In survival analysis, data are frequently collected by some complex sampling schemes, e.g., length biased sampling, case-cohort sampling and so on. In this paper, we consider the additive hazards model for the general biased survival data. A simple and unified estimating equation method is developed to estimate the regression parameters and baseline hazard function. The asymptotic properties of the resulting estimators are also derived. Furthermore, to check the adequacy of the fitted model with general biased survival data, we present a test statistic based on the cumulative sum of the martingale-type residuals. Simulation studies are conducted to evaluate the performance of proposed methods, and applications to the shrub and Welsh Nickel Refiners datasets are given to illustrate the methodology.

This paper is devoted to the partial regularity of suitable weak solutions to the system of the incompressible shear-thinning flow in a bounded domain Ω ⊂ Rⁿ, n ≥ 2. It is proved that there exists a suitable weak solution of the shear-thinning fluid in the n-D smooth bounded domain (for n ≥ 2). For 3D model, it is proved that the singular points are concentrated on a closed set whose 1 dimensional Hausdorff measure is zero.

In this paper, we propose a new nonmonotone trust region Barzilai-Borwein (BB for short) method for solving unconstrained optimization problems. The proposed method is given by a novel combination of a modified Metropolis criterion, BB-stepsize and trust region method. The new method uses the reciprocal of BB-stepsize to approximate the Hessian matrix of the objective function in the trust region subproblems, and accepts some bad solutions according to the modified Metropolis criterion based on simulated annealing idea. Under some suitable assumptions, the global convergence of the new method is established. Some preliminary numerical results indicate that, the new method is more efficient compared with the existing trust region BB method.

Stochastic gradient descent (SGD) is one of the most common optimization algorithms used in pattern recognition and machine learning. This algorithm and its variants are the preferred algorithm while optimizing parameters of deep neural network for their advantages of low storage space requirement and fast computation speed. Previous studies on convergence of these algorithms were based on some traditional assumptions in optimization problems. However, the deep neural network has its unique properties. Some assumptions are inappropriate in the actual optimization process of this kind of model. In this paper, we modify the assumptions to make them more consistent with the actual optimization process of deep neural network. Based on new assumptions, we studied the convergence and convergence rate of SGD and its two common variant algorithms. In addition, we carried out numerical experiments with LeNet-5, a common network framework, on the data set MNIST to verify the rationality of our assumptions.

In this paper, we study a nonlinear Petrovsky type equation with nonlinear weak damping, a superlinear source and time-dependent coefficients
utt + △²u + k₁(t)|u_t|^m-2u_t = k₂(t)|u|^p-2u, x ∈ Ω, t > 0,
where Ω is a bounded domain in Rⁿ. Under certain conditions on k₁(t), k₂(t) and the initial-boundary data, the upper bound for blow-up time of the solution with negative initial energy function is given by means of an auxiliary functional and an energy estimate method if p > m. Also, a lower bound of blow-up time are obtained by using a Sobolev-type inequality and a first order differential inequality technique for n = 2, 3 and n > 4.

In this paper, we present a primal-dual interior point algorithm for semidefinite optimization problems based on a new class of kernel functions. These functions constitute a combination of the classic kernel function and a barrier term.
We derive the complexity bounds for large and small-update methods respectively. We show that the best result of iteration bounds for large and small-update methods can be achieved, namely O(q√n(log √n) ^q+1/q log n/ε) for large-update methods and O(q^3/2 (log √q)^q+1/q √n log n/ε) for small-update methods.
We test the efficiency and the validity of our algorithm by running some computational tests, then we compare our numerical results with results obtained by algorithms based on different kernel functions.

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.

The UK is the most important partner of the EU in terms of economic and other fields due to the geographical proximity. It was one of the largest economies in the EU and its per capita income is higher than the EU average, so it is a net contributor to the EU. With UKs membership of the EU ended on 31 January 2019, there are concerns that the Brexit may have a significant impact on the EU, resulting in social, economic, political, and institutional changes, etc. in EU. While the impact of Brexit on the UK has always been the subject of considerable scholarly interest in recent years, there is relatively little literature on the impact of Brexit on the EU. This paper focuses on the evaluation of the impact of Brexit on the EU economy and other relevant aspects along three dimensions: GDP, PPP, Quarterly GDP growth. Employing powerful quantitative analysis technology that includes vector autoregression model, multivariate time series model with intervention variables, and autoregression integrated moving average, this paper obtains the important and novel evidence about the potential impact of Brexit on the EU economy, pointing out that Brexit is of far-reaching significance to the EU. This analysis uses several statistical models to screen out several key influencing factors, which can be used to predict the total GDP of EU in the next five years. The results show that EU economy will react negatively to "no-deal" Brexit, and its growth rate of economy will slow down significantly in next 5 years. Finally, we put forward relevant policy suggestions on how to deal with the negative impact of Brexit on EU.

