speakers of different languages have paid attention to the phonetics of their language since ancient times. muslim scientists have also taken into consideration the value of phonological studies and among them there are linguists who are not less important than any western ones like saussure, chomsky and jacobson. some of them are even counted more major than westerners; one of these muslim sch...

Let R be a nil ring. We prove that primitive ideals in the polynomial ring R[x] in one indeterminate over R are of the form I [x] for some ideals I of R. All considered rings are associative but not necessarily have identities. Köthe’s conjecture states that a ring without nil ideals has no one-sided nil ideals. It is equivalent [4] to the assertion that polynomial rings over nil rings are Jaco...

In this paper we introduce certain classes of graphs that generalize φ-tolerance chain graphs. In a rank-tolerance representation of a graph, each vertex is assigned two parameters: a rank, which represents the size of that vertex, and a tolerance which represents an allowed extent of conflict with other vertices. Two vertices are adjacent if and only if their joint rank exceeds (or equals) the...

We consider a ring R = K[∂; id, θ] of differential operators over a differential field (K, θ). This is a (left and right) principal ideal domain. Hence, if R is simple, then for every matrix M ∈ R n n there exist unimodular matrices S and T ∈ R n n and f ∈ R such that SMT = diag(1, . . . , 1, f, 0, . . . , 0). The proof of the existence goes back to Jacobson, Nakayama and Teichmüller. Therefore...

A use case is a description of a sequence of actions constituting a complete task or transaction in an application. Use cases were first proposed by Jacobson (1987) and have since been incorporated as one of the key modeling constructs in UML (Booch, Jacobson, & Rumbaugh, 1999) and the Unified Software Development Process (Jacobson, Booch, & Rumbaugh, 1999). This paper traces the development of...

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the annihilating-ideal graph of a commutative ring $r$ is denoted by $ag(r)$, whose vertices are all nonzero ideals of $r$ with nonzero annihilators and two distinct vertices $i$ and $j$ are adjacent if and only if $ij=0$. in this article, we completely characterize rings $r$ when $gr(ag(r))neq 3$.

a connected graph g is said to be neighbourly irregular graph if no two adjacent vertices of g have same degree. in this paper we obtain neighbourly irregular derived graphs such as semitotal-point graph, k^{tℎ} semitotal-point graph, semitotal-line graph, paraline graph, quasi-total graph and quasivertex-total graph and also neighbourly irregular of some graph products.