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.

In this paper, we study a class of time-periodic population model with dispersal. It is well known that the existence of the periodic traveling fronts has been established. However, the uniqueness and stability of such fronts remain unsolved. In this paper, we first prove the uniqueness of non-critical periodic traveling fronts. Then, we show that all non-critical periodic traveling fronts are exponentially asymptotically stable.

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.

This article proves the existence and uniqueness conditions of the solution of two-dimensional time-space tempered fractional diffusion-wave equation. We find analytical solution of the equation via the two-step Adomian decomposition method (TSADM). The existence result is obtained with the help of some fixed point theorems, while the uniqueness of the solution is a consequence of the Banach contraction principle. Additionally, we study the stability via the Ulam-Hyers stability for the considered problem. The existing techniques use numerical algorithms for solving the two-dimensional time-space tempered fractional diffusion-wave equation, and thus, the results obtained from them are the approximate solution of the problem with high computational and time complexity. In comparison, our proposed method eliminates all the difficulties arising from numerical methods and gives an analytical solution with a straightforward process in just one iteration.

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, 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.

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.

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 $\mathcal{R}_{c}>1$.

We couple together existing ideas, existing results, special structure and novel ideas to accomplish the exact limits and improved decay estimates with sharp rates for all order derivatives of the global weak solutions of the Cauchy problem for an $n$-dimensional incompressible Navier-Stokes equations. We also use the global smooth solution of the corresponding heat equation to approximate the global weak solutions of the incompressible Navier-Stokes equations.

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.

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.

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 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 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.

In this paper, a stochastic SEITR model is formulated to describe the transmission dynamics of tuberculosis with incompletely treatment. Sufficient conditions for the existence of a stationary distribution and extinction are obtained. In addition, numerical simulations are given to illustrate these analytical results. Theoretical and numerical results show that large environmental perturbations can inhibit the spread of tuberculosis.

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 obtain the thickness for some complete $k-$partite graphs for $k=2,3.$ We first compute the thickness of $K_{n,n+8}$ by giving a planar decomposition of $K_{4k-1,4k+7}$ for $k\geq 3$. Then, two planar decompositions for $K_{1,g,g(g-1)}$ when $g$ is even and for $K_{1,g,\frac{1}{2}(g-1)^2}$ when $g$ is odd are obtained. Using a recursive construction, we also obtain the thickness for some complete tripartite graphs. The results here support the long-standing conjecture that the thickness of $K_{m,n}$ is $\big\lceil\frac{mn}{2(m+n-2)}\big\rceil$ for any positive integers $ m, n$.

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.

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.

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.

As an extension of linear regression in functional data analysis, functional linear regression has been studied by many researchers and applied in various fields. However, in many cases, data is collected sequentially over time, for example the financial series, so it is necessary to consider the autocorrelated structure of errors in functional regression background. To this end, this paper considers a multiple functional linear model with autoregressive errors. Based on the functional principal component analysis, we apply the least square procedure to estimate the functional coefficients and autoregression coefficients. Under some regular conditions, we establish the asymptotic properties of the proposed estimators. A simulation study is conducted to investigate the finite sample performance of our estimators. A real example on China's weather data is applied to illustrate the validity of our model.

In this paper, the diffusive nutrient-microorganism model subject to Neumann boundary conditions is considered. The Hopf bifurcations and steady state bifurcations which bifurcate from the positive constant equilibrium of the system are investigated in details. In addition, the formulae to determine the direction of Hopf and steady state bifurcations are derived. Our results show the existence of spatially homogeneous/nonhomogeneous periodic orbits and steady state solutions, which indicates the spatiotemporal dynamics of the system. Some numerical simulations are also presented to support the analytical results.

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.

