In this short note we present a new Harnack expression for the Gaussian curvature flow, which is modeled from the shrinking self similiar solutions. As applications we give alternate proofs of Chow's Harnack inequality and entropy estimate.

This paper is concerned with an optimal model averaging estimation for linear regression model with right censored data. The weights for model averaging are picked up via minimizing the Mallows criterion. Under some mild conditions, it is shown that the identified weights possess the property of asymptotic optimality, that is, the model averaging estimator corresponding to these weights achieves the lowest squared error asymptotically. Some numerical studies are conducted to evaluate the finite-sample performance of our method and make comparisons with its intuitive competitors, while an application to the PBC dataset is provided to serve as an illustration.

Space-filling designs are widely used in various fields because of their nice space-filling properties. Uniform designs are one of space-filling designs, which desires the experimental points to scatter uniformly over the experimental area. For practical need, the construction and their properties of nine-level uniform designs are discussed via two code mappings in this paper. Firstly, the algorithm of constructing nine-level uniform designs is presented from an initial three-level design by the Type-I code mapping and tripling technique. Secondly, the algorithm of constructing nine-level uniform designs is presented from a three-level base design by the Type-II code mapping and generalized orthogonal arrays. Moreover, relative properties are discussed based on the two code mappings. Finally, some numerical examples are given out for supporting our theoretical results.

A matching is extendable in a graph G if G has a perfect matching containing it. A distance q matching is a matching such that the distance between any two distinct matching edges is at least q. In this paper, we prove that any distance 2k-3 matching is extendable in a connected and locally (k-1)-connected K_{1, k}-free graph of even order. Furthermore, we also prove that any distance q matching M in an r-connected and locally (k-1)-connected K_{1, k}-free graph of even order is extendable provided that M is bounded by a function on r, k and q. Our results improve some results in [J. Graph Theory 93 (2020), 5-C20].

In this paper, we introduce for the first time a new eligible kernel function with a hyperbolic barrier term for semidefinite programming (SDP). This add a new type of functions to the class of eligible kernel functions. We prove that the interior-point algorithm based on the new kernel function meets O(n^3/4 log n/ε) iterations as the worst case complexity bound for the large-update method. This coincides with the complexity bound obtained by the first kernel function with a trigonometric barrier term proposed by El Ghami et al. in 2012, and improves with a factor n^1/4 the obtained iteration bound based on the classic kernel function. We present some numerical simulations which show the effectiveness of the algorithm developed in this paper.

The effectiveness of this paper lies in the influence of the discretization step on the asymptotic stability of the positive two-dimensional fractional linear systems. It aims at investigating whether, how and when this step affects the asymptotically stable two-dimensional positive fractional linear continuous-discrete systems. To accomplish this study, a new test was outlined and used so that the asymptotic stability of the system was measured both before and after being exposed to the sampling step. Furthermore, the conditions of that stability were assessed. As a result, the outcome of the approximation shows that the stability is preserved under a particular set of conditions. On this basis, the newly proposed approach is recommended for testing the intended stability of such systems. A numerical example is tested to show the accuracy and the applicability of the proposed tests.

Several tests for multivariate mean vector have been proposed in the recent literature. Generally, these tests are directly concerned with the mean vector of a high-dimensional distribution. The paper presents two new test procedures for testing mean vector in large dimension and small samples. We do not focus on the mean vector directly, which is a different framework from the existing choices. The first test procedure is based on the asymptotic distribution of the test statistic, where the dimension increases with the sample size. The second test procedure is based on the permutation distribution of the test statistic, where the sample size is fixed and the dimension grows to infinity. Simulations are carried out to examine the finite-sample performance of the tests and to compare them with some popular nonparametric tests available in the literature.

Empirical likelihood in generalized linear models with multivariate responses and working covariance matrix is discussed. Under the weakest assumption on eigenvalues of Fisher's information matrix and some other regular conditions, we prove that the non-parametric Wilk's property still holds, that is, the empirical log-likelihood ratio at the true parameter values converges to the standard chi-square distribution. Numerical simulations are given to verify our theoretical result.

