In this paper, we study the irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as the approximate sequences.

In this paper, we are concerned with the necessary and sufficient condition of the global existence of smooth solutions of the Cauchy problem of the multi-dimensional scalar conservation law with source-term, where the initial data lies in $W^{1,\infty}(\mathbb{R}^n) \cap C^1(\mathbb{R}^n)$. We obtain the solution formula for smooth solution, and then apply it to establish and prove the necessary and sufficient condition for the global existence of smooth solution. Moreover, if the smooth solution blows up at a finite time, the exact lifespan of the smooth solution can be obtained. In particular, when the source term vanishes, the corresponding theorem for the homogeneous case is obtained too. Finally, we give two examples as its applications, one for the global existence of the smooth solution and the other one for the blowup of the smooth solutions at any given positive time.

In this paper, it is proved that the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear viscosity, different far field states and periodic perturbations, not only exists globally in time, but also converges towards the viscous shock wave of the corresponding Riemann problem as time goes to infinity. Furthermore, the decay rate is shown. The proof is given by a technical energy method.

In this paper, the large time behavior of solutions of 1-D isentropic Navier-Stokes system is investigated. It is shown that a composite wave consisting of two viscous shock waves is stable for the Cauchy problem provided that the two waves are initially far away from each other. Moreover the strengths of two waves could be arbitrarily large.

In the paper, we establish the existence of steady boundary layer solution of Boltzmann equation with specular boundary condition in L_x,v² ∩L_x,v^∞ in half-space. The uniqueness, continuity and exponential decay of the solution are obtained, and such estimates are important to prove the Hilbert expansion of Boltzmann equation for half-space problem with specular boundary condition.

In 2003, Gasser-Hsiao-Li [JDE (2003)] showed that the solution to the bipolar hydrodynamic model for semiconductors (HD model) without doping function time-asymptotically converges to the diffusion wave of the porous media equation (PME) for the switch-off case. Motivated by the work of Huang-Wu [arXiv:2210.13157], we will confirm that the time-asymptotic expansion proposed by Geng-Huang-Jin-Wu [arXiv:2202.13385] around the diffusion wave is a better asymptotic profile for the HD model in this paper, where we mainly adopt the approximate Green function method and the energy method.

The ideal reaction chromatography model can be regarded as a semi-coupled system of two hyperbolic partial differential equations, in which, one is a self-closed nonlinear equation for the reactant concentration and another is a linear equation coupling the reactant concentration for the resultant concentration. This paper is concerned with the initial-boundary value problem for the above model. By the characteristic method and the truncation method, we construct the global weak entropy solution of this initial initial-boundary value problem for Riemann type of initial-boundary data. Moreover, as examples, we apply the obtained results to the cases of head-on and wide pulse injections and give the expression of the global weak entropy solution.

In this paper, we will investigate the incompressible Navier-Stokes-Landau-Lifshitz equations, which is a system of the incompressible Navier-Stokes equations coupled with the Landau-Lifshitz-Gilbert equations. We will prove global existence of the smooth solution to the incompressible Navier-Stokes-Landau-Lifshitz equation with small initial data in $\mathbb{T}^2$ or $\mathbb{R}^2$ and $\mathbb{R}^3$.

We study the connection between the compressible Navier-Stokes equations coupled by the Qtensor equation for liquid crystals with the incompressible system in the periodic case, when the Mach number is low. To be more specific, the convergence of the weak solutions of the compressible nematic liquid crystal model to the incompressible one is proved as the Mach number approaches zero, and we also obtain the similar results in the stochastic setting when the equations are driven by a stochastic force. Our approach is based on the uniform estimates of the weak solutions and the martingale solutions, then we justify the limits using various compactness criteria.

We study the L²-supercritical nonlinear Schrödinger equation (NLS) with a partial confinement, which is the limit case of the cigar-shaped model in Bose-Einstein condensate (BEC). By constructing a cross constrained variational problem and establishing the invariant manifolds of the evolution flow, we show a sharp condition for global existence.

A four-dimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number ${R}_{0}$. Specifically, it shows that the disease-free equilibrium ${E}^{0}$ is globally asymptotically stable (GAS) for ${R}_{0}<1$, and globally attractive (GA) for ${R}_{0}=1$, while the endemic equilibrium $E^{\ast}$ is GAS and ${E}^{0}$ is unstable for ${R}_{0}>1$. Especially, to obtain the global stability of the equilibrium $E^{\ast}$ for $R_{0}>1$, the weak persistence of the model is proved by some analysis techniques.