In this paper, we consider the initial value problem of a class of fractional differential equations. Firstly, we obtain the existence and uniqueness of the solutions by using Picard’s method of successive approximation. Then we discuss the dependence of the solutions on the initial value.

Hartsfield and Ringel conjectured that every connected graph other than K₂ is antimagic. Since then, many classes of graphs have been proved to be antimagic. But few is known about the antimagicness of lexicographic product graphs. In this paper, via the construction of a directed Eulerian circuit, the Siamese method, and some modification on graph labeling, the antimagicness of lexicographic product graph G[P_n] is obtained.

In 2005, Flandrin et al. proved that if G is a k-connected graph of order n and V(G)=X₁ ∪ X₂ ∪ … ∪ X_k such that d(x)+ d(y) ≥ n for each pair of nonadjacent vertices x, y ∈ X_i and each i with i=1, 2, …, k, then G is hamiltonian. In order to get more sufficient conditions for hamiltonicity of graphs, Zhu, Li and Deng proposed the definitions of two kinds of implicit degree of a vertex v, denoted by id₁(v) and id₂(v), respectively. In this paper, we are going to prove that if G is a k-connected graph of order n and V (G)=X₁ ∪ X₂ ∪ … ∪ X_k such that id₂(x) + id₂(y) ≥ n for each pair of nonadjacent vertices x, y ∈ X_i and each i with i=1, 2, …, k, then G is hamiltonian.

In this paper, Wang's Harnack and shift Harnack inequality for a class of stochastic differential equations driven by G-Brownian motion are established. The results generalize the ones in the linear expectation setting. Moreover, some applications are also given.

A proper edge coloring of a graph G is acyclic if there is no 2-colored cycle in G. The acyclic chromatic index of G is the least number of colors such that G has an acyclic edge coloring and denoted by χ'_a(G). An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. In this paper, it is proved that χ'_a(G) ≤ △(G) + 10, if G is an IC-planar graph without adjacent triangles and χ'_a(G) ≤ △(G) + 8, if G is a triangle-free IC-planar graph.

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.

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.

Frequentist model averaging has received much attention from econometricians and statisticians in recent years. A key problem with frequentist model average estimators is the choice of weights. This paper develops a new approach of choosing weights based on an approximation of generalized cross validation. The resultant least squares model average estimators are proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors. Especially, the optimality is built under both discrete and continuous weigh sets. Compared with the existing approach based on Mallows criterion, the conditions required for the asymptotic optimality of the proposed method are more reasonable. Simulation studies and real data application show good performance of the proposed estimators.

In this paper, we make use of stochastic theta method to study the existence of the numerical approximation of random periodic solution. We prove that the error between the exact random periodic solution and the approximated one is at the 1/4 order time step in mean sense when the initial time tends to ∞.

An n×n matrix A consisting of nonnegative integers is a general magic square of order n if the sum of elements in each row, column, and main diagonal is the same. A general magic square A of order n is called a magic square, denoted by MS(n), if the entries of A are distinct. A magic square A of order n is normal if the entries of A are n² consecutive integers. Let A^*d denote the matrix obtained by raising each element of A to the d-th power. The matrix A is a d-multimagic square, denoted by MS(n, d), if A^*e is an MS(n) for 1 ≤ e ≤ d. In this paper we investigate the existence of normal bimagic squares of order 2u and prove that there exists a normal bimagic square of order 2u, where u and 6 are coprime and u ≥ 5.

This paper deals with an initial boundary value problem for a class of nonlinear wave equation with nonlinear damping and source terms whose solution may blow up in finite time. An explicit lower bound for blow up time is determined by means of a differential inequality argument if blow up occurs.

The admissibility of the initial-boundary data, which characterizes the existence of solution for the initial-boundary value problem, is important. Based on the Fokas method and the inverse scattering transformation, an approach is developed to solve the initial-boundary value problem of the nonlinear Schrödinger equation on a finite interval. A necessary and sufficient condition for the admissibility of the initial-boundary data is given, and the reconstruction of the potential is obtained.

