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 $0<\delta<1$.

This paper concerned with the orbital stability of solitary waves for the mKdV-Schrödinger system with cubic-quintic nonlinear terms through detailed spectral analysis and abstract stability theorem. First, we derived the explicit solitary wave solutions by assuming the solution expression. Then, through using the orbital stability theory developed by Grillakis et al., we established a general criteria for assessing the orbital stability for solitary waves of this system.

In this paper we consider a kind of predator-prey model named Holling-Tanner model. Firstly, we prove all solutions of this model to be bounded from above. Secondly, we find a positive invariant set of the model, and prove the existence of stable limit cycle in this invariant set by Poincaré-Bendixson theorem for the unstable equilibrium. Thirdly, we get the region of parameters in which the corresponding stable equilibrium are also globally asymptotically stable. Lastly, we give a bifurcation diagram and illustration with two limit cycles for special parameters through numerical simulation. By our knowledge, the invariant set constructed in this paper is better than that in the book written by Murray.

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.

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 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 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.

The initial boundary value problem of a class of coupled hyperbolic systems with logarithmic source terms is considered. In this article, we classify the initial data for the global existence, finite time blow-up and long time decay of the solution. By using potential well method combined with Sobolev embedding theorem, the sufficient initial conditions of global existence, asymptotic behavior, the upper and lower bounds of blow-up time are derived at low energy level $E(0) < d$. These results are extended in parallel to the critical case $E(0) = d$. Besides, with additional assumptions on initial data, the finite time blow up result is given with arbitrary positive initial energy $E(0) > 0$.

This paper is concerned with the non-isentropic compressible Navier-Stokes/Allen-Cahn equations with the diffusion interface, which is an important mathematical model in the numerical simulation of compressible immiscible two-phase flow. When the space-asymptotic states $(v_\pm, u_\pm, \theta_\pm)$ lie in the rarefaction curve of the Riemann problem of the compressible Euler equations, we prove that the time-asymptotic state of solutions to the 1-D Cauchy problem is the rarefaction wave, that is, the stability of the rarefaction wave, where the strength of the rarefaction wave is not required to be small. Moreover, we consider the general gases including ideal polytropic gas and allow the different space-asymptotic states $\chi_\pm$ for the concentration difference of the mixture fluids. The proof is mainly based on a basic energy method. By product, we give the proof of the uniqueness of the global solutions to the 1-D Cauchy problem.

This paper aims to construct a two-grid scheme of fully discretized expanded mixed finite element methods for optimal control problems governed by parabolic integro-differential equations and discuss a priori error estimates. The state variables and co-state variables are discretized by the lowest order Raviart-Thomas mixed finite element, and the control variable is approximated by piecewise constant functions. The time derivative is discretized by the backward Euler method. Firstly, we define some new mixed elliptic projections and prove the corresponding error estimates which play an important role in subsequent convergence analysis. Secondly, we derive a priori error estimates for all variables. Thirdly, we present a two-grid scheme and analyze its convergence. In the two-grid scheme, the solution of the parabolic optimal control problem on a fine grid is reduced to the solution of the parabolic optimal control problem on a much coarser grid and the solution of a decoupled linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy. At last, a numerical example is presented to verify the theoretical results.

The Cauchy problem of the fifth-order nonlinear Schrödinger (foNLS) equation is investigated with nonzero boundary conditions in detailed. Firstly, the spectral analysis of the scattering problem is carried out. A Riemann surface and affine parameters are introduced to transform the original spectral parameter to a new spectral parameter in order to avoid the multi-valued problem. Based on Lax pair of the foNLS equation, the Jost functions are obtained, and their analytical, asymptotic, symmetric properties, as well as the corresponding properties of the scattering matrix are established systematically. For the inverse scattering problem, we discuss the cases that the scattering coefficients have simple zeros and double zeros, respectively, and we further derive their corresponding exact solutions via solving a suitable Riemann-Hilbert problem. Moreover, some interesting phenomena are found when we choose some appropriate parameters for these exact solutions, which are helpful to study the propagation behavior of these solutions.

In this paper we prove vanishing viscosity limits of a coupled chemotaxis-fluid model in a bounded domain $\Omega\subset\mathbb{R}^3$. The proof is based on the Banach's fixed point theorem and the $L^p$-energy method. In addition, the $L^\infty$-estimates and gradient estimates of the heat equations also play a crucial role.

