Tensor data have been widely used in many fields, e.g., modern biomedical imaging, chemometrics, and economics, but often suffer from some common issues as in high dimensional statistics. How to find their low-dimensional latent structure has been of great interest for statisticians. To this end, we develop two efficient tensor sufficient dimension reduction methods based on the sliced average variance estimation (SAVE) to estimate the corresponding dimension reduction subspaces. The first one, entitled tensor sliced average variance estimation (TSAVE), works well when the response is discrete or takes finite values, but is not $\sqrt{n}$ consistent for continuous response; the second one, named bias-correction tensor sliced average variance estimation (CTSAVE), is a de-biased version of the TSAVE method. The asymptotic properties of both methods are derived under mild conditions. Simulations and real data examples are also provided to show the superiority of the efficiency of the developed methods.

This paper studies the global phase portraits of uniform isochronous centers system of degree six with polynomial commutator. Such systems have the form $\dot{x}=-y+xf(x,y),\ \dot{y}=x+yf(x,y)$, where $f(x,y)=a_{1}x+a_{2}xy+a_{3}xy^{2}+a_{4}xy^{3}+a_{5}xy^4=x\sigma(y)$, and any zero of $1+a_{1}y+a_{2}y^2+a_{3}y^{3}+a_{4}y^{4}+a_{5}y^{5}$, $y=\overline{y}$ is an invariant straight line. At last, all global phase portraits are drawn on the Poincarédisk.

Let $\mathbf{G}=\{G_i: i\in[n]\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$, where $\mathbf{G}$ can be seen as an edge-colored (multi)graph and each $G_i$ is the set of edges with color $i$. A graph $F$ on $V$ is called rainbow if any two edges of $F$ come from different $G_i$s'. We say that $\mathbf{G}$ is rainbow pancyclic if there is a rainbow cycle $C_{\ell}$ of length $\ell$ in $\mathbf{G}$ for each integer $\ell\in [3,n]$. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem: If $\delta(G_i)\geq\frac{n}{2}$ for each $i\in[n]$, then there is a rainbow Hamiltonian cycle in $\mathbf{G}$. In this paper, under the same condition, we show that $\mathbf{G}$ is rainbow pancyclic except that $n$ is even and $\mathbf{G}$ consists of $n$ copies of $K_{\frac{n}{2},\frac{n}{2}}$. This result supports the famous meta-conjecture posed by Bondy.

The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper. We first show that there exist both continuous and discontinuous stationary solutions. Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition. In addition, we demonstrate the influences of the diffusion coefficient on stationary solutions. The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem. Finally, some numerical simulations are given to illustrate the theoretical results.

Given two graphs $G$ and $H$, the Ramsey number $R(G,H)$ is the minimum integer $N$ such that any two-coloring of the edges of $K_{N}$ in red or blue yields a red $G$ or a blue $H$. Let $v(G)$ be the number of vertices of $G$ and $\chi(G)$ be the chromatic number of $G$. Let $s(G)$ denote the chromatic surplus of $G$, the number of vertices in a minimum color class among all proper $\chi(G)$-colorings of $G$. Burr showed that $R(G,H)\geq (v(G)-1)(\chi(H)-1)+s(H)$ if $G$ is connected and $v(G)\geq s(H)$. A connected graph $G$ is $H$-good if $R(G,H)=(v(G)-1)(\chi(H)-1)+s(H)$. %Ramsey goodness is a special property of graph. Let $tH$ denote the disjoint union of $t$ copies of graph $H$, and let $G\vee H$ denote the join of $G$ and $H$. Denote a complete graph on $n$ vertices by $K_n$, and a tree on $n$ vertices by $T_n$. Denote a book with $n$ pages by $B_n$, i.e., the join $K_2\vee \overline{K_n}$. Erdös, Faudree, Rousseau and Schelp proved that $T_n$ is $B_m$-good if $n\geq 3m-3$. In this paper, we obtain the exact Ramsey number of $T_n$ versus $2B_2$. Our result implies that $T_n$ is $2B_2$-good if $n\geq5$.

