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.

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.

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].

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.

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.

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, 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.

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.

In this paper, by using the concentration-compactness principle and a version of symmetry mountain pass theorem, we establish the existence and multiplicity of solutions to the following $p$-biharmonic problem with critical nonlinearity: $$\Bigg\{\begin{array}{ll} \Delta_p^2u=f(x,u)+\mu|u|^{p^*-2}u ~&\text{in}~\Omega, \\ u=\dfrac{\partial u}{\partial \nu}=0 ~&\text{on}~\partial \Omega, \end{array}$$ where $\Omega$ is a bounded domain in $\mathbb{R}^N$ $(N\geq 3)$ with smooth boundary, $\Delta_p^2u=\Delta(|\Delta u|^{p-2}\Delta u),$ $1 < p< \frac{N}{2}$, $p^*=\frac{Np}{N-2p},$ $\frac{\partial u}{\partial \nu}$ is the outer normal derivative, $\mu$ is a positive parameter and $f:\Omega\times \mathbb{R}\rightarrow \mathbb{R}$ is a Carathéodory function.

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.

Given a simple graph $G=(V, E)$ and its (proper) total coloring $\phi$ with elements of the set $\{1, 2,\cdots, k\}$, let $w_{\phi}(v)$ denote the sum of the color of $v$ and the colors of all edges incident with $v$. If for each edge $uv\in E$, $w_{\phi}(u)\neq w_{\phi}(v)$, we call $\phi$ a neighbor sum distinguishing total coloring of $G$. Let $L=\{L_x\, |\, x\in V\cup E\}$ be a set of lists of real numbers, each of size $k$. The neighbor sum distinguishing total choosability of $G$ is the smallest $k$ for which for any specified collection of such lists, there exists a neighbor sum distinguishing total coloring using colors from $L_x$ for each $x\in V\cup E$, and we denote it by ${\rm ch}''_{\sum}(G)$. The known results of neighbor sum distinguishing total choosability are mainly about planar graphs. In this paper, we focus on $1$-planar graphs. A graph is $1$-planar if it can be drawn on the plane so that each edge is crossed by at most one other edge. We prove that ${\rm ch}''_{\sum}(G)\leq \Delta+4$ for any $1$-planar graph $G$ with $\Delta\geq 15$, where $\Delta$ is the maximum degree of $G$.

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.

For an $r$-uniform hypergraph $F$, the anti-Ramsey number ${\rm ar}(n,r,F)$ is the minimum number $c$ of colors such that an $n$-vertex $r$-uniform complete hypergraph equipped any edge-coloring with at least $c$ colors unavoidably contains a rainbow copy of $F$. In this paper, we determine the anti-Ramsey number for cycles of length three in $r$-uniform hypergraphs for $r\geq 3$, including linear cycles, loose cycles and Berge cycles.

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.

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.

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.

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.

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.

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.

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.

This paper introduces the new class of periodic multivariate GARCH models in their periodic BEKK specification. Semi-polynomial Markov chains combined with algebraic geometry are used to obtain some properties like irreducibility. We impose weak conditions to obtain the strict periodic stationarity and the geometric ergodicity of the process, via the theory of positive linear operators on a cone : it is supposed that zero belongs to the support of the driving noise density which is absolutely continuous with respect to the Lebesgue measure and the spectral radius of a matrix built from the periodic coefficients of the model is smaller than one.

In this paper, we propose a new method for spread option pricing under the multivariate irreducible diffusions without jumps and with different types of jumps by the expansion of the transition density function. By the quasi-Lamperti transform, which unitizes the diffusion matrix at the initial time, and applying the small-time It$\mathrm{\hat{o}}$-Taylor expansion method, we derive explicit recursive formulas for the expansion coefficients of transition densities and spread option prices for multivariate diffusions with jumps in return. It is worth mentioning that we also give the closed-form formula of spread option price whose underlying asset price processes contain a Merton jump and a double exponential jump, which is innovative compared with current literature. The theoretical proof of convergence is presented in detail.

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.

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.

Let $\Gamma$ denote a bipartite Q-polynomial distance-regular graph with vertex set $X$, valency $k\geq 3$ and diameter $D\geq 3$. Let $A$ be the adjacency matrix of $\Gamma$ and let $A^*:=A^*(x)$ be the dual adjacency matrix of $\Gamma$ with respect to a fixed vertex $x \in X$. Let $T:=T(x)$ denote the Terwilliger algebra of $\Gamma$ generated by $A$ and $A^*$. In this paper, we first describe the relations between $A$ and $A^*$. Then we determine the dimensions of both $T$ and the center of $T$, and moreover we give a basis of $T$.