In this paper, we focus on the immiscible compressible two-phase flow described by the coupled compressible Navier-Stokes system and the modified Allen-Cahn equations. The generalized Navier boundary condition and the relaxation boundary condition are established in order to solve the problem of moving contact lines on the solid boundary by using the principle of minimum energy dissipation. The existence and uniqueness for local strong solution in three dimensional bounded domain for this type of boundary value problem is obtained by the elementary energy method and the maximum principle.

In this paper, we consider the one dimensional third order p-Laplacian equation (Φ_p(u"))'+ h(t)f(t, u(t))=0 with integral boundary conditions u(0)-αu'(0)= ∫t₀¹g₁(s)u(s)ds, u(1)+β u'(1)= ∫t₀¹g₂(s)u(s)ds, u"(0)=0. By using kernel functions and the Avery-Peterson fixed point theorem, we establish the existence of at least three positive solutions.

We investigate a diffusive, stage-structured epidemic model with the maturation delay and freelymoving delay. Choosing delays and diffusive rates as bifurcation parameters, the only possible way to destabilize the endemic equilibrium is through Hopf bifurcation. The normal forms of Hopf bifurcations on the center manifold are calculated, and explicit formulae determining the criticality of bifurcations are derived. There are two different kinds of stable oscillations near the first bifurcation: on one hand, we theoretically prove that when the diffusion rate of infected immature individuals is sufficiently small or sufficiently large, the first branch of Hopf bifurcating solutions is always spatially homogeneous; on the other, fixing this diffusion rate at an appropriate size, stable oscillations with different spatial profiles are observed, and the conditions to guarantee the existence of such solutions are given by calculating the corresponding eigenfunction of the Laplacian at the first Hopf bifurcation point. These bifurcation behaviors indicate that spatial diffusion in the epidemic model may lead to spatially inhomogeneous distribution of individuals.

A graph G is said to be p-factor-critical if G-u₁-u₂-...-u_p has a perfect matching for any u₁, u₂, ..., u_p∈ V(G). The concept of p-factor-critical is a generalization of the concepts of factor-critical and bicritical for p=1 and p=2, respectively. Heping Zhang and Fuji Zhang[Construction for bicritical graphs and k-extendable bipartite graphs, Discrete Math., 306(2006) 1415–1423] gave a concise structure characterization of bicritical graphs. In this paper, we present the characterizations of p-factor-critical graphs and minimal p-factor-critical graphs for p ≥ 2. As an application, we also obtain a class of graphs which are minimal p-factor-critical for p ≥ 1.

In this paper a new class of orthogonal arrays (OAs), i.e., OAs without interaction columns, are proposed which are applicable in factor screening, interaction detection and other cases. With the tools of difference matrices, we present some general recursive methods for constructing OAs of such type. Several families of OAs with high percent saturation are constructed. In particular, for any integer λ ≥ 3, such a two-level OA of run 4λ can always be obtained if the corresponding Hadamard matrix exists.

A graph is 1-planar if it can be drawn on the Euclidean plane so that each edge is crossed by at most one other edge. A proper vertex k-coloring of a graph G is defined as a vertex coloring from a set of k colors such that no two adjacent vertices have the same color. A graph that can be assigned a proper k-coloring is k-colorable. A cycle is a path of edges and vertices wherein a vertex is reachable from itself. A cycle contains k vertices and k edges is a k-cycle. In this paper, it is proved that 1-planar graphs without 4-cycles or 5-cycles are 5-colorable.

A spanning subgraph F of a graph G is called a path factor of G if each component of F is a path. A P_≥k-factor means a path factor with each component having at least k vertices, where k≥2 is an integer. Bazgan, Benhamdine, Li and Wozniak [C. Bazgan, A. H. Benhamdine, H. Li, M. Wozniak, Partitioning vertices of 1-tough graph into paths, Theoret. Comput. Sci. 263(2001)255--261.] obtained a toughness condition for a graph to have a P_≥3-factor. We introduce the concept of a P_≥k-factor deleted graph, that is, if a graph G has a P_≥k-factor excluding e for every e∈ E(G), then we say that G is a P_≥k-factor deleted graph. In this paper, we show four sufficient conditions for a graph to be a P_≥3-factor deleted graph. Furthermore, it is shown that four results are best possible in some sense.

