scholarly journals Functions of positive and negative type, and their connection with the theory of integral equations

1909 ◽  
Vol 83 (559) ◽  
pp. 69-70 ◽  
Keyword(s):  
Integral Equations ◽  
Symmetric Function ◽  
Symmetric Functions ◽  
Closed Interval ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Double Integral ◽  
Positive Type ◽  
Negative Type ◽  
Necessary And Sufficient

This memoir is concerned with continuous symmetric functions k ( s, t ) for which the double integral ∫ a b ∫ a b k ( s, t ) θ ( s ) θ ( t ) dsdt is either not negative, or not positive, for each function θ( s ) which is continuous in the interval ( a, b ) ; in the former case the function k ( s, t ) is said to be of positive type, while in the latter it is said to be of negative type. The importance of these classes of functions in the theory of integral equations will be gathered from Part I. The greater portion of the second part is devoted to a proof of the theorem that the necessary and sufficient condition, under which a continuous symmetric function, k ( s, t ), is of positive type, is that the functions k ( s 1 , s 1 ), k ( s 1 , s 2 s 1 , s 2 ),....., k ( s 1 , s 2 ...., s n s 1 , s 2 , ......, s n ), ... should never be negative, when the variable s 1 , s 2 , ...., s n , .... are each confined to the closed interval ( a, b ). This leads to several interesting properties of such a function ; for instance, if k ( a 1 , a 1 ) = 0, the functions k ( s , a 1 ), k ( a 1 , t ) are identically zero.

scholarly journals On classes of summable functions and their Fourier Series

1912 ◽  
Vol 87 (594) ◽  
pp. 225-229 ◽  
Keyword(s):  
Fourier Series ◽  
Integral Equations ◽  
Bounded Variation ◽  
Summable Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Suitable Constant ◽  
Necessary And Sufficient ◽  
Class Of Functions ◽  
Formal Statement

1. Functions which are summable may be such that certain functions of them are themselves summable. When this is the case they will possess certain special properties additional to those which the mere summability involves. A remarkable instance where this has been recognised is in the case of summable functions whose squares also are summable. The—in its formal statement almost self-evident—Theorem of Parseval which asserts that the sum of the squares of the coefficients of a Fourier series of a function f ( x ) is equal to the integral of the square of f ( x ), taken between suitable limits and multiplied by a suitable constant, has been recognised as true for all functions whose squares are summable. Moreover, not only has the converse of this been shown to be true, but writers have been led to develop a whole theory of this class of functions, in connection more especially with what are known as integral equations. That functions whose (1 + p )th power is summable, where p >0, but is not necessarily unity, should next be considered, was, of course, inevitable. As was to be expected, it was rather the integrals of such functions than the functions themselves whose properties were required. Lebesgue had already given the necessary and sufficient condition that a function should be an integral of a summable function. F. Riesz then showed that the necessary and sufficient condition that a function should be the integral of a function whose (1 + p )th power is summable had a form which constituted rather the generalisation of tire expression of the fact that such a function has bounded variation, than one which included the condition of Lebesgue as a particular case.

scholarly journals On the ordinary convergence of restricted fourier series

1917 ◽  
Vol 93 (651) ◽  
pp. 276-292 ◽  
Keyword(s):  
Fourier Series ◽  
General Type ◽  
Closed Interval ◽  
Single Point ◽  
Interval Of Periodicity ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Trigonometrical Series ◽  
Derived Series ◽  
Necessary And Sufficient

1. The necessary and sufficient condition that a trigonometrical series should be a Fourier series is that the integrated series should converge to an integral throughout the closed interval of periodicity, and should be the Courier series, accordingly, of an integral. Conversely, starting with the Courier series of an integral and differentiating it term by term, we obtain the Courier series of the most general type, namely, one associated with any function possessing an absolutely convergent integral. If the Courier series which is differentiated is not the Courier series of an integral, but of a function which fails to be an integral, at even a single point, the derived series will not lie a Courier series.

scholarly journals Balanced labellings of affine permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Hwanchul Yoo ◽  
Taedong Yun
Keyword(s):  
Symmetric Functions ◽  
Simple Algorithm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reduced Decompositions ◽  
Finite Case ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Affine Permutations ◽  
Necessary And Sufficient

International audience We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.

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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