Cost effective sampling design is a problem of major concern in some experiments especially when the measurement of the characteristic of interest is costly or painful or time consuming. In the current paper, a modification of ranked set sampling (RSS) called moving extremes RSS (MERSS) is considered for the estimation of the location parameter for location family. A maximum likelihood estimator (MLE) of the location parameter for this family is studied and its properties are obtained. We prove that the MLE is an equivariant estimator under location transformation. In order to give more insight into the performance of MERSS with respect to (w.r.t.) simple random sampling (SRS), the asymptotic efficiency of the MLE using MERSS w.r.t. that using SRS is computed for some usual location distributions. The relative results show that the MLE using MERSS can be real competitors to the MLE using SRS.

In the paper, we characterize a necessary and sufficient condition which ensures the continuities of the non-centered Hardy-Littlewood maximal function Mf and the centered Hardy-Littlewood maximal function M_cf on Rⁿ. As two applications, we can easily deduce that M_cf and Mf are continuous if f is continuous, and Mf is continuous if f is of local bounded variation on R.

A matching M of a graph G is an induced matching if no two edges in M are joined by an edge of G. Let iz(G) denote the total number of induced matchings of G, named iz-index. It is well known that the Hosoya index of a graph is the total number of matchings and the Hosoya index of a path can be calculated by the Fibonacci sequence. In this paper, we investigate the iz-index of graphs by using the Fibonacci-Narayana sequence and characterize some types of graphs with minimum and maximum iz-index, respectively.

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.

In recent years, there has been a large amount of literature on missing data. Most of them focus on situations where there is only missingness in response or covariate.
In this paper, we consider the adequacy check for the linear regression model with the response and covariates missing simultaneously.
We apply model adjustment and inverse probability weighting methods to deal with the missingness of response and covariate, respectively. In order to avoid the curse of dimension, we propose an empirical process test with the linear indicator weighting function. The asymptotic properties of the proposed test under the null, local and global alternative hypothetical models are rigorously investigated. A consistent wild bootstrap method is developed to approximate the critical value. Finally, simulation studies and real data analysis are performed to show that the proposed method performed well.

A graph is outer-1-planar if it can be drawn in the plane so that all vertices are on the outer face and each edge is crossed at most once. It is known that the list edge chromatic number χ'_l(G) of any outer-1-planar graph G with maximum degree △(G) ≥ 5 is exactly its maximum degree. In this paper, we prove χ'_l(G)=△(G) for outer-1-planar graphs G with △(G)=4 and with the crossing distance being at least 3.

Published auxiliary information can be helpful in conducting statistical inference in a new study. In this paper, we synthesize the auxiliary information with semiparametric likelihood-based inference for censoring data with the total sample size is available. We express the auxiliary information as constraints on the regression coefficients and the covariate distribution, then use empirical likelihood method for general estimating equations to improve the efficiency of the interested parameters in the specified model. The consistency and asymptotic normality of the resulting regression parameter estimators established. Also numerical simulation and application with different supposed conditions show that the proposed method yields a substantial gain in efficiency of the interested parameters.

This paper is concerned with the existence and stability of steady state solutions for the SKT biological competition model with cross-diffusion.
By applying the detailed spectral analysis and in virtue of the bifurcating direction to the limiting system as the cross diffusion rate tends to infinity, it is proved the stability/instability of the nontrivial positive steady states with some special bifurcating structure.
Further, the existence and stability/instability of the corresponding nontrivial positive steady states for the original cross-diffusion system are proved by applying perturbation argument.

Let H=(V, E) be an n-balanced k-partite k-graph with partition classes V₁, …, V_k. Suppose for every legal (k-1)-tuple f contained in V \ V₁ and for every legal (k-1)-tuple g contained in V \ V_k such that f ∪ g∉ E(H), we have d(f) + d(g) ≥ n + 1. In this paper, we prove that under this condition H must have a perfect matching. Another result of this paper is about the perfect matching in 3-uniform hm-bipartite hypergraphs. Let G be a 3-uniform hm-bipartite hypergraph with one of whose sides V₁ has the size n, the another side V₂ has size 2n. If for all the legal 2-tuple f with|f ∩ V₁|=1 and for all the legal 2-tuple g with|g ∩ V₁|=0, we have d(f) ≥ n-2 and d(g) > n/2, then G has a perfect matching.

We consider a nonlocal boundary value problem for a viscoelastic equation with a Bessel operator and a weighted integral condition and we prove a general decay result. We also give an example to show that our general result gives the optimal decay rate for ceratin polynomially decaying relaxation functions. This result improves some other results in the literature.

