This article proves the existence and uniqueness conditions of the solution of two-dimensional time-space tempered fractional diffusion-wave equation. We find analytical solution of the equation via the two-step Adomian decomposition method (TSADM). The existence result is obtained with the help of some fixed point theorems, while the uniqueness of the solution is a consequence of the Banach contraction principle. Additionally, we study the stability via the Ulam-Hyers stability for the considered problem. The existing techniques use numerical algorithms for solving the two-dimensional time-space tempered fractional diffusion-wave equation, and thus, the results obtained from them are the approximate solution of the problem with high computational and time complexity. In comparison, our proposed method eliminates all the difficulties arising from numerical methods and gives an analytical solution with a straightforward process in just one iteration.

In this paper, we study a class of time-periodic population model with dispersal. It is well known that the existence of the periodic traveling fronts has been established. However, the uniqueness and stability of such fronts remain unsolved. In this paper, we first prove the uniqueness of non-critical periodic traveling fronts. Then, we show that all non-critical periodic traveling fronts are exponentially asymptotically stable.

We couple together existing ideas, existing results, special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an $n$-dimensional incompressible Navier-Stokes equations. We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.

Inspired by the idea of stochastic quantization proposed by Parisi and Wu, we reconstruct the transition probability function that has a central role in the renormalization group using a stochastic differential equation. From a probabilistic perspective, the renormalization procedure can be characterized by a discrete-time Markov chain. Therefore, we focus on this stochastic dynamic, and establish the local Poincaré inequality by calculating the Bakry-Émery curvature for two point functions. Finally, we choose an appropriate coupling relationship between parameters $K$ and $T$ to obtain the Poincaré inequality of two point functions for the limiting system. Our method extends the classic Bakry-Émery criterion, and the results provide a new perspective to characterize the renormalization procedure.

The purpose of this work is to investigate the existence and uniqueness of weak solutions to the initial-boundary value problem for a coupled system of an incompressible non-Newtonian fluid and the Vlasov equation. The coupling arises from the acceleration in the Vlasov equation and the drag force in the incompressible viscous non-Newtonian fluid with the stress tensor of a power-law structure for $p\geqslant\frac{11}{5}$. The main idea of the existence analysis is to reformulate the coupled system by means of a so-called truncation function. The advantage of the new formulation is to control the external force term $G=-\mathbf{\int}_{\mathbb{R}^d}(\mathbf{u}-\mathbf{v})fd\mathbf{v}~(d=2,3)$. The global existence of weak solutions to the reformulated system is shown by using the Faedo-Galerkin method and weak compactness techniques. We further prove the uniqueness of weak solutions to the considered system.

In this paper, a stochastic SEITR model is formulated to describe the transmission dynamics of tuberculosis with incompletely treatment. Sufficient conditions for the existence of a stationary distribution and extinction are obtained. In addition, numerical simulations are given to illustrate these analytical results. Theoretical and numerical results show that large environmental perturbations can inhibit the spread of tuberculosis.

The multicolor Ramsey number $r_k(C_4)$ is the smallest integer $N$ such that any $k$-edge coloring of $K_N$ contains a monochromatic $C_4$. The current best upper bound of $r_k(C_4)$ was obtained by Chung (1974) and independently by Irving (1974), i.e., $r_k(C_4)\le k^2+k+1$ for all $k\ge2$. There is no progress on the upper bound since then. In this paper, we improve the upper bound of $r_k(C_4)$ by showing that $r_k(C_4)\le k^2+k-1$ for even $k\ge 6$. The improvement is based on the upper bound of the Turán number $\mathrm{ex}(n,C_4)$, in which we mainly use the double counting method and many novel ideas from Firke, Kosek, Nash, and Williford [J. Combin. Theory, Ser. B 103 (2013), 327-336].

