We investigate the generalized second-order Arnoldi (GSOAR) method, a generalization of the SOAR method proposed by Bai and Su [SIAM J. Matrix Anal. Appl., 26 (2005): 640--659.], and the Refined GSOAR (RGSOAR) method for the quadratic eigenvalue problem (QEP). The two methods use the GSOAR procedure to generate an orthonormal basis of a given generalized second-order Krylov subspace, and with such basis they project the QEP onto the subspace and compute the Ritz pairs and the refined Ritz pairs, respectively. We develop implicitly restarted GSOAR and RGSOAR algorithms, in which we propose certain exact and refined shifts for respective use within the two algorithms. Numerical experiments on real-world problems illustrate the efficiency of the restarted algorithms and the superiority of the restarted RGSOAR to the restarted GSOAR. The experiments also demonstrate that both IGSOAR and IRGSOAR generally perform much better than the implicitly restarted Arnoldi method applied to the corresponding linearization problems, in terms of the accuracy and the computational efficiency.

In this paper, we introduce dual Toeplitz operators on the orthogonal complement of the Hardy space of the polydisk and establish their main algebraic properties using an auxiliary transformation of operators. As a byproduct, we exploit this mysterious transformation in the investigation of boundedness and compactness of Hankel products and mixed Toeplitz-Hankel products on the Hardy space of the polydisk.

In this paper, a smoothing homotopy method for solving the nonlinear complementarity problem is considered. The homotopy equation is constructed based on Chen-Harker-Kanzow-Smale smooth function. Under certain mild nonmonotone condition, the global convergence result is obtained. Furthermore, the initial point can be chosen almost everywhere in $R^n$ but not just in $R^n_+$. The numerical experimental results show that the method is effective.

Let $m_T[0,2)$ be the number of Laplacian eigenvalues of a tree $T$ in $[0,2)$, multiplicities included. We give best possible upper bounds for $m_T[0,2)$ using the parameters such as the number of pendant vertices, diameter, matching number, and domination number, and characterize the trees $T$ of order $n$ with $m_T[0,2)=n-1$, $n-2$, and $\left \lceil \frac{n}{2} \right \rceil$, respectively, and in particular, show that $m_T[0,2)=\left \lceil \frac{n}{2} \right \rceil$ if and only if the matching number of $T$ is $\left \lfloor \frac{n}{2} \right \rfloor$.

The initial value problem for a matrix Riccati differential equation associated with an $M$-matrix is known to have a global solution $X(t)$ on $[0, \infty)$ when $X(0)$ takes values from a suitable set of nonnegative matrices. It is also known, except for the critical case, that as $t$ goes to infinity $X(t)$ converges to the minimal nonnegative solution of the corresponding algebraic Riccati equation. In this paper we present a new approach for proving the convergence, which is based on the doubling procedure and is also valid for the critical case. The approach also provides a way for solving the initial value problem and a new doubling algorithm for computing the minimal nonnegative solution of the algebraic Riccati equation.

Let the metric $g$ of a three-dimensional $\eta$-Einstein almost Kenmotsu manifold $M$ be a Ricci soliton, we prove that $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the soliton is expanding.

Asai and Friedberg studied the imaginary Doi-Naganuma lifting which sends elliptic modular forms to automorphic forms over an imaginary quadratic field. In this paper we extend this lifting to weak Maass forms by using regularized integral. We construct an automorphic object with singularities on the quadratic upper half-plane $\mathfrak{H}^{1}$ by the regularized theta lifting of a weak Maass form. We also give the convergence region and describe its singularity type. Finally we compute the Fourier coefficients of the lifted form explicitly and present the case of Poincaré series as an example.

Let $G_1$ be a planar graph such that all cycles of length at most 4 are independent and let $G_2$ be a planar graph without 3-cycles and adjacent 4-cycles. It is proved that the set of vertices of $G_1$ and $G_2$ can be equitably partitioned into $t$ subsets for every $t\geq 3$ so that each subset induces a forest. These results partially confirm a conjecture of Wu, Zhang and Li [5].

