Let $n$ be a positive integer. An $n$-manifold $M$ is a Hausdorff topological space with a countable base of open sets such that $M$ is locally the Euclidean space $\mathbb{R}^n$, that is, for each $x \in M$ there exists an open neighborhood $U$ of $x$ and a homeomorphism $U \stackrel{\varphi}{\to} V$ from $U$ onto an open subset $V$ of $\mathbb{R}^n$. A compact $n$-manifold is an $n$-manifold which is compact as a topological space. It is clear that any $n$-manifold is locally path-connected and so is a path-connected space if it is a connected space. It is also clear that a compact $n$-manifold has only a finite number of path components, each of these being a compact path-connected $n$-manifold.

$\mathbb{R}^n$ and any of its non-empty open subsets are examples of $n$-manifolds. These are not compact manifolds. Let \[ S^n = \{ x = (x_1,\dotsc,x_{n+1}) \in \mathbb{R}^{n+1} |\ \|x\| = \sqrt{x^2_1+\cdots+x^2_{n+1}} = r \gt 0 \}, \] called the $n$-sphere (of radius $r$). $S^n$ is a path-connected compact $n$-manifold (note that $n \ge 1$). It is easy to see that any two $n$-spheres of different radii are homeomorphic.

Recently, Chan, Chyan and Srivastava [1] introduced and systematically investigated the Lagrange polynomials in several variables. The main object of this paper is to study a basic (or $q-$) analogue of the Chan-Chyan-Srivastava multivariable polynomials, which we define here. Several results involving generating functions are presented in this paper for the multivariable $q-$Lagrange polynomials.

On a foliated Riemannian manifold with a transverse spin structure, we give a lower bound of the square for the eigenvalues of the basic Dirac operator in terms of the smallest eigenvalue of the basic Yamabe operator and of the norm of an appropriate endomorphism of the normal bundle $Q$ of $\mathcal F$. We study the limiting case.

A (partial) Latin square is a table of multiplication of a (partial) quasigroup. Multiplication of a (partial) quasigroup may be considered as a set of triples. We give a necessary and sufficient condition for a set of triples to be a quotient of a (partial) latin square.

In this paper, we obtain some new sufficient conditions for starlikeness of order $\alpha$ of biholomorphic mappings on the unit ball in $C^n$ or a complex Hilbert space $X$ by using differential inequalities. We also obtain a distortion theorem and a covering theorem. As their special case, we obtain some sufficient conditions for starlikeness of order $\alpha$ of analytic functions on the unit disc in the complex plane $C$, which generalize some results of P. T. Mocanu and G. Oros.

Let $R$ be a prime ring without nonzero nil one-sided ideals. Suppose that $g$ is a generalized derivation of $R$ and that $f(X_{1},\cdots,X_{k})$ is a multilinear polynomial not central-valued on $R$ such that $g(f(x_{1},\cdots,x_{k}))$ is nilpotent for all $x_{1},\cdots,x_{k}$ in some nonzero ideal of $R$. Then $g=0$.

For a $\gamma$-th order Cesàro mean bounded linear operator $T$ on a Banach space $X$, we characterize the range $R(A)$ of the operator $A = T-I$, by using an $A$-ergodic net and its companion net which were introduced by Dotson and developed by Shaw. Similarly, if $A$ is the generator of a $\gamma$-th order Cesàro mean bounded $C_{0}$-semigroup (or strongly continuous cosine operator function) of bounded linear operators on $X$, then we characterize the range $R(A)$.

In this paper, the notion of crosstalk-free rearrangeability (CF-rearrangeability) of multistage interconnection networks (MINs) is formally defined. Using the concept of line digraphs from graph theory, we show that the problem of crosstalk-free routing on any bit permutation network (BPN) is always equivalent to the classical permutation routing problem on a BPN of smaller size and with fewer stages. We also show the CF-rearrangeability and minimality (in stage number) of three families of BPNs, including the dilated Benes network. Some necessary conditions for a BPN to be CF-rearrangeable are given, and a brief discussion of CF-rearrangeable networks with dilation in time or space is included.