In this paper, we consider the time-harmonic electromagnetic scattering by a perfect conductor in a homogeneous chiral environment. For three-dimensional cylindrical structures, it can be simplified as a two-dimensional model problem, which can be modeled by two scalar Helmholtz equations via coupled boundary conditions. The boundary integral equation method is used to prove the unique existence of the weak solution to this problem. Then we apply the linear sampling method to recover the scatterer from one of the far field pattern of wave fields. Some numerical examples are shown to verify the correctness and effectiveness of the proposed method.

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.

Let $\tau\in\mathbb{C}\setminus\{0\},$ let $p$ and $q$ be distinct positive integers, and let $a,$ $b,$ $c$ be meromorphic functions such that at least one of $b$ and $c$ is not identically equal to zero. The main purpose of this paper is to study the logistic delay differential equations of the Lotka-Volterra type $$w'(z)=w(z)[a(z)+b(z)w^{p}(z-\tau)+c(z)w^{q}(z-\tau)].$$ We prove that any admissible meromorphic solution $w$ of the equation satisfies that the counting function $N(r, w)$ of poles and the characteristic function $T(r, w)$ have the same growth category. Furthermore, we obtain that ``most" of admissible meromorphic solutions of a more general delay differential equation \begin{eqnarray*} w'(z)=w(z)\Big[a(z)+\sum_{j=1}^{k}b_{j}(z)w^{j}(z-\tau)\Big], \qquad k\in \mathbb{N}, \end{eqnarray*} have a pole at least.

Recurrent event data with a terminal event are commonly encountered in longitudinal follow-up studies. In this paper, we investigate regression analysis of the weighted composite endpoint of recurrent and terminal events with a semiparametric mixed model. Particularly, the weighted composite endpoint is constructed by the severity of all events while leaving the dependence structure among the recurrent and terminal events unspecified. The semiparametric mixed model is flexible since it allows the covariate effects on the rate function of the weighted composite endpoint to be proportional or convergent. For inference on the model parameters, the estimating equation approach and the inverse probability weighting technique are developed. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed procedure is evaluated through Monte Carlo simulation studies. We apply the proposed method to a real data set on a medical cost study of chronic heart failure patients for illustration.

A non-increasing sequence $\pi=(d_1,\cdots,d_n)$ of nonnegative integers is said to be a graphic sequence if it is realizable by a simple graph $G$ on $n$ vertices. In this case, $G$ is referred to as a realization of $\pi$. In terms of graphic sequences, the Loebl-Komlós-Sós conjecture states that for any integers $k$ and $n$, if $\pi=(d_1,\cdots,d_n)$ is a graphic sequence with $d_{\lceil\frac{n}{2}\rceil}\ge k$, then every realization of $\pi$ contains all trees with $k$ edges as subgraphs. This problem can be viewed as a forcible degree sequence problem. In this paper, we consider a potential degree sequence problem of the Loebl-Komlós-Sós conjecture, that is, we prove that for any integers $k$ and $n$, if $\pi=(d_1,\cdots,d_n)$ is a graphic sequence with $d_{\lceil\frac{n}{2}\rceil}\ge k$, then there is a realization of $\pi$ containing all trees with $k$ edges as subgraphs.

The eccentricity matrix $\mathcal E(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by leaving unchanged the largest nonzero entries in each row and each column, and replacing the remaining ones with zeros. In this paper, we consider the set $\mathcal C \mathcal T$ of clique trees whose blocks contain at most two cut-vertices of the clique tree. Along with studying the structural properties of a clique tree in $\mathcal C \mathcal T$, we prove its eccentricity matrix to be irreducible, and then determine its inertia showing that every graph in $\mathcal C \mathcal T$ with more than four vertices and odd diameter has two positive and two negative $\mathcal E$-eigenvalues. Positive $\mathcal E$-eigenvalues and negative $\mathcal E$-eigenvalues turn out to be equal in number even for graphs in $\mathcal C \mathcal T$ with even diameter; that shared cardinality also counts the `diametrally distinguished' vertices. Finally, we prove that the spectrum of the eccentricity matrix of a clique tree $G$ in $\mathcal C \mathcal T$ is symmetric with respect to the origin if and only if $G$ has an odd diameter and exactly two adjacent central vertices. Our results generalize those achieved on trees by I. Mahato and M. R. Kannan in 2022.

