In this paper, we shall study the uniqueness problems on meromorphic functions sharing nonzero finite value or fixed point. We have answered some questions posed by Dyavanal. Our results improve or generalize a few of known results.

This survey article on Dr. Ky Fan summarizes his versatile achievements and fundamental contributions in the fields of topological groups, nonlinear and convex analysis, operator theory, linear algebra and matrix theory, mathematical programming, and approximation theory, etc., and as well reveals Fan’s exemplary mathematical formation opening up the beauty of pure mathematics, with natural conditions, concise statements and elegant proofs. This article contains a brief biography of Dr. Fan and epitomizes his life. He was not only a great mathematician, but also a very serious teacher known to be extremely strict to his students. He loved his motherland and made generous donations for promoting mathematical development in China. He devoted his life to mathematics, continued his research and published papers till 85 years old.

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.

The necessary and sufficient conditions for the small norm perturbation of a Drazin invertible operator to be still Drazin invertible and the sufficient conditions for the finite rank perturbation of a Drazin invertible operator to be still Drazin invertible are established.

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.

In the setting of the Heisenberg group, based on the rotation method, we obtain the sharp (p, p) estimate for the Hardy operator. It will be shown that the norm of the Hardy operator on L^p(Hⁿ) is still p/(p−1). This goes some way to imply that the L^p norms of the Hardy operator are the same despite the domains are intervals on ℝ, balls in ℝⁿ, or ‘ellipsoids’ on the Heisenberg group Hⁿ. By constructing a special function, we find the best constant in the weak type (1,1) inequality for the Hardy operator. Using the translation approach, we establish the boundedness for the Hardy operator from H¹ to L¹. Moreover, we describe the difference between M^p weights and A^p weights and obtain the characterizations of such weights using the weighted Hardy inequalities.

We show that a connected uniform hypergraph G is odd-bipartite if and only if G has the same Laplacian and signless Laplacian Z-eigenvalues. We obtain some bounds for the largest (signless) Laplacian Z-eigenvalue of a hypergraph. For a k-uniform hyperstar with d edges (2d≥k≥3), we show that its largest (signless) Laplacian Z-eigenvalue is d.

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This paper is devoted to investigating the bounded behaviors of the oscillation and variation operators for Calderón-Zygmund singular integrals and the corresponding commutators on the weighted Morrey spaces. We establish several criterions of boundedness, which are applied to obtain the corresponding bounds for the oscillation and variation operators of Hilbert transform, Hermitian Riesz transform and their commutators with BMO functions, or Lipschitz functions on weighted Morrey spaces.

We consider the problem of variable selection for single-index varying-coefficient model, and present a regularized variable selection procedure by combining basis function approximations with SCAD penalty. The proposed procedure simultaneously selects significant covariates with functional coefficients and local significant variables with parametric coefficients. With appropriate selection of the tuning parameters, the consistency of the variable selection procedure and the oracle property of the estimators are established. The proposed method can naturally be applied to deal with pure single-index model and varying-coefficient model. Finite sample performances of the proposed method are illustrated by a simulation study and the real data analysis.

The Gilmore-Lawler bound (GLB) is one of the well-known lower bound of quadratic assignment problem (QAP). Checking whether GLB is tight is an NP-complete problem. In this article, based on Xia and Yuan linearization technique, we provide an upper bound of the complexity of this problem, which makes it pseudo-polynomial solvable. We also pseudo-polynomially solve a class of QAP whose GLB is equal to the optimal objective function value, which was shown to remain NP-hard.

We give a survey of results on the construction of and obstructions to metrics of almost nonnegative curvature operator on closed manifolds and results on the cohomology rings of closed, simply-connected manifolds with a lower curvature and upper diameter bound. The latter is motivated by a question of Grove whether these condition imply finiteness of rational homotopy types. This question has answers by F. Fang–X. Rong, B. Totaro and recently A. Dessai and the present author.

In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S-a)+m, a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and a>0 is a constant. Numerical results are given to show the accuracy of this approximation method.

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.

We consider the blow-up behavior of Hammerstein-type delay Volterra integral equations (DVIEs). Two types of delays, i.e., vanishing delay (pantograph delay) and non-vanishing delay (constant delay), are considered. With the same assumptions of Volterra integral equations (VIEs), in a similar technology to VIEs, the blow-up conditions of the two types of DVIEs are given. he blow-up behaviors of DVIEs with non-vanishing delay vary with different nitial functions and the length of the lag, while DVIEs with pantograph delay wn the same blow-up behavior of VIEs. Some examples and applications to elay differential equations illustrate this influence.

