Functional completeness in one variable

1968 ◽  
Vol 33 (1) ◽  
pp. 105-106 ◽  
Author(s):  
James Rosenberg
Keyword(s):  
Group Theory ◽  
Sufficient Condition ◽  
Functional Completeness ◽  
Necessary And Sufficient Condition ◽  
The Subject ◽  
Complete Sets ◽  
Necessary And Sufficient

It is extremely difficult to make general statements about functional completeness. (For the main reference on the subject see Post [2].) In this paper we restrict ourselves to the case of unary functions in a finite valued logic, and prove a result concerning minimal functionally complete sets, along with a necessary and sufficient condition for completeness. A basic familiarity with group theory would be helpful.

scholarly journals On the formation of usually convergent Fourier series

1913 ◽  
Vol 88 (602) ◽  
pp. 178-188
Keyword(s):  
Fourier Series ◽  
First Principles ◽  
The Other ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Theoretical Interest ◽  
The Subject ◽  
Necessary And Sufficient ◽  
Sufficient Test ◽  
Suitable Modification

1. Series which converge expect at a set of content zero, or, using the expression very commonly adopted, series which converge usually, posses many of the properties which appertain to series which converge every-where. It becomes, therefore, of importance to device circumstances under which we can assert the consequence that a series converge in this manner. The subject has recently received considerable attention. so far as Fourier Series are concerned no result of even an approximately final character has been obtained. It may be supposed, indeed, that the result* of Jerosch and Weyl were at first so regarded, but, if we examine them closely in the light of the Riesz-Fischer theorems, which was known previously to the result of these authors, it becomes evident that they are merely equivalent to the statement that the Fourier Series of a function, whose square is summable, is changed into one which converges usually, if the typical coefficient a n and b n are divided by the sixth root of the integer n denoting their place in the series. Now it is difficult to believe that the question of the usual convergences of a Fourier Series can depended on the degree of the summability of the function with which it is associated and it is still more difficult to see how precisely the sixth root of n can have anything to do it. On the other hand Weyl's method, which itself marks an advance on that of Jerosch, does not obviously lend itself to any suitable modification which would secure a greater degree of generality in the result. The mistake is frequently made of confusing theoretical interest with pracitcal importance in the matter of a necessary and sufficient test. Tests which are only sufficient, but not necessary, are often much more convenient. Still more frequent it is convenient to work from first principles, and not to use any test at all. Instead of employing Weyl's necessary and sufficient condition that a series should converge usually, I have attacked the problem directly. The principles I have employed do not differ essentially from those already exposed in previous communication to this Society, but the generality and interest of the result obtained in the matter in hand seem to justify a further communication.

Animalhood, interests, and rights*

2020 ◽  
Vol 11 (2) ◽  
pp. 156-172
Author(s):  
Juan Pablo Mañalich R.
Keyword(s):  
Legal Rights ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moral Requirement ◽  
Intentional States ◽  
Nonhuman Animals ◽  
The Subject ◽  
The One ◽  
Necessary And Sufficient ◽  
Normative Status

A being to which intentional states – such as desires or preferences – may be ascribed is a being capable of having (actual) interests, whereas to be the subject of interests of some kind is both a necessary and sufficient condition to be the holder of individual rights. After clarifying the sense in which, according to the ‘interest-theory’, the notion of a rights-subject specifies a distinctive normative status, this article will highlight the importance of distinguishing between subjectivity-dependent interests capable of being attributed to conscious beings, on the one hand, and biologically structured needs of conscious and nonconscious living beings, on the other. This distinction allows one to see that the moral requirement of recognizing legal rights for (individual) animals ought not to be conflated with biocentric demands of ecological justice. However, the argument thus delineated will not, without more, answer the crucial question of which specific legal rights ought to be ascribed to nonhuman animals. The article closes with an exploration of the need for holding onto the distinction between rights-subjecthood and personhood by analyzing some implications of Tooley's ‘particular-interest principle’.

Complete Sets of Observables and Pure States

1968 ◽  
Vol 20 ◽  
pp. 1276-1280 ◽  
Author(s):  
Stanley P. Gudder
Keyword(s):  
Weak Topology ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Complete Set ◽  
Pure States ◽  
Complete Sets ◽  
Necessary And Sufficient

It was shown in (1) that a complete set of bounded observables is metrically complete. However, an extra axiom was needed to prove this result (1, footnote, p. 436). In this note we prove the above-mentioned result without the extra axiom. We also show that there is an abundance of pure states if M is closed in the weak topology and give a necessary and sufficient condition for the latter to be the case.

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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