scholarly journals (1, 2) AND WEAK (1, 3) HOMOTOPIES ON KNOT PROJECTIONS

2013 ◽  
Vol 22 (14) ◽  
pp. 1350085 ◽  
Author(s):  
NOBORU ITO ◽  
YUSUKE TAKIMURA
Keyword(s):  
Equivalence Relation ◽  
Finite Sequence ◽  
Reidemeister Move ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Homotopy Classes ◽  
The Third ◽  
Reidemeister Moves ◽  
Necessary And Sufficient

In this paper, we obtain the necessary and sufficient condition that two knot projections are related by a finite sequence of the first and second flat Reidemeister moves (Theorem 2.2). We also consider an equivalence relation that is called weak (1, 3) homotopy. This equivalence relation occurs by the first flat Reidemeister move and one of the third flat Reidemeister moves. We introduce a map sending weak (1, 3) homotopy classes to knot isotopy classes (Sec. 3). Using the map, we determine which knot projections are trivialized under weak (1, 3) homotopy (Corollary 4.1).

Theory of models with generalized atomic formulas

1960 ◽  
Vol 25 (1) ◽  
pp. 1-26 ◽  
Author(s):  
H. Jerome Keisler
Keyword(s):  
Natural Number ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Elementary Extension ◽  
Necessary And Sufficient Condition ◽  
Elementary Class ◽  
Relational Systems ◽  
Necessary And Sufficient

IntroductionWe shall prove the following theorem, which gives a necessary and sufficient condition for an elementary class to be characterized by a set of sentences having a prescribed number of alternations of quantifiers. A finite sequence of relational systems is said to be a sandwich of order n if each is an elementary extension of (i ≦ n—2), and each is an extension of (i ≦ n—2). If K is an elementary class, then the statements (i) and (ii) are equivalent for each fixed natural number n.

Quasi-standard C*-algebras

1990 ◽  
Vol 107 (2) ◽  
pp. 349-360 ◽  
Author(s):  
R. J. Archbold ◽  
D. W. B. Somerset
Keyword(s):  
Cross Sections ◽  
Equivalence Relation ◽  
Dense Subset ◽  
Base Space ◽  
Ideal Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
C Algebras ◽  
Necessary And Sufficient ◽  
Full Algebra

AbstactA necessary and sufficient condition is given for a separable C*-algebra to be *-isomorphic to a maximal full algebra of cross-sections over a base space such that the fibre algebras are primitive throughout a dense subset. The condition is that the relation of inseparability for pairs of points in the primitive ideal space should be an open equivalence relation.

A note on exchangeable events

1979 ◽  
Vol 16 (03) ◽  
pp. 662-664
Author(s):  
Yu-Sheng Hsu
Keyword(s):  
Infinite Sequence ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Finetti ◽  
Necessary And Sufficient

Kendall [2] gave a thorough discussion of De Finetti constants for a sequence (finite or infinite) of exchangeable events. Galambos [1] and Ridler-Rowe [3] also found some interesting results in this area. In this paper, we intend to give a necessary and sufficient condition for a finite sequence of exchangeable events to be extendable to an infinite sequence of exchangeable events.

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.

scholarly journals Quotient supermanifolds

1998 ◽  
Vol 58 (1) ◽  
pp. 107-120 ◽  
Author(s):  
Claudio Bartocci ◽  
Ugo Bruzzo ◽  
Daniel Hernández Ruipérez ◽  
Vladimir Pestov
Keyword(s):  
Equivalence Relation ◽  
Smooth Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

A necessary and sufficient condition for the existence of a supermanifold structure on a quotient defined by an equivalence relation is established. Furthermore, we show that an equivalence relation R on a Berezin-Leĭtes-Kostant supermanifold X determines a quotient supermanifold X/R if and only if the restriction R0 of R to the underlying smooth manifold X0 of X determines a quotient smooth manifold X0/R0.

Characterizing rosy theories

2007 ◽  
Vol 72 (3) ◽  
pp. 919-940 ◽  
Author(s):  
Clifton Ealy ◽  
Alf Onshuus
Keyword(s):  
Equivalence Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Rosy Theories ◽  
Relation Rank

AbstractWe examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

scholarly journals A UNIVERSALITY RESULT FOR THE GLOBAL FLUCTUATIONS OF THE EIGENVECTORS OF WIGNER MATRICES

2012 ◽  
Vol 01 (04) ◽  
pp. 1250011 ◽  
Author(s):  
FLORENT BENAYCH-GEORGES
Keyword(s):  
Random Process ◽  
Limit Distribution ◽  
Brownian Bridge ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Wigner Matrix ◽  
The Third ◽  
Necessary And Sufficient ◽  
Global Fluctuations ◽  
Third Moment

We prove that for [Formula: see text] the eigenvectors matrix of a Wigner matrix, under some moments conditions, the bivariate random process [Formula: see text] converges in distribution to a bivariate Brownian bridge. This result has already been proved for GOE and GUE matrices. It is conjectured here that the necessary and sufficient condition, for the result to be true for a general Wigner matrix, is the matching of the moments of orders 1, 2 and 4 of the entries of the Wigner with the ones of a GOE or GUE matrix. Surprisingly, the third moment of the entries of the Wigner matrix has no influence on the limit distribution.

scholarly journals Seven-dimensional considerations of Einstein's connection. III. The seven-dimensional Einstein's connection

2003 ◽  
Vol 2003 (15) ◽  
pp. 947-958
Author(s):  
Mi Ae Kim ◽  
Keum Sook So ◽  
Chung Hyun Cho ◽  
Kyung Tae Chung
Keyword(s):  
Recurrence Relations ◽  
Sufficient Condition ◽  
Field Tensor ◽  
Necessary And Sufficient Condition ◽  
The Third ◽  
Unified Field ◽  
Necessary And Sufficient

The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in7-g-UFT and to display a surveyable tensorial representation of seven-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations of the third kind obtained in earlier papers. All considerations in this paper are restricted to the first and second classes ofX7, since the case of the third class, the simplest case, was already studied by many authors.

A note on exchangeable events

1979 ◽  
Vol 16 (3) ◽  
pp. 662-664 ◽  
Author(s):  
Yu-Sheng Hsu
Keyword(s):  
Infinite Sequence ◽  
Finite Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
De Finetti ◽  
Necessary And Sufficient

Kendall [2] gave a thorough discussion of De Finetti constants for a sequence (finite or infinite) of exchangeable events. Galambos [1] and Ridler-Rowe [3] also found some interesting results in this area. In this paper, we intend to give a necessary and sufficient condition for a finite sequence of exchangeable events to be extendable to an infinite sequence of exchangeable events.

scholarly journals Unions of Cockcroft two-complexes

1994 ◽  
Vol 37 (2) ◽  
pp. 317-324
Author(s):  
W. A. Bogley
Keyword(s):  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Third ◽  
Necessary And Sufficient

A combinatorial hypothesis is presented that serves as a necessary and sufficient condition for a union of connected Cockcroft two-complexes to be Cockcroft. This combinatorial hypothesis has a component that can be expressed in terms of the second homology of groups. The hypothesis is applied to the study of the third homology of groups.

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