We introduce the notions of approximate Connes-amenability and approximate strong Connes-amenability for dual Banach algebras. Then we characterize these two types of algebras in terms of approximate normal virtual diagonals and approximate $\sigma WC-$virtual diagonals. We investigate these properties for von Neumann algebras, measure algebra and the algebra of $p$-pseudomeasures on locally compact groups. In particular we show that a von Neumann algebra is approximately Connes-amenable if and only if it has an approximate normal virtual diagonal. This is the ``approximate'' analog of the main result of Effros in [10]. We show that in general the concepts of approximate Connes-amenability and Connes-amenability are distinct, but for measure algebras these two concepts coincide. Moreover cases where approximate Connes-amenability of $\mathcal{A}^{**}$ implies approximate Connes-amenability or approximate amenability of $\mathcal{A}$ are also discussed.

In the present paper, we introduce a numerical solution algorithm based on a Petrov-Galerkin method in which the element shape functions are cubic B-splines and the weight functions quadratic B-splines . The motion of a single solitary wave and interaction of two solitary waves are studied. Accuracy and efficiency of the proposed method are discussed by computing the numerical conserved laws and $L_{2}$ , $L_{\infty }$ error norms. The obtained results show that the present method is a remarkably successful numerical technique for solving the modified equal width wave(MEW) equation. A linear stability analysis of the scheme shows that it is unconditionally stable.

In this paper, we prove that if $(M,g)$ is a closed orientable Riemannian manifold with a ”transversely oriented harmonic $g$-Riemannian foliation of codimension $q$ on $M$ and if there exists a parallel basic 2-form on $M$ and a positive constant $k$ such that the transversal Ricci curvature satisfies $Ric_{\nabla}(Z,Z)\geq k(q-1)|Z|^2$ for every transverse vector field $Z$, then the smallest nonzero eigenvalue $\lambda_B$ of the basic Laplacian $\Delta_B$ satisfies $\lambda_B\geq 2k(q-1).$

In this paper we prove that every Banach lattice having the Fatou property and having its $\sigma$-order continuous part as an order dense subset, can be represented as the space $L_w^1(\nu)$ of weakly integrable functions with respect to some vector measure $\nu$ defined on a $\delta$-ring.

We establish the existence of local solutions for the Cauchy problem for one-dimensional $p$-Laplacian equation with a concentrated nonlinear source.

Based on Fischer-Burmeister function, we propose a new smoothing function. Using this function,the existence and continuity of the smooth path for solving the nonlinear complementarity problem with a $P_0$ function are discussed. Then we present a one-step smoothing inexact Newton method for nonlinear complementarity problem with a $P_0$ function. The proposed method solves the corresponding linear system approximately in each iteration. Furthermore, we investigate the boundedness of the sequence generated by our algorithm and prove the global convergence and local superlinear convergence under mild conditions.

In this article we study surfaces in $\mathbb{S}^3(1)\times\mathbb{R}$ for which the $\mathbb{R}$-direction makes a constant angle with the normal plane. We give a complete classification for such surfaces with parallel mean curvature vector.

We consider the nonautonomous differential equation of second order $x''+ a(t)x-b(t) x^l+c(t)x^{2k+1}=0$, where $a(t),b(t),c(t)$ are $T$-periodic functions and $2\leq l < 2k+1$. This is a generalization of a biomathematical model of an aneurysm in the circle of Willis. We prove the existence of a $T$-periodic solution for this equation, using a saddle-point theorem.

In this paper, we use new concepts of subcompatibility and subsequential continuity contained in (Bouhadjera, Godet-Thobie, Common fixed theorems for pairs of subcompatible maps, 17 June 2009. [math.FA]) to prove common fixed point theorems for a pair of maps in metric as well as convex metric spaces which are essentially patterned after a theorem of Huang and Li (Fixed point theorems of compatible mappings in convex metric spaces, Soochow J. Math. 22(3) (1996), 439--447). We also prove some related fixed point theorems and utilize certain such results to prove theorems on best approximation.

We characterize Banach lattices under which each AM-compact operator (resp. the second power of a positive AM-compact operator) is weakly compact. Also, we give some interesting results about b-weakly compact operators and operators of strong type B.

Let $G$ be a locally compact group, $\omega$ be a weight function on $G$ and $0<p<1$. We investigate the space $L^p(G,\omega){\star}L^p(G,\omega)$ as a subset of some special spaces such as $L^p(G,\omega)$, $L^{\infty}(G,1/\widetilde{\omega})$ and $C_0(G,1/\widetilde{\omega})$. As the main result we show that all of the mentioned inclusions are equivalent to the discreteness of $G$.

Buildings of types $\textsf{C}_n$ and $\textsf{D}_n$ are defined by rank $n$ polar spaces. The associated building Grassmannians are polar and half-spin Grassmannians. Apartments in dual polar spaces and half-spin Grassmannians were characterized in [4]. We characterize apartments in all polar Grassmannians consisting of non-maximal singular subspaces. This characterization is a partial case of more general results concerning embeddings of polar Johnson graphs in polar Grassmann graphs.

Let $K$ be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. We show that if the Wronskian of two entire functions in $K$ is a polynomial, then both functions are polynomials. As a consequence, if a meromorphic function $f$ on all $K$ is transcendental and has finitely many multiple poles, then $f'$ takes all values in $K$ infinitely many times. We then study applications to a meromorphic function $f$ such that $f'+bf^2$ has finitely many zeros, a problem linked to the Hayman conjecture on a p-adic field.

The so-called subgroup commutativity degree $sd(G)$ of a finite group $G$ is the number of permuting subgroups $(H,K) \in \mathrm{L}(G) \times \mathrm{L}(G)$, where $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided by $|\mathrm{L}(G)|^2$. It allows to measure how $G$ is far from the celebrated classification of quasihamiltonian groups of K. Iwasawa. Here we generalize $sd(G)$, looking at suitable sublattices of $\mathrm{L}(G)$, and show some new lower bounds. More precisely, we define and study the subgroup S-commutativity degree of a group, which measures the probability that subnormal subgroups commute with maximal subgroups..