For a function ? satisfying some suitable growth conditions, consider the following general dispersive equation defined by{i?_tu+?(√-?)u = 0, (x, t)∈ Rⁿ×R,
u(x, 0)=f(x), f∈S(Rⁿ), (*)where ?(√-?) is a pseudo-differential operator with symbol ?(ξ). In the present paper, when the initial data f belongs to Sobolev space, we give the local and global weighted L^q estimate for the global maximal operator S_?^** defined by S_?^**f(x)=sup_t∈RS_{t, ?}f(x), whereS_{t, ?}f(x) = (2π)^-n∫_Rⁿe^{ix·ξ+it?(ξ)}f(ξ)dξis a formal solution of the equation (*).

We investigate the family of vertex-transitive graphs with diameter 2. Let Γ be such a graph. Suppose that its automorphism group is transitive on the set of ordered non-adjacent vertex pairs. Then either Γ is distance-transitive or Γ has girth at most 4. Moreover, if Γ has valency 2, then Γ ? C₄ or C₅; and for any integer n ≥ 3, there exist such graphs Γ of valency n such that its automorphism group is not transitive on the set of arcs. Also, we determine this family of graphs of valency less than 5. Finally, the family of diameter 2 circulants is characterized.

A graphic sequence π=(d₁, d₂, ..., d_n) is said to be forcibly k-edge-connected if every realization of π is k-edge-connected. In this paper, we obtain a new sufficient degree condition for π to be forcibly k-edge-connected. We also show that this new sufficient degree condition implies a strongest monotone degree condition for π to be forcibly 2-edge-connected and a conjecture about a strongest monotone degree condition for π to be forcibly 3-edge-connected due to Bauer et al. (Networks, 54(2) (2009) 95-98), and also implies a strongest monotone degree condition for π to be forcibly 4-edge-connected.

The Turán number of a k-uniform hypergraph H, denoted by ex_k (n; H), is the maximum number of edges in any k-uniform hypergraph F on n vertices which does not contain H as a subgraph. Let C_e^(k) denote the family of all k-uniform minimal cycles of length e, S(e₁, ..., e_r) denote the family of hypergraphs consisting of unions of r vertex disjoint minimal cycles of length e₁, ..., e_r, respectively, and C_e^(k) denote a k-uniform linear cycle of length e. We determine precisely ex_k(n; S(e₁, ..., e_r)) and ex_k(n; C_e1^(k), ..., C_er^(k)) for sufficiently large n. Our results extend recent results of Füredi and Jiang who determined the Turán numbers for single k-uniform minimal cycles and linear cycles.

The traditional approaches to false discovery rate (FDR) control in multiple hypothesis testing are usually based on the null distribution of a test statistic. However, all types of null distributions, including the theoretical, permutation-based and empirical ones, have some inherent drawbacks. For example, the theoretical null might fail because of improper assumptions on the sample distribution. Here, we propose a null distribution-free approach to FDR control for multiple hypothesis testing in the case-control study. This approach, named target-decoy procedure, simply builds on the ordering of tests by some statistic or score, the null distribution of which is not required to be known. Competitive decoy tests are constructed from permutations of original samples and are used to estimate the false target discoveries. We prove that this approach controls the FDR when the score function is symmetric and the scores are independent between different tests. Simulation demonstrates that it is more stable and powerful than two popular traditional approaches, even in the existence of dependency. Evaluation is also made on two real datasets, including an arabidopsis genomics dataset and a COVID-19 proteomics dataset.

This paper proposes an approximate analytical solution method to calculate counterparty credit risk exposures. Compared with the Standard Approach for measuring Counterparty Credit Risk and the Internal Modeling Method provided by Basel Committee, the proposed method significantly improves the calculation efficiency based on sacrificing a little accuracy. Taking Forward Rate Agreement as an example, this article derives the exact expression for Expected Exposure. By approximating the distribution of Forward Rate Agreement’s future value to a normal distribution, the approximate analytical expression for Potential Future Exposure is derived. Numerical results show that this method is reliable and is robust under different parameters.

