This paper introduces an algorithm for the nonnegative matrix factorization-and-completion problem, which aims to find nonnegative low-rank matrices X and Y so that the product XY approximates a nonnegative data matrix M whose elements are partially known (to a certain accuracy). This problem aggregates two existing problems: (i) nonnegative matrix factorization where all entries of M are given, and (ii) low-rank matrix completion where nonnegativity is not required. By taking the advantages of both nonnegativity and low-rankness, one can generally obtain superior results than those of just using one of the two properties. We propose to solve the non-convex constrained least-squares problem using an algorithm based on the classical alternating direction augmented Lagrangian method. Preliminary convergence properties of the algorithm and numerical simulation results are presented. Compared to a recent algorithm for nonnegative matrix factorization, the proposed algorithm produces factorizations of similar quality using only about half of the matrix entries. On tasks of recovering incomplete grayscale and hyperspectral images, the proposed algorithm yields overall better qualities than those produced by two recent matrix-completion algorithms that do not exploit nonnegativity.

Abundant exact interaction solutions, including lump-soliton, lumpkink, and lump-periodic solutions, are computed for the Hirota-Satsuma-Ito equation in (2+1)-dimensions, through conducting symbolic computations with Maple. The basic starting point is a Hirota bilinear form of the Hirota-Satsuma-Ito equation. A few three-dimensional plots and contour plots of three special presented solutions are made to shed light on the characteristic of interaction solutions.

A (2+ 1)-dimensional generalized Bogoyavlensky-Konopelchenko equation that possesses a Hirota bilinear form is considered. Starting with its Hirota bilinear form, a class of explicit lump solutions is computed through conducting symbolic computations with Maple, and a few plots of a specific presented lump solution are made to shed light on the characteristics of lumps. The result provides a new example of (2+ 1)-dimensional nonlinear partial differential equations which possess lump solutions.

A class of trilinear differential operators is introduced through a technique of assigning signs to derivatives and used to create trilinear differential equations. The resulting trilinear differential operators and equations are characterized by the Bell polynomials, and the superposition principle is applied to the construction of resonant solutions of exponential waves. Two illustrative examples are made by an algorithm using weights of dependent variables.

In this paper, we consider a stochastic lattice differential equation with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and multiplicative white noise at each node. We prove the existence of a compact global random attractor which, pulled back, attracts tempered random bounded sets.

An algorithm for the direct inversion of the linear systems arising from Nystr?m discretization of integral equations on one-dimensional domains is described. The method typically has O(N) complexity when applied to boundary integral equations (BIEs) in the plane with non-oscillatory kernels such as those associated with the Laplace and Stokes’ equations. The scaling coefficient suppressed by the “big-O” notation depends logarithmically on the requested accuracy. The method can also be applied to BIEs with oscillatory kernels such as those associated with the Helmholtz and time-harmonic Maxwell equations; it is efficient at long and intermediate wave-lengths, but will eventually become prohibitively slow as the wave-length decreases. To achieve linear complexity, rank deficiencies in the off-diagonal blocks of the coefficient matrix are exploited. The technique is conceptually related to the H – and H ²-matrix arithmetic of Hackbusch and co-workers, and is closely related to previous work on Hierarchically Semi-Separable matrices.

Numerical multilinear algebra (or called tensor computation), in which instead of matrices and vectors the higher-order tensors are considered in numerical viewpoint, is a new branch of computational mathematics. Although it is an extension of numerical linear algebra, it has many essential differences from numerical linear algebra and more diffculties than it. In this paper, we present a survey on the state of the art knowledge on this topic, which is incomplete, and indicate some new trends for further research. Our survey also contains a detailed bibliography as its important part. We hope that this new area will be receiving more attention of more scholars.

The core inverse for a complex matrix was introduced by O. M. Baksalary and G. Trenkler. D. S. Rakíc, N.Č. Diňcíc and D. S. Djordjevíc generalized the core inverse of a complex matrix to the case of an element in a ring. They also proved that the core inverse of an element in a ring can be characterized by five equations and every core invertible element is group invertible. It is natural to ask when a group invertible element is core invertible. In this paper, we will answer this question. Let R be a ring with involution, we will use three equations to characterize the core inverse of an element. That is, let a, b ∈ R. Then a ∈ R^# with a^# = b if and only if (ab)∗ = ab, ba²= a, and ab²= b. Finally, we investigate the additive property of two core invertible elements. Moreover, the formulae of the sum of two core invertible elements are presented.

To ?nd a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To ?nd cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.

We analyze the convergence properties of the spectral method when used to approximate smooth solutions of delay differential or integral equations with two or more vanishing delays. It is shown that for the pantograph-type functional equations the spectral methods yield the familiar exponential order of convergence. Various numerical examples are used to illustrate these results.

