scholarly journals On Orbits of the Ring Zmn under Action of the Group SL(m, Zn)

10.14311/768 ◽  
2005 ◽  
Vol 45 (5) ◽  
Author(s):  
P. Novotný ◽  
J. Hrivnák
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Matrix Group ◽  
Arbitrary Natural Number ◽  
Finite Matrix

We consider the action of the finite matrix group SL(m,Zn ) on the ring Zmn. We determine orbits of this action for n arbitrary natural number. It is a generalization of the task which was studied by A. A. Kirillov for m = 2 and n prime number.

On a conjecture about the quasirecognition by prime graph of some finite simple groups

2019 ◽  
Vol 18 (04) ◽  
pp. 1950070
Author(s):  
Ali Mahmoudifar
Keyword(s):  
Computer Program ◽  
Natural Number ◽  
Simple Group ◽  
Prime Number ◽  
Prime Power ◽  
Prime Graph ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Number Formula ◽  
Finite Linear

It is proved that some finite simple groups are quasirecognizable by prime graph. In [A. Mahmoudifar and B. Khosravi, On quasirecognition by prime graph of the simple groups [Formula: see text] and [Formula: see text], J. Algebra Appl. 14(1) (2015) 12pp], the authors proved that if [Formula: see text] is a prime number and [Formula: see text], then there exists a natural number [Formula: see text] such that for all [Formula: see text], the simple group [Formula: see text] (where [Formula: see text] is a linear or unitary simple group) is quasirecognizable by prime graph. Also[Formula: see text] in that paper[Formula: see text] the author posed the following conjecture: Conjecture. For every prime power [Formula: see text] there exists a natural number [Formula: see text] such that for all [Formula: see text] the simple group [Formula: see text] is quasirecognizable by prime graph. In this paper [Formula: see text] as the main theorem we prove that if [Formula: see text] is a prime power and satisfies some especial conditions [Formula: see text] then there exists a number [Formula: see text] associated to [Formula: see text] such that for all [Formula: see text] the finite linear simple group [Formula: see text] is quasirecognizable by prime graph. Finally [Formula: see text] by a calculation via a computer program [Formula: see text] we conclude that the above conjecture is valid for the simple group [Formula: see text] where [Formula: see text] [Formula: see text] is an odd number and [Formula: see text].

scholarly journals Integral almost square-free modular categories

2017 ◽  
Vol 16 (06) ◽  
pp. 1750104 ◽  
Author(s):  
Jingcheng Dong ◽  
Libin Li ◽  
Li Dai
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Partial Answer ◽  
Dimension Formula ◽  
Modular Category ◽  
Number Formula

We study integral almost square-free modular categories; i.e., integral modular categories of Frobenius–Perron dimension [Formula: see text], where [Formula: see text] is a prime number, [Formula: see text] is a square-free natural number and [Formula: see text]. We prove that, if [Formula: see text] or [Formula: see text] is prime with [Formula: see text], then they are group-theoretical. This generalizes several results in the literature and gives a partial answer to the question posed by the first author and Tucker. As an application, we prove that an integral modular category whose Frobenius–Perron dimension is odd and less than [Formula: see text] is group-theoretical.

scholarly journals Robin Criterion on Divisibility

2021 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Riemann Hypothesis ◽  
Sum Of Divisors

Robin criterion states that the Riemann Hypothesis is true if and only if the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all $n > 5040$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. We show that the Robin inequality is true for all $n > 5040$ which are not divisible by any prime number between $2$ and $953$. We prove that the Robin inequality holds when $\frac{\pi^{2}}{6} \times \log\log n' \leq \log\log n$ for some $n>5040$ where $n'$ is the square free kernel of the natural number $n$. The possible smallest counterexample $n > 5040$ of the Robin inequality complies that necessarily $(\log n)^{\beta} < 1.2592\times\log(N_{m})$, where $N_{m} = \prod_{i = 1}^{m} q_{i}$ is the primorial number of order $m$ and $\beta = \prod_{i = 1}^{m} \frac{q_{i}^{a_{i}+1}}{q_{i}^{a_{i}+1}-1}$ when $n$ is an Hardy-Ramanujan integer of the form $\prod_{i=1}^{m} q_{i}^{a_{i}}$.

scholarly journals Quantum Multiple Construction of Subfactors

2008 ◽  
Vol 51 (3) ◽  
pp. 321-333
Author(s):  
Marta Asaeda
Keyword(s):  
Natural Number ◽  
Finite Index ◽  
Tensor Category ◽  
Quantum Double ◽  
Arbitrary Natural Number

AbstractWe construct the quantum s-tuple subfactors for an AFD II1 subfactor with finite index and depth, for an arbitrary natural number s. This is a generalization of the quantum multiple subfactors by Erlijman and Wenzl, which in turn generalized the quantum double construction of a subfactor for the case that the original subfactor gives rise to a braided tensor category. In this paper we give a multiple construction for a subfactor with a weaker condition than braidedness of the bimodule system.