A tournament is an orientation of the complete graph. Tournaments form perhaps the most interesting class of digraphs and it has a great potential for application. Tournaments provide a model of the statistical technique called the method of paired comparisons and they have also been studied in connection with sociometric relations in small groups. In this paper, we investigate disjoint cycles of the same length in tournaments. In 2010, Lichiardopol conjectured that for given integers l ≥ 3 and k ≥ 1, any tournament with minimum out-degree at least (l - 1)k - 1 contains k disjoint l-cycles, where an l-cycle is a cycle of order l. Bang-Jensen et al. verified the conjecture for l = 3 and Ma et al. proved that it also holds for l ≥ 10. This paper provides a proof of the conjecture for the case of 9 ≥ l ≥ 4

For some infectious diseases such as mumps, HBV, there is evidence showing that vaccinated individuals always lose their immunity at different rates depending on the inoculation time. In this paper, we propose an age-structured epidemic model using a step function to describe the rate at which vaccinated individuals lose immunity and reduce the age-structured epidemic model to the delay differential model. For the age-structured model, we consider the positivity, boundedness, and compactness of the semiflow and study global stability of equilibria by constructing appropriate Lyapunov functionals. Moreover, for the reduced delay differential equation model, we study the existence of the endemic equilibrium and prove the global stability of equilibria. Finally, some numerical simulations are provided to support our theoretical results and a brief discussion is given.

A graph G is called a fractional [a, b]-covered graph if for each e ∈ E(G), G contains a fractional [a,b]-factor covering e. A graph G is called a fractional (a,b,k)-critical covered graph if for any W ⊆ V (G) with |W| = k, G - W is fractional [a, b]-covered, which was first defined and investigated by Zhou, Xu and Sun [S. Zhou, Y. Xu, Z. Sun, Degree conditions for fractional (a,b,k)-critical covered graphs, Information Processing Letters 152(2019)105838]. In this work, we proceed to study fractional (a,b,k)-critical covered graphs and derive a result on fractional (a,b,k)-critical covered graphs depending on minimum degree and neighborhoods of independent sets.

As biological studies become more expensive to conduct, it is a frequently encountered question that how to take advantage of the available auxiliary covariate information when the exposure variable is not measured. In this paper, we propose an induced cure rate mean residual life time regression model to accommodate the survival data with cure fraction and auxiliary covariate, in which the exposure variable is only assessed in a validation set, but a corresponding continuous auxiliary covariate is ascertained for all subjects in the study cohort. Simulation studies elucidate the practical performance of the proposed method under finite samples. As an illustration, we apply the proposed method to a heart disease data from the Study of Left Ventricular Dysfunction.

This paper considers an on-off fluid queue model. The on and off states of the system appear alternately, and the sojourn times at these two different states are independent, and each one follows an exponential distribution. The fluid flows into the system buffer with some strategies to wait for the system service under the first-come first-served discipline. Here the system can process the fluid in the buffer only when the system is on state. With given utility functions such as an expected average social profit, and an individual expected profit, the equilibrium strategies are characterized under both the fully unobservable case and the partially observable case.

In this paper, we investigate dual problems for nonconvex set-valued vector optimization via abstract subdifferential. We first introduce a generalized augmented Lagrangian function induced by a coupling vector-valued function for set-valued vector optimization problem and construct related set-valued dual map and dual optimization problem on the basic of weak efficiency, which used by the concepts of supremum and infimum of a set. We then establish the weak and strong duality results under this augmented Lagrangian and present sufficient conditions for exact penalization via an abstract subdifferential of the object map. Finally, we define the sub-optimal path related to the dual problem and show that every cluster point of this sub-optimal path is a primal optimal solution of the object optimization problem. In addition, we consider a generalized vector variational inequality as an application of abstract subdifferential.

We study the Schwarzschild spacetime solutions to the Einstein-Euler equations. In our analysis, we aim to show local stability under small perturbations. To resolve this problem, we use the Nash-Moser (-Hamilton) theorem. The work was originally developed for the nonrelativistic Euler-Poisson equations.

n this paper, we deal with the nonlinear second-order differential equation with damped vibration term involving p-Laplacian operator. Of particular interest is the resolution of an open problem. An interesting outcome from our result is that we can obtain the fast homoclinic solution with general superlinear growth assumption in suitable Sobolev space. To our knowledge, our theorems appear to be the first such result about damped vibration problem with p-Laplacian operator.