In this paper, we shall establish some inequalities for differentiable co-ordinated convex and concave functions on a rectangle from the plane. They are connected with the left side of extended Hermite-Hadamard inequality in two variables. Also, these inequalities are able to be applied to some special means and cubature formulae.

In this paper we discuss the Bures distance between $\alpha$-CP maps on a $C^*$-algebra and the transition probability between $P$-functionals on a *-algebra. We first review the notion of $\alpha$-CP maps and the representation theorem associated to $\alpha$-CP maps. Using the Krein space representation and the set of intertwiners between Krein space representations, we study the Bures distance between $\alpha$-CP maps. We prove that the transition probability between $P$-functionals can be estimated by some functionals using $J$-representations on Krein spaces.

We study non-paraSasakian paracontact metric $(\kappa,\mu)$-spaces with $\kappa=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of $h$. We will also give explicit examples of every possible constant rank of $h$.

In this paper, we introduce a modified Chebyshev-like's method with order four and study the semilocal convergence of the method by using majorizing functions for solving nonlinear equations in Banach spaces. We prove an existence-uniqueness theorem and give a priori error bounds which demonstrates the R-order of the method. Moveover, the local convergence of this method is also analyzed. Finally, numerical application on nonlinear integral equations is given to show our approach.

Let $(R,m)$ be a complete Noetherian local ring, $I$ a proper ideal of R and $M$, $N$ two finitely generated R-modules such that Supp $(N)\subseteq V(I)$. Let $t\geq 0$ be an integer such that for each $0\leq i\leq t$, the $R$-module $H^i_I(M)$ is in dimension $\lt n$. Then we show that each element $L$ of the set $\mathfrak{J}$, which is defined as: $$ \{\operatorname{Ext}^j_R(N,H^i_I(M)):j\ge0 \ and \ 0\le i \le t\} \\ \cup \{\operatorname{Hom}_R(N,H^{t+1}_I(M)), \operatorname{Ext}^1_R(N,H^{t+1}_I(M))\} $$ is in dimension $\lt n-2$ and so as a consequence, it follows that the set $$ \operatorname{Ass}_R(L) \cap \{\mathfrak{p} \in \operatorname{Spec}(R): \dim(R/\mathfrak{p}) \ge n − 2\} $$ is finite. In particular, the set $$ \operatorname{Ass}_R(\oplus_{i=0}^{t+1} H^i_I(R)) \cap \{\mathfrak{p} \in \operatorname{Spec}(R): \dim(R/\mathfrak{p}) \geq n-2\} $$ is finite. Also, as an immediately consequence of this result it follows that the $R$-modules $\operatorname{Ext}^j_R(N,H^i_I(M))$ are in dimension $\lt n-1$, for all integers $i,j \geq 0$, whenever $\dim(M/IM) = n$. These results generalizes the main results of Huneke-Koh [17], Delfino [10], Chiriacescu [9], Asadollahi-Naghipour [1], Quy [18], Brodmann-Lashgari [7], Bahmanpour-Naghipour [5] and Bahmanpour et al. [6] in thecase of complete local rings.

