Extremality and stationarity of collections of sets : metric, slope and normal cone characterisations
- Bui, Hoa
- Authors: Bui, Hoa
- Date: 2019
- Type: Text , Thesis , PhD
- Full Text:
- Description: Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.
- Description: Doctor of Philosophy
- Authors: Bui, Hoa
- Date: 2019
- Type: Text , Thesis , PhD
- Full Text:
- Description: Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.
- Description: Doctor of Philosophy
Subgraph adaptive structure-aware graph contrastive learning
- Chen, Zhikui, Peng, Yin, Yu, Shuo, Cao, Chen, Xia, Feng
- Authors: Chen, Zhikui , Peng, Yin , Yu, Shuo , Cao, Chen , Xia, Feng
- Date: 2022
- Type: Text , Journal article
- Relation: Mathematics (Basel) Vol. 10, no. 17 (2022), p. 3047
- Full Text:
- Reviewed:
- Description: Graph contrastive learning (GCL) has been subject to more attention and been widely applied to numerous graph learning tasks such as node classification and link prediction. Although it has achieved great success and even performed better than supervised methods in some tasks, most of them depend on node-level comparison, while ignoring the rich semantic information contained in graph topology, especially for social networks. However, a higher-level comparison requires subgraph construction and encoding, which remain unsolved. To address this problem, we propose a subgraph adaptive structure-aware graph contrastive learning method (PASCAL) in this work, which is a subgraph-level GCL method. In PASCAL, we construct subgraphs by merging all motifs that contain the target node. Then we encode them on the basis of motif number distribution to capture the rich information hidden in subgraphs. By incorporating motif information, PASCAL can capture richer semantic information hidden in local structures compared with other GCL methods. Extensive experiments on six benchmark datasets show that PASCAL outperforms state-of-art graph contrastive learning and supervised methods in most cases.
- Authors: Chen, Zhikui , Peng, Yin , Yu, Shuo , Cao, Chen , Xia, Feng
- Date: 2022
- Type: Text , Journal article
- Relation: Mathematics (Basel) Vol. 10, no. 17 (2022), p. 3047
- Full Text:
- Reviewed:
- Description: Graph contrastive learning (GCL) has been subject to more attention and been widely applied to numerous graph learning tasks such as node classification and link prediction. Although it has achieved great success and even performed better than supervised methods in some tasks, most of them depend on node-level comparison, while ignoring the rich semantic information contained in graph topology, especially for social networks. However, a higher-level comparison requires subgraph construction and encoding, which remain unsolved. To address this problem, we propose a subgraph adaptive structure-aware graph contrastive learning method (PASCAL) in this work, which is a subgraph-level GCL method. In PASCAL, we construct subgraphs by merging all motifs that contain the target node. Then we encode them on the basis of motif number distribution to capture the rich information hidden in subgraphs. By incorporating motif information, PASCAL can capture richer semantic information hidden in local structures compared with other GCL methods. Extensive experiments on six benchmark datasets show that PASCAL outperforms state-of-art graph contrastive learning and supervised methods in most cases.
Visual tools for analysing evolution, emergence, and error in data streams
- Hart, Sol, Yearwood, John, Bagirov, Adil
- Authors: Hart, Sol , Yearwood, John , Bagirov, Adil
- Date: 2007
- Type: Text , Conference paper
- Relation: Paper presented at 6th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2007, Melbourne, Victoria : 11th-13th July 2007 p. 987-992
- Full Text:
- Description: The relatively new field of stream mining has necessitated the development of robust drift-aware algorithms that provide accurate, real time, data handling capabilities. Tools are needed to assess and diagnose important trends and investigate drift evolution parameters. In this paper, we present two new and novel visualisation techniques, Pixie and Luna graphs, which incorporate salient group statistics coupled with intuitive visual representations of multidimensional groupings over time. Through the novel representations presented here, spatial interactions between temporal divisions can be diagnosed and overall distribution patterns identified. It provides a means of evaluating in non-constrained capacity, commonly constrained evolutionary problems.
- Description: 2003005432
- Authors: Hart, Sol , Yearwood, John , Bagirov, Adil
- Date: 2007
- Type: Text , Conference paper
- Relation: Paper presented at 6th IEEE/ACIS International Conference on Computer and Information Science, ICIS 2007, Melbourne, Victoria : 11th-13th July 2007 p. 987-992
- Full Text:
- Description: The relatively new field of stream mining has necessitated the development of robust drift-aware algorithms that provide accurate, real time, data handling capabilities. Tools are needed to assess and diagnose important trends and investigate drift evolution parameters. In this paper, we present two new and novel visualisation techniques, Pixie and Luna graphs, which incorporate salient group statistics coupled with intuitive visual representations of multidimensional groupings over time. Through the novel representations presented here, spatial interactions between temporal divisions can be diagnosed and overall distribution patterns identified. It provides a means of evaluating in non-constrained capacity, commonly constrained evolutionary problems.
- Description: 2003005432
A framework for monitoring progress and planning teaching towards the effective use of computer algebra systems
- Authors: Pierce, Robyn , Stacey, Kaye
- Date: 2004
- Type: Text , Journal article
- Relation: International Journal of Computers for Mathematical Learning Vol. 9, no. 1 (2004), p. 59-93
- Full Text:
- Reviewed:
- Description: This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.
- Description: C1
- Description: 2003000923
- Authors: Pierce, Robyn , Stacey, Kaye
- Date: 2004
- Type: Text , Journal article
- Relation: International Journal of Computers for Mathematical Learning Vol. 9, no. 1 (2004), p. 59-93
- Full Text:
- Reviewed:
- Description: This article suggests a framework to organise a cluster of variables that are associated with students' effective use of computer algebra systems (CAS) in mathematics learning. Based on a review of the literature and from the authors' own teaching experience, the framework identifies the main characteristics of students' interactions with CAS technology and how these may be used to monitor students' developing use of CAS; from this, the framework may be used to plan teaching in order to gain greater benefit from the availability of CAS. Four case studies describing students' development over a semester are reported. These demonstrate a variety of combinations of technical competencies and personal attributes. They indicate the importance of both the technical and personal aspects but suggest that negative attitudes rather than technical difficulties can limit the effective use of CAS. Finally practical suggestions are given for teaching strategies which may promote effective use of CAS.
- Description: C1
- Description: 2003000923
Magic and antimagic labeling of graphs
- Authors: Sugeng, Kiki Ariyanti
- Date: 2005
- Type: Text , Thesis , PhD
- Full Text:
- Description: "A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."
- Description: Doctor of Philosophy
- Authors: Sugeng, Kiki Ariyanti
- Date: 2005
- Type: Text , Thesis , PhD
- Full Text:
- Description: "A bijection mapping that assigns natural numbers to vertices and/or edges of a graph is called a labeling. In this thesis, we consider graph labelings that have weights associated with each edge and/or vertex. If all the vertex weights (respectively, edge weights) have the same value then the labeling is called magic. If the weight is different for every vertex (respectively, every edge) then we called the labeling antimagic. In this thesis we introduce some variations of magic and antimagic labelings and discuss their properties and provide corresponding labeling schemes. There are two main parts in this thesis. One main part is on vertex labeling and the other main part is on edge labeling."
- Description: Doctor of Philosophy
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