On some punctured codes of ℤq-Simplex codes

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

Quasi-clean rings and strongly quasi-clean rings

An element [Formula: see text] of a ring [Formula: see text] is called a quasi-idempotent if [Formula: see text] for some central unit [Formula: see text] of [Formula: see text], or equivalently, [Formula: see text], where [Formula: see text] is a central unit and [Formula: see text] is an idempotent of [Formula: see text]. A ring [Formula: see text] is called a quasi-Boolean ring if every element of [Formula: see text] is quasi-idempotent. A ring [Formula: see text] is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or [Formula: see text] has no image isomorphic to [Formula: see text]; For an indecomposable commutative semilocal ring [Formula: see text] with at least two maximal ideals, [Formula: see text]([Formula: see text]) is strongly quasi-clean if and only if [Formula: see text] is quasi-clean if and only if [Formula: see text], [Formula: see text] is a maximal ideal of [Formula: see text]. For a prime [Formula: see text] and a positive integer [Formula: see text], [Formula: see text] is strongly quasi-clean if and only if [Formula: see text]. Some open questions are also posed.

On the skew Laplacian spectral radius of a digraph

Let [Formula: see text] be an orientation of a simple graph [Formula: see text] with [Formula: see text] vertices and [Formula: see text] edges. The skew Laplacian matrix [Formula: see text] of the digraph [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the imaginary unit, [Formula: see text] is the diagonal matrix with oriented degrees [Formula: see text] as diagonal entries and [Formula: see text] is the skew matrix of the digraph [Formula: see text]. The largest eigenvalue of the matrix [Formula: see text] is called skew Laplacian spectral radius of the digraph [Formula: see text]. In this paper, we study the skew Laplacian spectral radius of the digraph [Formula: see text]. We obtain some sharp lower and upper bounds for the skew Laplacian spectral radius of a digraph [Formula: see text], in terms of different structural parameters of the digraph and the underlying graph. We characterize the extremal digraphs attaining these bounds in some cases. Further, we end the paper with some problems for the future research in this direction.

Universal method to describe multiple localized modes in photonic crystal heterojunction

A general method is proposed to describe the energy levels of the interface states in one-dimensional photonic crystal (PC) heterojunction [Formula: see text] containing dispersive or non-dispersion materials. We found that the finite energy levels of the interface states for the finite configuration can be described totally by the dispersion relation of the PC with a periodic unit [Formula: see text]. It is further found that this method is also applicable for the case of defect modes. We believe our method can be used to guide the practical application.

Structure of block quantum dynamical semigroups and their product systems

Paschke’s version of Stinespring’s theorem associates a Hilbert [Formula: see text]-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a [Formula: see text]-algebra [Formula: see text] one may associate an inclusion system [Formula: see text] of Hilbert [Formula: see text]-[Formula: see text]-modules with a generating unit [Formula: see text]. Suppose [Formula: see text] is a von Neumann algebra, consider [Formula: see text], the von Neumann algebra of [Formula: see text] matrices with entries from [Formula: see text]. Suppose [Formula: see text] with [Formula: see text] is a QDS on [Formula: see text] which acts block-wise and let [Formula: see text] be the inclusion system associated to the diagonal QDS [Formula: see text] with the generating unit [Formula: see text] It is shown that there is a contractive (bilinear) morphism [Formula: see text] from [Formula: see text] to [Formula: see text] such that [Formula: see text] for all [Formula: see text] We also prove that any contractive morphism between inclusion systems of von Neumann [Formula: see text]-[Formula: see text]-modules can be lifted as a morphism between the product systems generated by them. We observe that the [Formula: see text]-dilation of a block quantum Markov semigroup (QMS) on a unital [Formula: see text]-algebra is again a semigroup of block maps.

Power commuting traces of bilinear maps on invertible elements

Let [Formula: see text] be a unital simple ring. Under a mild technical restriction on [Formula: see text], we characterize bilinear mappings [Formula: see text] satisfying [Formula: see text], and [Formula: see text] for all unit [Formula: see text] and [Formula: see text], where [Formula: see text]. As an application we describe bijective linear maps [Formula: see text] satisfying [Formula: see text] for all invertible [Formula: see text]. Precisely, we will show that [Formula: see text] is an isomorphism.

The semantic indications of the wording (afeal) in the Noble Qur’an

This study aims at investigating the Quranic words which are formed using the metrical unit "Formula afeal function on Sensual meanings”. It is well-known that understanding semantics in Quran is based upon understanding the connotation of its lexical items in addition to the structures used in it. The current study also enriches the Quranic and linguistic studies. The significance of this study lies in the fact that it locates the words which are formed using the metrical unit "Formula 'afeal" then reveals the semantic and morphological meanings of this group of words. It also sheds light on some linguistic phenomena that are usually related to such kind of studies. The study throws light on the role of context in determining the semantic meaning, regarding attentively both synonymy and polysemy and referring to some examples of collocation.

Special clean elements in rings

A clean decomposition [Formula: see text] in a ring [Formula: see text] (with idempotent [Formula: see text] and unit [Formula: see text]) is said to be special if [Formula: see text]. We show that this is a left-right symmetric condition. Special clean elements (with such decompositions) exist in abundance, and are generally quite accessible to computations. Besides being both clean and unit-regular, they have many remarkable properties with respect to element-wise operations in rings. Several characterizations of special clean elements are obtained in terms of exchange equations, Bott–Duffin invertibility, and unit-regular factorizations. Such characterizations lead to some interesting constructions of families of special clean elements. Decompositions that are both special clean and strongly clean are precisely spectral decompositions of the group invertible elements. The paper also introduces a natural involution structure on the set of special clean decompositions, and describes the fixed point set of this involution.

Unital locally matrix algebras and Steinitz numbers

An [Formula: see text]-algebra [Formula: see text] with unit [Formula: see text] is said to be a locally matrix algebra if an arbitrary finite collection of elements [Formula: see text] from [Formula: see text] lies in a subalgebra [Formula: see text] with [Formula: see text] of the algebra [Formula: see text], that is isomorphic to a matrix algebra [Formula: see text], [Formula: see text]. To an arbitrary unital locally matrix algebra [Formula: see text], we assign a Steinitz number [Formula: see text] and study a relationship between [Formula: see text] and [Formula: see text].

Generalized inverses and their relations with clean decompositions

An element [Formula: see text] in a ring [Formula: see text] is called clean if it is the sum of an idempotent [Formula: see text] and a unit [Formula: see text]. Such a clean decomposition [Formula: see text] is said to be strongly clean if [Formula: see text] and special clean if [Formula: see text]. In this paper, we prove that [Formula: see text] is Drazin invertible if and only if there exists an idempotent [Formula: see text] and a unit [Formula: see text] such that [Formula: see text] is both a strongly clean decomposition and a special clean decomposition, for some positive integer [Formula: see text]. Also, the existence of the Moore–Penrose and group inverses is related to the existence of certain ∗-clean decompositions.