The strong ellipticity condition plays an important role in nonlinear elasticity and in materials. In this paper, we de?ne M-eigenvalues for an elasticity tensor. The strong ellipticity condition holds if and only if the smallest M-eigenvalue of the elasticity tensor is positive. If the strong ellipticity condition holds, then the elasticity tensor is rank-one positive de?nite. The elasticity tensor is rank-one positive de?nite if and only if the smallest Z-eigenvalue of the elasticity tensor is positive. A Z-eigenvalue of the elasticity tensor is an M-eigenvalue but not vice versa. If the elasticity tensor is second-order positive de?nite, then the strong ellipticity condition holds. The converse conclusion is not right. Computational methods for ?nding M-eigenvalues are presented.

We study iterative methods for solving a set of sparse non-negative tensor equations (multivariate polynomial systems) arising from data mining applications such as information retrieval by query search and community discovery in multi-dimensional networks. By making use of sparse and non-negative tensor structure, we develop Jacobi and Gauss-Seidel methods for solving tensor equations. The multiplication of tensors with vectors are required at each iteration of these iterative methods, the cost per iteration depends on the number of non-zeros in the sparse tensors. We show linear convergence of the Jacobi and Gauss-Seidel methods under suitable conditions, and therefore, the set of sparse non-negative tensor equations can be solved very efficiently. Experimental results on information retrieval by query search and community discovery in multi-dimensional networks are presented to illustrate the application of tensor equations and the effectiveness of the proposed methods.

We consider the spectrally negative Lévy processes and determine the joint laws for the quantities such as the first and last passage times over a fixed level, the overshoots and undershoots at first passage, the minimum, the maximum, and the duration of negative values. We apply our results to insurance risk theory to find an explicit expression for the generalized expected discounted penalty function in terms of scale functions. Furthermore, a new expression for the generalized Dickson’s formula is provided.

A total [k]-coloring of a graph G is a mapping ?: V (G) ∪E(G) → {1,2, ..., k} such that any two adjacent elements in V (G)∪E(G) receive different colors. Let f(v) denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv ∈E(G),f(u)≠f(v). By χnsd″(G), we denote the smallest value k in such a coloring of G. Pil?niak and Wo?niak conjectured χnsd″(G)≤?(G)+3 for any simple graph with maximum degree Δ(G). This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for K4-minor free graphs. Furthermore, we show that if G is a K4-minor free graph with ?(G)≥4, then χnsd″(G)≤?(G)+2 The bound Δ(G) + 2 is sharp.

Stimulated by the study of sufficient matrices in linear complementarity problems, we study column sufficient tensors and tensor complementarity problems. Column sufficient tensors constitute a wide range of tensors that include positive semi-definite tensors as special cases. The inheritance property and invariant property of column sufficient tensors are presented. Then, various spectral properties of symmetric column sufficient tensors are given. It is proved that all H-eigenvalues of an even-order symmetric column sufficient tensor are nonnegative, and all its Z-eigenvalues are nonnegative even in the odd order case. After that, a new subclass of column sufficient tensors and the handicap of tensors are defined. We prove that a tensor belongs to the subclass if and only if its handicap is a finite number. Moreover, several optimization models that are equivalent with the handicap of tensors are presented. Finally, as an application of column sufficient tensors, several results on tensor complementarity problems are established.

This work is to analyze a spectral Jacobi-collocation approximation for Volterra integral equations with singular kernel ?(t, s) = (t - s)^-μ. In an earlier work of Y. Chen and T. Tang [J. Comput. Appl. Math., 2009, 233: 938-950], the error analysis for this approach is carried out for 0<μ<1/2 under the assumption that the underlying solution is smooth. It is noted that there is a technical problem to extend the result to the case of Abel-type, i.e., μ = 1/2. In this work, we will not only extend the convergence analysis by Chen and Tang to the Abel-type but also establish the error estimates under a more general regularity assumption on the exact solution.

This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or to have general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one more having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which, the criterion for the positivity of the rate and an approximating procedure of estimating the rate are deduced. As the first step of the approximation, the ratio of the resulting bounds is usually no more than 2. The criteria as well as basic estimates for more general types of stability are also presented. Even though the paper contributes mainly to the non-ergodic case, there are some improvements in the ergodic one. To illustrate the power of the results, a large number of examples are included.

H-tensor is a new developed concept which plays an important role in tensor analysis and computing. In this paper, we explore the properties of H-tensors and establish some new criteria for strong H-tensors. In particular, based on the principal subtensor, we provide a new necessary and sufficient condition of strong H-tensors, and based on a type of generalized diagonal product dominance, we establish some new criteria for identifying strong H-tensors. The results obtained in this paper extend the corresponding conclusions for strong H-matrices and improve the existing results for strong H-tensors.

