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46Miller, Mirka
25Morris, Sidney
24Kruger, Alexander
17Pineda-Villavicencio, Guillermo
15Sugeng, Kiki Ariyanti
14Lin, Yuqing
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13Rubinov, Alex
13Ryan, Joe
11Hofmann, Karl
11Outrata, Jiri
10Yost, David
9Kelarev, Andrei
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8Ugon, Julien
7Gabriyelyan, Saak
7Théra, Michel
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6Yearwood, John

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270102 Applied Mathematics
140802 Computation Theory and Mathematics
12Graph theory
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9Normal cone
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7Subtransversality
60103 Numerical and Computational Mathematics
6Connectivity
6Data mining
6Metric regularity
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6Transversality
6Variational analysis
50906 Electrical and Electronic Engineering
5Moore bound
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A new proof of Balinski's theorem on the connectivity of polytopes

- Pineda-Villavicencio, Guillermo

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2021**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 344, no. 7 (2021), p.**Full Text:****Reviewed:****Description:**Balinski (1961) proved that the graph of a d-dimensional convex polytope is d-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly d vertices; some of which appear to be new. © 2021 Elsevier B.V.

**Authors:**Pineda-Villavicencio, Guillermo**Date:**2021**Type:**Text , Journal article**Relation:**Discrete Mathematics Vol. 344, no. 7 (2021), p.**Full Text:****Reviewed:****Description:**Balinski (1961) proved that the graph of a d-dimensional convex polytope is d-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly d vertices; some of which appear to be new. © 2021 Elsevier B.V.

A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure

- Gala, Sadek, Ragusa, Maria, Théra, Michel

**Authors:**Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Bolletino dell Unione Matematica Italiana Vol. 14, no. 2 (2021), p. 331-337**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana.

**Authors:**Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Bolletino dell Unione Matematica Italiana Vol. 14, no. 2 (2021), p. 331-337**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana.

Alternative representations of the normal cone to the domain of supremum functions and subdifferential calculus

- Correa, R., Hantoute, A., López, Marco

**Authors:**Correa, R. , Hantoute, A. , López, Marco**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

**Authors:**Correa, R. , Hantoute, A. , López, Marco**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 683-699**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**The first part of the paper provides new characterizations of the normal cone to the effective domain of the supremum of an arbitrary family of convex functions. These results are applied in the second part to give new formulas for the subdifferential of the supremum function, which use both the active and nonactive functions at the reference point. Only the data functions are involved in these characterizations, the active ones from one side, together with the nonactive functions multiplied by some appropriate parameters. In contrast with previous works in the literature, the main feature of our subdifferential characterization is that the normal cone to the effective domain of the supremum (or to finite-dimensional sections of this domain) does not appear. A new type of optimality conditions for convex optimization is established at the end of the paper. © 2021, The Author(s), under exclusive licence to Springer Nature B.V.

An adaptive splitting algorithm for the sum of two generalized monotone operators and one cocoercive operator

**Authors:**Dao, Minh , Phan, Hung**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p.**Full Text:****Reviewed:****Description:**Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

**Authors:**Dao, Minh , Phan, Hung**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (2021), p.**Full Text:****Reviewed:****Description:**Splitting algorithms for finding a zero of sum of operators often involve multiple steps which are referred to as forward or backward steps. Forward steps are the explicit use of the operators and backward steps involve the operators implicitly via their resolvents. In this paper, we study an adaptive splitting algorithm for finding a zero of the sum of three operators. We assume that two of the operators are generalized monotone and their resolvents are computable, while the other operator is cocoercive but its resolvent is missing or costly to compute. Our splitting algorithm adapts new parameters to the generalized monotonicity of the operators and, at the same time, combines appropriate forward and backward steps to guarantee convergence to a solution of the problem. © 2021, The Author(s).

Enlargements of the moreau–rockafellar subdifferential

- Abbasi, Malek, Kruger, Alexander, Théra, Michel

**Authors:**Abbasi, Malek , Kruger, Alexander , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 701-719**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup

**Authors:**Abbasi, Malek , Kruger, Alexander , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 701-719**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup

Hilbert 13: Are there are any genuine continuous multivariate real-valued functions?