In this paper, the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description, numerical simulation and theoretical analysis. Two important factors, heat and magnetic influences are involved. The mathematical model is formulated by four nonlinear partial differential equations (PDEs), determining four major physical variables. The influences of magnetic fields are supposed to be weak, and the strength is parallel to the $z$-axis. The elliptic equation is treated by a block-centered method, and the law of conservation is preserved. The computational accuracy is improved one order. Other equations are convection-dominated, thus are approximated by upwind block-centered differences. Upwind difference can eliminate numerical dispersion and nonphysical oscillation. The diffusion is approximated by the block-centered difference, while the convection term is treated by upwind approximation. Furthermore, the unknowns and adjoint functions are computed at the same time. These characters play important roles in numerical computations of conductor device problems. Using the theories of priori analysis such as energy estimates, the principle of duality and mathematical inductions, an optimal estimates result is obtained. Then a composite numerical method is shown for solving this problem.

In this paper, we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject. The $k$-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data, and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered. In addition, we also consider two other clustering methods, $k$-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis. Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index. The approaches are further illustrated through empirical analysis of human mortality data.

Let $a$ and $b$ be positive integers such that $a\leq b$ and $a\equiv b\pmod 2$. We say that $G$ has all $(a, b)$-parity factors if $G$ has an $h$-factor for every function $h: V(G) \rightarrow \{a,a+2,\cdots,b-2,b\}$ with $b|V(G)|$ even and $h(v)\equiv b\pmod 2$ for all $v\in V(G)$. In this paper, we prove that every graph $G$ with $n\geq 2(b+1)(a+b)$ vertices has all $(a,b)$-parity factors if $\delta(G)\geq (b^2-b)/a$, and for any two nonadjacent vertices $u,v \in V(G)$, $\max\{d_G(u),d_G(v)\}\geq \frac{bn}{a+b}$. Moreover, we show that this result is best possible in some sense.

Inspired by the transmission characteristics of the coronavirus disease 2019 (COVID-19), an epidemic model with quarantine and standard incidence rate is first developed, then a novel analysis approach is proposed for finding the ultimate lower bound of the number of infected individuals, which means that the epidemic is uniformly persistent if the control reproduction number $\mathcal{R}_{c}>1$. This approach can be applied to the related biomathematical models, and some existing works can be improved by using that. In addition, the infection-free equilibrium $V^0$ of the model is locally asymptotically stable (LAS) if $\mathcal{R}_{c}<1$ and linearly stable if $\mathcal{R}_{c}=1$; while $V^0$ is unstable if $\mathcal{R}_{c}>1$.

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

In this article, we study strong limit theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. We establish general strong law and complete convergence theorems for weighted sums of extended negatively dependent random variables under the sub-linear expectations. Our results of strong limit theorems are more general than some related results previously obtained by Thrum (1987), Li et al. (1995) and Wu (2010) in classical probability space.

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\cdots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N, N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $1\leq i\leq r$. We show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum\limits^{j+1}_{i=1}\alpha_i$ for $j=1,2,\cdots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\cdots,C_{2\lfloor \alpha_r n\rfloor})=\big(\sum\limits^r_{j=1} \alpha_j+o(1)\big)n.$

This paper considers the random coefficient autoregressive model with time-functional variance noises, hereafter the RCA-TFV model. We first establish the consistency and asymptotic normality of the conditional least squares estimator for the constant coefficient. The semiparametric least squares estimator for the variance of the random coefficient and the nonparametric estimator for the variance function are constructed, and their asymptotic results are reported. A simulation study is presented along with an analysis of real data to assess the performance of our method in finite samples.

This paper considers a multi-state repairable system that is composed of two classes of components, one of which has a priority for repair. First, we investigate the well-posedenss of the system by applying the operator semigroup theory. Then, using Greiner's idea and the spectral properties of the corresponding operator, we obtain that the time-dependent solution of the system converges strongly to its steady-state solution.

The classical Ambarzumyan's theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator $-\frac{d^2}{dx^2}+q$ with an integrable real-valued potential $q$ on $[0,\pi]$ are $\{n^2:n\geq 0\}$, then $q=0$ for almost all $x\in [0,\pi]$. In this work, the classical Ambarzumyan's theorem is extended to the Dirac operator on equilateral tree graphs. We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case, then the potential is identically zero.

In this paper, we study the following quasi-linear elliptic equation: \begin{equation}\nonumber \left\{ \begin{aligned} &- {\rm div} (\phi(|\nabla u|)\nabla u)=\lambda \psi(|u|)u + \varphi(|u|)u, ~~~\text{in } \Omega,\\ & u= 0, ~~~\text {on} \partial \Omega, \end{aligned} \right. \end{equation} where $\Omega \subset \mathbb R^N$ is a bounded domain, $\lambda > 0$ is a parameter. The function $\psi(|t|)t$ is the subcritical term, and $\varphi(|t|)t$ is the critical Orlicz-Sobolev growth term with respect to $\phi$. Under appropriate conditions on $\phi$, $\psi$ and $\varphi$, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for $\lambda \in (0,\lambda_0)$, where $\lambda_0>0$ is a fixed constant.