In the current paper, the best linear unbiased estimators (BLUEs) of location and scale parameters from location-scale family will be respectively proposed in cases when one parameter is known and when both are unknown under moving extremes ranked set sampling (MERSS). Explicit mathematical expressions of these estimators and their variances are derived. Their relative efficiencies with respect to the minimum variance unbiased estimators (MVUEs) under simple random sampling (SRS) are compared for the cases of some usual distributions. The numerical results show that the BLUEs under MERSS are significantly more efficient than the MVUEs under SRS.

A path-factor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\geq2$ be an integer. A $P_{\geq k}$-factor of $G$ means a path factor in which each component is a path with at least $k$ vertices. A graph $G$ is a $P_{\geq k}$-factor covered graph if for any $e\in E(G)$, $G$ has a $P_{\geq k}$-factor including $e$. Let $\beta$ be a real number with $\frac{1}{3}\leq\beta\leq1$ and $k$ be a positive integer. We verify that (\romannumeral1) a $k$-connected graph $G$ of order $n$ with $n\geq5k+2$ has a $P_{\geq3}$-factor if $|N_G(I)|>\beta(n-3k-1)+k$ for every independent set $I$ of $G$ with $|I|=\lfloor\beta(2k+1)\rfloor$; (\romannumeral2) a $(k+1)$-connected graph $G$ of order $n$ with $n\geq5k+2$ is a $P_{\geq3}$-factor covered graph if $|N_G(I)|>\beta(n-3k-1)+k+1$ for every independent set $I$ of $G$ with $|I|=\lfloor\beta(2k+1)\rfloor$.

Let $\lambda K_{m,n}$ be a complete bipartite multigraph with two partite sets having $m$ and $n$ vertices, respectively. A $K_{p,q}$-factorization of $\lambda K_{m,n}$ is a set of $K_{p,q}$-factors of $\lambda K_{m,n}$ which partition the set of edges of $\lambda K_{m,n}$. When $\lambda =1$, Martin, in [Complete bipartite factorizations by complete bipartite graphs, Discrete Math., 167/168 (1997), 461-480], gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will study the $K_{p,q}$-factorization of $\lambda K_{m,n}$ for $p=1$, to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.

This paper deals with the existence, uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions. The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.

In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk $(X_n)$ on $\mathbb Z$. If $M_n$ is its maximal position at time $n$, we prove that there is a constant $\alpha>0$ such that $M_n/n$ converges to $\alpha$ almost surely on the set of infinite number of visits to the set of catalysts. We also derive the asymptotic law of the centered process $M_n-\alpha n$ as $n\to \infty$. Our results are similar to those in [13]. However, our results are proved under the assumption of finite $L\log L$ moment instead of finite second moment. We also study the limit of $(X_n)$ as a measure-valued Markov process. For any function $f$ with compact support, we prove a strong law of large numbers for the process $X_n(f)$.

This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems, in which linear parts might admit certain degeneracy. By introducing Fredholm operators and coincidence degree theory, sufficient conditions for nonlinear discrete systems to be controllable are presented. In addition, applications are given to illustrate main results.

This paper aims to prove the asymptotic behavior of the solution for the thermo-elastic von Karman system where the thermal conduction is given by Gurtin-Pipkins law. Existence and uniqueness of the solution are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional. Therefore, the stability result clarified that the solutions energy functional decays exponentially at infinite time.

Estimation of treatment effects is one of the crucial mainstays in economics and sociology studies. The problem will become more serious and complicated if the treatment variable is endogenous for the presence of unobserved confounding. The estimation and conclusion are likely to be biased and misleading if the endogeny of treatment variable is ignored. In this article, we propose the pseudo maximum likelihood method to estimate treatment effects in nonlinear models. The proposed method allows the unobserved confounding and random error terms to exist in an arbitrary relationship (such as, add or multiply), and the unobserved confounding have different influence directions on treatment variables and outcome variables. The proposed estimator is consistent and asymptotically normally distributed. Simulation studies show that the proposed estimator performs better than the special regression estimator, and the proposed method is stable for various distribution of error terms. Finally, the proposed method is applied to the real data that studies the influence of individuals have health insurance on an individual’s decision to visit a doctor.

