Super Edge-connectivity and Zeroth-order General Randi? Index for -1 ≤ α < 0

Zhi-hong HE, Mei LU

Let G be a connected graph with order n, minimum degree δ=δ(G) and edge-connectivity λ=λ(G). A graph G is maximally edge-connected if λ=δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randi? index R_{α}^{0}(G)=???20180401???d_{G}^{α}(x), where d_{G}(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randi? index for -1 ≤ α < 0, respectively.

Asymptotic Theory for Relative-Risk Models with Missing Time-Dependent Covariates

Zai-ying ZHOU, Peng-cheng ZHANG, Ying YANG

Relative-risk models are often used to characterize the relationship between survival time and timedependent covariates. When the covariates are observed, the estimation and asymptotic theory for parameters of interest are available; challenges remain when missingness occurs. A popular approach at hand is to jointly model survival data and longitudinal data. This seems efficient, in making use of more information, but the rigorous theoretical studies have long been ignored. For both additive risk models and relative-risk models, we consider the missing data nonignorable. Under general regularity conditions, we prove asymptotic normality for the nonparametric maximum likelihood estimators.

Let N={0, 1,…, n-1}. A strongly idempotent self-orthogonal row Latin magic array of order n (SISORLMA(n) for short) based on N is an n×n array M satisfying the following properties:(1) each row of M is a permutation of N, and at least one column is not a permutation of N; (2) the sums of the n numbers in every row and every column are the same; (3) M is orthogonal to its transpose; (4) the main diagonal and the back diagonal of M are 0, 1,…, n-1 from left to right. In this paper, it is proved that an SISORLMA(n) exists if and only if n∉{2, 3}. As an application, it is proved that a nonelementary rational diagonally ordered magic square exists if and only if n∉{2, 3}, and a rational diagonally ordered magic square exists if and only if n≠2.

A Classification of Spectrum-determined Circulant Digraphs

Zheng-xin QIN, Ji-xiang MENG, Qiong-xiang HUANG

Some researchers have proved that Ádám's conjecture is wrong. However, under special conditions, it is right. Let Z_{n} be a cyclic group of order n and C_{n}(S) be the circulant digraph of Z_{n} with respect to S ⊆ Z_{n}\{0}. In the literature, some people have used a spectral method to solve the isomorphism for the circulants of prime-power order. In this paper, we also use the spectral method to characterize the circulants of order p^{a}q^{b}w^{c}(where p, q and w are all distinct primes), and to make Ádám's conjecture right.

On the Strong n-partite Tournaments with Exactly Two Cycles of Length n-1

Qiao-ping GUO, Yu-bao GUO, Sheng-jia LI, Chun-fang LI

Gutin and Rafiey (Australas J. Combin. 34 (2006), 17-21) provided an example of an n-partite tournament with exactly n-m + 1 cycles of length of m for any given m with 4 ≤ m ≤ n, and posed the following question. Let 3 ≤ m ≤ n and n ≥ 4. Are there strong n-partite tournaments, which are not themselves tournaments, with exactly n-m + 1 cycles of length m for two values of m? In the same paper, they showed that this question has a negative answer for two values n-1 and n. In this paper, we prove that a strong n-partite tournament with exactly two cycles of length n-1 must contain some given multipartite tournament as subdigraph. As a corollary, we also show that the above question has a negative answer for two values n-1 and any l with 3 ≤ l ≤ n and l≠n-1.

In this paper, we analyze ovarian cancer cases from six hospitals in China, screen the prognostic factors and predict the survival rate. The data has the feature that all the covariates are categorical. We use three methods to estimate the survival rate-the traditional Cox regression, the two-step Cox regression and a method based on conditional inference tree. By comparison, we know that they are all effective and can predict the survival curve reasonably. The analysis results show that the survival rate is determined by a combination of risk factors, where clinical stage is the most important prognosis factor.