In this paper, we investigate the following $p$-Kirchhoff equation \begin{eqnarray*} \left\{\begin{array}{ll} \big(a+b\int_{\mathbb{R}^{N}}(|\nabla u|^{p}+|u|^{p})dx\big)\big(-\Delta_{p}u+|u|^{p-2}u\big)=|u|^{s-2}u+\mu u,~x\in \mathbb{R}^{N},\\ \int_{\mathbb{R}^{N}}|u|^{2}dx=\rho, \end{array} \right. \end{eqnarray*} where $a> 0, \,b \geq 0 , \,\rho>0$ are constants, $p^{\ast}=\frac{Np}{N-p}$ is the critical Sobolev exponent, $\mu$ is a Lagrange multiplier, $-\Delta_{p}u=-{\rm div}(|\nabla u|^{p-2}\nabla u), \ 2<p<N<2p, \ \mu\in\mathbb{R}$, and $ s\in(2\frac{N+2}{N}p-2,~p^{\ast})$. We demonstrate that the $p$-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.

In this work, we mainly consider the Cauchy problem for the reverse space-time nonlocal Hirota equation with the initial data rapidly decaying in the solitonless sector. Start from the Lax pair, we first construct the basis Riemann-Hilbert problem for the reverse space-time nonlocal Hirota equation. Furthermore, using the approach of Deift-Zhou nonlinear steepest descent, the explicit long-time asymptotics for the reverse space-time nonlocal Hirota is derived. For the reverse space-time nonlocal Hirota equation, since the symmetries of its scattering matrix are different with the local Hirota equation, the $\vartheta(\lambda_{i}) \ (i=0, 1)$ would like to be imaginary, which results in the $\delta_{\lambda_{i}}^{0}$ contains an increasing $t^{\frac{\pm Im\vartheta(\lambda_{i})}{2}}$, and then the asymptotic behavior for nonlocal Hirota equation becomes differently.

The law of the iterated logarithm (LIL) for the performance measures of a two-station queueing network with arrivals modulated by independent queues is developed by a strong approximation method. For convenience, two arrival processes modulated by queues comprise the external system, all others are belong to the internal system. It is well known that the exogenous arrival has a great influence on the asymptotic variability of performance measures in queues. For the considered queueing network in heavy traffic, we get all the LILs for the queue length, workload, busy time, idle time and departure processes, and present them by some simple functions of the primitive data. The LILs tell us some interesting insights, such as, the LILs of busy and idle times are zero and they reflect a small variability around their fluid approximations, the LIL of departure has nothing to do with the arrival process, both of the two phenomena well explain the service station's situation of being busy all the time. The external system shows us a distinguishing effect on the performance measures: an underloaded (overloaded, critically loaded) external system affects the internal system through its arrival (departure, arrival and departure together). In addition, we also get the strong approximation of the network as an auxiliary result.

In this paper, we consider a system which has $k$ statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements with doubly type-II censored scheme. These elements $(X_1, Y_1 ),$ $(X_2, Y_2 ),$ $\cdots$, $(X_k, Y_k)$ follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress $T$ which follows a Kumaraswamy distribution. The system is regarded as operating only if at least $s$ out of $k$ ($1\leq s\leq k$) strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}$=$P$(at least $s$ of the $(Z_1, \cdots, Z_k)$ exceed $T)$ where $Z_i=\min(X_i, Y_i ), \ i=1,\cdots, k.$ The Bayes estimates of $R_{s,k}$ have been developed by using the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and exact Bayes estimates of $R_{s,k}$ are obtained analytically when the common second shape parameter is known. The asymptotic confidence interval and the highest probability density credible interval are constructed for $R_{s,k}$. The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation $$ u_{t}(x,t)=\Delta u(x,t) +au(x,t)\ln u(x,t)+ bu^{\alpha}(x,t), $$ on $\mathbf{M}\times (-\infty, \infty)$ with $\alpha\in\mathbf{R}$, where $a$ and $b$ are constants. As application, the Harnack inequalities are derived.

