scholarly journals Differential principal factors and Pólya property of pure metacyclic fields

2019 ◽  
Vol 15 (10) ◽  
pp. 1983-2025
Author(s):  
Daniel C. Mayer
Keyword(s):  
Cyclotomic Field ◽  
Unit Group ◽  
Index Formula ◽  
Principal Ideals ◽  
Cubic Fields ◽  
Principal Factors ◽  
Number Formula ◽  
Galois Closures ◽  
Theoretical Results ◽  
Unit Norm

Barrucand and Cohn’s theory of principal factorizations in pure cubic fields [Formula: see text] and their Galois closures [Formula: see text] with [Formula: see text] types is generalized to pure quintic fields [Formula: see text] and pure metacyclic fields [Formula: see text] with [Formula: see text] possible types. The classification is based on the Galois cohomology of the unit group [Formula: see text], viewed as a module over the automorphism group [Formula: see text] of [Formula: see text] over the cyclotomic field [Formula: see text], by making use of theorems by Hasse and Iwasawa on the Herbrand quotient of the unit norm index [Formula: see text] by the number [Formula: see text] of primitive ambiguous principal ideals, which can be interpreted as principal factors of the different [Formula: see text]. The precise structure of the group of differential principal factors is determined with the aid of kernels of norm homomorphisms and central orthogonal idempotents. A connection with integral representation theory is established via class number relations by Parry and Walter involving the index of subfield units [Formula: see text]. Generalizing criteria for the Pólya property of Galois closures [Formula: see text] of pure cubic fields [Formula: see text] by Leriche and Zantema, we prove that pure metacyclic fields [Formula: see text] of only one type cannot be Pólya fields. All theoretical results are underpinned by extensive numerical verifications of the [Formula: see text] possible types and their statistical distribution in the range [Formula: see text] of [Formula: see text] normalized radicands.

Pólya S3-extensions of ℚ

2018 ◽  
Vol 149 (6) ◽  
pp. 1421-1433 ◽  
Author(s):  
Abbas Maarefparvar ◽  
Ali Rajaei
Keyword(s):  
Number Field ◽  
Upper Bounds ◽  
Ring Of Integers ◽  
Cubic Fields ◽  
Galois Closures ◽  
Unit Norm

AbstractA number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.

scholarly journals Application of the Strong Artin Conjecture to the Class Number Problem

2013 ◽  
Vol 65 (6) ◽  
pp. 1201-1216 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Class Number ◽  
Number Fields ◽  
Class Numbers ◽  
Density Result ◽  
Class Number Formula ◽  
Number Problem ◽  
Number Formula ◽  
Group A ◽  
Galois Closures ◽  
Extremal Value

AbstractWe construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group A4; S4, and S5. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski–Michel, we choose subfamilies of L-functions that are zero-free close to 1. For these subfamilies, the L-functions have the extremal value at s = 1, and by the class number formula, we obtain the extreme class numbers.

A systematic investigation of the gap in various Barabási–Albert and Erdös–Rényi networks

2017 ◽  
Vol 28 (02) ◽  
pp. 1750022 ◽  
Author(s):  
Muneer A. Sumour ◽  
F. W. S. Lima
Keyword(s):  
Random Graphs ◽  
Systematic Investigation ◽  
Index Formula ◽  
Initial Number ◽  
Number Formula ◽  
Structure Formula ◽  
Versus Node

On Barabási–Albert networks (BA) and variations as well as on Erdös–Rényi (ER) random graphs, we study the occurrence of a gap in the neighbor numbers [Formula: see text] versus node index [Formula: see text] (with [Formula: see text]) at [Formula: see text], 4, 6, 10, 50, 100 and 150 and with up to [Formula: see text] nodes. Here, we call “gap” a jump in the neighbor numbers [Formula: see text] when [Formula: see text] equals the initial number [Formula: see text] of neighbors; [Formula: see text] is also the number of neighbors randomly selected by a newly added node. The size of the gap depends on the value of [Formula: see text] and causes a deformation in the structure [Formula: see text] of the networks studied here. We give a systematic investigation of the gap in all types of BA networks known to us, and only the undirected BA (UBA) network and the ER graphs show no gap for [Formula: see text].

A note on Dedekind Criterion

2020 ◽  
pp. 2150066
Author(s):  
Anuj Jakhar ◽  
Sudesh K. Khanduja
Keyword(s):  
Prime Number ◽  
Minimal Polynomial ◽  
Algebraic Integer ◽  
Algebraic Number Field ◽  
Index Formula ◽  
Valuation Theory ◽  
Valuation Rings ◽  
Number Formula ◽  
Ring Of Algebraic Integers ◽  
Polynomial Formula