In this paper, we investigate the stationary distribution of a novel stochastic hybrid predator-prey model with fear effect and Beddington-DeAngelis functional response. Based on Markov semigroup theory, the existence of asymptotically stable stationary distribution of stochastic hybrid system is established, which can converge in $L^1$ to an invariant density under appropriate conditions. Moreover, we derive sufficient conditions for extinction and persistence of prey and predator populations in the stochastic hybrid model.

We consider a system of $k$-Hessian equations: $$ \left\{ \begin{aligned} &S_{k}(\lambda (D^{2}u))=(-u)^{\alpha_{1}}+(-v)^{\beta_{1}},&\quad &{\rm in}\ B,\\ &S_{k}(\lambda (D^{2}v))=(-u)^{\alpha_{2}}, &\quad &{\rm in}\ B,\\ &u=v=0, &\quad \ &{\rm on}\ \partial B,\\ \end{aligned} \right. $$ where $1\leq k\leq n\ (n\geq2),\ \alpha_{1},\alpha_{2}$ and $\beta_{1}$ are positive constants, $B=\{x\in \mathbb{R}^{n}:|x|<1\}$. By giving the complete classification for the constants $\alpha_1$, $\alpha_2$ and $\beta_1$ according to the value of $k$, some sharp conditions are obtained for the existence, uniqueness and nonexistence results of $k$-convex solutions to the above problem.

In this paper, we explore the existence of analytic solutions for an iterative functional equation of the form $$g(g(x))+x=f(g(x))$$ that originates from Painlevé equations. By an invertible transformation, we study the analytic solutions of an auxiliary equation under three different cases, and obtain the invertible analytic solutions for the original equation.

A general model of insider trading on a dynamic asset in a finite time interval is proposed, in which an insider possesses the whole information on the dynamic values, noise traders without any information submit orders randomly as a martingale with volatility following a stochastic process, and market makers observe partial information when setting price in a semi-strong efficient way. With the help of filtering theory, BSDE method and dynamic programming principle, we establish a market equilibrium consisting of linear insider trading strategy and linear pricing rule, with the later characterized by price pressure on market orders and price pressure on asset observations. It shows that in the equilibrium, all the information on the risky asset is incorporated into the market price at the end of the transaction, and price pressure on market orders is a submartingale while market depth process is a martingale. Furthermore, as market makers’ information precision on the asset tends to zero, the equilibrium with partial observation of market makers on the risky asset converges to the one without partial observation of market makers, while when market makers observe almost all of information on the asset, the expected profit earned by the insider makes almost zero, which is in accord with our economic intuition. Our results cover some classical results about continuous-time insider trading on a static asset.

This study delves into the Hénon-type weighted elliptic equation, given by $-\Delta_g u = (\sinh r)^\alpha e^u$, within the context of hyperbolic space $\mathbb{H}^n$, where $\alpha > 0$ and $n > 2$. Our research reveals notable distinctions in the stability of solutions when compared to the Euclidean case.

We propose two nonmonotone retraction-based proximal gradient methods for solving a class of nonconvex nonsmooth optimization problems over the Stiefel manifold. The proposed methods are equipped with the descent direction obtained by a proximal mapping restricted in tangent space of the manifold and the Barzilai-Borwein stepsizes determined by two recent iteration points and the corresponding descent directions. By employing, respectively, the Grippo-Lampariello-Lucidi nonmonotone line search strategy and the Dai-Fletcher nonmonotone line search strategy, our proposed methods are proved to be globally convergent. Analysis on the iteration complexity for obtaining an $\epsilon$-stationary solution is provided. Numerical results on the sparse principle component analysis problems demonstrate the efficiency of our methods.

This paper studies a model of $n$ insiders with perfect information trading on a risky asset with value normally distributed and disclosed at a random deadline. We propose a concept of information protocol equilibrium under semi-strong efficient pricing, consisting of an $n-$profile of insider trading strategies with terminal residual information protocols, and find that if a common protocol on terminal residual information before trading is obeyed by all insiders, then, in the market with more than two insiders there exists a uique equilibrium only when it requires to release common partial information eventually, or it does not exist if it requires to release all or not any; but in the market with a single insider, the insider may release all private information eventually to make a maximal profit. Thereby, the existence and uniqueness of information protocol equilibrium among $n$ insiders are deduced. Finally, numerical results illustrate some market characteristics of equilibria with different information protocols required before trading.