Composite Quantile Regression Estimation for Left Censored Response Longitudinal Data

Li-qun XIAO, Zhan-feng WANG, Yao-hua WU

For left censored response longitudinal data, we propose a composite quantile regression estimator (CQR) of regression parameter. Statistical properties such as consistency and asymptotic normality of CQR are studied under relaxable assumptions of correlation structure of error terms. The performance of CQR is investigated via simulation studies and a real dataset analysis.

Matrix Image Method for Ranking Nonregular Fractional Factorial Designs

Xue-ping CHEN, Jin-guan LIN, Xuan-qing CHEN, Xiao-di WANG

Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. In this paper, we study matrix image theory and present a new method for distinguishing and assessing nonregular designs with complex alias structure, which works for all symmetrical and asymmetrical, regular and nonregular orthogonal arrays. Based on the matrix image theory, our proposed method captures orthogonality and projection properties. Empirical studies show that the proposed method has a more precise differentiation capacity when comparing with some other criteria.

A Lower Bound for the Algebraic Connectivity of a Graph in Terms of the Domination Number

Yi-Zheng FAN, Ying-Ying TAN

We investigate how the algebraic connectivity of a graph changes by relocating a connected branch from one vertex to another vertex, and then minimize the algebraic connectivity among all connected graphs of order n with fixed domination number γ ≤ (n+2)/3, and finally present a lower bound for the algebraic connectivity in terms of the domination number. We also characterize the minimum algebraic connectivity of graphs with domination number half their order.

Free Boundary Value Problem for the Cylindrically Symmetric Compressible Navier-Stokes Equations with a Constant Exterior Pressure

Ru-xu LIAN, Jun-li WANG

This paper is concerned with the free boundary value problem (FBVP) for the cylindrically symmetric barotropic compressible Navier-Stokes equations (CNS) with density-dependent viscosity coefficients in the case that across the free surface stress tensor is balanced by a constant exterior pressure. Under certain assumptions imposed on the initial data, the unique cylindrically symmetric strong solution is shown to exist globally in time and tend to a non-vacuum equilibrium state exponentially as time tends to infinity.

A Neighborhood Union Condition for Fractional ID-[a, b]-factor-critical Graphs

Yuan YUAN, Rong-Xia HAO

Let a, b, r be nonnegative integers with 1 ≤ a ≤ b and r ≥ 2. Let G be a graph of order n with n > ((a+2b)(r(a+b)-2))/b. In this paper, we prove that G is fractional ID-[a, b]-factor-critical if δ(G) ≥ (bn)/(a+2b) +a(r-1) and|N_{G}(x_{1}) ∪ N_{G}(x_{2}) ∪…∪ N_{G}(x_{r})| ≥ ((a+b)n)/(a+2b) for any independent subset {x_{1}, x_{2},…, x_{r}} in G. It is a generalization of Zhou et al.'s previous result[Discussiones Mathematicae Graph Theory, 36:409-418 (2016)] in which r=2 is discussed. Furthermore, we show that this result is best possible in some sense.

Some physicists depicted the molecular structure SnCl_{2}·2(H_{2}O) by a piece of an Archimedean tiling (4.8.8) that is a partial cube. Inspired by this fact, we determine Archimedean tilings whose connected subgraphs are all partial cubes. Actually there are only four Archimedean tilings, (4.4.4.4), (6.6.6), (4.8.8) and (4.6.12), which have this property. Furthermore, we obtain analytical expressions for Wiener numbers of some connected subgraphs of (4.8.8) and (4.6.12) tilings. In addition, we also discuss their asymptotic behaviors.

