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.
This paper is an attempt to study the minimization problem of the risk probability of piecewise deterministic Markov decision processes (PDMDPs) with unbounded transition rates and Borel spaces. Different from the expected discounted and average criteria in the existing literature, we consider the risk probability that the total rewards produced by a system do not exceed a prescribed goal during a first passage time to some target set, and aim to find a policy that minimizes the risk probability over the class of all history-dependent policies. Under suitable conditions, we derive the optimality equation (OE) for the probability criterion, prove that the value function of the minimization problem is the unique solution to the OE, and establish the existence of ε(≥ 0)-optimal policies. Finally, we provide two examples to illustrate our results.
In this paper, the analytical blowup solutions of the N-dimensional radial symmetric compressible Euler equations are constructed. Some previous results of the blowup solutions for the compressible Euler equations with constant damping are generalized to the time-depending damping case. The generalization is untrivial because that the damp coefficient is a nonlinear function of time t.
In the paper, we want to derive a few of nonlinear Schröodinger equations with various formats and investigate their properties, such as symmetries, single soliton solutions, multi-soliton solutions, and so on. First of all, we propose an efficient and straightforward scheme for generating nonisospectral integrable hierarchies of evolution equations for which a generalized nonisospectral integrable Schrödinger hierarchy (briefly GNISH) singles out, from which we get a derivative nonlinear Schrödinger equation, a generalized nonlocal Schrödinger integrable system and furthermore we investigate the symmetries and conserved qualities of the GNISH. Next, we apply the dbar method to obtain a generalized nonlinear Schrödinger-Maxwell-Bloch (GNLS-MB) equation and its hierarchy by introducing a generalized Zakhrov-Shabat spectral problem, whose soliton solutions and gauge transformations are obtained.
In this paper, a compact finite difference scheme for the nonlinear fractional integro-differential equation with weak singularity at the initial time is developed, with $O(N^{-(2-\alpha)}+M^{-4})$ accuracy order, where $N,~M$ denote the numbers of grids in temporal and spatial direction, $\alpha\in(0,1)$ is the fractional order. To recover the full accuracy based on the regularity requirement of the solution, we adopt the $L1$ method and the trapezoidal product integration (PI) rule with graded meshes to discretize the Caputo derivative and the Riemann-Liouville integral, respectively, further handle the nonlinear term carefully by the Newton linearized method. Based on the discrete fractional Grönwall inequality and preserved discrete coefficients of Riemann-Liouville fractional integral, the stability and convergence of the proposed scheme are analyzed by the energy method. Theoretical results are also confirmed by a numerical example.
A $k$-coloring of a graph $G$ is a mapping $c:V(G)\rightarrow\{1,2,\cdots,k\}$. The coloring $c$ is called injective if any two vertices have a common neighbor get distinct colors. A graph $G$ is injectively $k$-choosable if for any color list $L$ of admissible colors on $V(G)$ of size $k$ allows an injective coloring $\varphi$ such that $\varphi(v)\in L(v)$ for each $v\in V(G)$. Let $\chi_{i}(G)$, $\chi_{i}^{l}(G)$ denote the injective chromatic number and injective choosability number of $G$, respectively. In this paper, we show that $\chi_{i}^{l}(G)\leq{\Delta+4}$ if $\Delta\ge22$ and $\chi_{i}^{l}(G) \leq{\Delta+5}$ if $\Delta\ge15$, where $G$ is a triangle-free planar graph and without intersecting $4$-cycles.
This paper mainly focus on the global existence of the strong solutions for the generalized Navier-Stokes equations with damping. We obtain the global existence and uniqueness when $\alpha\geq\frac{5}{4}$ for $\beta\geq1$ and when $\frac{1}{2}+\frac{2}{\beta}\leq \alpha\leq \frac{5}{4}$ for $\frac{8}{3}\leq\beta<+\infty.$
This paper proposes a method for modelling volatilities (conditional covariance matrices) of high dimensional dynamic data. We combine the ideas of approximate factor models for dimension reduction and multivariate GARCH models to establish a model to describe the dynamics of high dimensional volatilities. Sparsity condition and thresholding technique are applied to the estimation of the error covariance matrices, and quasi maximum likelihood estimation (QMLE) method is used to estimate the parameters of the common factor conditional covariance matrix. Asymptotic theories are developed for the proposed estimation. Monte Carlo simulation studies and real data examples are presented to support the methodology.
