance of our method in finite samples.

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\cdots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N, N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $1\leq i\leq r$. We show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum\limits^{j+1}_{i=1}\alpha_i$ for $j=1,2,\cdots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\cdots,C_{2\lfloor \alpha_r n\rfloor})=\big(\sum\limits^r_{j=1} \alpha_j+o(1)\big)n.$

The classical Ambarzumyan's theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator $-\frac{d^2}{dx^2}+q$ with an integrable real-valued potential $q$ on $[0,\pi]$ are $\{n^2:n\geq 0\}$, then $q=0$ for almost all $x\in [0,\pi]$. In this work, the classical Ambarzumyan's theorem is extended to the Dirac operator on equilateral tree graphs. We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case, then the potential is identically zero.

This paper mainly studies the contact extension of conservative or dissipative systems, including some old and new results for wholeness. Then extension of contact system is corresponding to the symplectification of contact Hamiltonian system. This is a reciprocal process and the relation between symplectic system and contact system has been discussed. We have an interesting discovery that by adding a pure variable $p$, the slope of the tangent of the orbit, every differential system can be regarded as an independent subsystem of contact Hamiltonian system defined on the projection space of contact phase space.

In this paper, we study the following quasi-linear elliptic equation: \begin{equation}\nonumber \left\{ \begin{aligned} &- {\rm div} (\phi(|\nabla u|)\nabla u)=\lambda \psi(|u|)u + \varphi(|u|)u, ~~~\text{in } \Omega,\\ & u= 0, ~~~\text {on} \partial \Omega, \end{aligned} \right. \end{equation} where $\Omega \subset \mathbb R^N$ is a bounded domain, $\lambda > 0$ is a parameter. The function $\psi(|t|)t$ is the subcritical term, and $\varphi(|t|)t$ is the critical Orlicz-Sobolev growth term with respect to $\phi$. Under appropriate conditions on $\phi$, $\psi$ and $\varphi$, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for $\lambda \in (0,\lambda_0)$, where $\lambda_0>0$ is a fixed constant.

We are concerned with singularity formation of strong solutions to the two-dimensional (2D) full compressible magnetohydrodynamic equations with zero resistivity in a bounded domain. By energy method and critical Sobolev inequalities of logarithmic type, we show that the strong solution exists globally if the temporal integral of the maximum norm of the deformation tensor is bounded. Our result is the same as Ponce's criterion for 3D incompressible Euler equations. In particular, it is independent of the magnetic field and temperature. Additionally, the initial vacuum states are allowed.

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation $$ u_{t}(x,t)=\Delta u(x,t) +au(x,t)\ln u(x,t)+ bu^{\alpha}(x,t), $$ on $\mathbf{M}\times (-\infty, \infty)$ with $\alpha\in\mathbf{R}$, where $a$ and $b$ are constants. As application, the Harnack inequalities are derived.

The law of the iterated logarithm (LIL) for the performance measures of a two-station queueing network with arrivals modulated by independent queues is developed by a strong approximation method. For convenience, two arrival processes modulated by queues comprise the external system, all others are belong to the internal system. It is well known that the exogenous arrival has a great influence on the asymptotic variability of performance measures in queues. For the considered queueing network in heavy traffic, we get all the LILs for the queue length, workload, busy time, idle time and departure processes, and present them by some simple functions of the primitive data. The LILs tell us some interesting insights, such as, the LILs of busy and idle times are zero and they reflect a small variability around their fluid approximations, the LIL of departure has nothing to do with the arrival process, both of the two phenomena well explain the service station's situation of being busy all the time. The external system shows us a distinguishing effect on the performance measures: an underloaded (overloaded, critically loaded) external system affects the internal system through its arrival (departure, arrival and departure together). In addition, we also get the strong approximation of the network as an auxiliary result.

In this paper, we consider a susceptible-infective-susceptible (SIS) reaction-diffusion epidemic model with spontaneous infection and logistic source in a periodically evolving domain.~Using the iterative technique, the uniform boundedness of solution is established.~In addition, the spatial-temporal risk index $\mathcal{R}_0(\rho)$ depending on the domain evolution rate $\rho(t)$ as well as its analytical properties are discussed.~The monotonicity of $\mathcal{R}_0(\rho)$ with respect to the diffusion coefficients of the infected $d_I$, the spontaneous infection rate $\eta(\rho(t)y)$ and interval length $L$ is investigated under appropriate conditions.~Further, the existence and asymptotic behavior of periodic endemic equilibria are explored by upper and lower solution method.~Finally, some numerical simulations are presented to illustrate our analytical results.~Our results provide valuable information for disease control and prevention.

We consider the relation between the direction of the vorticity and the global regularity of 3D shear thickening fluids. It is showed that a weak solution to the non-Newtonian incompressible fluid in the whole space is strong if the direction of the vorticity is 11-5p/2-Hölder continuous with respect to the space variables when 2

<11/5.