‎for any group $g$‎, ‎we define an equivalence relation $thicksim$ as below‎: ‎[forall g‎, ‎h in g gthicksim h longleftrightarrow |g|=|h|]‎ ‎the set of sizes of equivalence classes with respect to this relation is called the same-order type of $g$ and denote by $alpha{(g)}$‎. ‎in this paper‎, ‎we give a partial answer to a conjecture raised by shen‎. ‎in fact‎, ‎we show that if $g$ is a nilpote...

We examine the problem of equivalence of discrete time auto-regressive representations (DTARRs) over a finite time interval. Two DTARRs are defined as fundamentally equivalent (FE) over a finite time interval [0, N ] if their solution spaces or behaviours are isomorphic. We generalise the concept of strict equivalence (SE) of matrix pencils to the case of general polynomial matrices and in turn...

We define an extended fuzzy equivalence relation, discuss some basic properties of extended fuzzy equivalence relations, find the extended fuzzy equivalence relation generated by a fuzzy relation in a set, and give some lattice theoretic properties of extended fuzzy equivalence relations.

A complementation operation on a vertex of a digraph changes all outgoing arcs into non-arcs, and outgoing non-arcs into arcs. This defines an equivalence relation where two digraphs are equivalent if one can be obtained from the other by a sequence of such operations. We show that given an adjacency-list representation of a digraph G, many fundamental graph algorithms can be carried out on any...

The largest class of algebraic hyper structures satisfying the module like axioms is the v H module. In this paper, we consider the category of v H -modules and prove that the direct limit always exists in this category. Direct limits are defined by a universal property, and so are unique. The most powerful tool in order to obtain a module from a given v H module is the quotient out procedure. ...

Lexical items play a key role in both language in general and translation in particular. Likewise, equivalence is a controversial concept discussed so widely in translation studies. Some theorists deem it to be fundamental in translation theory and define translation in terms of equivalence. The aim of this study is to identify the problems of lexical gaps in two translations of Nahj al-ba...

Languages such as ML or Lisp permit the use of recursively defined function expressions with locally declared storage locations. Although this can be very convenient from a programming point of view it severely complicates the properties of program equivalence even for relatively simple fragments of such languages—such as the simply typed fragment of Standard ML with integer-valued references c...

‎Let $G$ be a non-abelian group and let $Gamma(G)$ be the non-commuting graph of $G$‎. ‎In this paper we define an equivalence relation $sim$ on the set of $V(Gamma(G))=Gsetminus Z(G)$ by taking $xsim y$ if and only if $N(x)=N(y)$‎, ‎where $ N(x)={uin G | x textrm{ and } u textrm{ are adjacent in }Gamma(G)}$ is the open neighborhood of $x$ in $Gamma(G)$‎. ‎We introduce a new graph determined ...

We deal with the new class of pre-ℐ-regular pre-ℐ-open sets in which the notion of pre-ℐ-open set is involved. We characterize these sets and study some of their fundamental properties. We also present other notions called extremally pre-ℐ-disconnectedness, locally pre-ℐ-indiscreetness, and pre-ℐ-regular sets by utilizing the notion of pre-ℐ-open and pre-ℐ-closed sets by which we obtain some eq...

Relative self-linking and linking " numbers " for pairs of knots and 2-component links in oriented 3-manifolds are defined in terms of intersection invariants of immersed surfaces in 4-manifolds. The resulting concordance invariants generalize the usual ho-mological notion of linking by taking into account the fundamental group of the ambient manifold and often map onto infinitely generated gro...