On varieties of Abelian topological groups with coproducts
- Authors: Gabriyelyan, Saak , Morris, Sidney
- Date: 2017
- Type: Text , Journal article
- Relation: Bulletin of the Australian Mathematical Society Vol. 95, no. 1 (2017), p. 54-65
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- Description: A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality. © 2016 Australian Mathematical Publishing Association Inc..
Open mapping theorem for topological groups
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2007
- Type: Text , Journal article
- Relation: Topology Proceedings Vol. 31, no. 2 (2007), p. 533-551
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- Description: We survey sufficient conditions that force a surjective continuous homomorphism between topological groups to be open. We present the shortest proof yet of an open mapping theorem between projective limits of finite dimensional Lie groups.
- Description: C1
- Description: 2003005915
Optimal rees matrix constructions for analysis of data
- Authors: Kelarev, Andrei , Yearwood, John , Zi, Lifang
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of the Australian Mathematical Society Vol. 92, no. 3 (2012), p. 357-366
- Relation: http://purl.org/au-research/grants/arc/LP0990908
- Relation: http://purl.org/au-research/grants/arc/DP0211866
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- Description: Abstract We introduce a new construction involving Rees matrix semigroups and max-plus algebras that is very convenient for generating sets of centroids. We describe completely all optimal sets of centroids for all Rees matrix semigroups without any restrictions on the sandwich matrices. © 2013 Australian Mathematical Publishing Association Inc.
- Description: 2003010862
Optimality conditions via weak subdifferentials in reflexive Banach spaces
- Authors: Hassani, Sara , Mammadov, Musa , Jamshidi, Mina
- Date: 2017
- Type: Text , Journal article
- Relation: Turkish Journal of Mathematics Vol. 41, no. 1 (2017), p. 1-8
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- Description: In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. © Tübitak.
Orthogonality in locally convex spaces : two nonlinear generalizations of Neumann's lemma
- Authors: Barbagallo, Annamaria , Ernst, Octavian-Emil , Théra, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 484, no. 1 (Apr 2020), p. 18
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- Description: In this note we prove a symmetric version of the Neumann lemma as well as a symmetric version of the Soderlind-Campanato lemma. We establish in this way two partial generalizations of the well-known Casazza-Christenses lemma. This work is related to the Birkhoff-James orthogonality and to the concept of near operators introduced by S. Campanato. (C) 2019 Published by Elsevier Inc.
Parallel algorithms for generalized clique transversal problems
- Authors: Miller, Mirka , Dahlhaus, Elias , Manuel, Paul
- Date: 2005
- Type: Text , Journal article
- Relation: Australasian Journal of Combinatorics Vol. 33, no. (2005), p. 3-14
- Full Text: false
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- Description: The K ` - clique transversal problem is to locate a minimum collection of cliques of size ` in a graph G such that every maximal clique of size ` in G contains at least one member of the collection. We give an NC algorithm to solve this problem on strongly chordal graphs. Keywords: balanced graphs, strongly chordal graphs, clique transversal, k-fold clique transversal, K ` - clique transversal. 1 Introduction A 0 Gamma 1 matrix is balanced if it does not contain as a submatrix, an edge - vertex incidence matrix of an odd cycle. A 0 Gamma 1 matrix is totally balanced if it does not contain as a submatrix, an edge - vertex incidence matrix of any cycle. A hypergraph H is an ordered pair (V; E) where V is a set of vertices and E is a family of subsets of V . The members of E are called hyperedges of H . Let V = fv 1 ; v 2 ; : : : ; v n g and E = fE 1 ; E 2 ; : : : ; Em g. Let A(H) denote the hyperedge - vertex incidence matrix of a hypergraph H .
- Description: C1
- Description: 2003001400
Polytopes close to being simple
- Authors: Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David
- Date: 2020
- Type: Text , Journal article
- Relation: Discrete and Computational Geometry Vol. 64, no. 1 (2020), p. 200-215
- Relation: http://purl.org/au-research/grants/arc/DP180100602
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- Description: It is known that polytopes with at most two nonsimple vertices are reconstructible from their graphs, and that d-polytopes with at most d- 2 nonsimple vertices are reconstructible from their 2-skeletons. Here we close the gap between 2 and d- 2 , showing that certain polytopes with more than two nonsimple vertices are reconstructible from their graphs. In particular, we prove that reconstructibility from graphs also holds for d-polytopes with d+ k vertices and at most d- k+ 3 nonsimple vertices, provided k
Primal necessary characterizations of transversality properties
- Authors: Cuong, Nguyen , Kruger, Alexander
- Date: 2021
- Type: Text , Journal article
- Relation: Positivity Vol. 25, no. 2 (2021), p. 531-558
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper continues the study of general nonlinear transversality properties of collections of sets and focuses on primal necessary (in some cases also sufficient) characterizations of the properties. We formulate geometric, metric and slope characterizations, particularly in the convex setting. The Hölder case is given a special attention. Quantitative relations between the nonlinear transversality properties of collections of sets and the corresponding regularity properties of set-valued mappings as well as two nonlinear transversality properties of a convex set-valued mapping to a convex set in the range space are discussed. © 2020, Springer Nature Switzerland AG.
