REPRESENTATIONS OF INTEGERS BY THE BINARY QUADRATIC FORM

2015 ◽  
Vol 100 (2) ◽  
pp. 182-191 ◽  
Author(s):  
BUMKYU CHO
Keyword(s):  
Field Theory ◽  
Quadratic Form ◽  
Binary Quadratic Form ◽  
Class Field Theory ◽  
Class Numbers ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Class Field ◽  
Representations Of Integers ◽  
Necessary And Sufficient

In terms of class field theory we give a necessary and sufficient condition for an integer to be representable by the quadratic form $x^{2}+xy+ny^{2}$ ($n\in \mathbb{N}$ arbitrary) under extra conditions $x\equiv 1\;\text{mod}\;m$, $y\equiv 0\;\text{mod}\;m$ on the variables. We also give some examples where their extended ring class numbers are less than or equal to $3$.

scholarly journals Six-dimensional considerations of Einstein's connection for the first two classes. II. The Einstein's connection in6-g-UFT

1999 ◽  
Vol 22 (3) ◽  
pp. 483-488
Author(s):  
Kyung Tae Chung ◽  
Gye Tak Yang ◽  
In Ho Hwang
Keyword(s):  
Field Theory ◽  
Recurrence Relations ◽  
Sufficient Condition ◽  
Field Tensor ◽  
Necessary And Sufficient Condition ◽  
Unified Field Theory ◽  
The Third ◽  
Unified Field ◽  
Necessary And Sufficient ◽  
Lower Dimensional

Lower dimensional cases of Einstein's connection were already investigated by many authors forn=2,3,4,5. In the following series of two papers, we present a surveyable tensorial representation of6-dimensional Einstein's connection in terms of the unified field tensor:I. The recurrence relations in6-g-UFT.II. The Einstein's connection in6-g-UFT.In our previous paper [2], we investigated some algebraic structure in Einstein's6-dimensional unified field theory (i.e.,6-g-UFT), with emphasis on the derivation of the recurrence relations of the third kind which hold in6-g-UFT. This paper is a direct continuation of [2]. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in6-g-UFT and to display a surveyable tensorial representation of6-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations of the third kind obtained in the first paper [2].All considerations in this paper are restricted to the first and second classes of the6-dimensional generalized Riemannian manifoldX6, since the case of the third class, the simplest case, was already studied by many authors.

Characterizations of r-potent matrices

1984 ◽  
Vol 96 (2) ◽  
pp. 213-222 ◽  
Author(s):  
Joseph P. McCloskey
Keyword(s):  
Quadratic Form ◽  
Random Vector ◽  
Symmetric Matrix ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Real Quadratic

A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

On Weakly Positive Matrices

1963 ◽  
Vol 15 ◽  
pp. 313-317 ◽  
Author(s):  
Eugene P. Wigner
Keyword(s):  
Quadratic Form ◽  
Present Article ◽  
Positive Definite ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Hermitian Quadratic Form ◽  
Positive Matrices ◽  
Characteristic Values ◽  
Image Position ◽  
Necessary And Sufficient

A matrix is said to be positive definite if it is hermitian and if all of its characteristic values are positive. It is well known, and easy to prove, that the necessary and sufficient condition for a matrix P to be positive definite is that its hermitian quadratic formwith any vector v ≠ 0 be positive. (This will imply, in the present article, that it is real.) It is easy to see from (1) that if P1 and P2 are positive definite, the same holds of a1P1 + a2P2 if a1 and a2 are positive numbers.

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.

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