We propose a line search exact penalty method with bi-object strategy for nonlinear semidefinite programming. At each iteration, we solve a linear semidefinite programming to test whether the linearized constraints are consistent or not. The search direction is generated by a piecewise quadratic-linear model of the exact penalty function. The penalty parameter is only related to the information of the current iterate point. The line search strategy is a penalty-free one. Global and local convergence are analyzed under suitable conditions. We finally report some numerical experiments to illustrate the behavior of the algorithm on various degeneracy situations.

In this paper we first establish the uniform regularity of smooth solutions with respect to the viscosity coefficients to the isentropic compressible magnetohydrodynamic system in a periodic domain $\mathbb{T}^n$. We then apply our result to obtain the isentropic compressible magnetohydrodynamic system with zero viscosity.

Let $m$, $t$, $r$ and $k_i$ $(1\leq i\leq m)$ be positive integers with $k_i\geq\frac{(t+3)r}{2}$, and $G$ be a digraph with vertex set $V(G)$ and arc set $E(G)$. Let $H_1,H_2,\cdots,H_t$ be $t$ vertex-disjoint subdigraphs of $G$ with $mr$ arcs. In this article, it is verified that every $[0,k_1+k_2+\cdots+k_m-(m-1)r]$-digraph $G$ has a $[0,k_i]_1^{m}$-factorization $r$-orthogonal to every $H_i$ for $1\leq i\leq t$.

In this paper, we study the well-posedness and the asymptotic stability of a one-dimensional thermoelastic microbeam system, where the heat conduction is given by Gurtin-Pipkin thermal law. We first establish the well-posedness of the system by using the semigroup arguments and Lumer-Phillips theorem. We then obtain an explicit and general formula for the energy decay rates through perturbed energy method and some properties of the convex functions.

In this paper, we investigate a delayed HIV infection model that considers the homeostatic proliferation of CD4⁺ T cells. The existence and stability of uninfected equilibrium and infected equilibria (smaller and larger ones) are studied by analyzing the characteristic equation of the system. The intracellular delay does not affect the stability of uninfected equilibrium, but it can change the stability of larger positive equilibrium and Hopf bifurcation appears inducing stable limit cycles. Furthermore, direction and stability of Hopf bifurcation are well investigated by using the central manifold theorem and the normal form theory. The numerical simulation results show that the stability region of larger positive equilibrium becomes smaller as the increase of time delay. Moreover, when the maximum homeostatic growth rate is very small, the larger positive equilibrium is always stable. On the contrary, when the rate of supply of T cells is very small, the larger positive equilibrium is always unstable.

In this paper, we study the existence of positive solution for the $p$-Laplacian equations with fractional critical nonlinearity \[ \begin{cases} (-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u=K(x)f(u)+P(x)|u|^{p^{*}_{s}-2}u, \qquad x\in \mathbb{R}^{N}, \\ u\in \mathcal {D}^{s,p}(\mathbb{R}^{N}), \end{cases} \] where $s\in(0,1), \ p^{*}_{s}=\frac{Np}{N-sp}, \ N>sp, \ p>1$ and $ V(x),K(x)$ are positive continuous functions which vanish at infinity, $f$ is a function with a subcritical growth, and $P(x)$ is bounded, nonnegative continuous function. By using variational method in the weighted spaces, we prove the above problem has at least one positive solution.

The authors present some new criteria for oscillation and asymptotic behavior of solutions of third-order nonlinear differential equations with a sublinear neutral term of the form $$\left(r(t)(z''(t))^{\alpha}\right)'+\int^{d}_{c}q(t,\xi)f\left(x\left(\sigma(t,\xi)\right)\right)d\xi=0, \qquad t\geq t_{0}$$ where $z(t)=x(t)+\int^{b}_{a}p(t,\xi)x^{\gamma}\left(\tau(t,\xi)\right)d\xi,~0<\gamma\leq1.$ Under the conditions $\int^{\infty}_{t_{0}}r^{-\frac{1}{\alpha}}(t)dt=\infty$ or $\int^{\infty}_{t_{0}}r^{-\frac{1}{\alpha}}(t)dt<\infty.$ The results obtained here extend, improve and complement to some known results in the literature. Examples are provided to illustrate the theorems.