We construct a sequence of transient random walks in random environments and prove that by proper scaling, it converges to a diffusion process with drifted Brownian potential. To this end, we prove a counterpart of convergence for transient random walk in non-random environment, which is interesting itself.

The concepts of strongly lifting modules and strongly dual Rickart modules are introduced and their properties are studied and relations between them are given in this paper. It is shown that a strongly lifting module has the strongly summand sum property and the generalized Hopfian property, and a ring R is a strongly regular ring if and only if R_R is a strongly dual Rickart module, if and only if aRis a fully invariant direct summand of R_Rfor every a ∈ R.

Let m, m^{\prime }, n be positive integers such that m\ne m^{\prime }. Let A be an mth order n-dimensional tensor, and let B be an m^{\prime }th order n-dimensional tensor. λ ∈ ? is called a B-eigenvalue of A if A{x}^{m-\mathrm{1}}=\lambda B{x}^{m^{\prime }-\mathrm{1}} and B{x}^{m^{\prime }}=\mathrm{1} for some x ∈{?}^n\backslash \{\mathrm{0}\}. In this paper, we propose a linear homotopy method for solving this eigenproblem. We prove that the method finds all isolated B-eigenpairs. Moreover, it is easy to implement. Numerical results are provided to show the efficiency of the proposed method.

We give an easy proof of Andrews and Clutterbuck’s main results [J. Amer. Math. Soc., 2011, 24(3): 899−916], which gives both a sharp lower bound for the spectral gap of a Schrödinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at the same estimates by the ‘double coordinate’ approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of Schödinger operator.

We consider a supercritical branching process (Z_n) in an independent and identically distributed random environment ξ, and present some recent results on the asymptotic properties of the limit variable W of the natural martingale Wn=Zn/E[Zn|ξ], the convergence rates of W-W_n(by considering the convergence in law with a suitable norming, the almost sure convergence, the convergence in L^P, and the convergence in probability), and limit theorems (such as central limit theorems, moderate and large deviations principles) on (log Z_n).

For a recollement (A ;ℬ; C ) of abelian categories, we show that torsion pairs in A and C can induce torsion pairs in ℬ; and the converse holds true under certain conditions.

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.

For a finitely triangulated closed surface M², let α_x be the sum of angles at a vertex x. By the well-known combinatorial version of the 2-dimensional Gauss-Bonnet Theorem, it holds ∑_x(2π−α_x) = 2πχ(M²), where χ denotes the Euler characteristic of M², α_x denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplexτ. Our main result is ∑_τ (−1)^dim(τ)δ(τ) =χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version ∑x∈K0κ(x) =χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B:χ(W) −12χ(B) = ∑_τ∈W−B(−1)^dim(τ)δ(τ) +∑_τ∈B(−1)^dim(τ)ρ(τ).

The main purpose of the paper is looking for a larger class of matrices which have real spectrum. The first well-known class having this property is the symmetric one, then is the Hermite one. This paper introduces a new class, called Hermitizable matrices. The closely related isospectral problem, not only for matrices but also for differential operators is also studied. The paper provides a way to describe the discrete spectrum, at least for tridiagonal matrices or one-dimensional differential operators. Especially, an unexpected result in the paper says that each Hermitizable matrix is isospectral to a birth–death type matrix (having positive sub-diagonal elements, in the irreducible case for instance). Besides, new efficient algorithms are proposed for computing the maximal eigenpairs of these class of matrices.

Let r= 2^d-1 + 1. We investigate the diophantine inequality|∑i=1rλiΦi(xi,yi)+η||<(max?1≤i≤r{|xi|,|yi|})-σwhere Φ_i(x, y) ∈Z[x, y] (1≤i≤r) are nondegenerate forms of degree d= 3 or 4.

We obtain the structure of the rings in which every element is either a sum or a difference of a nilpotent and an idempotent that commute. This extends the structure theorems of a commutative weakly nil-clean ring, of an abelian weakly nil-clean ring, and of a strongly nil-clean ring. As applications, this result is used to determine the 2-primal rings R such that the matrix ring Mn(R) is weakly nil-clean, and to show that the endomorphism ring End_D(V ) over a vector space V_D is weakly nil-clean if and only if it is nil-clean or dim(V ) = 1 with D≅Z3.

The near-group rings are an important class of fusion rings in the theory of tensor categories. In this paper, the irreducible {?}_{+}-modules over the near-group fusion ring K({?}_{\mathrm{3}}, 3) are explicitly classified. It turns out that there are only four inequivalent irreducible {?}_{+}-modules of rank 2 and two inequivalent irreducible {?}_{+}-modules of rank 4 over K({?}_{\mathrm{3}}, 3).

We present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.