8th Annual International Conference 2024 [45]

Characterization of Automorphisms of Zero Divisor Graph of Nilradicals of Some Classes of Local Rings

2026-02-24T10:02:08Z

Characterization of Automorphisms of Zero Divisor Graph of Nilradicals of Some Classes of Local Rings KIPLAGAT P. The study of zero divisor graphs G(R) of rings R was first introduced by Beck in 1988. Beck isolated all Z (R) of the ring R and considered them as the vertex set of G(R). Beck’s focus was on the chromatic number χ(G(R)) of the graph G(R). Beck illustrated that G(R) is simple, connected, and diam(G(R)) ≤ 2. Besides, Beck further established that χ(G(R)) < 4. In 1999, Anderson and Livingstone standard- ized Beck’s graph G(R) by Γ(R). They considered Z (R)∗ and showed that Γ(R) is simple, connected, and that diam(Γ(R)) ≤ 3. The study on automorphisms of Γ(R) have helped in creating a link between the commutative algebra and graph theory. Various studies have been carried out by researchers to visualize and better under- stand algebraic structures of rings in graph theory. This approach is most effective when applied to commutative local rings. The objective of this study is to char- acterize automorphisms of Γ (R) of nilradicals of some classes of local commutative finite rings utilizing either Livingstone’s and Anderson’s zero divisor graph or Mu- lay’s condensed zero divisor graph. The methodology involves using combinatorial algorithmic technique to divide the rings into disjoint subsets of nilpotent and non- nilpotent elements, then constructing Z (R)∗ graphs and using case-by-case analysis to investigate maps that preserve structure within the graph. Structure-preserving maps are then identified using mathematical induction. Results and structure of Z (N (R)) will depend primarily on the order of nilpotent elements and their connec- tions, which will give a clear picture of how these nilradicals interact. Even though automorphisms can reflect many properties of the ring, it is still unknown whether the automorphism group can distinguish between such rings, especially in higher- dimensional contexts. This study will help determine how such automorphisms are useful in categorizing different local rings.

The Conditions under Which the Norm of Elementary Operator of Length Two Is Expressible In Terms Of Its Co-Efficient Operators in Tensor Product of -Algebra: An Application of Stampflis Maximal Numerical Range

2026-02-24T09:55:41Z

The Conditions under Which the Norm of Elementary Operator of Length Two Is Expressible In Terms Of Its Co-Efficient Operators in Tensor Product of -Algebra: An Application of Stampflis Maximal Numerical Range Kimeu B.., Kinyanjui J., Kaunda Z. Many researchers have studied the properties of elementary operators, including numerical ranges, spectrum, compactness and rank, in great depth, and numerous results have been obtained using different approaches such as spectral resolution theorem, tracial geometric mean, finite rank operator and Stampfli's maximal numerical range. The conditions under which the norm property can be expressed in terms of its coefficient operators, has also been considered in many studies but remains unresolved. This study sought to determine the application of Stampfli's maximal numerical range by determining the conditions under which the norm of the elementary operator of length two can be expressed in terms of its coefficient operators. The study used Stampfli's maximal numerical range and techniques of tensor products of -Algebras to express the norm of an elementary operator of length two in terms of its coefficient operators. Finally, the study used Cauchy Schwartz's inequality to determine the upper bound. In this study, the norm of the elementary operator is expressed in terms of coefficient operators and shown to be, . where are fixed elements in the set of bounded linear operator of tensor product of Hilbert space and . Theme: Integration of innovation and technology for sustainable interdisciplinary collaboration

On Counting the Number of Cyclic Codes of Length N over Prime Fields

2026-02-24T09:48:40Z

On Counting the Number of Cyclic Codes of Length N over Prime Fields Ongili P., Mude H. Cyclic codes are a cornerstone of coding theory. Many studies have extensively explored specific finite prime fields, GF(p). However, a generalized framework for GF(p) remains unexplored. Previous research has enumerated cyclic codes over GF(13), GF(17), GF(19), GF(23), GF(31), and GF(37), deriving significant find- ings for special cases where n assumes specific forms like n = pm or n = am · pm . Despite these advances, existing studies are often restricted to individual prime fields and lack a unified framework applicable across all GF(p). This research de- velops a mathematical framework for counting cyclic codes over any prime field GF(p). The methodology involves use of rigorous mathematical derivations and proofs to establish the relationship between number of cyclic codes, cyclotomic cosets and the factorization of xn − 1 into irreducible polynomials over GF(p). This framework provides a comprehensive understanding of how the structures of n and p influence the number of cyclic codes. Results show that the number of cyclic codes, N, over the prime fields can be expressed as N = (py + 1)C , where C is derived from the number of cyclotomic cosets modulo k ∀ n = k · py . This work contributes to efficient design of error-correcting codes, strengthens the theoretical foundation for secure communication systems, and bridges theoretical concepts of pure mathematics.

Study of W 𝐖𝟑 - Curvature Tensor on Lorentzian Para-Kenmotsu Manifolds

2026-02-24T09:42:54Z

Study of W 𝐖𝟑 - Curvature Tensor on Lorentzian Para-Kenmotsu Manifolds Mburu F.., Njori P.., Gitonga C. This study undertakes a comprehensive analysis of curvature tensors on Lorentzian Para Kenmotsu manifolds focusing on W3-Curvature tensor. Properties of this Curvature tensor under various conditions on these manifolds are explored and their geometric implications examined. The study also includes investigations of W3-flatness, (𝜉 - W3) flatness, (𝜙 − 𝑊3) flatness and 𝑊3 Semisymmetric on Lorentzian Para Kenmotsu manifolds and their connections to 𝜂 - Einstein, Einstein and special 𝜂 -Einstein. Additionally, The Ricci operator's behaviors on Lorentzian Para Kenmotsu manifolds under the conditions W3. 𝑄 = 0, 𝑄. 𝑊3 = 0 are analyzed. Expressions for this curvature tensor while considering the condition W3(𝜉, 𝑋). W3 = 0, 𝑅. W 3 = 0 and 𝑊3. 𝑊3 = 0 are derived. Proves to determine whether these manifolds are flat will be provided. The findings of this study enhance understanding of the geometric properties of Lorentzian Para Kenmotsu manifolds in relation to W3 curvature tensors