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TitleOn the Connectedness of Square Element Graphs over Arbitrary Rings
Vol.43(2) (2019) page: 153-164
Author(s): B. Biswas; R.S. Gupta; M.K. Sen; S. Kar  

In this paper, we continue our study of the square element graph which is defined as follows: if R is a ring then the square element graph \mathbb Sq(R) is the simple undirected graph whose vertex set consists of all non-zero elements of R, and two vertices u, v are adjacent if and only if u+v=x2 for some x ∈ R−{0}. Here we mainly consider the connectedness of \mathbb Sq(R) over different rings. We particularly look at \mathbb Sq(R) taken over some infinite rings which contain \mathbb Z as a subring. For a ring R and an ideal I of R, the relation between the connectedness of \mathbb Sq(R) and \mathbb Sq(R/I) is studied. Finally, connectedness of \mathbb Sq(R) over various polynomial rings and matrix rings are considered.

Keywords: Square element graph; Infinite graph; Connectedness; Diameter.
Size:145KB   download
TitleDifferential Identities and Generalized Derivations in Prime Rings with Involution
Vol.43(2) (2019) page: 165-181
Author(s): A. Boua; M. Ashraf  

Let R be a ∗-ring with the center Z(R). In the present paper we study commutativity of a prime ring which admits derivations d, g satisfying any one of the properties:

(i)] d([x, x]) ±[x, x] ∈ Z(R),
(ii)]d(x°x) ±x°x ∈ Z(R),
(iii)]d([x, x]) ±x°x ∈ Z(R),
(iv)]d(x°x) ±[x,x] ∈ Z(R),
(v)] d(x)°d(x) ±x°x ∈ Z(R),
(vi)]d(x)g(x) ±[x,x] ∈ Z(R) and
(vii)] d(x)g(x) ±x°x ∈ Z(R),

for all x ∈ R. Further, some related results are also obtained if R admits a generalized derivation F satisfying either of the properties

F(xx)±F(x)F(x) ∈ Z(R)

for all x ∈ R or

F(xx)±F(x)F(x) ∈ Z(R)

for all x ∈ R. Finally, an example has been provided to justify the hypotheses in various results.

Keywords: Prime rings; Involution; Derivation; Generalized derivations.
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TitleOperators Induced by Toeplitz and Hankel Operators
Vol.43(2) (2019) page: 183-192
Author(s): G. Datt; A. Mittal  

In this paper, we introduce a new class of operators called Toep-Hank operators on the space H2(\BbbT) with the help of already existing class of operators namely Hankel and Toeplitz operators and discuss its structural properties. A matrix characterization for an operator to be Toep-Hank operator is obtained. Commutators of Toep-Hank operators and Toeplitz operators are computed.

Keywords: Toeplitz operators; Hankel operators; Hilbert-Schmidt operator; Hyponormal operator.
Size:136KB   download
TitleSimplicity of Special Algebras over Laurent Polynomial Algebra
Vol.43(2) (2019) page: 193-206
Author(s): W. Jeon; K.-B. Nam  

Special algebras of Cartan type have been generalized in several directions. In this paper, we define them based on the Laurent polynomial algebra over a field of characteristic 0. These algebras are infinite dimensional Lie algebras and spanned by Dijh(f)=h{∂j(f)∂i−∂i(f)∂j} where f and h are polynomials. First we extend polynomials to Laurent polynomials and to rational functions later. We derive necessary and sufficient conditions for simple Lie algebras over Laurent polynomials, and we investigate the unique minimal ideal for the case of a field of rational function.

Keywords: Special Lie algebras; Generalized Cartan type; Lie algebras of Cartan type S or S*; Infinite dimensional simple Lie algebras.
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TitleOn Projection Algebras
Vol.43(2) (2019) page: 207-215
Author(s): Z. Khanjanzadeh; A. Madanshekaf  

Projection algebras (spaces) can be interpreted as an algebraic version of ultrametric algebras. Computer scientists use these algebras as a convenient means for algebraic specification of process algebras. A topos-theoretic notion related to the topos PRO of projection algebras is the ideal topology which has been investigated by the authors. In this manuscript, first, we show that for any element m of the monoid \mathbbN of extended natural numbers with the binary operation m·n = min{m, n} the notion of `jm-separated' is the same as `m-separated' in PRO. Then using this fact the associated sheaf functor with respect to jm is explicitly described. Furthermore, for any m ∈ \mathbbN, we provide another form of jm in terms of the implication map ↓(m+1)⇒(−) on the Heyting algebra of (right) ideals of \mathbbN. Finally, we establish a strict chain of De Morgan (sheaf) topoi in PRO.

