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.

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].

TWO-SCALE DISTRIBUTION OF PRIME NUMBER AND ITS DIFFUSION EQUATION MODEL

Fractals ◽  
2017 ◽  
Vol 25 (02) ◽  
pp. 1750022 ◽  
Author(s):  
YINGJIE LIANG ◽  
WEN CHEN
Keyword(s):  
Brownian Motion ◽  
Diffusion Equation ◽  
Fractional Derivative ◽  
Natural Number ◽  
Prime Number ◽  
Power Law ◽  
Large Scale ◽  
Equation Model ◽  
Small Scale ◽  
Number Formula

This study reveals the two-scale characteristics in prime number distribution. It is observed a sub-diffusion process of power law decay at the small scale of the natural number [Formula: see text], but is found to obey the classical Brownian motion of an exponential decay at the large scale [Formula: see text]. Such two-scale mechanism gives rise to the multi-fractal scaling from the power law to the exponential law distributions in a transition region of the natural number [Formula: see text]. In the small range, the sub-diffusion of prime number distribution is well depicted by the fractional derivative equation model, and in the large scale, exponential decay distribution can accurately be described by a classical diffusion equation model. The Riemann diffusion equation proposed recently by the present authors can accurately model the prime distribution from small to moderate to large scales and is reduced to the fractional derivative sub-diffusion equation at small scale and the classical Brownian motion diffusion equation at large scale, respectively.

Symbolic computation study on exact solutions to a generalized (3+1)-dimensional Kadomtsev-Petviashvili-type equation

2020 ◽  
pp. 2150116
Author(s):  
Cheng-Cheng Zhou ◽  
Xing Lü ◽  
Hai-Tao Xu
Keyword(s):  
Numerical Simulation ◽  
Exact Solutions ◽  
Prime Number ◽  
Symbolic Computation ◽  
Three Dimensional ◽  
Type Equation ◽  
Two Dimensional ◽  
Type Solution ◽  
Bilinear Operators ◽  
Number Formula

Based on the prime number [Formula: see text], a generalized (3+1)-dimensional Kadomtsev-Petviashvili (KP)-type equation is proposed, where the bilinear operators are redefined through introducing some prime number. Computerized symbolic computation provides a powerful tool to solve the generalized (3+1)-dimensional KP-type equation, and some exact solutions are obtained including lump-type solution and interaction solution. With numerical simulation, three-dimensional plots, density plots, and two-dimensional curves are given for particular choices of the involved parameters in the solutions to show the evolutionary characteristics.

Indices in a number field II

2019 ◽  
Vol 15 (01) ◽  
pp. 89-103
Author(s):  
Mohamed Ayad ◽  
Rachid Bouchenna ◽  
Omar Kihel
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Additive Group ◽  
Necessary Condition ◽  
Ring Of Integers ◽  
Number Formula

Let [Formula: see text] be a number field of degree [Formula: see text] over [Formula: see text] and [Formula: see text] its ring of integers. For a prime number [Formula: see text], we determine the types of splittings of [Formula: see text] in [Formula: see text] for which the set [Formula: see text] is of cardinality a power of [Formula: see text]. We prove that this necessary condition is also sufficient for [Formula: see text] to be a subgroup of the additive group [Formula: see text]. Consequently, we show that, in this case, the subset of [Formula: see text], [Formula: see text] is an order of the number field.

scholarly journals Period and toroidal knot mosaics

2017 ◽  
Vol 26 (05) ◽  
pp. 1750031 ◽  
Author(s):  
Seungsang Oh ◽  
Kyungpyo Hong ◽  
Ho Lee ◽  
Hwa Jeong Lee ◽  
Mi Jeong Yeon
Keyword(s):  
Quantum System ◽  
Prime Number ◽  
Growth Rates ◽  
Exact Enumeration ◽  
Common Features ◽  
System A ◽  
Number Formula ◽  
And Mathematics ◽  
Definition Of ◽  
Positive Integers

Knot mosaic theory was introduced by Lomonaco and Kauffman in the paper on ‘Quantum knots and mosaics’ to give a precise and workable definition of quantum knots, intended to represent an actual physical quantum system. A knot [Formula: see text]-mosaic is an [Formula: see text] matrix whose entries are eleven mosaic tiles, representing a knot or a link by adjoining properly. In this paper, we introduce two variants of knot mosaics: period knot mosaics and toroidal knot mosaics, which are common features in physics and mathematics. We present an algorithm producing the exact enumeration of period knot [Formula: see text]-mosaics for any positive integers [Formula: see text] and [Formula: see text], toroidal knot [Formula: see text]-mosaics for co-prime integers [Formula: see text] and [Formula: see text], and furthermore toroidal knot [Formula: see text]-mosaics for a prime number [Formula: see text]. We also analyze the asymptotics of the growth rates of their cardinality.

Maximal inverse subsemigroups of the semigroup of all full transformations satisfying x ≈ x3

2016 ◽  
Vol 09 (01) ◽  
pp. 1650042
Author(s):  
Somnuek Worawiset
Keyword(s):  
Natural Number ◽  
Transformation Semigroup ◽  
Inverse Semigroups ◽  
Full Transformation ◽  
Full Transformation Semigroup ◽  
Inverse Subsemigroup ◽  
Number Formula ◽  
Inverse Subsemigroups ◽  
Text Element ◽  
Element Set

We classify the maximal Clifford inverse subsemigroups [Formula: see text] of the full transformation semigroup [Formula: see text] on an [Formula: see text]-element set with [Formula: see text] for all [Formula: see text]. This classification differs from the already known classifications of Clifford inverse semigroups, it provides an algorithm for its construction. For a given natural number [Formula: see text], we find also the largest size of an inverse subsemigroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] with least rank [Formula: see text] for any element in [Formula: see text].

scholarly journals A search for c-Wieferich primes

2020 ◽  
pp. 1-18
Author(s):  
Alex Samuel Bamunoba ◽  
Jonas Bergström
Keyword(s):  
Finite Field ◽  
Prime Number ◽  
Number Formula ◽  
Existence Criterion

Let [Formula: see text] be a power of a prime number [Formula: see text], [Formula: see text] be a finite field with [Formula: see text] elements and [Formula: see text] be a subgroup of [Formula: see text] of order [Formula: see text]. We give an existence criterion and an algorithm for computing maximally [Formula: see text]-fixed c-Wieferich primes in [Formula: see text]. Using the criterion, we study how c-Wieferich primes behave in [Formula: see text] extensions.

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