Internet security applications of the Munn rings
- Authors: Kelarev, Andrei , Yearwood, John , Watters, Paul , Wu, Xinwen , Abawajy, Jemal , Pan, L.
- Date: 2010
- Type: Text , Journal article
- Relation: Semigroup Forum Vol. 81, no. 1 (2010), p. 162-171
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- Description: Effective multiple clustering systems, or clusterers, have important applications in information security. The aim of the present article is to introduce a new method of designing multiple clusterers based on the Munn rings and describe a class of optimal clusterers which can be obtained in this construction.
Internet security applications of Grobner-Shirvov bases
- Authors: Kelarev, Andrei , Yearwood, John , Watters, Paul
- Date: 2010
- Type: Text , Journal article
- Relation: Asian-European Journal of Mathematics Vol. 3, no. 3 (2010), p. 435-442
- Relation: http://purl.org/au-research/grants/arc/DP0211866
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A Grobner-Shirshov Algorithm for Applications in Internet Security
- Authors: Kelarev, Andrei , Yearwood, John , Watters, Paul , Wu, Xinwen , Ma, Liping , Abawajy, Jemal , Pan, L.
- Date: 2011
- Type: Text , Journal article
- Relation: Southeast Asian Bulletin of Mathematics Vol. 35, no. (2011), p. 807-820
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- Description: The design of multiple classication and clustering systems for the detection of malware is an important problem in internet security. Grobner-Shirshov bases have been used recently by Dazeley et al. [15] to develop an algorithm for constructions with certain restrictions on the sandwich-matrices. We develop a new Grobner-Shirshov algorithm which applies to a larger variety of constructions based on combinatorial Rees matrix semigroups without any restrictions on the sandwich-matrices.
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
Teaching with CAS in a time of transition
- Authors: Kendal, Margaret , Stacey, Kaye , Pierce, Robyn
- Date: 2002
- Type: Text , Journal article
- Relation: International Journal of Computer Algebra in Mathematics Education Vol. 9, no. 2 (2002), p. 113-128
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- Description: Integrating a powerful instrument such as CAS into teaching and learning mathematics requires changes to many aspects of the classroom, which teachers will make from the base of their prior teaching styles and their beliefs about mathematics and how it should be taught. The paper describes the different ways in which two pioneering Australian teachers adapted their teaching to use CAS. One teacher used CAS with the primary goal of increasing understanding but restricted students’ use. The other teacher adopted CAS as an extra technique for solving standard problems, emphasising timesaving routines by hand and with CAS. Through these case studies we comment on the following issues related to teaching with CAS: different ways of organising the classroom, variety in approaches to teaching the use of CAS, the increased range of methods for solving problems and for teaching, the contrast between using of graphics calculators and CAS, the challenge of finding the place of by-hand skills and CAS use, and the curriculum and assessment changes required in schools.
- Description: C1
- Description: 2003000120
From convergence principles to stability and optimality conditions
- Authors: Klatte, Diethard , Kruger, Alexander , Kummer, Bernd
- Date: 2012
- Type: Text , Journal article
- Relation: Journal of Convex Analysis Vol. 19, no. 4 (2012), p. 1043-1072
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- Description: We show in a rather general setting that Hoelder and Lipschitz stability properties of solutions to variational problems can be characterized by convergence of more or less abstract iteration schemes. Depending on the principle of convergence, new and intrinsic stability conditions can be derived. Our most abstract models are (multi-) functions on complete metric spaces. The relevance of this approach is illustrated by deriving both classical and new results on existence and optimality conditions, stability of feasible and solution sets and convergence behavior of solution procedures. © Heldermann Verlag.
- Description: 2003010677
Attracting sets for increasing co-radiant and topical operators
- Authors: Kloeden, Peter , Rubinov, Alex
- Date: 2002
- Type: Text , Journal article
- Relation: Mathematische Nachrichten Vol. 243, no. (2002), p. 134-145
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- Description: A generalization of the Perron-Frobenius theorem to increasing positively homogeneous of degree one operators is extended to increasing co-radiant and topical operators, which are of interest in mathematical economics. In particular, small attracting sets containing the limit points of all sequences generated by iteration of such operators are determined.
