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Research Article
On the Socle of Finite Primitive Permutation Groups Having Frobenious Structure
Danbaba Adamu*,
Momoh Umoru Sunday
Issue:
Volume 13, Issue 6, December 2024
Pages:
79-83
Received:
7 August 2024
Accepted:
5 September 2024
Published:
13 December 2024
DOI:
10.11648/j.pamj.20241306.11
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Abstract: The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socle of the nilpotent groups whose structures is in conformity with D were classified based on the classification scheme for the finite primitive groups in relation to socle type. The socle type described in the classification scheme was in condition (1) was in line with the structure of D, as such it pave way in determining the socle with nilpotency class having same or similar structure with D. Further investigations showed that Frobenious group's were 2-transitive and the structure of D gave the conditions it being regular elementary abelian and so is nilpotent. It was observed the stabiliser of the groups in a finite primitive groups were paramount in the determination of the socle of the groups, as such much attention was given to the stabilizer of each group under consideration in a quest to determine the socle and the nilpotency class. The other conditions for the classification of finite primitive groups based of the socle type were not given much attention as it could give the needed condition for the existence of nilpotency class of the groups, as groups of such types were either almost simple, diagonal, product or twisted wreath product type. Therefore finite primitive group's under those conditions which could not give the expected nilpotency class and order were not give much attention. The degree of homogeneity was not given much priority as the article intended to discuss only the socle type and it nilpotency class or order.
Abstract: The nilpotentcy class for the Frobenius was determined based on the structure theorem. The socle of the groups were observed to be regular normal and elementary abelian such features were the conditions for the nilpotency classes, as they were the basis on which the socle of these groups constructed were nilpotent of some classes or order. The socl...
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Research Article
On the Least Common Multiple of Polynomials over a Number Field
Ilaria Viglino*
Issue:
Volume 13, Issue 6, December 2024
Pages:
84-99
Received:
7 September 2024
Accepted:
17 October 2024
Published:
18 December 2024
DOI:
10.11648/j.pamj.20241306.12
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Abstract: For an irreducible integral polynomial f of degree n, Cilleruelo’s conjecture states an asymptotic formula for the logarithm of the least common multiple of the first M values f(1) to f(M). It’s well-known for n = 1 as a consequence of Dirichlet’s Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomials. Recently the conjecture was shown by Rudnick and Zehavi for a large family of polynomials of any degree. We want to investigate an average version of the conjecture for Sn-polynomials with integral coefficients over a fixed extension K=Q by considering the least common multiple of ideals of OK. The case of linear polynomials is dealt with separately by exploiting Dirichlet’s Theorem for primes in arithmetic progression, to get an asymptotic estimate. In our case, to achieve explicit error terms, we want effective versions of the asymptotics. We will state here both a conditional and unconditional results proved by Lagarias and Odlyzko. For degree-2 polynomials, it is possible to obtain explicit asymptotics for the least common multiple, analogously to the ones achieved for polynomials in Z[X]. However, the latter is not a subject of the current paper.
Abstract: For an irreducible integral polynomial f of degree n, Cilleruelo’s conjecture states an asymptotic formula for the logarithm of the least common multiple of the first M values f(1) to f(M). It’s well-known for n = 1 as a consequence of Dirichlet’s Theorem for primes in arithmetic progression, and it was proved by Cilleruelo for quadratic polynomial...
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Review Article
Fixed Point Theorem in Fuzzy b-Metric Space Using Compatible Mapping of Type (A)
Issue:
Volume 13, Issue 6, December 2024
Pages:
100-108
Received:
29 September 2024
Accepted:
1 November 2024
Published:
18 December 2024
DOI:
10.11648/j.pamj.20241306.13
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Abstract: One of the most active and developing fields in both pure and applied mathematics is the theory of fixed points. It is possible to formulate a large number of nonlinear issues that arise in many scientific domains as fixed point problems. Since Zadeh first introduced the concept of fuzzy mathematics in 1965, the interest in fuzzy metrics has grown to the point that several studies have concentrated on examining their topological characteristics and applying them to mathematical issues. This was primarily because, in certain situations, fuzziness rather than randomization was the cause of uncertainty in the distance between two spots. Many mathematicians have examined and developed the concept of distance in relation to fuzzy frameworks because it is a naturalist concept. Generally speaking, it is impossible to determine the precise distance between any two locations. Thus, we deduce that if we measure the same distance between two locations at different times, the results will differ. There are two approaches that can be used to manage this situation: statistical and probabilistic. But instead of employing non-negative real numbers, the probabilistic approach makes use of the concept of a distribution function. Since fuzziness, rather than randomness, is the cause of the uncertainty in the distance between two places. Because of the positive real number b ≥ 1, the area of fuzzy b-metric space is larger than fuzzy metric space. Thus, this field is the source of our concern. This study aims to use the notion of compatible mappings and semicompatible mappings of type (A) to develop some common fixed point theorems in fuzzy b- metric space. A few ramifications of our primary discovery are also provided. Included are pertinent examples to highlight the importance of these key findings. Our results add to a number of previously published findings in the literature.
Abstract: One of the most active and developing fields in both pure and applied mathematics is the theory of fixed points. It is possible to formulate a large number of nonlinear issues that arise in many scientific domains as fixed point problems. Since Zadeh first introduced the concept of fuzzy mathematics in 1965, the interest in fuzzy metrics has grown ...
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Research Article
On a Certain Subclass of Multivalent Function Defined by Generalized Ruscheweyh Derivative
Shivani Indora*,
Sushil Kumar Bissu,
Manisha Summerwar
Issue:
Volume 13, Issue 6, December 2024
Pages:
109-118
Received:
3 November 2024
Accepted:
25 November 2024
Published:
18 December 2024
DOI:
10.11648/j.pamj.20241306.14
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Abstract: Fractional calculus is the prominent branch of applied mathematics, it deals with a lot of diverse possibility of finding differentiation as well as integration of function f(z) when the order of differentiation operator ‘D’ and integration operator ‘J’ is a real number or a complex number. The combination of fractional calculus with geometric function theory is the dynamic field of the current research scenario. It has many applications not only in the field of mathematics but also in the different fields like modern mathematical physics, electrochemistry, viscoelasticity, fluid dynamics, electromagnetic, the theory of partial differential equations systems, Mathematical modeling. Various new subclasses of univalent and multivalent functions defined by using different operators. In this research paper, we work on the formation of new subclass of analytic and multivalent functions defined under the open unit disk. By using Generalized Ruscheweyh derivative operator we define a new subclass of analytic and multivalent functions. The main aim of this research article is to derive interesting characteristics of new subclass of multivalent functions, which mainly include coefficient bound, growth and distortion bounds for function and its first derivative, extreme point and obtain unidirectional results for the multivalent functions which are belonging to this new subclass.
Abstract: Fractional calculus is the prominent branch of applied mathematics, it deals with a lot of diverse possibility of finding differentiation as well as integration of function f(z) when the order of differentiation operator ‘D’ and integration operator ‘J’ is a real number or a complex number. The combination of fractional calculus with geometric func...
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