In this paper, we design the differentially private variants of the classical Frank-Wolfe algorithm with shuffle model in the optimization of machine learning. Under weak assumptions and the generalized linear loss (GLL) structure, we propose a noisy Frank-Wolfe with shuffle model algorithm (NoisyFWS) and a noisy variance-reduced Frank-Wolfe with the shuffle model algorithm (NoisyVRFWS) by adding calibrated laplace noise under shuffling scheme in the $\ell_{p} (p\in [1,2])$-case, and study their privacy as well as utility guarantees for the Hölder smoothness GLL. In particular, the privacy guarantees are mainly achieved by using advanced composition and privacy amplification by shuffling. The utility bounds of the NoisyFWS and NoisyVRFWS are analyzed and obtained the optimal excess population risks $\mathcal{O}(n^{-\frac{1+\alpha}{4\alpha}}+\frac{\log(d)\sqrt{\log (1/\delta)}}{n\epsilon})$ and $\mathcal{O}(n^{-\frac{1+\alpha}{4\alpha}}+\frac{\log(d)\sqrt{\log (1/\delta)}}{n^{2}\epsilon})$ with gradient complexity $\mathcal{O}(n^{\frac{(1+\alpha)^{2}}{4\alpha^{2}}})$ for $\alpha \in [1/\sqrt{3},1]$. It turns out that the risk rates under shuffling scheme are a nearly-dimension independent rate, which is consistent with the previous work in some cases. In addition, there is a vital tradeoff between $(\alpha,L)$-Hölder smoothness GLL and the gradient complexity. The linear gradient complexity $\mathcal{O}(n)$ is showed by the parameter $\alpha=1$.

In this paper, we obtain the thickness for some complete $k-$partite graphs for $k=2,3.$ We first compute the thickness of $K_{n,n+8}$ by giving a planar decomposition of $K_{4k-1,4k+7}$ for $k\geq 3$. Then, two planar decompositions for $K_{1,g,g(g-1)}$ when $g$ is even and for $K_{1,g,\frac{1}{2}(g-1)^2}$ when $g$ is odd are obtained. Using a recursive construction, we also obtain the thickness for some complete tripartite graphs. The results here support the long-standing conjecture that the thickness of $K_{m,n}$ is $\big\lceil\frac{mn}{2(m+n-2)}\big\rceil$ for any positive integers $ m, n$.

Based on the martingale difference divergence, a recently proposed metric for quantifying conditional mean dependence, we introduce a consistent test of U-type for the goodness-of-fit of linear models under conditional mean restriction. Methodologically, our test allows heteroscedastic regression models without imposing any condition on the distribution of the error, utilizes effectively important information contained in the distance of the vector of covariates, has a simple form, is easy to implement, and is free of the subjective choice of parameters. Theoretically, our mathematical analysis is of own interest since it does not take advantage of the empirical process theory and provides some insights on the asymptotic behavior of U-statistic in the framework of model diagnostics. The asymptotic null distribution of the proposed test statistic is derived and its asymptotic power behavior against fixed alternatives and local alternatives converging to the null at the parametric rate is also presented. In particular, we show that its asymptotic null distribution is very different from that obtained for the true error and their differences are interestingly related to the form expression for the estimated parameter vector embodied in regression function and a martingale difference divergence matrix. Since the asymptotic null distribution of the test statistic depends on data generating process, we propose a wild bootstrap scheme to approximate its null distribution. The consistency of the bootstrap scheme is justified. Numerical studies are undertaken to show the good performance of the new test.

Let $X=\{ X(t), t\in $$\mathbb{R}_{+}$} be a centered space anisotropic Gaussian process values in $\mathbb{R}^d$ with non-stationary increments, whose components are independent but may not be identically distributed. Under certain conditions, then almost surely $c_1 \leq \phi-m(X([0,1])) \leq c_2$, where $\phi$ denotes the exact Hausdorff measure associated with function $ \phi(s)= s^{\frac{ 1}{α_k} +\sum\limits_{i=1}^k(1- \frac{α_i}{α_k})} \log\log\frac{1}{s}$ for some $1\leq k\leq d$, $ (α_1, \cdots, α_d)\in (0,1]^d$. We also obtain the exact Hausdorff measure of the graph of $X$ on $[0,1]$.