In this work, we investigate a class of nonlinear boundary value problems within the framework of Sobolev spaces with variable exponents. After establishing precise formulations of these problems, we reformulate them as nonlinear hyperbolic-type equations. For each of the six problems considered, we establish both existence and uniqueness results. Furthermore, we address the associated stationary problems and prove the existence of solutions by applying a suitable variant of Brouwer's fixed-point theorem.

In this paper, an HIV-1 infection model with intracellular delay, humoral immunity and immune impairment is investigated, in which both virus-to-cell infection and cell-to-cell transmission are considered. The basic reproduction ratio is calculated and the existence of feasible equilibria is established. By analyzing the distributions of roots of the corresponding characteristic equations, the local asymptotic stability of each of feasible equilibria is established. With the help of appropriate Lyapunov functionals and LaSalle's invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable, and the virus is eventually eliminated; if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable. Finally, numerical simulations are carried out to illustrate the effects of some parameters on HIV-1 infection dynamics.

This paper is concerned with the energy decay rate of the total energy to a wave equation with $p(x)$-Laplacian damping (nonlinear strong damping) and nonlinear source. With some suitable restrictions on variable growth exponents $r(x)$ and $p(x)$, first, we prove that the local solution can be extended to exist globally. Second, by using suitable and weighted multiplier techniques, it is proved that the total energy decays logarithmically. The key and main difficulty is to give a prior estimate for the wighted integral $\int_{\Omega}\chi^{p(x)-1}(\tau)|\nabla u(\tau)|^{p(x)}dx$ by some differential inequality techniques. In the proof of energy decay, the traditional method to eliminate the lower-order terms by exploiting the unique continuation and compactness arguments is not needed in our energy decay estimate.

In this paper, we studied a stochastic predator-prey model of two predators with stage structure. By constructing a suitable stochastic Lyapunov function, the condition of stationary distribution is verified, and we get the sufficient condition for the model to have ergodic stationary distribution. Then, by using the Itô’s formula for the model, the sufficient conditions for the extinction of the predator population are given. Finally, some examples and numerical simulations are illustrated to verify the theoretical results.

The Cauchy problem for non-isentropic compressible Navier-Stokes/Allen-Cahn system with degenerate heat-conductivity $\kappa(\theta)=\tilde{\kappa}\theta^\beta$ in 1-D is discussed in this paper. This system is widely used to describe the motion of immiscible two-phase flow with diffused interface. The well-posedness for strong solution of this problem is established with the $H^1$ initial data for density, temperature, velocity, and the $H^2$ initial data for phase field. The result shows that no discontinuity of the phase field, vacuum, shock wave, mass or heat concentration will be developed at any finite time in the whole space. From the hydrodynamic point of view, this means that no matter how complex the interaction between the hydrodynamic and phase-field effects, phase separation will not occur, but the phase transition is possible.

In this paper, we present some new regularity criteria for suitable weak solutions to the 3D corotational Beris-Edwards system. First, we prove that suitable weak solutions are regular if the scaled $L^{p ; q}$-norm of the velocity field or gradient of velocity is small with $\frac{2}{p}+\frac{3}{q}=2,1＜p \leq \infty$. Next, we give $\varepsilon$-regularity criteria in terms of velocity field $\mathbf{u}$ and director field $\mathbf{Q}$ in Lorentz spaces, which extends the results obtained by Wang et al (J. Evol. Equ. 21: 1627-1650, 2021) for Navier-Stokes equations.

Let $w(z)$ be non-rational meromorphic solutions with hyper-order less than $1$ to a family of higher order nonlinear delay differential equations \begin{align*} w(z+1)w(z-1)+a(z)\frac{w^{(k)}(z)}{w(z)}=R(z,w(z)), \qquad k\in\mathbb{N^{+}}, \end{align*} where $a(z)$ is rational, $R(z,w(z))=\frac{P(z,w(z))}{Q(z,w(z))}$ is an irreducible rational function in $w$ with rational coefficients in $z$. This paper mainly show the relationships of the degree of $P(z,w(z))$ and $Q(z,w(z))$ when the above equations exist such solutions $w(z)$. There are also some examples to show that our results are sharp.

