Applications of the low-basis theorem in arithmetic

1985 ◽  
pp. 65-88 ◽  
Author(s):  
P. Clote
Keyword(s):  
Basis Theorem

scholarly journals A generalization of the Burnside basis theorem

2014 ◽  
Vol 400 ◽  
pp. 8-16 ◽  
Author(s):  
Paul Apisa ◽  
Benjamin Klopsch
Keyword(s):  
Basis Theorem

The Weak Basis Theorem Fails in Non-Locally Convex F-Spaces

1977 ◽  
Vol 29 (5) ◽  
pp. 1069-1071 ◽  
Author(s):  
L. Drewnowski
Keyword(s):  
Weak Basis ◽  
Basis Theorem ◽  
Locally Convex

W. J. Stiles showed in [10, Corollary 4.5] that Banach's weak basis theorem fails in the spaces lp, 0 < p < 1. Then, J. H. Shapiro [9] indicated certain general classes of non-locally convex F-spaces with the same property, and asked whether the weak basis theorem fails in every non-locally convex F-space with a weak basis. Our purpose is to answer this question in the affirmative. In [3] we observed that, essentially, the only case that remained open is that of an F-space with irregular basis (en), i.e. such that snen →0 for any scalar sequence (sn).

On the Basis and Chromatic Number of a Graph

1966 ◽  
Vol 18 ◽  
pp. 969-973
Author(s):  
C. J. Everett
Keyword(s):  
Chromatic Number ◽  
Directed Graphs ◽  
Cardinal Number ◽  
Ordered Sets ◽  
Basis Theorem ◽  
Arbitrary Collection

The basis theorem for directed graphs is, in effect, a result on weakly ordered sets, and, in §1, a proof is given, based on Zorn's lemma, that generalizes, and perhaps clarifies the exposition in (1, Chapter 2). In §2, a graph G* is defined, on an arbitrary collection Q of non-void subsets of a set X (which includes all its one-element subsets), in such a way that the partitions of X into Q-sets correspond to the kernels of G*. Applied to the collection Q of non-null internally stable subsets of a graph G without loops, this identifies the chromatic number of G with the least cardinal number of any kernel of G*.

The Normal Basis Theorem

1979 ◽  
Vol 86 (3) ◽  
pp. 212-212
Author(s):  
William C. Waterhouse
Keyword(s):  
Normal Basis ◽  
Basis Theorem

scholarly journals A proof of Lyndon's finite basis theorem

1980 ◽  
Vol 29 (3) ◽  
pp. 229-233 ◽  
Author(s):  
Joel Berman
Keyword(s):  
Finite Basis ◽  
Basis Theorem

The Normal Basis Theorem

1950 ◽  
Vol s1-25 (4) ◽  
pp. 259-264 ◽  
Author(s):  
J. W. S. Cassels ◽  
G. E. Wall
Keyword(s):  
Normal Basis ◽  
Basis Theorem

An effective selection theorem

1982 ◽  
Vol 47 (2) ◽  
pp. 388-394 ◽  
Author(s):  
Ashok Maitra
Keyword(s):  
Polish Space ◽  
Local Version ◽  
Compact Set ◽  
Selection Theorem ◽  
Analytic Subset ◽  
Effective Version ◽  
Basis Theorem ◽  
Measurable Selector ◽  
Effective Selection ◽  
Borel Measurable

A recent result of J.P. Burgess [1] states:Theorem 0. Let F be a multifunction from an analytic subset T of a Polish space to a Polish space X. If F is Borel measurable, Graph(F) is coanalytic in T × X and F(t) is nonmeager in its closure for each t Є T, then F admits a Borel measurable selector.The above result unifies and significantly extends earlier results of H. Sarbadhikari [8], S.M. Srivastava [9] and G. Debs (unpublished). The reader is referred to [1] for details.The aim of this article is to give an effective version of Theorem 0. We do this by proving a basis theorem for Π11 sets which are nonmeager in their closure and satisfy a local version of the measurability condition in Theorem 0. Our basis theorem generalizes a well-known result of P.G. Hinman [4] and S.K. Thomason [10] (see also [5] and [7, 4F.20]). Our methods are similar to those used by A. Louveau to prove that a , σ-compact set is contained in a , σ-compact set (see [7, 4F.18]).The paper is organized as follows. §2 is devoted to preliminaries. In §3, we prove the basis theorem and deduce as a consequence an effective version of Theorem 0. We show in §4 how our methods can be used to give alternative proofs of some known results.Discussions with R. Barua, B.V. Rao and V.V. Srivatsa are gratefully acknowledged. I am indebted to J.P. Burgess for drawing my attention to an error in an earlier draft of this paper.

A Hilbert Basis Theorem for Quantum Groups

1997 ◽  
Vol 29 (2) ◽  
pp. 150-158 ◽  
Author(s):  
K. A. Brown ◽  
K. R. Goodearl
Keyword(s):  
Quantum Groups ◽  
Hilbert Basis ◽  
Basis Theorem
Sign in / Sign up

Export Citation Format

Share Document