For the functional partially linear models including flexible nonparametric part and functional linear part, the estimators of the nonlinear function and the slope function have been studied in existing literature. How to test the correlation between response and explanatory variables, however, still seems to be missing. Therefore, a test procedure for testing the linearity in the functional partially linear models will be proposed in this paper. A test statistic is constructed based on the existing estimators of the nonlinear and the slope functions. Further, we prove that the approximately asymptotic distribution of the proposed statistic is a chi-squared distribution under some regularity conditions. Finally, some simulation studies and a real data application are presented to demonstrate the performance of the proposed test statistic.

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

In this paper, we analyze the relationship between the equilibrium reinsurance strategy and the tail of the distribution of the risk. Since Mean Residual Life (MRL) has a close relationship with the tail of the distribution, we consider two classes of risk distributions, Decreasing Mean Residual Life (DMRL) and Increasing Mean Residual Life (IMRL) distributions, which can be used to classify light-tailed and heavy-tailed distributions, respectively. We assume that the underlying risk process is modelled by the classical Cramér-Lundberg model process. Under the mean-variance criterion, by solving the extended Hamilton-Jacobi-Bellman equation, we derive the equilibrium reinsurance strategy for the insurer and the reinsurer under DMRL and IMRL, respectively. Furthermore, we analyze how to choose the reinsurance premium to make the insurer and the reinsurer agree with the same reinsurance strategy. We find that under the case of DMRL, if the distribution and the risk aversions satisfy certain conditions, the insurer and the reinsurer can adopt a reinsurance premium to agree on a reinsurance strategy, and under the case of IMRL, the insurer and the reinsurer can only agree with each other that the insurer do not purchase the reinsurance.

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

In this paper, we study the optimal timing to convert the risk of business for an insurance company in order to improve its solvency. The cash flow of company evolves according to a jump-diffusion process. Business conversion option offers the company an opportunity to transfer the jump risk business out. In exchange for this option, the company needs to pay both fixed and proportional transaction costs. The proportional cost can also be seen as the profit loading of the jump risk business. We formulated this problem as an optimal stopping problem. By solving this stopping problem, we find that the optimal timing of business conversion mainly depends on the profit loading of the jump risk business. A larger profit loading would make the conversion option valueless. The fixed cost, however, only delays the optimal timing of business conversion. In the end, numerical results are provided to illustrate the impacts of transaction costs and environmental parameters to the optimal strategies.

This paper is concerned with the valuation of single and double barrier knock-out call options in a Markovian regime switching model with specific rebates. The integral formulas of the rebates are derived via matrix Wiener-Hopf factorizations and Fourier transform techniques, also, the integral representations of the option prices are constructed. Moreover, the first-passage time density functions in two-state regime model are derived. As applications, several numerical algorithms and numerical examples are presented.

Considering that HBV belongs to the DNA virus family and is hepatotropic, we model the HBV DNA-containing capsids as a compartment. In this paper, a delayed HBV infection model is established, where the general incidence function and two infection routes including cell-virus infection and cell-cell infection are introduced. According to some preliminaries, including well-posedness, basic reproduction number and existence of two equilibria, we obtain the threshold dynamics for the model. We illustrate numerical simulations to verify the above theoretical results, and furthermore explore the impacts of intracellular delay and cell-cell infection on the global dynamics of the model.

The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color. An outerplane graph with independent crossings is a graph embedded in the plane in such a way that all vertices are on the outer face and two pairs of crossing edges share no common end vertex. It is proved that every outerplane graph with independent crossings and maximum degree $\Delta$ has strong chromatic index at most $4\Delta-6$ if $\Delta\geq 4$, and at most 8 if $\Delta\leq 3$. Both bounds are sharp.

This article analyzes the Pareto optimal allocations, agreeable trades and agreeable bets under the maxmin Choquet expected utility (MCEU) model. We provide several useful characterizations for Pareto optimal allocations for risk averse agents. We derive the formulation descriptions for non-existence agreeable trades or agreeable bets for risk neutral agents. We build some relationships between ex-ante stage and interim stage on agreeable trades or bets when new information arrives.

Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number $b(G)$ of a graph $G$, is defined as the smallest integer $k$ for which there are vertices $x_1,\cdots, x_k$ such that for every vertex $u$ of $G$, there exists $i\in \{1,\cdots, k\}$ with $d_G(u, x_i)\le k-i$, and $d_G(x_i, x_j) \ge j-i$ for any $1\le i<j \le k$. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.