In this paper, we derive a time-dependent Ginzburg-Landau model for liquid ⁴He coupling with an applied magnetic field basing on the Le Châtlier principle.
We also obtain the existence and uniqueness of global weak solution for this model. In addition, by utilizing the regularity estimates for linear semigroup, we prove that the model possesses a global classical solution.

In this paper, we propose a criterion based on the variance variation of the sample eigenvalues to correctly estimate the number of significant components in high-dimensional principal component analysis (PCA), and it corresponds to the number of significant eigenvalues of the covariance matrix for p-dimensional variables. Using the random matrix theory, we derive that the consistent properties of the proposed criterion for the situations that the significant eigenvalues tend to infinity, as well as that the bounded significant population eigenvalues. Numerical simulation shows that the probability of estimator is correct by our variance variation criterion converges to 1 is faster than that by criterion of Passemier and Yao [Estimation of the number of spikes, possibly equal, in the high-dimensional case. J. Multivariate Anal., (2014)](PYC), AIC and BIC under the finite fourth moment condition as the dominant population eigenvalues tend to infinity. Moreover, in the case of the maximum eigenvalue bounded, once the gap condition is satisfied, the rate of convergence to 1 is faster than that of PYC and AIC, especially the effect is better than AIC when the sample size is small. It is worth noting that the variance variation criterion significantly improves the accuracy of model selection compared with PYC and AIC when the random variable is a heavy-tailed distribution or finite fourth moment not exists.

Let $H=(V,E)$ be a hypergraph, where $V$ is a set of vertices and $E$ is a set of non-empty subsets of $V$ called edges. If all edges of $H$ have the same cardinality $r$, then $H$ is an $r$-uniform hypergraph; if $E$ consists of all $r$-subsets of $V$, then $H$ is a complete $r$-uniform hypergraph, denoted by $K_n^r$, where $n=|V|$. A hypergraph $H'=(V',E')$ is called a subhypergraph of $H=(V,E)$ if $V'\subseteq V$ and $E'\subseteq E$. The edge-connectivity of a hypergraph $H$ is the cardinality of a minimum edge set $F\subseteq E$ such that $H-F$ is not connected, where $H-F=(V,E\setminus F)$. An $r$-uniform hypergraph $H=(V,E)$ is $k$-edge-maximal if every subhypergraph of $H$ has edge-connectivity at most $k$, but for any edge $e\in E(K_n^r)\setminus E(H)$, $H+e$ contains at least one subhypergraph with edge-connectivity at least $k+1$.
Let $k$ and $r$ be integers with $k\geq2$ and $r\geq2$, and let $t=t(k,r)$ be the largest integer such that $(^{t-1}_{r-1})\leq k$. That is, $t$ is the integer satisfying $(^{t-1}_{r-1})\leq k<(^{\ \ t}_{r-1})$. We prove that if $H$ is an $r$-uniform $k$-edge-maximal hypergraph such that $n=|V(H)|\geq t$, then (i) $|E(H)|\leq (^{t}_{r})+(n-t)k$, and this bound is best possible; (ii)\ $|E(H)|\geq (n-1)k -((t-1)k-(^{t}_{r}))\lfloor\frac{n}{t}\rfloor$, and this bound is best possible.

A 2-distance coloring of a graph is a coloring of the vertices such that two vertices at distance at most two receive distinct colors. A list assignment of a graph $G$ is a mapping $L$ which assigns to each vertex $v$ a set $L(v)$ of positive integers. The list 2-distance chromatic number of $G$ denoted by $\chi_{2}^{l}(G)$ is the least integer $k$ for which $G$ is list 2-distance $k$-colorable. In this paper, we prove that every planar graph with $g(G)\geq 5$ and $\Delta(G)\geq 40$ is list 2-distance ($\Delta(G)+4$)-colorable.