scholarly journals Uniqueness of the Kadomtsev-Petviashvili and Boussinesq Equations

2011 ◽  
Vol 66 (6-7) ◽  
pp. 377-382 ◽  
Author(s):  
Wen-Xiu Ma ◽  
Aslı Pekcan
Keyword(s):  
Natural Number ◽  
Soliton Solution ◽  
Boussinesq Equations ◽  
Uniqueness Property ◽  
Integrable Equations ◽  
Completely Integrable ◽  
Arbitrary Natural Number ◽  
Integrability Theory

The Kadomtsev-Petviashvili and Boussinesq equations (uxxx -6uux)x -utx ±uyy = 0; (uxxx - 6uux)x +uxx ±utt = 0; are completely integrable, and in particular, they possess the three-soliton solution. This article aims to expose a uniqueness property of the Kadomtsev-Petviashvili (KP) and Boussinesq equations in the integrability theory. It is shown that the Kadomtsev-Petviashvili and Boussinesq equations and their dimensional reductions are the only integrable equations among a class of generalized Kadomtsev-Petviashvili and Boussinesq equations (ux1x1x1 - 6uux1 )x1 + ΣMi;j=1aijuxixj = 0; where the aij’s are arbitrary constants and M is an arbitrary natural number, if the existence of the three-soliton solution is required

scholarly journals Permutations that preserve asymptotically null sets and statistical convergence

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4821-4827
Author(s):  
Jeff Connor
Keyword(s):  
Natural Number ◽  
Statistical Convergence ◽  
Asymptotic Density ◽  
Arbitrary Natural Number ◽  
Convergent Sequences

The main result of this article is a characterization of the permutations ?: N ? N that map a set with zero asymptotic density into a set with zero asymptotic density; a permutation has this property if and only if the lower asymptotic density of Cp tends to 1 as p ? ? where p is an arbitrary natural number and Cp = {l : ?-1(l)? lp}. We then show that a permutation has this property if and only if it maps statistically convergent sequences into statistically convergent sequences.

scholarly journals On the maximal abelian ℓ-extension of a finite algebraic number field with given ramification

1978 ◽  
Vol 70 ◽  
pp. 183-202 ◽  
Author(s):  
Hiroo Miki
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Algebraic Number Field

Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively : (1) the class number of the cyclotomic Zℓ-extension of k,(2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,(3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and(4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

1988 ◽  
Vol 53 (2) ◽  
pp. 470-480 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Algebraic Extension ◽  
Algebraic Integers ◽  
Nonstandard Models ◽  
Ring Of Algebraic Integers

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofK(x)within *K. For eachxЄ *K–Kand each natural numberd, YK(x,d)is defined to be the number of algebraic extensions ofK(x)of degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK(x,d)= 0 for alldЄ N,d> 1; in other words,K(x)has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω(ω3+ 1) are Hilbertian elements in*Q, where pωis theωth prime number.

scholarly journals Prime Numbers

2017 ◽  
Vol 5 (6) ◽  
pp. 41-66
Author(s):  
Mady Ndiaye
Keyword(s):  
Information Technology ◽  
Natural Number ◽  
Prime Number ◽  
Mathematical Reasoning ◽  
Prime Numbers ◽  
The Other ◽  
Natural Numbers ◽  
Mathematical Formulas ◽  
The Universe

A prime number is a natural number that has Just two divisors: one and itself. From antiquity until our time, scientists are researching mathematical reasoning to understand the prime numbers; eminent scholars had worked on this field before it is abandoned. Mathematicians considered the prime numbers like « building blocs in building natural numbers » and the field of mathematics the most difficult. Everything is about numbers, everything is about measure, The understanding of the natural numbers and more general the understanding of the numbers depend on the understanding of the prime numbers. This understanding of the prime will gives us greater ease to understand the other sciences. The prime numbers play a very important role for securing information technology hence promotion of the NTIC, Every year, there is a price for persons who will discover the biggest prime “it‟s the hunt for the big prime” This first part of this article about the prime numbers has taken a weight off the scientists „s shoulders by highlighting the universe of the prime numbers and has bring the problem of the prime numbers to an end. The mathematical formulas set out in this article allow us to determine all the biggest prime numbers compared to the capacity of our machines.

A propositional calculus with denumerable matrix

1959 ◽  
Vol 24 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Michael Dummett
Keyword(s):  
Natural Number ◽  
Propositional Calculus ◽  
Finite Matrix ◽  
Image Position ◽  
Matrix Characteristic

§1. In [1] Gödel proves the non-existence of a finite matrix characteristic for the intuitionist propositional calculus IC by the use of the finite matrices , where n is a natural number and

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