Let $G$ be a finite group generated by $S$ and $C(G,S)$ the Cayley digraphs of $G$ with connection set $S$. In this paper, we give some sufficient conditions for the existence of hamiltonian circuit in $C(G,S)$, where $G=Z_m\rtimes H$ is a semiproduct of $Z_m$ by a subgroup $H$ of $G$. In particular, if $m$ is a prime, then the Cayley digraph of $G$ has a hamiltonian circuit unless $G=Z_m\times H$. In addition, we introduce a new digraph operation, called $\varphi$-semiproduct of $\Gamma_1$ by $\Gamma_2$ and denoted by $\Gamma_1 \rtimes_\varphi\Gamma_2$, in terms of mapping $\varphi:V(\Gamma_2)\longrightarrow\{1,-1\}$. Furthermore we prove that $C(Z_m, \{a\}) \rtimes_{\varphi}C(H,S)$ is also a Cayley digraph if $\varphi$ is a homomorphism from $H$ to $\{1,-1\}\le Z_m^*$, which produces some classes of Cayley digraphs that have hamiltonian circuits.

Aiming at the problem that the asset's fluctuation influences the borrower's repayment ability, a loan with a new and flexible repayment method is designed, which depends on the asset value of the borrower. The repayment method can reduce the loan default probability, but it causes the uncertainty of the pay off time. Because the repayment term is related to the regular repayment amount in this method, a boundary for the regular repayment amount is set up in order to avoid too long repayment term. This will balance the benefit of borrowers and lenders and improve the applicability of this method. By establishing a mathematical model of the residual value of the loan, this model can be transformed into an initial-boundary problem of a partial differential equation. The analytic solution and the expected time to pay off the loan are obtained. Finally, numerical analysis are shown.

A scheme for generating nonisospectral integrable hierarchies is introduced. Based on the method, we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem. It follows that the corresponding expanded isospectral and nonisospectral integrable hierarchies are deduced based on a 6 dimensional complex linear space $\widetilde{\mathbb{C}}^6$. By reducing these integrable hierarchies, we obtain the expanded isospectral and nonisospectral derivative nonlinear Schrödinger equation. By using the trace identity, the bi-Hamiltonian structure of these two hierarchies are also obtained. Moreover, some symmetries and conserved quantities of the resulting hierarchy are discussed.

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

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

The Gerdjikov-Ivanov (GI) hierarchy is derived via recursion operator, in this article, we mainly investigate the third-order flow GI equation. In the framework of the Riemann-Hilbert method, the soliton matrices of the third-order flow GI equation with simple zeros and elementary high-order zeros of Riemann-Hilbert problem are constructed through the standard dressing process. Taking advantage of this result, some properties and asymptotic analysis of single soliton solution and two soliton solution are discussed, and the simple elastic interaction of two soliton are proved. Compared with soliton solution of the classical second-order flow, we find that the higher-order dispersion term affects the propagation velocity, propagation direction and amplitude of the soliton. Finally, by means of a certain limit technique, the high-order soliton solution matrix for the third-order flow GI equation is derived.

This paper concerns a global optimality principle for fully coupled mean-field control systems. Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear relation is introduced, with which we successfully decouple the fully coupled first-order variational equations. We give a new second-order expansion of $Y^\varepsilon$ that can work well in mean-field framework. Based on this result, the stochastic maximum principle is proved. The comparison with the stochastic maximum principle for controlled mean-field stochastic differential equations is supplied.

For a 2-station and 3-class reentrant line under first-buffer first-served (FBFS) service discipline in light traffic, we firstly construct the strong approximations for performance measures including the queue length, workload, busy time and idle time processes. Based on the obtained strong approximations, we use a strong approximation method to find all the law of the iterated logarithms (LILs) for the above four performance measures, which are expressed as some functions of system parameters: means and variances of interarrival and service times, and characterize the fluctuations around their fluid approximations.

A graph is outer-1-planar if it can be drawn in the plane so that all vertices lie on the outer-face and each edge crosses at most one another edge. It is known that every outer-1-planar graph is a planar partial 3-tree. In this paper, we conjecture that every planar graph $G$ has a proper incidence $(\Delta(G)+2)$-coloring and confirm it for outer-1-planar graphs with maximum degree at least $8$ or with girth at least $4$. Specifically, we prove that every outer-$1$-planar graph $G$ has an incidence $(\Delta(G)+3,2)$-coloring, and every outer-$1$-planar graph $G$ with maximum degree at least $8$ or with girth at least $4$ has an incidence $(\Delta(G)+2,2)$-coloring.