The paper deals with a Cauchy problem for the chemotaxis system with the effect of fluid \begin{eqnarray*}\label{1.2} \left\{ \begin{array}{ll} u_t^{\epsilon}+u^{\epsilon}\cdot\nabla u^{\epsilon}-\Delta u^{\epsilon}+\nabla\mathbf{P}^{\epsilon}= n^{\epsilon}\nabla c^{\epsilon}, \ \ \ &{\rm in}\ \mathbb{R}^{d}\times (0, \infty), \\[6pt] \nabla\cdot u^{\epsilon}=0, \ \ \ &{\rm in}\ \mathbb{R}^{d}\times (0, \infty), \\[6pt] n_t^{\epsilon}+u^{\epsilon}\cdot\nabla n^{\epsilon}-\Delta n^{\epsilon}=-\nabla\cdot(n^{\epsilon}\nabla c^{\epsilon}), &{\rm in}\ \mathbb{R}^{d}\times (0, \infty),\\[6pt] \frac{1}{\epsilon}c_t^{\epsilon}-\Delta c^{\epsilon}= n^{\epsilon}, &{\rm in}\ \mathbb{R}^{d}\times (0, \infty),\\[6pt] (u^{\epsilon}, n^{\epsilon}, c^{\epsilon})|_{t=0}= (u_{0}, n_{0}, c_{0}), &{\rm in}\ \mathbb{R}^{d},\\[6pt] \end{array} \right. \end{eqnarray*} where $d\geq2$. It is known that for each $\epsilon>0$ and all sufficiently small initial data $(u_{0},n_{0},c_{0})$ belongs to certain Fourier space, the problem possesses a unique global solution $(u^{\epsilon},n^{\epsilon},c^{\epsilon})$ in Fourier space. The present work asserts that these solutions stabilize to $(u^{\infty},n^{\infty},c^{\infty})$ as $\epsilon^{-1}\rightarrow 0$. Moreover, we show that $c^{\epsilon}(t)$ has the initial layer as $\epsilon^{-1}\rightarrow 0$. As one expects its limit behavior maybe give a new viewlook to understand the system.

In this work, a novel time-stepping $\overline{L1}$ formula is developed for a hidden-memory variable-order Caputo's fractional derivative with an initial singularity. This formula can obtain second-order accuracy and an error estimate is analyzed strictly. As an application, a fully discrete difference scheme is established for the initial-boundary value problem of a hidden-memory variable-order time fractional diffusion model. Numerical experiments are provided to support our theoretical results.

Kawasaki disease (KD) is an acute, febrile, systemic vasculitis that mainly affects children under five years of age. In this paper, we propose and study a class of 5-dimensional ordinary differential equation model describing the vascular endothelial cell injury in the lesion area of KD. This model exhibits forward/backward bifurcation. It is shown that the vascular injury-free equilibrium is locally asymptotically stable if the basic reproduction number $R_{0}<1$. Further, we obtain two types of sufficient conditions for the global asymptotic stability of the vascular injury-free equilibrium, which can be applied to both the forward and backward bifurcation cases. In addition, the local and global asymptotic stability of the vascular injury equilibria and the presence of Hopf bifurcation are studied. It is also shown that the model is permanent if the basic reproduction number $R_{0}>1$, and some explicit analytic expressions of ultimate lower bounds of the solutions of the model are given. Our results suggest that the control of vascular injury in the lesion area of KD is not only correlated with the basic reproduction number $R_0$, but also with the growth rate of normal vascular endothelial cells promoted by the vascular endothelial growth factor.

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.

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.

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.

This research paper addresses a topic of interest to many researchers and engineers due to its effective applications in various industrial areas. It focuses on the thermoelastic laminated beam model with nonlinear structural damping, nonlinear time-varying delay, and microtemperature effects. Our primary goal is to establish the stability of the solution. To achieve this, and under suitable hypotheses, we demonstrate energy decay and construct a Lyapunov functional that leads to our results.

The multicolor Ramsey number $r_k(C_4)$ is the smallest integer $N$ such that any $k$-edge coloring of $K_N$ contains a monochromatic $C_4$. The current best upper bound of $r_k(C_4)$ was obtained by Chung (1974) and independently by Irving (1974), i.e., $r_k(C_4)\le k^2+k+1$ for all $k\ge2$. There is no progress on the upper bound since then. In this paper, we improve the upper bound of $r_k(C_4)$ by showing that $r_k(C_4)\le k^2+k-1$ for even $k\ge 6$. The improvement is based on the upper bound of the Turán number $\mathrm{ex}(n,C_4)$, in which we mainly use the double counting method and many novel ideas from Firke, Kosek, Nash, and Williford [J. Combin. Theory, Ser. B 103 (2013), 327-336].

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.

