On the q-capability of groups

2020 ◽  
pp. 2150071
Author(s):  
Forough Gharibi Monfared ◽  
Saeed Kayvanfar ◽  
Farangis Johari
Keyword(s):  
Positive Integer ◽  
Special Formula ◽  
The Relationship

Given a positive integer [Formula: see text] in this paper, we investigate some more properties of the [Formula: see text]-capability of groups. For instance, the relationship between [Formula: see text]-capability and the varietal capability is determined. Moreover, we introduce the notion of [Formula: see text]-epicenter for a group and then we obtain some criteria for the [Formula: see text]-capability of groups. Finally, as an application, we characterize all [Formula: see text]-capable extra-special [Formula: see text]-groups when [Formula: see text] is a power of [Formula: see text]

scholarly journals Conceptions of Topological Transitivity on Symmetric Products

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 61-80
Author(s):  
Franco Barragán ◽  
Sergio Macías ◽  
Anahí Rojas
Keyword(s):  
Positive Integer ◽  
Topological Space ◽  
Symmetric Product ◽  
Topological Transitivity ◽  
Symmetric Products ◽  
Weakly Mixing ◽  
The Relationship

Let X be a topological space. For any positive integer n , we consider the n -fold symmetric product of X , ℱ n ( X ), consisting of all nonempty subsets of X with at most n points; and for a given function ƒ : X → X , we consider the induced functions ℱ n ( ƒ ): ℱ n ( X ) → ℱ n ( X ). Let M be one of the following classes of functions: exact, transitive, ℤ-transitive, ℤ + -transitive, mixing, weakly mixing, chaotic, turbulent, strongly transitive, totally transitive, orbit-transitive, strictly orbit-transitive, ω-transitive, minimal, I N, T T ++ , semi-open and irreducible. In this paper we study the relationship between the following statements: ƒ ∈ M and ℱ n ( ƒ ) ∈ M .

scholarly journals 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity

2021 ◽  
Author(s):  
Li Jiang
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Basic Theorem ◽  
The Relationship ◽  
The Way

The 3x+1 problem is a problem of continuous iteration for integers. According to the basic theorem of arithmetic and the way of iteration, we derive a general formula for continuous iteration for odd integers. Through this formula, we can construct a loop iteration equation and obtain the result of the equation: the equation has only one positive integer solution. In addition, this general formula can be converted into a linear indeterminate equation. The process of solving this equation shows that the relationship between the iteration result and the odd number being iterated is linear. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question.

On étale covers of curves

1999 ◽  
Vol 127 (1) ◽  
pp. 1-5
Author(s):  
S. KAMIENNY ◽  
J. L. WETHERELL
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Abelian Varieties ◽  
Ring Of Integers ◽  
The Relationship

Let K be a number field with ring of integers R. For each integer g>1 we consider the collection of abelian, étale R-coverings f[ratio ]Y→X, where X and Y are connected proper curves over R and the genus of X is g. We ask the following question: is there a positive integer B = B(K, g) which bounds the degree of such coverings? In this note we provide partial results towards such a bound and study the relationship with bounds on torsion in abelian varieties.

scholarly journals 3x + 1 Problem: Iterative Operations Neither Cause Loops nor Diverge to Infinity

2021 ◽  
Author(s):  
Li Jiang
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Basic Theorem ◽  
The Relationship ◽  
The Way

The 3x+1 problem is a problem of continuous iteration for integers. According to the basic theorem of arithmetic and the way of iteration, we derive a general formula for continuous iteration for odd integers. Through this formula, we can construct a loop iteration equation and obtain the result of the equation: the equation has only one positive integer solution. In addition, this general formula can be converted into a linear indeterminate equation. The process of solving this equation shows that the relationship between the iteration result and the odd number being iterated is linear. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question.

scholarly journals Sampling parts of random integer partitions: a probabilistic and asymptotic analysis

2015 ◽  
Vol 25 (1) ◽  
pp. 79-95
Author(s):  
Ljuben Mutafchiev
Keyword(s):  
Asymptotic Behavior ◽  
Positive Integer ◽  
Asymptotic Analysis ◽  
Integer Partitions ◽  
Random Partition ◽  
Limiting Distributions ◽  
Sampling Plans ◽  
Random Integer ◽  
Sampling Procedures ◽  
The Relationship