In this article, we study $L_k$-finite-type hypersurfaces $M^n$ of a hyperbolic space $\mathbb{H}^{n+1}\subset\mathbb{R}^{n+2}_1$, for $k\geq 1$. In the 3-dimensional case, we obtain the following classification result. Let $\psi:M^3\rightarrow\mathbb{H}^{4}\subset\mathbb{R}^5_1$ be an orientable hypersurface with constant $k$-th mean curvature $H_k$, which is not totally umbilical. Then $M^3$ is of $L_k$-2-type if and only if $M^3$ is an open portion of a standard Riemannian product $\mathbb{H}^1(r_1)\times\mathbb{S}^{2}(r_2)$ or $\mathbb{H}^2(r_1)\times\mathbb{S}^{1}(r_2)$, with $-r_1^2+r_2^2=-1$. In the $n$-dimensional case, we show that a hypersurface $M^n\subset\mathbb{H}^{n+1}$, with constant $k$-th mean curvature $H_k$ and having at most two distinct principal curvatures, is of $L_k$-2-type if and only if $M^n$ is an open portion of a Riemannian product $\mathbb{H}^m(r_1)\times\mathbb{S}^{n-m}(r_2)$, with $-r_1^2+r_2^2=-1$, for some integer $m\in\{1,\dots,n-1\}$. In the case $k=n-1$ we drop the condition on the principal curvatures of the hypersurface $M^n$, and prove that if $M^n\subset\mathbb{H}^{n+1}$ is an orientable $H_{n-1}$-hypersurface of $L_{n-1}$-2-type then its Gauss-Kronecker curvature $H_{n}$ is a nonzero constant.

In this paper we investigate maximization and minimization problems related to a $p$-Laplacian equation on a multiply connected domain in $\mathbb{R}^2$, where the admissible set is a rearrangement class of a fixed function. We prove existence and representation of the maximizers and existence, uniqueness and representation of the minimizer.

This article aims to demonstrate the existence of new solutions for the vector quasi-equilibrium problems. Firstly we prove the existence of the equilibrium for the generalized abstract economy model under upper semicontinuity assumptions. By using these results, we solve the announced problem in case of multivalued trifunctions. Secondly, we consider the generalized strong vector quasi-equilibrium problem and prove the existence of its solutions in case of correspondences being weakly naturally quasi-concave or weakly biconvex and also in case of weak-continuity assumptions. In all situations, our theoretical analysis is based on fixed-point theorems. Our study indicates that the refinement of the hypotheses concerning the equilibrium problems plays an important role in the developing of this theory and it improves, by its novelty, the existent results obtained so far in literature.

Let $\varphi : \mathbb{R}^n\times [0,\,\infty)\to[0,\infty)$ be a Musielak-Orlicz function and $A$ an expansive dilation. Let $H^\varphi_A({\mathbb {R}}^n)$ be the anisotropic Hardy space of Musielak-Orlicz type defined via the grand maximal function. Its atomic characterization and some other maximal function characterizations of $H^\varphi_A({\mathbb {R}^n})$, in terms of the radial, the non-tangential and the tangential maximal functions, are known. In this article, the authors further obtain their characterizations in terms of the Lusin-area function, the $g$-function or the $g^\ast_\lambda$-function via first establishing an anisotropic Peetre's inequality of Musielak-Orlicz type. Moreover, the range of $\lambda$ in the $g^\ast_\lambda$-function characterization of $H^\varphi_A({\mathbb {R}}^n)$ coincides with the known best conclusions in the case when $H^\varphi_A({\mathbb {R}}^n)$ is the classical Hardy space $H^p({\mathbb{R}}^n)$ or the anisotropic Hardy space $H^p_A({\mathbb {R}}^n)$ or their weighted variants, where $p\in(0,\,1]$.

In this paper, we introduce the new concept called partial generalized convex contractions and partial generalized convex contractions of order 2. Also, we establish some approximate fixed point theorems for such mappings in $\alpha$-complete metric spaces. Our results extend and unify the results of Miandaragh et al. [M. A. Miandaragh, M. Postolache, S. Rezapour, Approximate fixed points of generalized convex contractions, Fixed Point Theory and Applications 2013, 2013 :255] and several well-known results in literature. We give some examples of a nonlinear contraction mapping, which is not applied to the existence of approximate fixed point and fixed point by using the results of Miandaragh et al. We also consider approximate fixed point results in metric space endowed with an arbitrary binary relation and approximate fixed point results in metric space endowed with graph.

In this paper, the authors establish the boundedness of commutators generated by Calderón-Zygmund singular integral operators and weighted Lipschitz functions on weighted Morrey-Herz spaces.