Projective limits of finite-dimensional Lie groups
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2003
- Type: Text , Journal article
- Relation: Proceedings of the London Mathematical Society Vol. 87, no. 3 (Nov 2003), p. 647-676
- Full Text: false
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- Description: For a topological group G we define N to be the set of all normal subgroups modulo which G is a finite-dimensional Lie group. Call G a pro-Lie group if, firstly, G is complete, secondly, N is a filter basis, and thirdly, every identity neighborhood of G contains some member of N. It is easy to see that every pro-Lie group G is a projective limit of the projective system of all quotients of G modulo subgroups from N. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup N of a pro-Lie group G, for any one parameter subgroup Y : R G/N there is a one parameter subgroup X : R G such that X(t) N = Y(t) for any real number t. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.
- Description: C1
- Description: 2003000376
Quadratic form representations via generalized continuants
- Authors: Delorme, Charles , Pineda-Villavicencio, Guillermo
- Date: 2015
- Type: Text , Journal article
- Relation: Journal of Integer Sequences Vol. 18, no. 6 (2015), p. Article number 15.6.4
- Full Text: false
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- Description: H. J. S. Smith proved Fermat’s two-square theorem using the notion of palindromic continuants. In this paper we extend Smith’s approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of integers and rings of polynomials over fields of odd characteristic. Also, we present new deterministic algorithms for finding the corresponding proper representations. © 2015 University of Waterloo. All rights reserved.
Qualitative stability of a class of non-monotone variational inclusions. Application in electronics
- Authors: Adly, Samir , Outrata, Jiri
- Date: 2013
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 20, no. 1 (2013), p. 43-66
- Full Text: false
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- Description: The main concern of this paper is to investigate some stability properties (namely Aubin property and isolated cahnness) of a special non-monotone variational inclusion. We provide a characterization of these properties in terms of the problem data and show their importance for the design of electrical circuits involving nonsmooth and non-monotone electronic devices Uke DIAC (Diode Alternating Current). Circuits with other devices like SCR (Silicon Controlled Rectifiers), Zener diodes, thyristors, varactors and transistors can be analyzed in the same way. © Heldermann Verlag.
- Description: 2003011029
Quasilinear Mappings, M-Ideals and Popyhedra
- Authors: Yost, David
- Date: 2012
- Type: Text , Conference paper
- Relation: Operators and Matrices Vol. 6, p. 279-286
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- Description: We survey the connection between two results from rather different areas: failure of the 3-space property for local convexity (and other properties) within the category of quasi-Banach spaces, and the irreducibility (in the sense of Minkowski difference) of large families of finite dimensional polytopes.
- Description: C1
Regularity of collections of sets and convergence of inexact alternating projections
- Authors: Kruger, Alexander , Thao, Nguyen
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 23, no. 3 (2016), p. 823-847
- Relation: http://purl.org/au-research/grants/arc/DP110102011
- Full Text: false
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- Description: We study the usage of regularity properties of collections of sets in convergence analysis of alternating projection methods for solving feasibility problems. Several equivalent characterizations of these properties are provided. Two settings of inexact alternating projections are considered and the corresponding convergence estimates are established and discussed.
Relationship between adjacency matrices and super (a,d)-edge-antimagic total labeling of graphs
- Authors: Miller, Mirka , Sugeng, Kiki Ariyanti
- Date: 2005
- Type: Text , Journal article
- Relation: The Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 55, no. (2005), p. 71-82
- Full Text: false
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- Description: C1
- Description: 2003001403
Sigma supporting cone and optimality conditions in non-convex problems
- Authors: Hassani, Sara , Mammadov, Musa
- Date: 2014
- Type: Text , Journal article
- Relation: Far East Journal of Mathematical Sciences Vol. 91, no. 2 (2014), p. 169-190
- Full Text: false
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- Description: In this paper, a new supporting function for characterizing non-convex sets is introduced. The notions of σ-supporting cone and maximal conic gap are proposed and some properties are investigated. By applying these new notions, we establish the optimality conditions considered in [7] for a broader class of finite dimensional normed spaces in terms of weak subdifferentials.