Let $\{X_i : i \in \mathbb{N} \}$ be a family of Banach space and let $Y_i \subseteq X_i$ be a closed subspace in $X_i$ for each $i \in \mathbb{N}$ such that at least two $Y_i'$s are non-trivial. Consider $X = (\oplus_{c_0} X_i)_{i \in \mathbb{N}}$ and $Y = (\oplus_{c_0} Y_i)_{i \in \mathbb{N}}$. We show that $Y$ is strongly proximinal in $X$ if and only if $P_Y$ is upper Hausdorff semi-continuous on $X$ if and only if $Y_i$ is strongly proximinal subspace in $X_i$ for each $i \in \mathbb{N}$. This shows that in [9, Theorem 3.4], strong proximinality of $Y_i$'s is a necessary assumption. We also show that lower semi-continuity of metric projections is stable in $c_0$-direct sums.

Let $I$ be a regular ideal of a ring $R$. It is shown that $I$ is a $QB$-ideal if and only if for all finitely generated projective right $R$-module $A$ with $AI=A$, if $B_1$ and $B_2$ are any right $R$-modules such that $A \oplus B_1 \cong A \oplus B_2$, then there exists a pair of orthogonal ideals $I_1$ and $I_2$ and $B_1 \oplus C_1 \cong B_2 \oplus C_2$ such that $C_1 I_1 \cong C_1$ and $CI_2 \cong C_2$.

Several inequalities of Jensen’s type for functions convex on the co-ordinates are given. Obtained results generalize the coresponding results of S. S. Dragomir given in [2].

In this paper we introduce an iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality problem for a monotone, Lipschitz continuous mapping. The iterative process is based on so-called extragradient method. We obtain a strong convergence theorem for two sequences generated by this process.

For a given connected graph $G$ of order $n$, a routing $R$ is a set of $n(n-1)$ elementary paths specified for every ordered pair of vertices in $G$. The vertex-forwarding index $\xi(G)$ (the edge-forwarding index $\pi(G)$) of $G$ is the maximum number of paths of $R$ passing through any vertex (resp. edge) in $G$. In this paper we consider the vertex- and the edge- forwarding indices of the cartesian product of $k$ ($\ge 2$) graphs. As applications of our results, we determine the vertex- and the edge- forwarding indices of some well-known graphs, such as the $n$-dimensional generalized hypercube, the undirected toroidal graph, the directed toroidal graph and the cartesian product of the Petersen graphs.

We continue the study of $\{k\}$-dominating functions in graphs (or integer domination as we shall also say) started by Domke, Hedetniemi, Laskar, and Fricke [5]. For $k \ge 1$ an integer, a function $f \colon V(G) \rightarrow \{0,1,\ldots,k\}$ defined on the vertices of a graph $G$ is called a $\{k\}$-dominating function if the sum of its function values over any closed neighborhood is at least~$k$. The weight of a $\{k\}$-dominating function is the sum of its function values over all vertices. The $\{k\}$-domination number of $G$ is the minimum weight of a $\{k\}$-dominating function of $G$. We study the $\{k\}$-domination number on the Cartesian product of graphs, mostly on problems related to the famous Vizing's conjecture. A connection between the $\{k\}$-domination number and other domination type parameters is also studied.

Consider the nonlinear matrix equation $X + A^* X^{-2}A = I$, where $A$ is an $n \times n$ complex matrix, $I$ the identity matrix and $A^*$ the conjugate transpose of the matrix $A$. In this paper a perturbation bound for a class of special solutions of this matrix equation is derived, and an explicit expression of its condition number is obtained. The results are illustrated by using some numerical examples.

We consider a model of oriented percolation on ${\mathbb Z}^d \times {\mathbb Z}$, $d \gt 2$, with long-range interactions, in which the bond occupation probability decays as the $\alpha$-stable distribution with $\alpha = 1$. We use the lace expansion to get an $L^1$ infrared bound estimate which implies several critical exponents via the triangle condition.

Let $\{ \rm X_i;\, -\infty \lt i \lt \infty \}$ be a stationary sequence of random variables. Let $\rm F_n(x)$ be the corresponding empirical distribution function of $\rm X_1,\ldots,\rm X_n$, and let $\bar{\rm X} = \sum^n_{i=1} \rm X_i/n$ be the sample mean. In this paper, we derive the asymptotic almost sure representation, the central limit theorem, a law of iterated logarithm, a Wiener precess embedding and an invariant principle for $\rm F_n(\bar{\rm X})$ under different $\phi$-mixing conditions.

In this paper, we study the singularities in contact 3-Manifolds. We give simple presentations for singularities, make an initial classification of them and show that the space of all singularities has continuous moduli. We also study the stabilities of singularities and obtain that a nondegenerate singular point is isolated. Finally, by means of the transversality theorem of Thom, we show that generically every immersion contains no degenerate singular points. Thus the singular set of a closed surface is finite.