scholarly journals Bell Numbers Modulo a Prime Number, Traces and Trinomials

10.37236/3532 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Luis H. Gallardo ◽  
Olivier Rahavandrainy
Keyword(s):  
Prime Number ◽  
Finite Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bell Numbers ◽  
Fixed Integer ◽  
Bell Number ◽  
Modulo P ◽  
Necessary And Sufficient

Given a prime number $p$, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition $\beta(n) =k\beta(0)$ in $\mathbb{F}_{p^p}$ in order to resolve the congruence $B(n) \equiv k \pmod{p}$, where $B(n)$ is the $n$-th Bell number, and $k$ is any fixed integer. Several applications of the formula and of the condition are included, in particular we give equivalent forms of the conjecture of Kurepa that $B(p-1)$ is $\neq 1$ modulo $p$.

scholarly journals Optimal quinary negacyclic codes with minimum distance four

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinmei Fan ◽  
Yanhai Zhang
Keyword(s):  
Finite Fields ◽  
Minimum Distance ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Negacyclic Codes ◽  
Generator Polynomial ◽  
Polynomials Over Finite Fields ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>Based on solutions of certain equations over finite yields, a necessary and sufficient condition for the quinary negacyclic codes with parameters <inline-formula><tex-math id="M1">\begin{document}$ [\frac{5^m-1}{2},\frac{5^m-1}{2}-2m,4] $\end{document}</tex-math></inline-formula> to have generator polynomial <inline-formula><tex-math id="M2">\begin{document}$ m_{\alpha^3}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> is provided. Several classes of new optimal quinary negacyclic codes with the same parameters are constructed by analyzing irreducible factors of certain polynomials over finite fields. Moreover, several classes of new optimal quinary negacyclic codes with these parameters and generator polynomial <inline-formula><tex-math id="M3">\begin{document}$ m_{\alpha}(x)m_{\alpha^e}(x) $\end{document}</tex-math></inline-formula> are also presented.</p>

Cyclic self-orthogonal codes over finite chain ring

2018 ◽  
Vol 11 (06) ◽  
pp. 1850078 ◽  
Author(s):  
Abhay Kumar Singh ◽  
Narendra Kumar ◽  
Kar Ping Shum
Keyword(s):  
Prime Number ◽  
Cyclic Codes ◽  
Sufficient Condition ◽  
Galois Rings ◽  
Finite Chain ◽  
Necessary And Sufficient Condition ◽  
Chain Ring ◽  
Finite Chain Ring ◽  
Orthogonal Codes ◽  
Necessary And Sufficient

In this paper, we study the cyclic self-orthogonal codes over a finite commutative chain ring [Formula: see text], where [Formula: see text] is a prime number. A generating polynomial of cyclic self-orthogonal codes over [Formula: see text] is obtained. We also provide a necessary and sufficient condition for the existence of nontrivial self-orthogonal codes over [Formula: see text]. Finally, we determine the number of the above codes with length [Formula: see text] over [Formula: see text] for any [Formula: see text]. The results are given by Zhe-Xian Wan on cyclic codes over Galois rings in [Z. Wan, Cyclic codes over Galois rings, Algebra Colloq. 6 (1999) 291–304] are extended and strengthened to cyclic self-orthogonal codes over [Formula: see text].

A Proposed Framework Emphasizing Auditor Reliability over Auditor Independence

2003 ◽  
Vol 17 (3) ◽  
pp. 257-266 ◽  
Author(s):  
Mark H. Taylor ◽  
F. Todd DeZoort ◽  
Edward Munn ◽  
Martha Wetterhall Thomas
Keyword(s):  
Public Interest ◽  
Auditor Independence ◽  
Public Confidence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accounting Profession ◽  
The Public ◽  
Necessary And Sufficient

This paper introduces an auditor reliability framework that repositions the role of auditor independence in the accounting profession. The framework is motivated in part by widespread confusion about independence and the auditing profession's continuing problems with managing independence and inspiring public confidence. We use philosophical, theoretical, and professional arguments to argue that the public interest will be best served by reprioritizing professional and ethical objectives to establish reliability in fact and appearance as the cornerstone of the profession, rather than relationship-based independence in fact and appearance. This revised framework requires three foundation elements to control subjectivity in auditors' judgments and decisions: independence, integrity, and expertise. Each element is a necessary but not sufficient condition for maximizing objectivity. Objectivity, in turn, is a necessary and sufficient condition for achieving and maintaining reliability in fact and appearance.

The Power of Public Positions

2018 ◽  
Author(s):  
Thomas Sinclair
Keyword(s):  
Detailed Analysis ◽  
Political Authority ◽  
The State ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
State Institutions ◽  
State Of Nature ◽  
Necessary And Sufficient

The Kantian account of political authority holds that the state is a necessary and sufficient condition of our freedom. We cannot be free outside the state, Kantians argue, because any attempt to have the “acquired rights” necessary for our freedom implicates us in objectionable relations of dependence on private judgment. Only in the state can this problem be overcome. But it is not clear how mere institutions could make the necessary difference, and contemporary Kantians have not offered compelling explanations. A detailed analysis is presented of the problems Kantians identify with the state of nature and the objections they face in claiming that the state overcomes them. A response is sketched on behalf of Kantians. The key idea is that under state institutions, a person can make claims of acquired right without presupposing that she is by nature exceptional in her capacity to bind others.

scholarly journals Lorentz Boosts and Wigner Rotations: Self-Adjoint Complexified Quaternions

Physics ◽  
2021 ◽  
Vol 3 (2) ◽  
pp. 352-366
Author(s):  
Thomas Berry ◽  
Matt Visser
Keyword(s):  
Point Of View ◽  
Sufficient Condition ◽  
Use Of Self ◽  
Necessary And Sufficient Condition ◽  
Wigner Rotation ◽  
Inertial Frames ◽  
Explicit Formulae ◽  
Specific Implementation ◽  
Necessary And Sufficient ◽  
Lorentz Boosts

In this paper, Lorentz boosts and Wigner rotations are considered from a (complexified) quaternionic point of view. It is demonstrated that, for a suitably defined self-adjoint complex quaternionic 4-velocity, pure Lorentz boosts can be phrased in terms of the quaternion square root of the relative 4-velocity connecting the two inertial frames. Straightforward computations then lead to quite explicit and relatively simple algebraic formulae for the composition of 4-velocities and the Wigner angle. The Wigner rotation is subsequently related to the generic non-associativity of the composition of three 4-velocities, and a necessary and sufficient condition is developed for the associativity to hold. Finally, the authors relate the composition of 4-velocities to a specific implementation of the Baker–Campbell–Hausdorff theorem. As compared to ordinary 4×4 Lorentz transformations, the use of self-adjoint complexified quaternions leads, from a computational view, to storage savings and more rapid computations, and from a pedagogical view to to relatively simple and explicit formulae.

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