Exponential Decay in Time of Density of One-dimensional Quantum Navier-Stokes Equations

Jian-wei DONG, Guang-pu LOU, Jun-hui ZHU, Yong YANG

The long-time behavior of the particle density of the compressible quantum Navier-Stokes equations in one space dimension is studied. It is shown that the particle density converges exponentially fast to the constant thermal equilibrium state as the time tends to infinity, the decay rate is also obtained. The results hold regardless of either the bigger of the scaled Planck constant or the viscosity constant. This improves the decay results of[5] by removing the crucial assumption that the scaled Planck constant is bigger than the viscosity constant. The proof is based on the entropy dissipation method and the Bresch-Desjardins type of entropy.

Improved Upper Bound for Generalized Acyclic Chromatic Number of Graphs

Jian-sheng CAI, Xu-ding ZHU

A vertex coloring of a graph G is called r-acyclic if it is a proper vertex coloring such that every cycle D receives at least min{|D|,r} colors. The r-acyclic chromatic number of G is the least number of colors in an r-acyclic coloring of G. We prove that for any number r ≥ 4, the r-acyclic chromatic number of any graph G with maximum degree △ ≥ 7 and with girth at least (r-1)△ is at most (4r-3)△.

A subset S ⊆ V in a graph G=(V, E) is a total[1, 2]-set if, for every vertex v ∈ V, 1 ≤ |N(v)∩S| ≤ 2. The minimum cardinality of a total[1, 2]-set of G is called the total[1, 2]-domination number, denoted by γ_{t}[1,2](G).
We establish two sharp upper bounds on the total[1,2]-domination number of a graph G in terms of its order and minimum degree, and characterize the corresponding extremal graphs achieving these bounds. Moreover, we give some sufficient conditions for a graph without total[1, 2]-set and for a graph with the same total[1, 2]-domination number,[1, 2]-domination number and domination number.

Stable and Total Fenchel Duality for Composed Convex Optimization Problems

Dong-hui FANG, Xian-yun WANG

In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate functions and the subdifferentials of convex functions, we give some new constraint qualifications which completely characterize the strong Fenchel duality and the total Fenchel duality for composed convex optimiztion problem in real locally convex Hausdorff topological vector spaces.

In this paper, high-order numerical analysis of finite element method (FEM) is presented for twodimensional multi-term time-fractional diffusion-wave equation (TFDWE). First of all, a fully-discrete approximate scheme for multi-term TFDWE is established, which is based on bilinear FEM in spatial direction and Crank-Nicolson approximation in temporal direction, respectively. Then the proposed scheme is proved to be unconditionally stable and convergent. And then, rigorous proofs are given here for superclose properties in H^{1}-norm and temporal convergence in L^{2}-norm with order O(h^{2} + τ^{3-α}), where h and τ are the spatial size and time step, respectively. At the same time, theoretical analysis of global superconvergence in H^{1}-norm is derived by interpolation postprocessing technique. At last, numerical example is provided to demonstrate the theoretical analysis.

Maximum Cliques of Hypergraphs and Polynomial Optimization

Yan-ming CHANG, Yue-jian PENG

A remarkable connection between the clique number and the Lagrangian of a graph was established by Motzkin and Straus. Later, Rota Buló and Pelillo extended the theorem of Motzkin-Straus to r-uniform hypergraphs by studying the relation of local (global) minimizers of a homogeneous polynomial function of degree r and the maximal (maximum) cliques of an r-uniform hypergraph. In this paper, we study polynomial optimization problems for non-uniform hypergraphs with four different types of edges and apply it to get an upper bound of Turán densities of complete non-uniform hypergraphs.

Upper Semicontinuity of Parametric Lexicographic Vector Equilibrium Problems

Zhi-miao FANG, Sheng-jie LI

In this paper, lexicographic vector equilibrium problems are investigated. By using the idea of sequential process, the upper semicontinuity and closedness of the solution set map are established for a parametric lexicographic strong vector equilibrium problem.

Inspired by the multiple recurrence and multiple ergodic theorems for measure preserving systems, we discuss an analogous question for measure preserving semigroups. In this note, we deal with the symmetric semigroups associated to reversible Markov chains.