A path in an edge-colored graph G is called a rainbow path if no two edges of the path are colored the same color. The minimum number of colors required to color the edges of G such that every pair of vertices are connected by at least k internally vertex-disjoint rainbow paths is called the rainbow k-connectivity of the graph G, denoted by rc_k(G). For the random graph G(n, p), He and Liang got a sharp threshold function for the property rc_k(G(n, p)) ≤ d. For the random equi-bipartite graph G(n, n, p), Fujita et. al. got a sharp threshold function for the property rc_k(G(n, n, p)) ≤ 3. They also posed the following problem: For d ≥ 2, determine a sharp threshold function for the property rc_k(G) ≤ d, where G is another random graph model. This paper is to give a solution to their problem in the general random bipartite graph model G(m, n, p).

The paper is devoted to the homogenization of elliptic systems in divergence form. We obtain uniform interior as well as boundary Lipschitz estimates in a bounded C^1,γ domain when the coefficients are Dini continuous, inhomogeneous terms are divergence of Dini continuous functions and the boundary functions have Dini continuous derivatives. The results extend Avellaneda and Lin's work[Comm. Pure Appl. Math., 40:803-847 (1987)], where Hölder continuity is the main assumption on smoothness of the data.

Let a, b and k be nonnegative integers with a ≥ 2 and b ≥ a(k + 1) + 2. A graph G is called a k-Hamiltonian graph if after deleting any k vertices of G the remaining graph of G has a Hamiltonian cycle. A graph G is said to have a k-Hamiltonian[a, b]-factor if after deleting any k vertices of G the remaining graph of G admits a Hamiltonian[a, b]-factor. Let G is a k-Hamiltonian graph of order n with n ≥ a + k + 2. In this paper, it is proved that G contains a k-Hamiltonian[a, b]-factor if δ(G) ≥ a + k and δ(G) ≥ I(G) ≥ a-1 + ((a(k+1))/b-2).

In this paper, we assume that the pest population is divided into susceptible pests and infected pests, and only susceptible pests do harm to crops. Considering the two methods of spraying pesticides and releasing infected pests and natural enemies to control susceptible pests (the former is applied more frequently), and assuming that only susceptible pests develop resistance to pesticides, a pest control model with resistance development is established. By using the basic theory of impulsive differential systems and analytical methods, the sufficient condition for the global attractiveness of the susceptible pest eradication periodic solution is given. Combined with numerical simulations, the effects of spraying frequency of pesticides on critical threshold conditions for eradicating susceptible pests are discussed. The results confirm that it is not that the more frequently the pesticides are sprayed, the better the result of the pest control is. Two control strategies for eradicating susceptible pests are proposed:switching pesticides and releasing natural enemies elastically. Finally, the parameters in the critical threshold are analyzed from the following two aspects:(1) The key factors affecting pest control are determined by parameter sensitivity analyses. The results indicate that the correlation of the critical threshold concerning the killing efficiency rate and the decay rate of pesticides to susceptible pests varies due to the resistance development of susceptible pests. (2) Three-dimensional graphs and contours of susceptible pest eradication critical threshold with two parameters are simulated, and the effects of the main parameters on the critical threshold are analyzed.

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.

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 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.

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.

A nonincreasing sequence π=(d₁, …, d_n) of nonnegative integers is a graphic sequence if it is realizable by a simple graph G on n vertices. In this case, G is referred to as a realization of π. Given a graph H, a graphic sequence π is potentially H-graphic if π has a realization containing H as a subgraph. For graphs G₁ and G₂, the potential-Ramsey number r_pot(G₁, G₂) is the smallest integer k such that for every k-term graphic sequence π, either π is potentially G₁-graphic or the complementary sequence π=(k-1-d_k, …, k-1-d₁) is potentially G₂-graphic. For 0 ≤ k ≤ ⎣ t/2 」, denote K_t^-k to be the graph obtained from K_t by deleting k independent edges. If k=0, Busch et al. (Graphs Combin., 30(2014)847-859) present a lower bound on r_pot(G, K_t) by using the 1-dependence number of G. In this paper, we utilize i-dependence number of G for i ≥ 1 to give a new lower bound on rpot(G, K_t^-k) for any k with 0 ≤ k ≤ ⎣ t/2 」. Moreover, we also determine the exact values of r_pot(K_n, K_t^-k) for 1 ≤ k ≤ 2.

In this paper, we concern the Klein-Gordon-Maxwell system with steep potential well
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Without global and local compactness, we can tell the difference of multiple solutions from their norms in L^p(R³).