**Authors:**Morris, Sidney**Date:**2021**Type:**Text , Journal article**Relation:**Bulletin of the American Mathematical Society Vol. 58, no. 1 (2021), p. 107-118**Full Text:**false**Reviewed:****Description:**This article begins with a provocative question: Are there any genuine continuous multivariate real-valued functions? This may seem to be a silly question, but it is in essence what David Hilbert asked as one of the 23 problems he posed at the second International Congress of Mathematicians, held in Paris in 1900. These problems guided a large portion of the research in mathematics of the 20th century. Hilbert’s 13th problem conjectured that there exists a continuous function (Formula presented), where (Formula presented), which cannot be expressed in terms of composition and addition of continuous functions from ℝ2 → ℝ, that is, as composition and addition of continuous real-valued functions of two variables. It took over 50 years to prove that Hilbert’s conjecture is false. This article discusses the solution. © 2021. American Mathematical Society.

On the regularity of weak solutions of the boussinesq equations in besov spaces

- Barbagallo, Annamaria, Gala, Sadek, Ragusa, Maria, Théra, Michel

**Authors:**Barbagallo, Annamaria , Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 49, no. 3 (2021), p. 637-649**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space Ḃ

**Authors:**Barbagallo, Annamaria , Gala, Sadek , Ragusa, Maria , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 49, no. 3 (2021), p. 637-649**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space Ḃ

Primal necessary characterizations of transversality properties

- Cuong, Nguyen, Kruger, Alexander

**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**Full Text:**false**Reviewed:****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.

Strictly convex banach algebras

**Authors:**Yost, David**Date:**2021**Type:**Text , Journal article**Relation:**Axioms Vol. 10, no. 3 (2021), p.**Full Text:****Reviewed:****Description:**We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C

**Authors:**Yost, David**Date:**2021**Type:**Text , Journal article**Relation:**Axioms Vol. 10, no. 3 (2021), p.**Full Text:****Reviewed:****Description:**We discuss two facets of the interaction between geometry and algebra in Banach algebras. In the class of unital Banach algebras, there is essentially one known example which is also strictly convex as a Banach space. We recall this example, which is finite-dimensional, and consider the open question of generalising it to infinite dimensions. In C

Strongly regular points of mappings

- Abbasi, Malek, Théra, Michel

**Authors:**Abbasi, Malek , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (Journal article 2021), p.**Full Text:****Reviewed:****Description:**In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound. © 2021, The Author(s).

**Authors:**Abbasi, Malek , Théra, Michel**Date:**2021**Type:**Text , Journal article**Relation:**Fixed Point Theory and Algorithms for Sciences and Engineering Vol. 2021, no. 1 (Journal article 2021), p.**Full Text:****Reviewed:****Description:**In this paper, we use a robust lower directional derivative and provide some sufficient conditions to ensure the strong regularity of a given mapping at a certain point. Then, we discuss the Hoffman estimation and achieve some results for the estimate of the distance to the set of solutions to a system of linear equalities. The advantage of our estimate is that it allows one to calculate the coefficient of the error bound. © 2021, The Author(s).

The linkedness of cubical polytopes: the cube

- Bui, Hoa, Pineda-Villavicencio, Guillermo, Ugon, Julien

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2021**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph G is strongly k-linked if it has at least 2k + 1 vertices and, for every vertex v of G, the subgraph G − v is k-linked. We show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2021**Type:**Text , Journal article**Relation:**Electronic Journal of Combinatorics Vol. 28, no. 3 (2021), p.**Relation:**http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**The paper is concerned with the linkedness of the graphs of cubical polytopes. A graph with at least 2k vertices is k-linked if, for every set of k disjoint pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We say that a polytope is k-linked if its graph is k-linked. We establish that the d-dimensional cube is [(d + 1)/2]-linked, for every d ≠ 3; this is the maximum possible linkedness of a d-polytope. This result implies that, for every d ≥ 1, a cubical d-polytope is [d/2]-linked, which answers a question of Wotzlaw (Incidence graphs and unneighborly polytopes, Ph.D. thesis, 2009). Finally, we introduce the notion of strong linkedness, which is slightly stronger than that of linkedness. A graph G is strongly k-linked if it has at least 2k + 1 vertices and, for every vertex v of G, the subgraph G − v is k-linked. We show that cubical 4-polytopes are strongly 2-linked and that, for each d ≥ 1, d-dimensional cubes are strongly