Sigma-porosity in monotonic analysis with applications to optimization
- Authors: Rubinov, Alex
- Date: 2005
- Type: Text , Journal article
- Relation: Abstract and Applied Analysis Vol. 2005, no. 3 (2005), p. 287-305
- Full Text: false
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- Description: We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are $sigma$-porous in corresponding spaces. Some applications to optimization are given.
- Description: C1
- Description: We introduce and study some metric spaces of increasing positively homogeneous (IPH) functions, decreasing functions, and conormal (upward) sets. We prove that the complements of the subset of strictly increasing IPH functions, of the subset of strictly decreasing functions, and of the subset of strictly conormal sets are $\sigma$-porous in corresponding spaces. Some applications to optimization are given.
- Description: 2003001421
Slopes of multifunctions and extensions of metric regularity
- Authors: Ngai, Huynh Van , Kruger, Alexander , Thera, Michel
- Date: 2012
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics (Tạp chí toán học) Vol. 40, no. 2/3 (2012), p. 355-369
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: This article aims to demonstrate how the definitions of slopes can be extended to multi-valued mappings between metric spaces and applied for characterizing metric regularity. Several kinds of local and nonlocal slopes are defined and several metric regularity properties for set-valued mappings between metric spaces are investigated.
Some examples of relations between non-stable integral cohomology operations
- Authors: Percy, Andrew
- Date: 2010
- Type: Text , Journal article
- Relation: Bulletin of the Korean Mathematical Society Vol. 47, no. 2 (2010), p. 275-286
- Full Text: false
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- Description: The algebraic structure of the natural integral cohomology operations is explored by means of examples. We decompose the generators of the groups with and into the operations of cup products, cross-cap products and compositions. Examination of these decompositions and comparison with other possible generators demonstrates the existence of relations between integral operations that have withheld formulation. The calculated groups and generators are collected in a table for practical reference.
Some new characterizations of intrinsic transversality in hilbert spaces
- Authors: Thao, Nguyen , Bui, Hoa , Cuong, Nguyen , Verhaegen, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 5-39
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- Description: Motivated by a number of questions concerning transversality-type properties of pairs of sets recently raised by Ioffe and Kruger, this paper reports several new characterizations of the intrinsic transversality property in Hilbert spaces. New results in terms of normal vectors clarify the picture of intrinsic transversality, its variants and sufficient conditions for subtransversality, and unify several of them. For the first time, intrinsic transversality is characterized by an equivalent condition which does not involve normal vectors. This characterization offers another perspective on intrinsic transversality. As a consequence, the obtained results allow us to answer a number of important questions about transversality-type properties. © 2020, The Author(s).
Sophus Lie's third fundamental theorem and the adjoint functor theorem
- Authors: Hofmann, Karl , Morris, Sidney
- Date: 2005
- Type: Text , Journal article
- Relation: Journal of Group Theory Vol. 8, no. 1 (2005), p. 115-133
- Full Text: false
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- Description: The essential attributes of a Lie group G are the associated Lie algebra LðGÞ and the exponential function exp : LðGÞ ! G. The prescription L operates not only on Lie groups but also on morphisms between them: it is a functor. Many features of Lie theory are shared by classes of topological groups which are much larger than that of Lie groups; these classes include the classes of compact groups, locally compact groups, and pro-Lie groups, that is, complete topological groups having arbitrarily small normal subgroups N such that G=N is a (finitedimensional) Lie group. Considering the functor L it is therefore appropriate to contemplate more general classes of topological groups. Certain functorial properties of the assignment of a Lie algebra to a topological group (where possible) will be essential. What is new here is that we will introduce a functorial assignment from Lie algebras to groups and investigate to what extent it is inverse to the Lie algebra functor L. While the Lie algebra functor is well known and is cited regularly, the existence of a Lie group functor available to be cited and applied appears less well known. Sophus Lie’s Third Fundamental Theorem says that for each finite-dimensional real Lie algebra there is a Lie group whose Lie algebra is (isomorphic to) the given one; but even in classical circumstances it is not commonly known that this happens in a functorial fashion and what the precise relationship between the Lie algebra functor and the Lie group functor is.
- Description: C1
- Description: 2003001415