Let $(Z_{n})$ be a supercritical bisexual branching process in a random environment $\xi$. We study the almost sure (a.s.) convergence rate of the submartingale $\overline{W}_{n} =Z_{n}/I_{n}$ to its limit $\overline{W}$, where $(I_n)$ is an usually used norming sequence. We prove that under a moment condition of order $p \in (1,2),\overline{W}-\overline{W}_{n}=o(e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming the logarithmic moment condition holds, we have $\overline{W}-\overline{W}_{n}=o(n^{-\alpha})$ a.s.. In order to obtain these results, we provide the $L^{p}-$ convergence of $(\overline{W}_{n})$; similar conclusions hold for a bisexual branching process in a varying environment.

The alternating direction method of multipliers (ADMM) is one of the most successful and powerful methods for separable minimization optimization. Based on the idea of symmetric ADMM in two-block optimization, we add an updating formula for the Lagrange multiplier without restricting its position for multiblock one. Then, combining with the Bregman distance, in this work, a Bregman-style partially symmetric ADMM is presented for nonconvex multi-block optimization with linear constraints, and the Lagrange multiplier is updated twice with different relaxation factors in the iteration scheme. Under the suitable conditions, the global convergence, strong convergence and convergence rate of the presented method are analyzed and obtained. Finally, some preliminary numerical results are reported to support the correctness of the theoretical assertions, and these show that the presented method is numerically effective.

We consider the following quasilinear Schrödinger equation involving $p$-Laplacian \begin{align*} -\Delta_p u +V(x)|u|^{p-2}u-\Delta_p(|u|^{2\eta})|u|^{2\eta-2}u=\lambda\frac{|u|^{q-2}u}{|x|^{\mu}}+\frac{|u|^{2\eta p^*(\nu)-2}u}{|x|^\nu}\quad\text{in}\ \mathbb{R}^N, \end{align*} where $ N> p>1,\ \eta\ge \frac{p}{2(p-1)}$, $p< q<2\eta p^*(\mu)$, $p^*(s)=\frac{p(N-s)}{N-p}$, and $\lambda, \mu, \nu$ are parameters with $\lambda>0$, $\mu, \nu \in [0,p)$. Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.

This paper studies a defined contribution (DC) pension fund investment problem with return of premiums clauses in a stochastic interest rate and stochastic volatility environment. In practice, most of pension plans were subject to the return of premiums clauses to protect the rights of pension members who died before retirement. In the mathematical modeling, we assume that a part of pension members could withdraw their premiums if they died before retirement and surviving members could equally share the difference between accumulated contributions and returned premiums. We suppose that the financial market consists of a risk-free asset, a stock, and a zero-coupon bond. The interest rate is driven by a stochastic affine interest rate model and the stock price follows the Heston’s stochastic volatility model with stochastic interest rates. Different fund managers have different risk preferences, and the hyperbolic absolute risk aversion (HARA) utility function is a general one including a power utility, an exponential utility, and a logarithm utility as special cases. We are concerned with an optimal portfolio to maximize the expected utility of terminal wealth by choosing the HARA utility function in the analysis. By using the principle of dynamic programming and Legendre transform-dual theory, we obtain explicit solutions of optimal strategies. Some special cases are also derived in detail. Finally, a numerical simulation is provided to illustrate our results.

In this paper, the approximate analytical oscillatory solutions to the generalized KolmogorovPetrovsky-Piskunov equation (gKPPE for short) are discussed by employing the theory of dynamical system and hypothesis undetermined method. According to the corresponding dynamical system of the bounded traveling wave solutions to the gKPPE, the number and qualitative properties of these bounded solutions are received. Furthermore, pulses (bell-shaped) and waves fronts (kink-shaped) of the gKPPE are given. In particular, two types of approximate analytical oscillatory solutions are constructed. Besides, the error estimations between the approximate analytical oscillatory solutions and the exact solutions of the gKPPE are obtained by the homogeneity principle. Finally, the approximate analytical oscillatory solutions are compared with the numerical solutions, which shows the two types of solutions are similar.

