The concept of Chen ideal submanifolds is illustrated by characterizing the complex hypercylinders in the complex Euclidean spaces and the complex hyperquadrics in the complex projective space forms in terms of equalities involving their intrinsic and normal scalar curvatures and simplest “delta” curvatures.

In this paper, we investigate the generalized Hyers-Ulam-Rassias stability of an $n$-dimensional functional equation \begin{equation*} f\Big(\sum_{i=1}^{n}x_i, \sum_{i=1}^{n}y_i\Big)+\!\sum_{1\le i \lt j\le n}f(x_i-x_j, y_i-y_j)=n\sum_{i=1}^{n}f(x_i, y_i), \,\,\, (n\ge2). \end{equation*}

The boundedness and compactness of the weighted composition operator between $H^\infty$ and the $\alpha$-Bloch space $\mathcal{B}^\alpha$ on the unit ball are discussed in this paper.

Let $R$ be a noncommutative prime ring and $a\in R$. Suppose that $f$ is a right generalized $\beta$-derivation of $R$ such that $a[f(x),x]_k=0$ for all $x\in R$, where $k$ is a fixed positive integer. Then $a=0$ or there exists $s\in C$ such that $f(x)=sx$ for all $x\in R$ except when $R=M_2(GF(2))$.

We consider a special class of the generalized Vandermonde matrices and obtain an LU factorization for its member by giving closed-form formulae of the entries of $L$ and $U$. Moreover, we express the matrices $L$ and $U$ as products of 1-banded (bidiagonal) matrices. Our result is applied to give the closed-form formula of the inverse of the considered matrix.

In this paper, we obtain the duration of deep earthquakes coming from the concept of pendulum motion. We consider the well-posedness and periodicity of the solutions for the seismic equation and get representation formula of duration for the equation. We revise the validity of our work by using numerical data gained from Center Weather Bureau in Taipei, Taiwan.

In this paper, by using a new maximum principle and the Fredholm alternative, the monotone method in the presence of lower and upper solutions for the periodic problem $$ \begin{array}{rl} u^{(iv)}(x) & \hspace{-0.2cm} = f(x,u(x),u^{\prime\prime}(x)) , \ \ 0\lt x \lt 2 \pi , \\ u^{(i)}(0)& \hspace{-0.2cm} =u^{(i)}(2 \pi) , \ \ i=0, 1, 2, 3 \end{array} $$ is developed, where $f:[0, 2 \pi] \times R^2 \longrightarrow R$ is a Caratheodory function.

This paper proposes a modified approximate proximal point algorithm to solve the problem of finding zeros of a maximal monotone operator in a Hilbert space. New accuracy criteria are imposed. Weak and strong convergence results are established.

In this paper we define the contact number of a pseudo-Riemannian submanifold into the pseudo-Euclidean space, and prove that this contact number is closely related to the notion of pseudo-isotropic submanifold. We give a classification of hypersurfaces into the pseudo-Euclidean space with contact number at least 3. A classification of the complete spacelike codimension-2 submanifolds of the Lorentz-Minkowski space with contact number at least 3 is also obtained.

Let $R$ be a ring with unity. It is shown that the formal power series ring $R[[x]]$ is right p.q.-Baer if and only if $R$ is right p.q.-Baer and every countable subset of right semicentral idempotents has a generalized countable join.

In this paper, we prove that if two distinct non-constant meromorphic functions $f$ and $g$ share four distinct values $a_1, a_2, a_3, a_4$ \DM \ such that each $a_i$-point is either a $(p,q)$-fold or $(q,p)$-fold point of $f$ and $g$, then $(p,q)$ is either $(1,2)$ or $(1,3)$ and $f, g$ are in some particular forms.

We give a loop group formulation for the problem of isometric immersions with flat normal bundle of a simply connected pseudo-Riemannian manifold $M_{c,r}^m$, of dimension $m$, constant sectional curvature $c \neq 0$, and signature $r$, into the pseudo-Euclidean space $\bf R_s^{m+k}$, of signature $s\geq r$. In fact these immersions are obtained canonically from the loop group maps corresponding to isometric immersions of the same manifold into a pseudo-Riemannian sphere or hyperbolic space $S_s^{m+k}$ or $H_s^{m+k}$, which have been known for some time. A simple formula is given for obtaining these immersions from those loop group maps.

The purpose of this paper is to extend the work of Jungck and Sessa [4] from single-valued maps to multivalued maps.

For $1\leq p\leq \infty , A_{w,\omega }^{p}\left( R^{d}\right) $ denotes the space (Banach space) of all functions in $L_{w}^{1}\left( R^{d}\right) $ a weighted $L^{1}-$space (Beurling algebra) with Fourier transforms $\overset{\wedge }{f}$ in\ $L_{\omega }^{p}\left( R^{d}\right) $ which is equipped with the sum norm \begin{equation*} \left\Vert f\right\Vert _{w,\omega }^{p}=\left\Vert f\right\Vert _{1,w}+\left\Vert \overset{\wedge }{f}\right\Vert _{p,\omega }, \end{equation*}% where $w$ and $\omega $ are Beurling weights on $R^{d}$.This space was defined in $\left[ 5\right] $ and generalized in $\left[ 6\right] .$ The present paper is a sequal to these works.In this paper we are going to discuss compact embeddings between the spaces $A_{w,\omega }^{p}\left( R^{d}\right) .$

Suppose $ A$ is a Banach algebra and $\epsilon$ is in A with $\|\epsilon\|\leq 1$. In this note we aim to study the algebraic properties of the Banach algebra $ A_\epsilon$, where the product on $ A_\epsilon$ is given by $a \odot b = a \epsilon b$, for $a, b \in A$. In particular we study the Arens regularity, amenability and derivations on $ A_\epsilon$. Also we prove that if $ A$ has an involution then $ A_\epsilon$ has the same involution just when $\epsilon = 1$ or $-1$.

In this paper, we obtain the representation for a reversed point martingale with respect to the reversed filtration generated by a point process. Besides, if a martingale can be expressed as a certain kind of stochastic integral with respect to some point martingale, then its reversed counterpart can also be expressed as a stochastic integral with respect to the corresponding reversed point martingale.

This paper compares the conditional metric entropy $h_{\mu}(T\mid G)$, with the topological entropy, $h_{top}(T\mid G)$, of a continuous map $T$, where $G$ is a closed fully $T$-invariant subset. The following Variational Inequality is proven, $$h_{top}(T\mid G)\leq \sup _{\mu\in M(X,T)}h_{\mu}(T\mid \langle G \rangle )\leq h_{top}(T\mid G)+h_{top}(T\mid cl(X\backslash G))$$ where $M(X,T)$ is the collection of all invariant measures of $X$, which is an extension of the usual variational principle when $G=X$.

We propose an algorithm for the pole assignment problem for descriptor systems with proportional and derivative state feedback. The algorithm is the first of its kind, making use of the Schur form and minimizing the departure from normality of the closed-loop poles by Newton’s method. Three illustrative examples are given.

We consider the general boundary value problem for a degenerate semilinear functional differential inclusion in a Banach space with infinite delay. We construct the multivalued integral operator whose fixed points are mild solutions of the above problem and study its properties. We apply the topological degree method to obtain the general existence principle and consider some particular cases, including Cauchy and periodic problems.

The purpose of this paper is to establish the well-posedness and the regularity of solutions of the Cauchy-Neumann problem for second order parabolic equations in cylinders with the base containing conical points.

Convergence criterion of the inexact methods is established for operators with h¨older continuous first derivatives. An application to a special nonlinear Hammerstein integral equation of the second kind is provided.