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.

Given a forbidden graph $H$ and a function $f(n)$, the Ramsey-Turán number $\textbf{RT}\left( {n,H,f\left( n \right)} \right)$ is the maximum number of edges of an $H$-free graph on $n$ vertices with independence number less than ${f\left( n \right)}$. For graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any red/blue edge coloring of the complete graph $K_N$ contains either a red $G$ or a blue $H$. Denote $G+H$ by the join graph obtained from disjoint $G$ and $H$ by adding all edges between them completely. We first show that for any fixed graph $H$, if there are two constants $p:=p(H)>0$ and $q:=q(H)>1$ such that $R(H,K_n)\le \frac{pn^q}{(\log n)^{q-1}}$, then $\textbf{RT}(n,K_2+H,o(n^{\frac{1}{q}}(\log n)^{1-\frac{1}{q}}))=o(n^2),$ which extends several previous results. Moreover, we show that for any fixed forest $F$ of order $k\ge3$, and for any $0<\delta<1$ and sufficiently large $n$, \begin{align*} \textbf{RT}( {n,F+F,n^\delta} )\le n^{2-(1-\delta)/\lceil\frac{(k-1)(2-\delta)}{1-\delta}\rceil}. \end{align*} As a corollary, we have an upper bound for ${\bf{RT}}( {n,K_{2,2,2},n^{\delta}})$ for any $0<\delta<1$.

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.

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, 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.

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 $G$.

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.

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.

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.

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 $T$.

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.

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.

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.

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 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.

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 $G$.

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.

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.

In this paper, the existence, uniqueness and Strichartz type estimates to solutions of multipoınt problem for abstract linear and nonlinear wave equations are obtained. The equation includes a linear operator $A$ defined in a Hilbert space $H$. We obtain the existence, uniqueness regularity properties, and Strichartz type estimates to solutions of a wide class of wave equations which appear in the fields of elastic rod, hydro-dynamical process, plasma, materials science and physics, by choosing the space $H$ and the operator $A$.

In the present paper, we study uniqueness and nondegeneracy of positive solutions to the general Kirchhoff type equations $$-M\Big(\int_{ R^N}|\nabla v|^2 dx\Big)\Delta v=g(v) \qquad \hbox{in} R^N, $$ where $M:[0,+\infty)\mapsto R$ is a continuous function satisfying some suitable conditions and $v\in H^1(R^N)$. Applying our results to the case $M(t)=at+b$, $a,b>0$, we make it clear all the positive solutions for all dimensions $N\geq 1$. Our results can be viewed as a generalization of the corresponding results of Li et al. [JDE, 2020, 268, Section 1.2].

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 $x,y\in V(G).$

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 consider a Dirichlet-Laplacian with a drift problem. We prove existence of weak solutions of it on the Nehari manifold. Then, we show that the associated energy functional has a minimizer on a rearrangement class generated by a determined function. Finally, we prove a duality theorem for this boundary value problem.

In this paper, we develop a robust variable selection procedure based on the exponential squared loss (ESL) function for the varying coefficient partially nonlinear model. Under certain conditions, some asymptotic properties of the proposed penalized ESL estimator are established. Meanwhile, the proposed procedure can automatically eliminate the irrelevant covariates, and simultaneously estimate the nonzero regression coefficients. Furthermore, we apply the local quadratic approximation (LQA) and minorization-maximization (MM) algorithm to calculate the estimates of non-parametric and parametric parts, and introduce a data-driven method to select the tuning parameters. Simulation studies illustrate that the proposed method is more robust than the classical least squares technique when there are outliers in the dataset. Finally, we apply the proposed procedure to analyze the Boston housing price data. The results reveal that the proposed method has a better prediction ability.

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.

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.

In this paper, we propose an efficient and accurate method for pricing Guaranteed Minimum Death Benefit (GMDB) under time-changed Lévy processes. Suppose that the GMDB payoff depends on a dollar cost averaging (DCA) style periodic investment, and the activity rate process in stochastic time change is modeled by a square-root process. We develop a recursive method to derive the closed form valuation formula by using the frame duality projection method. Numerical examples are reported for demonstrating the effectiveness of our approach and illustrating the interplay between contract parameters and the valuation.