Empirical likelihood inference for partially linear errors-in-variables models with longitudinal data is investigated. Under regularity conditions, it is shown that the empirical log-likelihood ratio at the true parameters converges to the standard Chi-squared distribution. Furthermore, we consider some estimates of the unknown parameter and the resulting estimators are shown to be asymptotically normal. Some simulations and a real data analysis are given to illustrate the performance of the proposed method.
The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on $\vec H$-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept ($\vec H$*-free digraph) to get a similar conclusion about the case of undirected graph.
In this paper, we investigate a deteriorating system with single vacation of a repairman. The system is described by infinite differential-integral equations with boundary conditions. Firstly, by using functional analysis methods, especially linear operator's C_{0}-semigroup theory, we prove the well-posedness of the system and the existence of a unique positive dynamic solution that satisfies probability condition. Next, by analyzing the spectral properties of the system operator, we prove that all points on the imaginary axis except zero belong to the resolvent set of the system operator. Lastly, we prove that zero is not an eigenvalue of the system operator, which implies that the steady-state solution of the system does not exist.
Let $G$ be a graph. We say that $G$ is 2-divisible if for each induced subgraph $H$ of $G$, either $V(H)$ is a stable set, or $V(H)$ can be partitioned into two sets $A$ and $B$ such that $\omega(H[A])<\omega(H)$ and $\omega(H[B])<\omega(H)$. A hole is an induced cycle of length at least 4, a bull is a graph consisting of a triangle with two disjoint pendant edges, a diamond is the graph obtained from $K_4$ by removing an edge, a dart denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 3, and a racket denotes the graph obtained from a diamond by adding a pendant edge to one vertex of degree 2. In this paper, we prove that every {odd hole, $H$}-free graph is 2-divisible, where $H$ is a dart, or a racket, or a bull. As corollaries, $\chi(G)\le $ min $\{2^{\omega(G)-1}, {\omega(G)+1\choose 2} \}$ if $G$ is {odd hole, dart}-free, or {odd hole, racket}-free, or {odd hole, bull}-free.
A subset $I$ of vertices of an undirected connected graph $G$ is a nonseparating independent set (NSIS) if no two vertices of $I$ are adjacent and $G-I$ is connected. Let $Z(G)$ denote the cardinality of a maximum NSIS of $G$. A nonseparating independent set containing $Z(G)$ vertices is called the maximum nonseparating independent set. In this paper, we firstly give an upper bound for $Z(G)$ of regular graphs and determine $Z(G)$ for some types of circular graphs. Secondly, we show a relationship between $Z(G)$ and the maximum genus $\gamma_{M}(G)$ of a general graph. Finally, an important formula is provided to compute $Z(G)$, i.e., $$Z(G)=\sum_{x\in I}d_{I}(x)+2 (\gamma_M(G-I)-\gamma_M(G))+(\xi(G-I)-\xi(G)),$$ where $I$ is the maximum nonseparating independent set and $\xi(G)$ is the Betti deficiency (Xuong, 1979) of $G$.
Supersaturated designs are common choice for screening experiments. This paper studies the properties of supersaturated designs. We give new lower bounds of $E(s^2)$-criterion and $E(f_{\rm NOD})$-criterion. Some linkages between the combined/double design and its original design are firstly provided, and the lower bounds of $E(s^2)$ and $E(f_{\rm NOD})$ for the combined/double design are also given. Furthermore, the close relationship between the minimum Lee-moment aberration criterion and the criteria for optimal supersaturated designs is revealed. These theoretical results can be used to construct or search for optimal supersaturated designs in practice. Numerical results are also provided, which lend further support to our theoretical findings.