In this paper, solutions of the Camassa-Holm equation near the soliton $Q$ is decomposed by pseudo-conformal transformation as follows: $\lambda^{1/2}(t)u(t,\lambda(t)y+x(t))=Q(y)+\varepsilon(t,y)$, and the estimation formula with respect to $\varepsilon(t,y)$ is obtained: $|\varepsilon(t,y)|\leq Ca_3Te^{-\theta|y|}+|\lambda^{1/2}(t)\varepsilon_0|$. For the CH equation, we prove that the solution of the Cauchy problem and the soliton $Q$ is sufficiently close as $y\rightarrow\infty$, and the approximation degree of the solution and $Q$ is the same as that of initial data and $Q$, besides the energy distribution of $\varepsilon$ is consistent with the distribution of the soliton $Q$ in $H^2$.

In this paper, we investigate the global solution and the structures of interaction between two dimensional non-selfsimilar shock wave and rarefaction wave of general two-dimensional scalar conservation law in which flux functions f(u) and g(u) do not need to be convex, and the initial value contains three constant states which are respectively separated by two general initial discontinuities. When initial value contains three constant states, the cases of selfsimilar shock wave and rarefaction wave had been studied before, but no results of the cases of neither non-selfsimilar shock wave or non-selfsimilar rarefaction wave. Under the assumption that Condition H which is generalization of one dimensional convex condition, and some weak conditions of initial discontinuity, according to all the kinds of combination of elementary waves respectively staring from two initial discontinuities, we get four cases of wave interactions as S + S, S + R, R + S and R + R. By studying these interactions between non-selfsimilar elementary waves, we obtain and prove all structures of non-selfsimilar global solutions for all cases.

This paper introduces two local conditional dependence matrices based on Spearman’s ρ and Kendall’s τ given the condition that the underlying random variables belong to the intervals determined by their quantiles. The robustness is studied by means of the influence functions of conditional Spearman’s ρ and Kendall’s τ. Using the two matrices, we construct the corresponding test statistics of local conditional dependence and derive their limit behavior including consistency, null and alternative asymptotic distributions. Simulation studies illustrate a superior power performance of the proposed Kendall-based test. Real data analysis with proposed methods provides a precise description and explanation of some financial phenomena in terms of mathematical statistics.

In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short time. The proof for large deviation principle is based on the weak convergence approach. For small time asymptotics we use the exponential equivalence to prove the result.

This paper investigates the stochastic dynamics of trophic cascade chemostat model perturbed by regime switching, Gaussian white noise and impulsive toxicant input. For the system with only white noise interference, sufficient conditions for stochastically ultimate boundedness and stochastically permanence are obtained, and we demonstrate that the stochastic system has at least one nontrivial positive periodic solution. For the system with Markov regime switching, sufficient conditions for extinction of the microorganisms are established. Then we prove the system is ergodic and has a stationary distribution. The results show that both impulsive toxins input and stochastic noise have great effects on the survival and extinction of the microorganisms. Finally, a series of numerical simulations are presented to illustrate the theoretical analysis.

Optical vortices arise as phase dislocations of light fields and they are of importance in modern optical physics. In this study, we employ the calculus of variations method to develop an existence theory for the steady state vortex solutions of a nonlinear Schrödinger type equation to model light waves that propagate in a medium with a new focusing-defocusing nonlinearity. First, we demonstrate the existence of positive radially symmetric solutions by constrained minimization, where we give some interesting explicit estimates related to vortex winding numbers and the wave propagation constant. Second, we establish the existence of saddle-point solutions through a mountain-pass argument.

We prove that the weak Morrey space $WM^{p}_{q}$ is contained in the Morrey space $M^{p}_{q_{1}}$ for $1\leq q_{1}< q\leq p<\infty$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\infty}$ for some $p\in (1,\infty)$, then $b\in \mathrm{BMO}$, where $T$ is a Calder\'on-Zygmund operator. Also, for $1

A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(\widetilde{G})$, and the negative inertia index, denoted by $n^{-}(\widetilde{G})$, of a mixed graph $\widetilde{G}$ are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of $\widetilde{G}$, respectively. In this paper, the positive and negative inertia indices of the mixed unicyclic graphs are studied. Moreover, the upper and lower bounds of the positive and negative inertia indices of the mixed graphs are investigated, and the mixed graphs which attain the upper and lower bounds are characterized respectively.

