On the construction of the linear Hull of a set in a convexity space using only the extension operation

1978 ◽  
Vol 11 (2) ◽  
pp. 106-109
Author(s):  
André Dessard
Keyword(s):  
Linear Hull ◽  
Convexity Space

scholarly journals Decomposable Convexities in Graphs and Hypergraphs

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Francesco M. Malvestuto
Keyword(s):  
Convex Hull ◽  
Convex Sets ◽  
Nonempty Subset ◽  
Convex Set ◽  
Convex Geometry ◽  
Convexity Space ◽  
Vertex Set ◽  
Convex Geometries ◽  
Acyclic Hypergraph ◽  
Maximal Clusters

Given a connected hypergraph with vertex set V, a convexity space on is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called convex sets. The convex hull of a subset X of V is the smallest convex set containing X. By a cluster of we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on is decomposable if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).

Linearization of a Convexity Space

1976 ◽  
Vol s2-13 (2) ◽  
pp. 209-214 ◽  
Author(s):  
P. Mah ◽  
S. A. Naimpally ◽  
J. H. M. Whitfield
Keyword(s):  
Convexity Space

scholarly journals Linear Cryptanalysis: Key Schedules and Tweakable Block Ciphers

2017 ◽  
pp. 474-505 ◽  
Author(s):  
Thorsten Kranz ◽  
Gregor Leander ◽  
Friedrich Wiemer
Keyword(s):  
Block Ciphers ◽  
Linear Cryptanalysis ◽  
Linear Hull ◽  
Key Schedule ◽  
Schedule Design ◽  
Key Scheduling

This paper serves as a systematization of knowledge of linear cryptanalysis and provides novel insights in the areas of key schedule design and tweakable block ciphers. We examine in a step by step manner the linear hull theorem in a general and consistent setting. Based on this, we study the influence of the choice of the key scheduling on linear cryptanalysis, a – notoriously difficult – but important subject. Moreover, we investigate how tweakable block ciphers can be analyzed with respect to linear cryptanalysis, a topic that surprisingly has not been scrutinized until now.

scholarly journals Topological convexity spaces

1974 ◽  
Vol 19 (2) ◽  
pp. 125-132 ◽  
Author(s):  
Victor Bryant
Keyword(s):  
Vector Space ◽  
Topological Vector Space ◽  
Topological Properties ◽  
Convexity Space ◽  
Basic Properties

We shall start by recalling the definition and some basic properties of a convexity space; a topological convexity space (tcs) will then be a convexity space together with an admissible topology, and will be a generalisation of a topological vector space (tvs). After showing that the usual tvs results connecting the linear and topological properties extend to this new setting we then prove a form of the Krein-Milman theorem in a tcs.

Subspaces with property (b) in locally convex spaces of quasi-barrelled type

1980 ◽  
Vol 88 (2) ◽  
pp. 331-337 ◽  
Author(s):  
Bella Tsirulnikov
Keyword(s):  
Convex Space ◽  
Locally Convex Space ◽  
Dense Subspace ◽  
Locally Convex Spaces ◽  
Linear Hull ◽  
Bounded Set ◽  
Property B ◽  
Locally Convex ◽  
Strong Dual ◽  
Convex Spaces

A subspace G of a locally convex space E has property (b) if for every bounded set B of E the codimension of G in the linear hull of G ∪ B is finite, (5). Extending the results of (5) and (14), we prove that, if the strong dual of E is complete, then subspaces with property (b) inherit the following properties of E: σ-evaluability, evaluability, the property of being Mazur, semibornological and bornological. We also prove that a dense subspace with property (b) of a Mazur space is sequentially dense, and of a semibornological space – dense in the sense of Mackey (locally dense, following M. Valdivia).

scholarly journals Identities on Maximal Subgroups of GLn(D)

2005 ◽  
Vol 12 (03) ◽  
pp. 461-470 ◽  
Author(s):  
D. Kiani ◽  
M. Mahdavi-Hezavehi
Keyword(s):  
Maximal Subgroup ◽  
Division Algebra ◽  
Division Ring ◽  
Group Identity ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
Polynomial Identities ◽  
Linear Hull ◽  
Finite Dimensional ◽  
Group Identities

Let D be a division ring with centre F. Assume that M is a maximal subgroup of GLn(D) (n≥1) such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is non-commutative and F is infinite, it is also proved that if M satisfies a group identity and F[M] is algebraic over F, then we have either M=K* where K is a field and [D:F]<∞, or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GLn(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite.

Sign in / Sign up

Export Citation Format

Share Document