In this paper, a new distribution named the Lindley-Weibull distribution which combines Lindley and Weibull distributions by using the method of T-X family is introduced. This distribution offers a more flexible model for lifetime data. We study its statistical properties include the shapes of density and hazard rate, residual and reversed residual lifetime, moment, moment generating functions, conditional moment, conditional moment generating, quantiles functions, mean deviations, Rényi entropy, Bonferroni and Loren curves. The distribution is capable of modeling increasing, decreasing, upside-down bathtub and decreasing-increasing-decreasing hazard rate functions. The method of maximum likelihood is adopted for estimating the model parameters. The potentiality of the new model is illustrated by means of one real data set.

Kawasaki disease (KD) is an acute, febrile, systemic vasculitis that mainly affects children under five years of age. In this paper, we propose and study a class of 5-dimensional ordinary differential equation model describing the vascular endothelial cell injury in the lesion area of KD. This model exhibits forward/backward bifurcation. It is shown that the vascular injury-free equilibrium is locally asymptotically stable if the basic reproduction number $R_{0}<1$. Further, we obtain two types of sufficient conditions for the global asymptotic stability of the vascular injury-free equilibrium, which can be applied to both the forward and backward bifurcation cases. In addition, the local and global asymptotic stability of the vascular injury equilibria and the presence of Hopf bifurcation are studied. It is also shown that the model is permanent if the basic reproduction number $R_{0}>1$, and some explicit analytic expressions of ultimate lower bounds of the solutions of the model are given. Our results suggest that the control of vascular injury in the lesion area of KD is not only correlated with the basic reproduction number $R_0$, but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor.

As an extension of linear regression in functional data analysis, functional linear regression has been studied by many researchers and applied in various fields. However, in many cases, data is collected sequentially over time, for example the financial series, so it is necessary to consider the autocorrelated structure of errors in functional regression background. To this end, this paper considers a multiple functional linear model with autoregressive errors. Based on the functional principal component analysis, we apply the least square procedure to estimate the functional coefficients and autoregression coefficients. Under some regular conditions, we establish the asymptotic properties of the proposed estimators. A simulation study is conducted to investigate the finite sample performance of our estimators. A real example on China's weather data is applied to illustrate the validity of our model.

In this paper, the diffusive nutrient-microorganism model subject to Neumann boundary conditions is considered. The Hopf bifurcations and steady state bifurcations which bifurcate from the positive constant equilibrium of the system are investigated in details. In addition, the formulae to determine the direction of Hopf and steady state bifurcations are derived. Our results show the existence of spatially homogeneous/nonhomogeneous periodic orbits and steady state solutions, which indicates the spatiotemporal dynamics of the system. Some numerical simulations are also presented to support the analytical results.

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices, and an HVN is a $K_4$ together with one more vertex which is adjacent to exactly two vertices of $K_4$. Combining with some known result, in this paper we show that if $G$ is $(P_5, \textit{HVN})$-free, then $\chi(G)\leq \max\{\min\{16, \omega(G)+3\}, \omega(G)+1\}$. This upper bound is almost sharp.

In this paper, we prove an existence and uniqueness theorem for backward doubly stochastic differential equations under a new kind of stochastic non-Lipschitz condition which involves stochastic and time-dependent condition. As an application, we use the result to obtain the existence of stochastic viscosity solution for some nonlinear stochastic partial differential equations under stochastic non-Lipschitz conditions.

An incidence of a graph $G$ is a vertex-edge pair $(v,e)$ such that $v$ is incidence with $e$. A conflict-free incidence coloring of a graph is a coloring of the incidences in such a way that two incidences $(u,e)$ and $(v,f)$ get distinct colors if and only if they conflict each other, i.e.,\, (i) $u=v$, (ii) $uv$ is $e$ or $f$, or (iii) there is a vertex $w$ such that $uw=e$ and $vw=f$. The minimum number of colors used among all conflict-free incidence colorings of a graph is the conflict-free incidence chromatic number. A graph is outer-1-planar if it can be drawn in the plane so that vertices are on the outer-boundary and each edge is crossed at most once. In this paper, we show that the conflict-free incidence chromatic number of an outer-1-planar graph with maximum degree $\Delta$ is either $2\Delta$ or $2\Delta+1$ unless the graph is a cycle on three vertices, and moreover, all outer-1-planar graphs with conflict-free incidence chromatic number $2\Delta$ or $2\Delta+1$ are completely characterized. An efficient algorithm for constructing an optimal conflict-free incidence coloring of a connected outer-1-planar graph is given.