We investigate a generalized (3+ 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.

In this paper, we prove the isomorphic criterion theorem for (n+2)- dimensional n-Lie algebras, and give a complete classification of (n + 2)- dimensional n-Lie algebras over an algebraically closed field of characteristic zero.

L. Addario-Berry et al. [Discrete Appl. Math., 2008, 156: 1168–1174] have shown that there exists a 16-edge-weighting such that the induced vertex coloring is proper. In this note, we improve their result and prove that there exists a 13-edge-weighting of a graph G, such that its induced vertex coloring of G is proper. This result is one step close to the original conjecture posed by M. Karónski et al. [J. Combin. Theory, Ser. B, 2004, 91: 151–157].

The numerical analysis of variational integrators relies on variational error analysis, which relates the order of accuracy of a variational integrator with the order of approximation of the exact discrete Lagrangian by a computable discrete Lagrangian. The exact discrete Lagrangian can either be characterized variationally, or in terms of Jacobi’s solution of the Hamilton–Jacobi equation. These two characterizations lead to the Galerkin and shooting constructions for discrete Lagrangians, which depend on a choice of a numerical quadrature formula, together with either a finite-dimensional function space or a one-step method. We prove that the properties of the quadrature formula, finite-dimensional function space, and underlying one-step method determine the order of accuracy and momentum-conservation properties of the associated variational integrators. We also illustrate these systematic methods for constructing variational integrators with numerical examples.

We study the long time behavior of solutions of the non-autonomous reaction-diffusion equation defined on the entire space Rn when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is established in L2(Rn) and H1(Rn), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.

Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and H_X({ℝ}^n) the associated Hardy-type space. In this article, we first establish the finite atomic characterization of H_X({ℝ}^n). As an application, we prove that the dual space of H_X({ℝ}^n) is the Campanato space associated with X. For any given \alpha \in (\mathrm{0},\mathrm{1}] and s\in {\mathbb{Z}}_{+}, using the atomic and the Littlewood–Paley function characterizations of H_X({ℝ}^n),we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function S_{\alpha ,s},the intrinsic g-function g_{\alpha ,s},and the intrinsic g_{\lambda }^{\ast }-function g_{\lambda ,\alpha ,s}^{\ast }, where λ coincides with the best known range.

We show that a best rank one approximation to a real symmetric tensor, which in principle can be nonsymmetric, can be chosen symmetric. Furthermore, a symmetric best rank one approximation to a symmetric tensor is unique if the tensor does not lie on a certain real algebraic variety.

The first Zagreb index M₁(G) is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index M₂(G) is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying molecular graph G. In this paper, we obtain lower and upper bounds on the first Zagreb index M₁(G) of G in terms of the number of vertices (n), number of edges (m), maximum vertex degree (Δ), and minimum vertex degree (δ). Using this result, we find lower and upper bounds on M₂(G). Also, we present lower and upper bounds on M₂(G) +M₂(G) in terms of n, m, Δ, and δ, where G denotes the complement of G. Moreover, we determine the bounds on first Zagreb coindex M₁(G) and second Zagreb coindex M₂(G). Finally, we give a relation between the first Zagreb index and the second Zagreb index of graph G.

For solving minimization problems whose objective function is the sum of two functions without coupled variables and the constrained function is linear, the alternating direction method of multipliers (ADMM) has exhibited its efficiency and its convergence is well understood. When either the involved number of separable functions is more than two, or there is a nonconvex function, ADMM or its direct extended version may not converge. In this paper, we consider the multi-block separable optimization problems with linear constraints and absence of convexity of the involved component functions. Under the assumption that the associated function satisfies the Kurdyka- Lojasiewicz inequality, we prove that any cluster point of the iterative sequence generated by ADMM is a critical point, under the mild condition that the penalty parameter is sufficiently large. We also present some sufficient conditions guaranteeing the sublinear and linear rate of convergence of the algorithm.

We introduce two new types of tensors called the strictly semi-monotone tensor and the range column Sufficient tensor and explore their structure properties. Based on the obtained results, we make a characterization to the solution of tensor complementarity problem.

This paper deals with the existence and uniqueness of mild solutions to neutral stochastic delay functional integro-differential equations perturbed by a fractional Brownian motion B^H, with Hurst parameter H∈ (1/2, 1). We use the theory of resolvent operators developed by R. Grimmer to show the existence of mild solutions. An example is provided to illustrate the results of this work.

We report two integrable peakon systems that have weak kink and kink-peakon interactional solutions. Both peakon systems are guaranteed integrable through providing their Lax pairs. The peakon and multi-peakon solutions of both equations are studied. In particular, the two-peakon dynamic systems are explicitly presented and their collisions are investigated. The weak kink solution is studied, and more interesting, the kink-peakon interactional solutions are proposed for the first time.