Abstract Let λ be a partition of the positive integer n, selected uniformly at random among all such partitions. Corteel et al. (1999) proposed three different procedures of sampling parts of λ at random. They obtained limiting distributions of the multiplicity μn = μn(λ) of the randomly-chosen part as n → ∞. The asymptotic behavior of the part size σn = σn(λ), under these sampling conditions, was found by Fristedt (1993) and Mutafchiev (2014). All these results motivated us to study the relationship between the size and the multiplicity of a randomly-selected part of a random partition. We describe it obtaining the joint limiting distributions of (μn; σn), as n → ∞, for all these three sampling procedures. It turns out that different sampling plans lead to different limiting distributions for (μn; σn). Our results generalize those obtained earlier and confirm the known expressions for the marginal limiting distributions of μn and σn.

scholarly journals ON n-VNL-modules and SVNL-modules

2020 ◽  
Vol 2 (4) ◽  
pp. 67
Author(s):  
Le Cheng
Keyword(s):  
Positive Integer ◽  
The Relationship

A ring R is called right n-VNL-ring if whenever  for some elements , there exists at least one element  regular. The aim of this paper is to generalize this concept into module classes, we define n-VNL-modules, study their properties and give some characterizations. It is proved that for any finite generated R- module M, M is an SVNL-module if and only if M is an n-VNL-module for every positive integer n. The locally projective n-VNL-modules are also be characterized. We discuss the relationship between n-VNL-modules and other modules under different conditions.

scholarly journals 3x + 1 problem: Iterative operations neither cause loops nor diverge to infinity

2021 ◽  
Author(s):  
Li Jiang
Keyword(s):  
Positive Integer ◽  
General Formula ◽  
Integer Solution ◽  
Positive Integer Solution ◽  
Basic Theorem ◽  
The Relationship ◽  
The Way

Abstract The 3x+1 problem is a problem of continuous iteration for integers. According to the basic theorem of arithmetic and the way of iteration, we derive a general formula for continuous iteration for odd integers. Through this formula, we can construct a loop iteration equation and obtain the result of the equation: the equation has only one positive integer solution. In addition, this general formula can be converted into a linear indeterminate equation. The process of solving this equation shows that the relationship between the iteration result and the odd number being iterated is linear. Extending this result to all positive even numbers, we get the answer to the 3x + 1 question.

GROUPS OF TWO-BRAID VIRTUAL KNOTS

2007 ◽  
Vol 16 (06) ◽  
pp. 671-697 ◽  
Author(s):  
TAIZO KANENOBU ◽  
KAZUNORI TSUJI
Keyword(s):  
Positive Integer ◽  
Virtual Knot ◽  
Virtual Knots ◽  
The Relationship

Giving a presentation of the group of a 2-braid virtual knot or link, we consider the groups of three families of 2-braid virtual knots. Each of them has a certain feature; for example, we can show: for any positive integer N, there exists a virtual knot group with an element of order N. It is known that the collection of the virtual knot groups is the same as that of the ribbon T2-knot groups. Using our examples we discuss the relationship among the virtual knot groups and other knot groups such as ribbon S2-knot groups, S2-knot groups, T2-knot groups, and S3-knot groups.

A Note on Root Decision Problems in Groups

1973 ◽  
Vol 25 (4) ◽  
pp. 702-705 ◽  
Author(s):  
Seymour Lipschutz ◽  
Martin Lipschutz
Keyword(s):  
Positive Integer ◽  
Decision Problems ◽  
Finitely Presented Group ◽  
The Relationship ◽  
Finitely Presented

Consider a positive integer r > 1. We say that the rth root problem is solvable for a group G if we can decide for any W ॉ G whether or not W has an rth root, i.e. whether or not there exists V ॉ G such that W = Vr.Baumslag, Boone and Neumann [1] proved that there exists a finitely presented group with all root problems unsolvable. Here we are concerned with the relationship between the different root problems.

Right-angled Artin groups, polyhedral products and the -generating function

2021 ◽  
pp. 1-25
Author(s):  
Jorge Aguilar-Guzmán ◽  
Jesús González ◽  
John Oprea
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Base Space ◽  
Topological Complexity ◽  
Product Spaces ◽  
Polyhedral Product ◽  
Right Angled Artin Groups ◽  
The Right ◽  
The Relationship ◽  
Mac Lane

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$ . We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$ . Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

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