- Description: C1
- Description: 2003000150
On Fréchet subdifferentials
- Authors: Kruger, Alexander
- Date: 2003
- Type: Text , Journal article
- Relation: Journal of Mathematical Sciences Vol. 116, no. 3 (2003), p. 3325-3358
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- Description: 2003002852
About regularity of collections of sets
- Authors: Kruger, Alexander
- Date: 2006
- Type: Text , Journal article
- Relation: Set-Valued Analysis Vol. 14, no. 2 (Jun 2006), p. 187-206
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- Description: The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular.
- Description: 2003001526
About intrinsic transversality of pairs of sets
- Authors: Kruger, Alexander
- Date: 2018
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.
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
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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.
On relaxing the Mangasarian-Fromovitz constraint qualification
- Authors: Kruger, Alexander , Minchenko, Leonld , Outrata, Jiri
- Date: 2014
- Type: Text , Journal article
- Relation: Positivity Vol. 18, no. 1 (2014), p. 171-189
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: For the classical nonlinear program, two new relaxations of the Mangasarian– Fromovitz constraint qualification are discussed and their relationship with some standard constraint qualifications is examined. In particular, we establish the equivalence of one of these constraint qualifications with the recently suggested by Andreani et al. Constant rank of the subspace component constraint qualification. As an application, we make use of this new constraint qualification in the local analysis of the solution map to a parameterized equilibrium problem, modeled by a generalized equation.
Error bounds and Hölder metric subregularity
- Authors: Kruger, Alexander
- Date: 2015
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 705-736
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- Description: The Holder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Holder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.
Error bounds for vector-valued funtions on metric spaces
- Authors: Kruger, Alexander , Bednarczuk, Ewa
- Date: 2012
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 40, no. 2/3 (2012), p. 165-180
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- Description: In this paper, we attempt to extend the definition and existing local error bound criteria to vector-valued functions, or more generally, to functions taking values in a normed linear space. Some new primal space derivative-like objects – slopes – are introduced and a classification scheme of error bound criteria is presented.
About [q]-regularity properties of collections of sets
- Authors: Kruger, Alexander , Thao, Nguyen
- Date: 2014
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 416, no. 2 (2014), p. 471-496
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.
- Description: We examine three primal space local Holder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed. (C) 2014 Elsevier Inc. All rights reserved.
About subtransversality of collections of sets
- Authors: Kruger, Alexander , Luke, Russell , Thao, Nguyen
- Date: 2017
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 25, no. 4 (2017), p. 701-729
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces.
Borwein-Preiss variational principle revisited
- Authors: Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn
- Date: 2016
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193
- Relation: http://purl.org/au-research/grants/arc/DP110102011
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- Description: In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.
Borwein–Preiss vector variational principle
- Authors: Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn
- Date: 2017
- Type: Text , Journal article
- Relation: Positivity Vol. 21, no. 4 (2017), p. 1273-1292
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.
Holder error bounds and holder calmness with applications to convex semi-infinite optimization
- Authors: Kruger, Alexander , Lopez, Marco , Yang, Xiaoqi , Zhu, Jiangxing
- Date: 2019
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 27, no. 4 (Dec 2019), p. 995-1023
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- Description: Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Holder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Holder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Holder calmness modulus of the argmin mapping in the framework of linear programming.
Embedding of the free abelian topological group A (X ⊕ X) into A (X)
- Authors: Krupski, Mikolaj , Leiderman, Arkady , Morris, Sidney
- Date: 2019
- Type: Text , Journal article
- Relation: Mathematika Vol. 65, no. 3 (2019), p. 708-718
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- Description: We consider the following question: for which metrizable separable spaces X does the free abelian topological group A (X ⊕ X) isomorphically embed into A (X). While for many natural spaces X such an embedding exists, our main result shows that if X is a Cook continuum or X is a rigid Bernstein set, then A(X ⊕ X) does not embed into A(X) as a topological subgroup. The analogous statement is true for the free boolean group B (X).
- Description: We consider the following question: for which metrizable separable spaces X does the free abelian topological group A (X