Linear Forms in Monic Integer Polynomials

2013 ◽  
Vol 56 (3) ◽  
pp. 510-519
Author(s):  
Artūras Dubickas
Keyword(s):  
Linear Form ◽  
Monic Polynomial ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Forms ◽  
Integer Polynomials ◽  
Necessary And Sufficient

Abstract. We prove a necessary and sufficient condition on the list of nonzero integers u1…, uk, k≥2, under which a monic polynomial f∊2ℤ[x] is expressible by a linear form u1 f1 + … + ukfk in monic polynomials f1…fk ∊ ℤ[x]. This condition is independent of f. We also show that if this condition holds, then the monic polynomials f1, … fk can be chosen to be irreducible in ℤ[x].

Notes on a theorem of Cochran

1952 ◽  
Vol 48 (3) ◽  
pp. 443-446 ◽  
Author(s):  
G. S. James
Keyword(s):  
Quadratic Forms ◽  
Degrees Of Freedom ◽  
Mathematical Statistics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linear Forms ◽  
Standard Normal ◽  
Real Linear ◽  
Necessary And Sufficient ◽  
Real Quadratic

1. General remarks. The theorem that has come to be known as Cochran's theorem in works on mathematical statistics was published in these Proceedings in 1934(1). If x1, …, xn are independently distributed standard normal deviates, and q1, …, qk are k real quadratic forms in the xi with ranks n1, …, nk respectively, and such that then Cochran's Theorem II states that a necessary and sufficient condition that q1 …, qk are independently distributed in χ2 forms with n1, …, nk degrees of freedom is that Σnj = n. The necessity of the condition is obvious. Cochxan proves its sufficiency by expressing each qj as a sum, involving nj squares of real linear forms in the xi; it follows easily that the coefficients ci are in fact + 1, and that the transformation is orthogonal. The theorem then follows immediately from the properties of orthogonal transformations in relation to independent normal deviates.

scholarly journals Symplectic quandles and parabolic representations of 2-bridge knots and links

2020 ◽  
Vol 31 (10) ◽  
pp. 2050081
Author(s):  
Kyeonghee Jo ◽  
Hyuk Kim
Keyword(s):  
Closed Form ◽  
Explicit Formula ◽  
Monic Polynomial ◽  
Integer Coefficient ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Arithmetic Properties ◽  
Polynomial Formula ◽  
Necessary And Sufficient ◽  
Knots And Links

In this paper, we study the parabolic representations of 2-bridge links by finiding arc coloring vectors on the Conway diagram. The method we use is to convert the system of conjugation quandle equations to that of symplectic quandle equations. In this approach, we have an integer coefficient monic polynomial [Formula: see text] for each 2-bridge link [Formula: see text], and each zero of this polynomial gives a set of arc coloring vectors on the diagram of [Formula: see text] satisfying the system of symplectic quandle equations, which gives an explicit formula for a parabolic representation of [Formula: see text]. We then explain how these arc coloring vectors give us the closed form formulas of the complex volume and the cusp shape of the representation. As other applications of this method, we show some interesting arithmetic properties of the Riley polynomial and of the trace field, and also describe a necessary and sufficient condition for the existence of epimorphisms between 2-bridge link groups in terms of divisibility of the corresponding Riley polynomials.

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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