Let $X=\{ X(t), t\in $$\mathbb{R}_{+}$} be a centered space anisotropic Gaussian process values in $\mathbb{R}^d$ with non-stationary increments, whose components are independent but may not be identically distributed. Under certain conditions, then almost surely $c_1 \leq \phi-m(X([0,1])) \leq c_2$, where $\phi$ denotes the exact Hausdorff measure associated with function $ \phi(s)= s^{\frac{ 1}{α_k} +\sum\limits_{i=1}^k(1- \frac{α_i}{α_k})} \log\log\frac{1}{s}$ for some $1\leq k\leq d$, $ (α_1, \cdots, α_d)\in (0,1]^d$. We also obtain the exact Hausdorff measure of the graph of $X$ on $[0,1]$.

Let $G$ be a graph. We use $\chi(G)$ and $\omega(G)$ to denote the chromatic number and clique number of $G$ respectively. A $P_5$ is a path on 5 vertices, and an HVN is a $K_4$ together with one more vertex which is adjacent to exactly two vertices of $K_4$. Combining with some known result, in this paper we show that if $G$ is $(P_5, \textit{HVN})$-free, then $\chi(G)\leq \max\{\min\{16, \omega(G)+3\}, \omega(G)+1\}$. This upper bound is almost sharp.

Follow-up experimental designs are widely applied to explore the relationship between factors and responses step by step in various fields such as science and engineering. When some additional resources or information become available after the initial design of experiment is carried out, some additional runs and/or factors may be added in the follow-up stage. In this paper, the issue of the uniform row augmented designs and column augmented designs with mixed two-, three- and four-level is investigated. The uniformity of augmented designs is discussed under the wrap-around $L_2$-discrepancy. Some lower bounds of wrap-around $L_2$-discrepancy for the augmented designs are obtained, which can be used to assess uniformity of augmented design. Numerical results show that augmented designs have high efficiency, which have low discrepancy and close to the proposed lower bounds.

This paper is devoted to solving a reflected backward stochastic differential equation (BSDE in short) with one continuous barrier and a quasi-linear growth generator $g$, which has a linear growth in $(y,z)$, except the upper direction in case of $y<0$, and is more general than the usual linear growth generator. By showing the convergence of a penalization scheme we prove existence and comparison theorem of the minimal $L^p\ (p>1)$ solutions for the reflected BSDEs. We also prove that the minimal $L^p$ solution can be approximated by a sequence of $L^p$ solutions of certain reflected BSDEs with Lipschitz generators.

In this paper, we study mulit-dimensional oblique reflected backward stochastic differential equations (RBSDEs) in a more general framework over finite or infinite time horizon, corresponding to the pricing problem for a type of real option. We prove that the equation can be solved uniquely in $L^p(1 < p \leq 2)$-space, when the generators are uniformly continuous but each component taking values independently. Furthermore, if the generator of this equation fulfills the infinite time version of Lipschitzian continuity, we can also conclude that the solution to the oblique RBSDE exists and is unique, despite the fact that the values of some generator components may affect one another.

Interior-point methods (IPMs) for linear programming (LP) are generally based on the logarithmic barrier function. Peng et al. (J. Comput. Technol. 6: 61-80, 2001) were the first to propose non-logarithmic kernel functions (KFs) for solving IPMs. These KFs are strongly convex and smoothly coercive on their domains. Later, Bai et al. (SIAM J. Optim. 15(1): 101-128, 2004) introduced the first KF with a trigonometric barrier term. Since then, no new type of KFs were proposed until 2020, when Touil and Chikouche (Filomat. 34(12): 3957-3969, 2020; Acta Math. Sin. (Engl. Ser.), 38(1): 44-67, 2022) introduced the first hyperbolic KFs for semidefinite programming (SDP). They established that the iteration complexities of algorithms based on their proposed KFs are $\mathcal{O}\left(n^{\frac{2}{3}}\log \frac{n}{\epsilon }\right)$ and $\mathcal{O}\left(n^{\frac{3}{4}}\log \frac{n}{\epsilon }\right)$ for large-update methods, respectively. The aim of this work is to improve the complexity result for large-update method. In fact, we present a new parametric KF with a hyperbolic barrier term. By simple tools, we show that the worst-case iteration complexity of our algorithm for the large-update method is $\mathcal{O}\left(\sqrt{n}\log n\log \frac{n}{\epsilon }\right)$ iterations. This coincides with the currently best-known iteration bounds for IPMs based on all existing kind of KFs.
The algorithm based on the proposed KF has been tested. Extensive numerical simulations on test problems with different sizes have shown that this KF has promising results.