In this paper, a linearized energy stable numerical scheme is used to solve the modified CahnHilliard-Navier-Stokes model, which is a phase-field model for two-phase incompressible flows. The time discretization is based on the convex splitting of the energy functional, which leads to a linearized system. In order to maintain the energy stability, the definition domain of energy function is extended to infinity. The stability of the scheme is proved and the error estimate is given. Numerical experiments are done to demonstrate the effectiveness for the proposed scheme.

This paper is concerned with the nonconstant positive steady states of the Shigesada-KawasakiTeramoto model for two competing species with double cross-diffusion terms. By applying the Lyapunov-Schmidt decomposition method, the higher order expansion and some detailed spectral analysis, we prove the existence, asymptotic behavior and spectral instability of non-trivial steady states in high-dimensional case when one of the two cross-diffusion coefficients is large enough.

Let $p, q$ be two positive integers. The 3-graph $F(p,q)$ is obtained from the complete 3-graph $K_p^3$ by adding $q$ new vertices and $p \binom{q}{2}$ new edges of the form $vxy$ for which $v\in V(K_p^3)$ and $\{x,y\}$ are new vertices. It frequently appears in many literatures on the Turán number or Turán density of hypergraphs. In this paper, we first construct a new class of $r$-graphs which can be regarded as a generalization of the 3-graph $F(p, q)$, and prove that these $r$-graphs have the same Turán density under some situations. Moreover, we investigate the Turán density of the $F(p,q)$ for small $p,q$ and obtain some new bounds on their Turán densities.

In this paper, we study the Cauchy problem of the 2D incompressible magnetohydrodynamic equations in Lei-Lin space. The global well-posedness of a strong solution in the Lei-Lin space $\chi^{-1}(\mathbb{R}^2)$ with any initial data in $\chi^{-1}(\mathbb{R}^2)\cap L^2(\mathbb{R}^2)$ is established. Furthermore, the uniqueness of the strong solution in $\chi^{-1}(\mathbb{R}^2)$ and the Leray-Hopf weak solution in $L^2(\mathbb{R}^2)$ is proved.

Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov\citeup{0007} proposed the $A_{\alpha}$-matrix:$A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where $\alpha\in[0, 1]$. The largest eigenvalue of this novel matrix is called the $A_\alpha$-index of $G$. In this paper, we characterize the graphs with minimum $A_\alpha$-index among $n$-vertex graphs with independence number $i$ for $\alpha\in[0,1)$, where $i=1, \ \lfloor\frac{n}{2}\rfloor,\lceil\frac{n}{2}\rceil,{\lfloor\frac{n}{2}\rfloor+1},n-3,n-2,n-1,$ whereas for $i=2$ we consider the same problem for $\alpha\in[0,\frac{3}{4}{]}.$ Furthermore, we determine the unique graph (resp. tree) on $n$ vertices with given independence number having the maximum $A_\alpha$-index with $\alpha\in[0,1)$, whereas for the $n$-vertex bipartite graphs with given independence number, we characterize the unique graph having the maximum $A_\alpha$-index with $\alpha\in[\frac{1}{2},1).$}

In this paper, we are concerned with a predator-prey model with Holling type II functional response and Allee effect in predator. We first mathematically explore how the Allee effect affects the existence and stability of the positive equilibrium for the system without diffusion. The explicit dependent condition of the existence of the positive equilibrium on the strength of Allee effect is determined. It has been shown that there exist two positive equilibria for some modulate strength of Allee effect. The influence of the strength of the Allee effect on the stability of the coexistence equilibrium corresponding to high predator biomass is completely investigated and the analytically critical values of Hopf bifurcations are theoretically determined. We have shown that there exists stability switches induced by Allee effect. Finally, the diffusion-driven Turing instability, which can not occur for the original system without Allee effect in predator, is explored, and it has been shown that there exists diffusion-driven Turing instability for the case when predator spread slower than prey because of the existence of Allee effect in predator.

We study the following quasilinear Schrödinger equation \begin{equation*}\label{for1f} -\Delta u+V(x)u-\Delta (u^2)u=K(x)g(u), \qquad x\in \mathbb{R}^3, \end{equation*} where the nonlinearity $g(u)$ is asymptotically cubic at infinity, the potential $V(x)$ may vanish at infinity. Under appropriate assumptions on $K(x)$, we establish the existence of a nontrivial solution by using the mountain pass theorem.