Let [Formula: see text] be an algebraic number field with [Formula: see text] an algebraic integer having minimal polynomial [Formula: see text] over the field [Formula: see text] of rational numbers and [Formula: see text] be the ring of algebraic integers of [Formula: see text]. For a fixed prime number [Formula: see text], let [Formula: see text] be the factorization of [Formula: see text] modulo [Formula: see text] as a product of powers of distinct irreducible polynomials over [Formula: see text] with [Formula: see text] monic. In 1878, Dedekind proved a significant result known as Dedekind Criterion which says that the prime number [Formula: see text] does not divide the index [Formula: see text] if and only if [Formula: see text] is coprime with [Formula: see text] where [Formula: see text]. This criterion has been widely used and generalized. In this paper, a simple proof of Generalized Dedekind Criterion [S. K. Khanduja and M. Kumar, On Dedekind criterion and simple extensions of valuation rings, Comm. Algebra 38 (2010) 684–696] using elementary valuation theory is given.

scholarly journals Stick number of spatial graphs

2017 ◽  
Vol 26 (14) ◽  
pp. 1750100 ◽  
Author(s):  
Minjung Lee ◽  
Sungjong No ◽  
Seungsang Oh
Keyword(s):  
Research Program ◽  
Upper Bound ◽  
Upper Bounds ◽  
Crossing Number ◽  
Index Formula ◽  
Spatial Graph ◽  
Spatial Graphs ◽  
Number Formula ◽  
Arc Index

For a nontrivial knot [Formula: see text], Negami found an upper bound on the stick number [Formula: see text] in terms of its crossing number [Formula: see text] which is [Formula: see text]. Later, Huh and Oh utilized the arc index [Formula: see text] to present a more precise upper bound [Formula: see text]. Furthermore, Kim, No and Oh found an upper bound on the equilateral stick number [Formula: see text] as follows; [Formula: see text]. As a sequel to this research program, we similarly define the stick number [Formula: see text] and the equilateral stick number [Formula: see text] of a spatial graph [Formula: see text], and present their upper bounds as follows; [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] are the number of edges and vertices of [Formula: see text], respectively, [Formula: see text] is the number of bouquet cut-components, and [Formula: see text] is the number of non-splittable components.

On the Mechanism of Multi-Valued Friction in Unsteady Sliding Line Contacts Operating in the Regime of Mixed-Film Lubrication

1997 ◽  
Vol 119 (1) ◽  
pp. 149-155 ◽  
Author(s):  
Xuejun Zhai ◽  
G. Needham ◽  
L. Chang
Keyword(s):  
Friction Model ◽  
Asperity Contact ◽  
Friction Behavior ◽  
Asperity Contacts ◽  
Line Contacts ◽  
Principal Factors ◽  
Mixed Film ◽  
Theoretical Results ◽  
Selection Of ◽  
Film Lubrication

Systematic analyses are presented to reveal the mechanism of multi-valued friction behavior in lubricated sliding contacts with time-varying velocities. The analyses are based on the theoretical results generated by a mixed-film friction model developed in this paper for line contacts. The model, which integrates theories of transient elastohydrodynamics and asperity contact mechanics, is validated by comparing its results with published experimental data. The results and the subsequent analyses disclose that strong multi-valued friction behavior can only be generated in the mixed-film lubrication regime with simultaneous presence of significant asperity contacts and hydrodynamic squeeze. Principal factors which influence the magnitude of dynamic friction are investigated in the paper. Being instructive to the design of tribocontacts in precise-motion control systems, the analyses suggest means to minimize the undesirable multi-valued friction behavior through proper selection of system parameters.

Theoretical assessment of the impact of desert aerosols on the dynamical transmission of meningitidis serogroup A

2019 ◽  
Vol 12 (05) ◽  
pp. 1950060
Author(s):  
A. Oumar Bah ◽  
M. Lam ◽  
A. Bah ◽  
S. Bowong
Keyword(s):  
Basic Reproduction Number ◽  
Reproduction Number ◽  
Globally Asymptotically Stable ◽  
Number Formula ◽  
High Level ◽  
The Impact ◽  
Theoretical Assessment ◽  
Theoretical Results ◽  
Disease Free ◽  
The Basic Reproduction Number

This paper has been motivated by the following biological question: how influential are desert aerosols in the transmission of meningitidis serogroup A (MenA)? A mathematical model for the dynamical transmission of MenA is considered, with the aim of investigating the impact of desert aerosols. Sensitivity analysis of the model has been performed in order to determine the impact of related parameters on meningitis outbreak. We derive the basic reproduction number [Formula: see text]. We prove that there exists a threshold parameter [Formula: see text] such that when [Formula: see text], the disease-free equilibrium is globally asymptotically stable (GAS). However, when [Formula: see text], the model exhibits the phenomenon of backward bifurcation. At the endemic level, we show that the number of infectious individuals in the presence of desert aerosols is larger than the corresponding number without the presence of desert aerosols. In conjunction with the inequality [Formula: see text] where [Formula: see text] is the basic reproduction number without desert aerosols, we found that the ingestion of aerosols by carriers will increase the endemic level, and the severity of the outbreak. This suggests that the control of MenA passes through a combination of a large coverage vaccination of young susceptible individuals and the production of a vaccine with a high level of efficacy as well as respecting the hygienic rules to avoid the inhalation of desert aerosols. Theoretical results are supported by numerical simulations.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

Sign in / Sign up

Export Citation Format

Share Document