Keywords: Projection algebra; (Lawvere-Tierney) topology; Associated sheaf functor; Topos.
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TitleA Note on Presentation of General Linear Groups over a Finite Field
Vol.43(2) (2019) page: 217-224
Author(s): S. Maheshwari; R.K. Sharma  

In this article we have given Lie regular generators of linear group GL(2,\mathbb\mathbbFq), where \mathbbFq is a finite field with q = pn elements. Using these generators we have obtained presentations of the linear groups GL(2,\mathbb\mathbbF2n) and GL(2,\mathbb\mathbbFpn) for each positive integer n.

Keywords: Lie regular units; General linear group; Presentation of a group; Finite field.
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TitleCharacterization of Modified Weibull Distribution Based on Dual Generalized Order Statistics
Vol.43(2) (2019) page: 225-242
Author(s): N.M. Mohamed  

In this paper we derive some new recurrence relations for single and product moments of Dual generalized order statistics from Modified Weibull distribution (MWD). Also, using a recurrence relation for single moment to characterize this distribution. Further, recurrence relations for marginal and joint moment generating functions of Dual generalized order statistics from MWD are established. The results for order statistics and lower record values are deduced. Finally, characterization by using joint moment generating functions of Dual generalized order statistics is discussed.

Keywords: Modified Weibull distribution; Dual generalized order statistics; Recurrence relations; Single and product moments; Moment generating function; Characterization.
Size:152KB   download
Title$\phi$-Classical Prime Submodules
Vol.43(2) (2019) page: 243-262
Author(s): H. Mostafanasab; E. Aslankarayigit Ugurlu  

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. A proper submodule N of M is called a classical prime submodule, if for each m ∈ M and elements a,b ∈ R, abm ∈ N implies that am ∈ N or bm ∈ N. Let ??????:S(M)→ S(M)∪{∅} be a function where S(M) is the set of all submodules of M. We introduce the concept of "??????-classical prime submodules". A proper submodule N of M is a ??????-classical prime submodule if whenever a,b ∈ R and m ∈ M with abm ∈ N\??????(N), then am ∈ N or bm ∈ N.

Keywords: Classical prime submodule; Weakly classical prime submodule; ??????-Classical prime submodule.
Size:194KB   download
TitleLocal Properties of Generalized Absolute Matrix Summability of Factored Fourier Series
Vol.43(2) (2019) page: 263-272
Author(s): H.S. Ozarslan  

In this paper, a theorem on local property of φ−|A,pn|k summability of factored Fourier series, which generalizes a known result, has been proved. This new theorem also includes several new and known results.

Keywords: Summability factors; Absolute matrix summability; Fourier series; Local property; Infinite series; H??????lder inequality; Minkowski inequality.
Size:116KB   download
TitleSome Recent Results of Fibonacci Numbers, Fibonacci Words and Sturmian Words
Vol.43(2) (2019) page: 273-286
Author(s): G. Pirillo  

The discovery of incommensurability is a fundamental accomplishment of the Pythagorean School, active in Crotone (Southern Italy) between the sixth and fourth centuries BC. A strong relation can be pointed out between the discovery and several concepts studied today in the field of discrete mathematics (Fibonacci Numbers, Fibonacci Word and Sturmian Words). In this paper, we will cover some of our previous results on Sturmian Words (a class of infinite words containing Fibonacci Word as main example) published in theoretical computer science journals, as well as some geometric constructions published in the Italian journal Nuova Secondaria, in 2005. Also, we will present some ideas that may be useful in mathematical education. Lastly, arguments supporting our thesis that Fibonacci Numbers were known by the Pythagorean School will be given.

Keywords: Incommensurability; Cutting sequence; Golden ratio; Fibonacci numbers; Fibonacci word.
Size:165KB   download
TitleOn Characterization of Quasi-complemented Almost Distributive Lattices
Vol.43(2) (2019) page: 287-295
Author(s): G.C. Rao; G. Nanaji Rao; A. Lakshmana  

Characterization of quasi-complemented ADLs in terms of its α-ideals are studied. Some sufficient conditions of weakly disjunctive ADL to become quasi-complemented ADLs are derived.

Keywords: Almost Distributive Lattice; Prime ideal; Annihilator ideal; ??-ideal; Weakly disjunctive ADL; Disjunctive ADL, quasi-complemented ADLs.
Size:109KB   download
TitleA Note on Morita Equivalence Preserving Functor from the Category of Semigroups to the Category of Semirings
Vol.43(2) (2019) page: 297-304
Author(s): S.K. Sardar; S. Gupta  

In this paper we have introduced a nice functorial connection between the category of semigroups and the category of semirings. Among other results we have proved that Morita equivalence is preserved by this functor.

Keywords: Power functor; Morita equivalence of semigroups (semirings); Morita context of semigroups (semirings).
Size:111KB   download
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