In this paper, we propose a pricing model of airbag options with discrete monitoring, time-varying barriers, early exercise opportunities, and other popular features simultaneously. We show that the option value is a viscosity solution of a PDE system. In particular, a closed-form solution is obtained in the classic Black-Scholes economy with no early exercise opportunities. For the general case, we develop a numerical algorithm and conduct an extensive numerical analysis after calibrating the model to the CSI 500 index in China. Greek letters, dynamic hedging, and assessment of investing in airbag options are also studied.

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.

In this paper, we consider the regular $s$-level fractional factorial split-plot (FFSP) designs when the subplot (SP) factors are more important. The idea of general minimum lower-order confounding criterion is applied to such designs, and the general minimum lower-order confounding criterion of type SP (SP-GMC) is proposed. Using a finite projective geometric formulation, we derive explicit formulae connecting the key terms for the criterion with the complementary set. These results are applied to choose optimal FFSP designs under the SP-GMC criterion. Some two- and three-level SP-GMC FFSP designs are constructed.

The link of (2+1)-dimensional Harry Dym equation with the modified Kadomtsev-Petviashvili equation by the reciprocal transformation is shown. With the help of Darboux transformation, exact solutions of the (2+1)-dimensional Harry Dym equation are constructed and represented in terms of Wronskians.

The stable switching phenomenon of a diffusion mussel-algae system with two delays and half saturation constant is studied. The stability of positive equilibrium and the existence of Hopf bifurcation on two-delay plane are investigated by calculating the stability switching curves. The normal form on the central manifold near Hopf bifurcation point is also derived. It can find that two different delays can induce the stable switches which cannot occur with the same delays. Through numerical simulations, the region of complete stability of system increases with the increase of half-saturation constant, indicating that the half-saturation constant contributes to the stability of system. In addition, double Hopf bifurcations may occur due to the intersection of bifurcation curves. These results show that two different delay and half-saturation constant have important effects on the system. The numerical simulation results validate the correctness of the theoretical analyses.

In this paper, we focus on the problem of nonparametric quantile regression with left-truncated and right-censored data. Based on Nadaraya-Watson (NW) Kernel smoother and the technique of local linear (LL) smoother, we construct the NW and LL estimators of the conditional quantile. Under strong mixing assumptions, we establish asymptotic representation and asymptotic normality of the estimators. Finite sample behavior of the estimators is investigated via simulation, and a real data example is used to illustrate the application of the proposed methods.

In this paper, we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term $f$ satisfies critical exponential growth and the external force $ g(x)\in L^{2}(\Omega)$. In the framework of time-dependent spaces, we verify the existence of absorbing sets and the asymptotic compactness of the process, then we obtain the existence of the time-dependent global attractor $\mathscr{A}=\{A_t\}_{t\in \mathbb{R}}$ in $\mathcal{M}_t$. Furthermore, we achieve the regularity of $\mathscr{A}$, that is, $A_t$ is bounded in $\mathcal{M}_t^1$ with a bound independent of $t$.

In this article, the maximum likelihood estimators (MLEs) of the scale and shape parameters β and λ from the Exponential-Poisson distribution will be considered in moving extremes ranked set sampling (MERSS). These MLEs will be compared in terms of asymptotic efficiencies. The numerical results show that the MLEs obtained via MERSS can serve as effective alternatives to those derived from simple random sampling.

This paper deals with numerical solutions for nonlinear first-order boundary value problems (BVPs) with time-variable delay. For solving this kind of delay BVPs, by combining Runge-Kutta methods with Lagrange interpolation, a class of adapted Runge-Kutta (ARK) methods are developed. Under the suitable conditions, it is proved that ARK methods are convergent of order $\min\{p,\mu\!+\!\nu\!+\!1\}$, where $p$ is the consistency order of ARK methods and $\mu,\nu$ are two given parameters in Lagrange interpolation. Moreover, a global stability criterion is derived for ARK methods. With some numerical experiments, the computational accuracy and global stability of ARK methods are further testified.

In the paper, by exploring Stampacchia truncation method, some comparison techniques and variational approaches, we study the existence and regularity of positive solutions for a boundary value problem involving the fractional p-Laplacian, where the nonlinear term satisfies some growth conditions but without the Ambrosetti and Rabinowitz condition.