scholarly journals On involutive division on monoids

2021 ◽  
Vol 29 (4) ◽  
pp. 387-398
Author(s):  
Oleg K. Kroytor ◽  
Mikhail D. Malykh
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Completely Continuous ◽  
Completely Continuous Operators ◽  
Finite Set ◽  
Definition Of ◽  
Original Domain ◽  
Necessary And Sufficient ◽  
Continuous Operators ◽  
Involutive Division

We consider an arbitrary monoid MM, on which an involutive division is introduced, and the set of all its finite subsets SetMM. Division is considered as a mapping d:SetMM{d:SetM \times M}, whose image d(U,m){d(U,m)} is the set of divisors of mm in UU. The properties of division and involutive division are defined axiomatically. Involutive division was introduced in accordance with the definition of involutive monomial division, introduced by V.P. Gerdt and Yu.A. Blinkov. New notation is proposed that provides brief but explicit allowance for the dependence of division on the SetMM element. The theory of involutive completion (closures) of sets is presented for arbitrary monoids, necessary and sufficient conditions for completeness (closedness) - for monoids generated by a finite set XX. The analogy between this theory and the theory of completely continuous operators is emphasized. In the last section, we discuss the possibility of solving the problem of replenishing a given set by successively expanding the original domain and its connection with the axioms used in the definition of division. All results are illustrated with examples of Thomas monomial division.

scholarly journals Generalized Timelike Mannheim Curves in Minkowski Space-Time

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
M. Akyig~it ◽  
S. Ersoy ◽  
İ. Özgür ◽  
M. Tosun
Keyword(s):  
Minkowski Space ◽  
Sufficient Conditions ◽  
Space Time ◽  
Necessary And Sufficient Conditions ◽  
Minkowski Space Time ◽  
Mannheim Curve ◽  
Definition Of ◽  
Necessary And Sufficient

We give the definition of generalized timelike Mannheim curve in Minkowski space-time . The necessary and sufficient conditions for the generalized timelike Mannheim curve are obtained. We show some characterizations of generalized Mannheim curve.

Rado's theorem for polymatroids

1975 ◽  
Vol 78 (2) ◽  
pp. 263-281 ◽  
Author(s):  
Colin J. H. McDiarmid
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Rank Function ◽  
Fundamental Importance ◽  
Matroid Theory ◽  
Continuous Analogue ◽  
Finite Set ◽  
Necessary And Sufficient ◽  
Transversal Theory

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

Fair bets and inductive probabilities

1955 ◽  
Vol 20 (3) ◽  
pp. 263-273 ◽  
Author(s):  
John G. Kemeny
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
The Other ◽  
Inductive Probability ◽  
Other Hand ◽  
Frequency Interpretation ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Degree Of Confirmation

The question of what constitutes fairness in betting quotients has been studied by Ramsey, deFinetti, and Shimony. Thanks to their combined efforts we now have a satisfactory definition of fairness.On the other hand, the explication of the concept of degree of confirmation (inductive probability) has progressed rapidly in recent years, thanks primarily to Carnap. This explication has usually proceeded by laying down the axioms for frequency-probabilities, and elaborating on these. While in the case where a frequency interpretation is intended these axioms are clearly justified, in our case they have been laid down without any justification. Carnap's presentation has been criticized for just this reason.The purpose of this paper is to show that the probability axioms are necessary and sufficient conditions to assure that the degrees of confirmation form a set of fair betting quotients. In addition it will be shown that one additional, highly controversial, axiom is precisely the condition needed to assure that not only deFinetti's weaker criterion but Shimony's criterion of fairness is also satisfied.

scholarly journals Square roots in finite full transformation semigroups

1982 ◽  
Vol 23 (2) ◽  
pp. 137-149 ◽  
Author(s):  
Mary Snowden ◽  
J. M. Howie
Keyword(s):  
Transformation Semigroup ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Square Root ◽  
Transformation Semigroups ◽  
Full Transformation ◽  
Square Roots ◽  
Finite Set ◽  
Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

scholarly journals Conceptualising strategic litigation

2021 ◽  
Author(s):  
Kris van der Pas
Keyword(s):  
Research Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Database Research ◽  
Conceptual Relations ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Exact Definition ◽  
Strategic Litigation