In this work, a novel time-stepping $\overline{L1}$ formula is developed for a hidden-memory variable-order Caputo's fractional derivative with an initial singularity. This formula can obtain second-order accuracy and an error estimate is analyzed strictly. As an application, a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model. Numerical experiments are provided to support our theoretical results.

The biased random walk on supercritical Galton-Watson trees is known to exhibit a multiscale phenomenon in the slow regime: the maximal displacement of the walk in the first $n$ steps is of order $(\log n)^3$, whereas the typical displacement of the walk at the $n$-th step is of order $(\log n)^2$. Our main result reveals another multiscale property of biased walks: the maximal potential energy of the biased walks is of order $(\log n)^2$ in contrast with its typical size, which is of order $\log n$. The proof relies on analyzing the intricate multiscale structure of the potential energy.

We examine a quasilinear system of viscoelastic equations in this study that have fractional boundary conditions, dispersion, source, and variable-exponents. We discovered that the solution of the system is global and constrained under the right assumptions about the relaxation functions and initial conditions. After that, it is demonstrated that the blow-up has negative initial energy. Subsequently, the growth of solutions is demonstrated with positive initial energy, and the general decay result in the absence of the source term is achieved by using an integral inequality due to Komornik.

The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid \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}= n^{\epsilon}\nabla c^{\epsilon}, \ \ \ &{\rm in}\ \mathbb{R}^{d}\times (0, \infty), \\[6pt] \nabla\cdot u^{\epsilon}=0, \ \ \ &{\rm in}\ \mathbb{R}^{d}\times (0, \infty), \\[6pt] n_t^{\epsilon}+u^{\epsilon}\cdot\nabla n^{\epsilon}-\Delta n^{\epsilon}=-\nabla\cdot(n^{\epsilon}\nabla c^{\epsilon}), &{\rm in}\ \mathbb{R}^{d}\times (0, \infty),\\[6pt] \frac{1}{\epsilon}c_t^{\epsilon}-\Delta c^{\epsilon}= n^{\epsilon}, &{\rm in}\ \mathbb{R}^{d}\times (0, \infty),\\[6pt] (u^{\epsilon}, n^{\epsilon}, c^{\epsilon})|_{t=0}= (u_{0}, n_{0}, c_{0}), &{\rm in}\ \mathbb{R}^{d},\\[6pt] \end{array} \right. \end{eqnarray*} where $d\geq2$. It is known that for each $\epsilon>0$ and all sufficiently small initial data $(u_{0},n_{0},c_{0})$ belongs to certain Fourier space, the problem possesses a unique global solution $(u^{\epsilon},n^{\epsilon},c^{\epsilon})$ in Fourier space. The present work asserts that these solutions stabilize to $(u^{\infty},n^{\infty},c^{\infty})$ as $\epsilon^{-1}\rightarrow 0$. Moreover, we show that $c^{\epsilon}(t)$ has the initial layer as $\epsilon^{-1}\rightarrow 0$. As one expects its limit behavior maybe give a new viewlook to understand the system.

A majority coloring of a directed graph is a vertex-coloring in which every vertex has the same color as at most half of its out-neighbors. Kreutzer et al. conjectured that every digraph is majority 3-colorable. For an integer $k\geq 2$, $\frac{1}{k}$-majority coloring of a directed graph is a vertex-coloring in which every vertex $v$ has the same color as at most $\frac{1}{k}d^+(v)$ of its out-neighbors. a $\frac{1}{k}$-majority coloring of a digraph is a coloring of the vertices such that each vertex receives the same color as at most a $\frac{1}{k}$ proportion of its out-neighbors. Girão et al. proved that every digraph admits a $\frac{1}{k}$-majority $2k$-coloring. In this paper, we prove that Kreutzer's conjecture is true for digraphs under some conditions, which improves Kreutzer's results, also we obtained some results of $\frac{1}{k}$-majority coloring of digraphs. Moreover, we discuss the majority 3-coloring of random digraphs with some conditions.