In this paper, we establish the exponential stability of the one-dimensional Euler-Bernoulli beam equation with local viscosity. On the one hand, we demonstrate that the Euler-Bernoulli beam equation system is exponentially stable by employing the multiplier method, which relies on an appropriately constructed Lyapunov function. On the other hand, we discretize the Euler-Bernoulli beam equation system using the finite volume difference method. For the resulting semi-discrete system, we construct a discretized multiplier based on the discretized Lyapunov function. Finally, we prove that the semi-discrete Euler-Bernoulli beam equation system is also uniformly exponentially stable.

A time-domain elastic scattering problem is considered in three dimensions. In problem setting, a rigid obstacle is immersed in an unbounded domain filled with homogeneous and isotropic elastic medium. In order to analyze the well-posedness of the target problem, we reduce the scattering problem into an initial boundary value problem in a bounded domain over a finite time interval by using a compressed coordinate transformation. The Galerkin method is adopted to prove the uniqueness results, and the energy method is used to prove the stability of the scattering problem. In addition, we derive a priori estimate with explicit time dependence.

The paper focuses on the ergodicity of a $\phi$-irreducible Markov chain $\{X_{n},n\geq0\}$ that is generated iteratively through the expression $X_{n+1}=f(X_{n})+\epsilon_{n+1}$. Here, $\{\epsilon_n,n\geq1\}$ is a sequence of independent identically distributed centered random variables, $f(\cdot)$ is an $\mathbb{R}$-valued continuous function, and $X_{0}$ is arbitrary but independent of $\{\epsilon_{n},n\geq1\}$. Our main contribution is to provide necessary and sufficient conditions for the ergodicity of this special class of Markov chains. We also present a generalized approach for $f(\cdot)$ in the end.

In this paper, we study bimagnetic monopoles which are topological solitons in three space dimensions. We prove the existence and uniqueness of solution of a static and radially symmetric Bogomol'nyi-Prasad-Sommerfield (BPS) bimagnetic monopoles formulated and presented in a recent study of Bazeia, Marques and Menezes. Our method is based on a dynamical shooting approach depending on two shooting parameters which provides an effective framework for constructing the BPS equations in magnetic core and magnetic shell. Furthermore, we obtain the relation between the BPS and non-BPS monopoles solutions, and properties of static BPS monopoles solutions.

Let $G$ be a simple connected graph with vertex set $V(G)$. For $S \subseteq V(G)$, let $\pi_G(S)$ denote the maximum cardinality of internally disjoint $S$-paths in $G$. For an integer $k$ with $k\ge 2$, the $k$-path-connectivity $\pi_k(G)$ is defined as the minimum $\pi_G(S)$ over all $k$-subsets $S$ of $V(G)$. It is proved that deciding whether $\pi_G(S) \ge r$ is NP-complete problem [Graphs Combin. 37 (2021) 2521-2533]. The hypercube $Q_n$ is the famous Cayley graph, which is widely studied in the research of developing multiprocessor systems. The hierarchical cubic network $HCN_n$ is given in [IEEE TPDS 6 (1995) 427-435] which takes $Q_n$ as building clusters and emulates the desirable properties very efficiently. In this paper, we consider the $3$-path-connectivity of $HCN_n$ and prove that $\pi_3(HCN_n)=\lfloor \frac{3n+2}{4} \rfloor$ for $n \ge 2$ by constructing multiple internally disjoint $S$-paths. This result improves the $3$-tree-connectivity [Discrete Appl. Math. 322 (2022) 203-209] from trees to paths.

In this paper, we consider a model which is derived from a class of the $2$-dimensional Kolmogorov systems. Our purpose is to investigate the continuity of periodic solutions for this model in coefficient functions with respect to weak topologies. Finally, we provide an example as an application to Lotka-Volterra systems.

In this paper, we investigate some asymptotic properties of the least squares estimator in nonlinear regression model with independent and identically distributed random errors under sub-linear expectations. The large deviation results for the estimator are established under some general conditions. As applications, the results on weak consistency and strong consistency are obtained under the meaning of capacity. A simulation study is also presented to verify the validity of the theoretical results.