Let $K$ be a field, $f(x)$ a monic polynomial in $K[x]$ and $KG$ the path algebra of a cyclic quiver $G$ with $s$ vertices and $s$ arrows. In this paper, we give a necessary and sufficient condition for the algebra $K\Gamma/(f(X))$ to be a symmetric algebra, where $X$ is the sum of all arrows in $K\Gamma$.

By defining new concepts like pseudocomplements in graphs a new class of graphs is obtained. They have very many properties in common with hypercubes and therefore they are called pseudocubes. Pseudocubes are Hasse diagram graphs (covering garphs) of finite lattices, where pseucomplements constitute a sublattice. As an application, the routing and fault tolerance properties of certain pseudocubes are determined.

In this paper, the second Gaussian and the second mean curvature of the helicoidal surfaces with light-like axis of type $IV^{+}$ is obtained in Minkowski 3-space. In addition, some relations between the mean, Gauss, the second Gaussian and the second mean curvature of the helicoidal surfaces with light-like axis of type $IV^{+}$ are given in Minkowski 3-space.

This paper provides a careful and accessible exposition of an $L^p$ approach to boundary value problems of nonlinear elastostatics in the case where solutions of the linearized problem correspond faithfully to those of the nonlinear problem, that is, in the case where there is no bifurcation. We prove that if the linearized problem has unique solutions, then so does the nonlinear one, nearby. This is done by using the linear $L^p$ theory and the inverse mapping theorem. The main theorem can be applied to the Saint Venant-Kirchhoff elastic material and the Hencky-Nadai elastoplastic material in a unified theory. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations.

In this paper, we shall discuss submetacompactness and weak submetacompactness in countable products of Čech-scattered spaces and prove the following: (1) If $\{ X_{n} : n \in \omega \}$ is a countable collection of submetacompact Čech-scattered spaces, then the product $\Prod_{n\in \omega} X_{n}$ is submetacompact. (2) If $Y$ is a hereditarily weakly submetacompact space and $\{X_{n} : n \in \omega \}$ is a countable collection of weakly submetacompact Čech-scattered spaces, then the product $Y \times \Prod_{n\in \omega}X_{n}$ is weakly submetacompact.

The concept of fuzzy sets was introduced by L. A. Zadeh in 1965. The concept of fuzzy graph was introduced by A. Rosenfeld [5] in 1975. Many of the crisp graph concepts have been extended to fuzzy graph theory. Here we define the fusion of two vertices in a fuzzy graph and investigate some properties and also give fusion algorithm for effective connectedness and for adjacency matrix.

In this paper $\sigma$-point-discrete weak bases are considered. Three necessary conditions that individually ensure that a space with a $\sigma$-point-discrete weak base has a s-compact-finite weak base are given. We show that $\sigma$-compact-finite weak bases are preserved by closed sequence-covering maps. It is shown that a space $X$ is metrizable if and only if $X^{\omega}$ has a $\sigma$-point-discrete weak base. Conditions are given to ensure when a paratopological group with $\sigma$-point-discrete weak base is metrizable. Several open questions are posed.

We introduce new notions of $p$-maps and $M$-maps, and investigate some of their basic properties, which are extensions of corresponding properties of $p$-spaces and $M$-spaces.

We determine periodic and aperiodic points of certain piecewise affine maps in the Euclidean plane. Using these maps, we prove for $\lambda \in \left\{ \frac{\pm 1 \pm \sqrt{5}}{2}, \pm \sqrt{2}, \pm{\sqrt{3}} \right \}$ that all integer sequences $(a_{k})_{k \in \mathbb{Z}}$ satisfying $0 \leq a_{k-1}+\lambda a_{k} + a_{k+1} \lt 1$ are periodic.