This research paper addresses a topic of interest to many researchers and engineers due to its effective applications in various industrial areas. It focuses on the thermoelastic laminated beam model with nonlinear structural damping, nonlinear time-varying delay, and microtemperature effects. Our primary goal is to establish the stability of the solution. To achieve this, and under suitable hypotheses, we demonstrate energy decay and construct a Lyapunov functional that leads to our results.

A graph $G$ is a fractional $(k,m)$-deleted graph if removing any $m$ edges from $G$, the resulting subgraph still admits a fractional $k$-factor. Let $k\ge2$ and $m\ge1$ be integers. Denote $\lfloor\frac{2m}{k}\rfloor^{*}=\lfloor\frac{2m}{k}\rfloor$ if $\frac{2m}{k}$ is not an integer, and $\lfloor\frac{2m}{k}\rfloor^{*}=\lfloor\frac{2m}{k}\rfloor-1$ if $\frac{2m}{k}$ is an integer. In this paper, we prove that $G$ is a fractional $(k,m)$-deleted graph if $\delta(G)\ge k+m$ and isolated toughness meets $$I(G)>\left\{\begin{array}{ll}3-\frac{1}{m},& \hbox{if $k=2$ and $m\ge3$,} \\k+\frac{\lfloor\frac{2m}{k}\rfloor^{*}}{m+1-\lfloor\frac{2m}{k}\rfloor^{*}},& \hbox{ otherwise.}\end{array}\right.$$ Furthermore, we show that the isolated toughness bound is tight.

Interior-point methods (IPMs) for linear programming (LP) are generally based on the logarithmic barrier function. Peng et al. (J. Comput. Technol. 6: 61-80, 2001) were the first to propose non-logarithmic kernel functions (KFs) for solving IPMs. These KFs are strongly convex and smoothly coercive on their domains. Later, Bai et al. (SIAM J. Optim. 15(1): 101-128, 2004) introduced the first KF with a trigonometric barrier term. Since then, no new type of KFs were proposed until 2020, when Touil and Chikouche (Filomat. 34(12): 3957-3969, 2020; Acta Math. Sin. (Engl. Ser.), 38(1): 44-67, 2022) introduced the first hyperbolic KFs for semidefinite programming (SDP). They established that the iteration complexities of algorithms based on their proposed KFs are $\mathcal{O}\left(n^{\frac{2}{3}}\log \frac{n}{\epsilon }\right)$ and $\mathcal{O}\left(n^{\frac{3}{4}}\log \frac{n}{\epsilon }\right)$ for large-update methods, respectively. The aim of this work is to improve the complexity result for large-update method. In fact, we present a new parametric KF with a hyperbolic barrier term. By simple tools, we show that the worst-case iteration complexity of our algorithm for the large-update method is $\mathcal{O}\left(\sqrt{n}\log n\log \frac{n}{\epsilon }\right)$ iterations. This coincides with the currently best-known iteration bounds for IPMs based on all existing kind of KFs.
The algorithm based on the proposed KF has been tested. Extensive numerical simulations on test problems with different sizes have shown that this KF has promising results.

This paper is devoted to solving a reflected backward stochastic differential equation (BSDE in short) with one continuous barrier and a quasi-linear growth generator $g$, which has a linear growth in $(y,z)$, except the upper direction in case of $y<0$, and is more general than the usual linear growth generator. By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal $L^p\ (p>1)$ solutions for the reflected BSDEs. We also prove that the minimal $L^p$ solution can be approximated by a sequence of $L^p$ solutions of certain reflected BSDEs with Lipschitz generators.

Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering. When some additional resources or information become available after the initial design of experiment is carried out, some additional runs and/or factors may be added in the follow-up stage. In this paper, the issue of the uniform row augmented designs and column augmented designs with mixed two-, three- and four-level is investigated. The uniformity of augmented designs is discussed under the wrap-around $L_2$-discrepancy. Some lower bounds of wrap-around $L_2$-discrepancy for the augmented designs are obtained, which can be used to assess uniformity of augmented design. Numerical results show that augmented designs have high efficiency, which have low discrepancy and close to the proposed lower bounds.

In this paper, $\bar{\partial}$-dressing method based on a local $3\times 3$ matrix $\bar{\partial}$-problem with non-normalization boundary conditions is used to investigate coupled two-component Kundu-Eckhaus equations. Firstly, we propose a new compatible system with singular dispersion relation, that is time spectral problem and spatial spectral problem of coupled two-component Kundu-Eckhaus equations via constraint equations. Then, we derive a hierarchy of nonlinear evolution equations by introducing a recursive operator. At last, by solving constraint matrixes, a spectral transform matrix is given which is sufficiently important for finding soliton solutions of potential function, and we obtain $N$-soliton solutions of coupled two-component Kundu-Eckhaus equations.

The focusing modified Korteweg-de Vries (mKdV) equation with multiple high-order poles under the nonzero boundary conditions is first investigated via developing a Riemann-Hilbert (RH) approach. We begin with the asymptotic property, symmetry and analyticity of the Jost solutions, and successfully construct the RH problem of the focusing mKdV equation. We solve the RH problem when $1/S_{11}(k)$ has a single high-order pole and multiple high-order poles. Furthermore, we derive the soliton solutions of the focusing mKdV equation which corresponding with a single high-order pole and multiple high-order poles, respectively. Finally, the dynamics of one- and two-soliton solutions are graphically discussed.

In this paper, we consider the symmetric cone linear complementarity problem with the Cartesian $P_0$-property and present a regularization smoothing method with a nonmonotone line search to solve this problem. It has been demonstrated that the proposed method exhibits global convergence under the condition that the solution set of the complementarity problem is nonempty. This condition is less stringent than those that have appeared in some existing literature. We also show that the method has locally quadratic convergence under appropriate conditions. Some experimental results are reported to illustrate the efficiency of the proposed method.

Let $F$, $G$ and $H$ be three graphs with $G\subseteq{H}$. We call $G$ an $F$-saturated graph relative to $H$, if there is no copy of $F$ in $G$ but there is a copy of $F$ in $G+e$ for any $e\in E(H)\setminus E(G)$. The $F$-saturation game on host graph $H$ consists of two players, named Max and Min, who alternately add edges of $H$ to $G$ such that each chosen edge avoids creating a copy of $F$ in $G$, and the players continue to choose edges until $G$ becomes $F$-saturated relative to $H$. Max wishes to maximize the length of the game, while Min wishes to minimize the process. Let ${\rm sat}_g(F,H)$ (resp. ${\rm sat}_{g}^{'}(F,H)$) denote the number of edges chosen when Max (resp. when Min) starts the game and both players play optimally. In this article, we show that ${\rm sat}_g(P_5,K_n) = {\rm sat}_g^{'}(P_5,K_n)= n+2$ for $n\ge 15$, and ${\rm sat}_g(P_5,K_{m,n})$, ${\rm sat}_g^{'}(P_5,K_{m,n})$ lie in $\{m+n-\lfloor \frac{m+2}{4}\rfloor, m+n-\lceil \frac{m-3}{4}\rceil \}$ if $n\ge\frac{5}{2}m$ and $m\ge 4$, respectively.

In this paper, we focus on a control-constrained stochastic LQ optimal control problem via backward stochastic differential equation (BSDE in short) with deterministic coefficients. One of the significant features in this framework, in contrast to the classical LQ issue, embodies that the admissible control set needs to satisfy more than the square integrability. By introducing two kinds of new generalized Riccati equations, we are able to announce the explicit optimal control and the solution to the corresponding H-J-B equation. A linear quadratic recursive utility portfolio optimization problem in the financial engineering is discussed as an explicitly illustrated example of the main result with short-selling prohibited. Feasibility of the mean-variance portfolio selection problem via BSDE for a financial market is characterized, and associated efficient portfolios are given in a closed form.