Given a distribution of pebbles on the vertices of a connected graph $G$, a pebbling move on $G$ consists of taking two pebbles off one vertex and placing one on an adjacent vertex. Rubbling is a version of pebbling where an additional move is allowed. In this new move, one pebble each is removed at vertices $u$ and $w$ that are adjacent to a vertex $v$, and an extra pebble is added at vertex $v$. The rubbling number of $G$, denoted by $\rho(G)$, is the smallest number $m$ such that for every distribution of $m$ pebbles on $G$ and every vertex $v$, at least one pebble can be moved to $v$ by a sequence of rubbling moves. The optimal rubbling number of $G$, denoted by $\rho_{opt}(G)$, is the smallest number $k$ such that for some distribution of $k$ pebbles on $G$, one pebble can be moved to any vertex of $G$. In this paper, we determine $\rho(G)$ for a non-complete bipartite graph $G\in B(s,t)$ with $\delta(G)\ge \lceil \frac{2s+1}{3}\rceil$, give an upper bound of $\rho(G)$ for $G\in B(s,t)$ with $\delta(G)\ge \lceil \frac{s+1}{2}\rceil$, and also obtain $\rho_{opt}(G)$ for a non-complete bipartite graph $G\in B(s,t)$ with $\delta(G)\ge \lceil \frac{s+1}{2}\rceil$, where $B(s,t)$ is the set of all connected bipartite graphs with partite sets of size $s$ and $t$ ($s\ge t$) and $\delta(G)$ is the minimum degree of $G$.

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.

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.

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.

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 the following fractional relativistic Schrödinger equation: \begin{equation*} (-\Delta+m^{2})^su+V(x)u=f(x,u), \qquad x\in \mathbb{R}^N, \end{equation*} where $(-\Delta+m^{2})^s$ is the fractional relativistic Schrödinger operator, $s\in (0, 1), m>0,$ $V : \mathbb{R}^N \to \mathbb{R}$ is a continuous potential and $f: \mathbb{R}^N\times\mathbb{R} \to \mathbb{R}$ is a superlinear continuous nonlinearity with subcritical growth. We consider the case where the potential $V$ is indefinite so that the relativistic Schrödinger operator $(-\Delta+m^{2})^s+V$ possesses a finite-dimensional negative space. With the help of extension method and Morse theory, the existence of a nontrivial solution for the above problem is obtained.

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.

In this paper, we study the problem of irreversible investment under endowment constraints. We first establish the existence and uniqueness of the result and then demonstrate the necessity and sufficient conditions for optimality. Based on this condition, we provide a characterization for optimal investment plans, which can be obtained by the so-called base capacity solving a backward equation. We may obtain explicit solutions for certain typical cases.

In this article, we study the empirical likelihood (EL) method for autoregressive models with spatial errors. The EL ratio statistics are constructed for the parameters of the models. It is shown that the limiting distributions of the EL ratio statistics are chi-square distributions, which are used to construct confidence intervals for the parameters of the models. A simulation study is conducted to compare the performances of the EL based and the normal approximation (NA) based confidence intervals. Simulation results show that the confidence intervals based on EL are superior to the NA based confidence intervals.

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.

The purpose of this work is to investigate the boundedness of the pullback attractors for the micropolar fluid flows in two-dimensional unbounded domains. Exactly, the $H^1$-boundedness and $H^2$-boundedness of the pullback attractors are established when the external force $F(t,x)$ has different regularity with respect to time variable, respectively.

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.

For a graph $G$ of order $n$ and a positive integer $k,$ a $k$-weak cycle partition of $G$, called $k$-WCP, is a sequence of vertex disjoint subgraphs $H_1,H_2,\cdots,H_k$ of $G$ with $\bigcup_{i=1}^{k}V(H_i)=V(G),$ where $H_i$ is isomorphic to $K_1,K_2$ or a cycle. Let $\sigma_2(G)=\min\{d(x)+d(y):xy\notin E(G),x,y\in V(G)\}.$ Hu and Li [Discrete Math. 307(2007)] proved that if $G$ is a graph of order $n\ge k+12$ with a $k$-WCP and $\sigma_2(G)\ge \frac{2n+k-4}{3},$ then $G$ contains a $k$-WCP with at most one subgraph isomorphic to $K_2.$ In this paper, we generalize their result on the analogy of Fan-type condition that $\max\{d(x),d(y)\}\ge \frac{2n+k-4}{6}$ for each pair of nonadjacent vertices $x,y\in V(G).$

The asymptotic analysis theory is a powerful mathematical tool employed in the study of complex systems. By exploring the behavior of mathematical models in the limit as certain parameters tend toward infinity or zero, the asymptotic analysis facilitates the extraction of simplified limit-equations, revealing fundamental principles governing the original complex dynamics. We will highlight the versatility of asymptotic methods in handling different scenarios, ranging from fluid mechanics to biological systems and economic mechanisms, with a greater focus on the financial markets models. This short overview aims to convey the broad applicability of the asymptotic analysis theory in advancing our comprehension of complex systems, making it an indispensable tool for researchers and practitioners across different disciplines. In particular, such a theory could be applied to reshape intricate financial models (e.g., stock market volatility models) into more manageable forms, which could be tackled with time-saving numerical implementations.