In this Article, we give a simple criterion for the regularity of a tri-linear mapping. We provide if f : X × Y × Z −→ W is a bounded tri-linear mapping and h : W −→ S is a bounded linear mapping, then f is regular if and only if hof is regular. We also shall give some necessary and sufficient conditions such that the fourth adjoint D^∗∗∗∗ of a tri-derivation D is again tri-derivation.

A mapping $f:V^n longrightarrow W$, where $V$ is a commutative semigroup, $W$ is a linear space and $n$ is a positive integer, is called multi-additive if it is additive in each variable. In this paper we prove the Hyers-Ulam stability of multi-additive mappings in 2-Banach spaces. The corollaries from our main results correct some outcomes from [W.-G. Park, Approximate additive mappings i...

a mapping $f:v^n longrightarrow w$, where $v$ is a commutative semigroup, $w$ is a linear space and $n$ is a positive integer, is called multi-additive if it is additive in each variable. in this paper we prove the hyers-ulam stability of multi-additive mappings in 2-banach spaces. the corollaries from our main results correct some outcomes from [w.-g. park, approximate additive mappings i...

In this paper, we obtain a necessary and sufficient condition for a conformal mapping between two Weyl manifolds to preserve Einstein tensor. Then we prove that some basic curvature tensors of $W_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. Also, we obtained the relation between the scalar curvatures of the Weyl manifolds r...

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $n$. A $k$-vertex weightings of a graph $G$ is a mapping $w: V(G) to {1, ldots, k}$. A $k$-vertex weighting induces an edge labeling $f_w: E(G) to N$ such that $f_w(uv)=w(u)+w(v)$. Such a labeling is called an {it edge-coloring k-vertex weightings} if $f_{w}(e)not= f_{w}(echr(chr(chr('39')39chr('39'))39chr(chr('39')39chr('39'...

in this paper, we obtain a necessary and sufficient condition for a conformal mapping between two weyl manifolds to preserve einstein tensor. then we prove that some basic curvature tensors of $w_n$ are preserved by such a conformal mapping if and only if the covector field of the mapping is locally a gradient. also, we obtained the relation between the scalar curvatures of the weyl manifolds r...

We analyze the mapping class group Hx(W ) of automorphisms of the exterior boundary W of a compression body (Q,V ) of dimension 3 or 4 which extend over the compression body. Here V is the interior boundary of the compression body (Q,V ). Those automorphisms which extend as automorphisms of (Q,V ) rel V are called discrepant automorphisms, forming the mapping class group Hd(W ) of discrepant au...

‎Let $mathcal{A}$ be a unital Banach algebra‎, ‎$mathcal{M}$ be a left $mathcal{A}$-module‎, ‎and $W$ in $mathcal{Z}(mathcal{A})$ be a left separating point of $mathcal{M}$‎. ‎We show that if $mathcal{M}$ is a unital left $mathcal{A}$-module and $delta$ is a linear mapping from $mathcal{A}$ into $mathcal{M}$‎, ‎then the following four conditions are equivalent‎: ‎(i) $delta$ is a Jordan left de...

This paper investigates the role of topological methods in the analysis of canardtype periodic and chaotic trajectories. In Sections 1 – 5 we apply topological degree [1, 2] to the analysis of multi-dimensional canards. This part of the paper was written mainly by the first and the last authors. Sections 6 – 7 are devoted to an application of a special corollary of the Poincaré–Bendixson theore...

In this paper, we define a new concept of factorization for a bounded bilinear mapping $f:Xtimes Yto Z$, depended on  a natural number $n$ and a cardinal number $kappa$; which is called $n$-factorization property of level $kappa$. Then we study the relation between $n$-factorization property of  level $kappa$ for $X^*$ with respect to $f$ and automatically boundedness and $w^*$-$w^*$-continuity...