This article conceptualises the term “strategic litigation” in order to provide for a definition of it. Strategic litigation is a tool increasingly used in Europe by individuals and organisations to attain different objectives. Next to that, there is increasing academic attention for the topic. Nevertheless, the exact definition of “strategic litigation” remains unclear. Therefore, this article uses the research method of conceptualisation as well as a database research and additional literature to provide for a definition. It looks firstly at the background concept, involving the range of meanings associated with the term “strategic litigation”, after which a systematised concept is formed. Thereby, use is made of the “necessary and sufficient conditions” (NSC) approach, to develop the conditions necessary and/or sufficient for a case to fit within the category of strategic litigation. Moreover, the external conceptual relations of the term are explored.

scholarly journals Isomorphism of Binary Operations in Differential Geometry

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1634
Author(s):  
Nikita E. Barabanov
Keyword(s):  
Sufficient Conditions ◽  
Canonical Representation ◽  
Necessary And Sufficient Conditions ◽  
Function Parameter ◽  
Binary Operations ◽  
Möbius Addition ◽  
Algebraic Automorphism ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Metric Tensors

We consider smooth binary operations invariant with respect to unitary transformations that generalize the operations of the Beltrami–Klein and Beltrami–Poincare ball models of hyperbolic geometry, known as Einstein addition and Möbius addition. It is shown that all such operations may be recovered from associated metric tensors that have a canonical form. Necessary and sufficient conditions for canonical metric tensors to generate binary operations are found. A definition of algebraic isomorphism of binary operations is given. Necessary and sufficient conditions for binary operations to be isomorphic are provided. It is proved that every algebraic automorphism gives rise to isomorphism of corresponding gyrogroups. Necessary and sufficient conditions in terms of metric tensors for binary operations to be isomorphic to Euclidean addition are given. The problem of binary operations to be isomorphic to Einstein addition is also solved in terms of necessary and sufficient conditions. We also obtain necessary and sufficient conditions for binary operations having the same function-parameter in the canonical representation of metric tensors to be isomorphic.

On extensions of the Riemann and Lebesgue Integrals by Nets

1975 ◽  
Vol 18 (1) ◽  
pp. 7-17 ◽  
Author(s):  
O. S. Bellamy ◽  
H. W. Ellis
Keyword(s):  
General Theory ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Set ◽  
Denjoy Integral ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Open Extension

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

Additivity of the Pn-Integral

1978 ◽  
Vol 30 (4) ◽  
pp. 783-796 ◽  
Author(s):  
George Cross
Keyword(s):  
Sufficient Conditions ◽  
Slight Modification ◽  
Necessary And Sufficient Conditions ◽  
Definition Of ◽  
Necessary And Sufficient

It is known that the Pn-tegral as originally defined is not additive on abutting intervals. This paper offers a slight modification in the definition of the integral and develops necessary and sufficient conditions for the integral to be additive.

scholarly journals Finding linking sets

2006 ◽  
Vol 2006 ◽  
pp. 1-13
Author(s):  
Martin Schechter ◽  
Kyril Tintarev
Keyword(s):  
Banach Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
General Definition ◽  
Definition Of ◽  
Necessary And Sufficient

We present the most general definition of the linking of sets in a Banach space and discuss necessary and sufficient conditions for sets to link.

*-Finite Graph and Its Applications

2013 ◽  
Vol 336-338 ◽  
pp. 2359-2362
Author(s):  
Dong Li Chen ◽  
Yan Zhang ◽  
Chun Hui Ma
Keyword(s):  
Nonstandard Analysis ◽  
Sufficient Conditions ◽  
Finite Graph ◽  
Infinite Graph ◽  
Necessary And Sufficient Conditions ◽  
Transfer Principle ◽  
Definition Of ◽  
Necessary And Sufficient

By the method of nonstandard analysis, the definition of *-finite graph is given, and necessary and sufficient conditions of *-finite graph are obtained. Further, by the Transfer Principle, we apply the theory of finite graph to *-finite graph, embed given infinite graph into some *-finite graph, and finally obtain the related results of infinite graph.

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