Several eigenvalue properties of the third-order boundary value problems with distributional potentials are investigated. Firstly, we prove that the operators associated with the problems are self-adjoint and the corresponding eigenvalues are real. Next, the continuity and differential properties of the eigenvalues of the problems are given, especially we find the differential expressions for the boundary conditions, the coefficient functions and the endpoints. Finally, we show a brief application to a kind of transmission boundary value problems of the problems studied here.

In this paper, a critical Galton-Watson branching process $\{Z_{n}\}$ is considered. Large deviation rates of $S_{Z_n}:=\sum\limits_{i=1}^{Z_n} X_i$ are obtained, where $\{X_i, \ i\geq 1\}$ is a sequence of independent and identically distributed random variables and $X_1$ is in the domain of attraction of an $\alpha$-stable law with $\alpha\in(0,2)$. One shall see that the convergence rate is determined by the tail index of $X_1$ and the variance of $Z_1$. Our results can be compared with those ones of the supercritical case.

In this paper, we consider the Cauchy problem of the $d$-dimensional damping incompressible magnetohydrodynamics system without dissipation. Precisely, this system includes a velocity damped term and a magnetic damped term. We establish the existence and uniqueness of global solutions to this damped system in the critical Besov spaces by means of the Fourier frequency localization and Bony paraproduct decomposition.

The Ananthakrishna model, seeking to explain the Portevin-Le Chatelier effect, is studied with or without non-synchronous perturbations. For the unperturbed model, Bogdanov-Takens bifurcation and zero-Hopf bifurcation are detected. For the perturbed model, rich dynamical behaviors are given by researching the Poincaré map, including solutions of different periods, quasi-periodic solutions, chaotic solutions, and bistability. Moreover, an augmented temperature-dependent perturbation amplitude induces a transition from non-serrated to serrated flow on the stress-time curve. Notably, on the stress-strain curve, the phenomenon of repeated yielding diminishes with an increase in the value of a temperature-dependent parameter, while it persists with an increase in the value of a temperature-independent parameter. Sensitivity analysis sheds light on the factors exerting the most significant influence on dislocation density.

Given two non-empty graphs $G,H$ and a positive integer $k$, the Gallai-Ramsey number $\operatorname{gr}_k(G:H)$ is defined as the minimum integer $N$ such that for all $n\geq N$, every exact $k$-edge-coloring of $K_n$ contains either a rainbow copy of $G$ or a monochromatic copy of $H$. Denote $\operatorname{gr}'_k(G:H)$ as the minimum integer $N$ such that for all $n\geq N$, every edge-coloring of $K_n$ using at most $k$ colors contains either a rainbow copy of $G$ or a monochromatic copy of $H$. In this paper, we get some exact values or bounds for $\operatorname{gr}_k(P_5:H)$ and $\operatorname{gr}'_k(P_5:H)$, where $H$ is a cycle or a book graph. In addition, our results support a conjecture of Li, Besse, Magnant, Wang and Watts in 2020.

We study a multivariate linear Hawkes process with random marks. In this paper, we establish that a central limit theorem, a moderate deviation principle and an upper bound of large deviation for multivariate marked Hawkes processes hold.

This paper is concerned with the attraction-repulsion Keller-Segel model with volume filling effect. We consider this problem in a bounded domain $\Omega\subset \mathbb R^3$ under zero-flux boundary condition, and it is shown that the volume filling effect will prevent overcrowding behavior, and no blow up phenomenon happen. In fact, we show that for any initial datum, the problem admits a unique global-in-time classical solution, which is bounded uniformly. Previous findings for the chemotaxis model with volume filling effect were derived under the assumption $0\le u_0(x)\le 1$ with $\rho(x,t)\equiv 1$. However, when the maximum size of the aggregate is not a constant but rather a function $\rho(x,t)$, ensuring the boundedness of the solutions becomes significantly challenging. This introduces a fundamental difficulty into the analysis.