Transversality properties : primal sufficient conditions

- Cuong, Nguyen, Kruger, Alexander

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2021**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 29, no. 2 (2021), p. 221-256**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V.

Almost simplicial polytopes : the lower and upper bound theorems

- Nevo, Eran, Pineda-Villavicencio, Guillermo, Ugon, Julien, Yost, David

**Authors:**Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2020**Type:**Text , Journal article**Relation:**Canadian Journal of Mathematics Vol. 72, no. 2 (2020), p. 537-556. http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest. © 2018 Canadian Mathematical Society.

**Authors:**Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2020**Type:**Text , Journal article**Relation:**Canadian Journal of Mathematics Vol. 72, no. 2 (2020), p. 537-556. http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We study -vertex -dimensional polytopes with at most one nonsimplex facet with, say, vertices, called almost simplicial polytopes. We provide tight lower and upper bound theorems for these polytopes as functions of, and, thus generalizing the classical Lower Bound Theorem by Barnette and the Upper Bound Theorem by McMullen, which treat the case where s = 0. We characterize the minimizers and provide examples of maximizers for any. Our construction of maximizers is a generalization of cyclic polytopes, based on a suitable variation of the moment curve, and is of independent interest. © 2018 Canadian Mathematical Society.

Assessing healthcare providers' performance with and without risk adjustment

**Authors:**Morales-Silva, Daniel**Date:**2020**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 102, no. 1 (AUG 2020), p. 172-173**Full Text:****Reviewed:**

**Authors:**Morales-Silva, Daniel**Date:**2020**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 102, no. 1 (AUG 2020), p. 172-173**Full Text:****Reviewed:**

**Authors:**Ali, Elaf**Date:**2020**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 101, no. 1 (Feb 2020), p. 172-173**Full Text:**false**Reviewed:**

Directional metric pseudo subregularity of set-valued mappings: a general model

- Van Ngai, Huynh, Tron, Nguyen, Van Vu, Nguyen, Théra, Michel

**Authors:**Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel**Date:**2020**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 61-87**Full Text:****Reviewed:****Description:**This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established. © 2019, Springer Nature B.V.

**Authors:**Van Ngai, Huynh , Tron, Nguyen , Van Vu, Nguyen , Théra, Michel**Date:**2020**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 61-87**Full Text:****Reviewed:****Description:**This paper investigates a new general pseudo subregularity model which unifies some important nonlinear (sub)regularity models studied recently in the literature. Some slope and abstract coderivative characterizations are established. © 2019, Springer Nature B.V.

Dual sufficient characterizations of transversality properties

- Cuong, Nguyen, Kruger, Alexander

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2020**Type:**Text , Journal article**Relation:**Positivity Vol. 24, no. 5 (2020), p. 1313-1359**Full Text:****Reviewed:****Description:**This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe. © 2020, Springer Nature Switzerland AG.**Description:**The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sk

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2020**Type:**Text , Journal article**Relation:**Positivity Vol. 24, no. 5 (2020), p. 1313-1359**Full Text:****Reviewed:****Description:**This paper continues the study of ‘good arrangements’ of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe. © 2020, Springer Nature Switzerland AG.**Description:**The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Sk

Embeddings of free topological vector spaces

- Leiderman, Arkady, Morris, Sidney

**Authors:**Leiderman, Arkady , Morris, Sidney**Date:**2020**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 101, no. 2 (2020), p. 311-324**Full Text:**false**Reviewed:****Description:**It is proved that the free topological vector space contains an isomorphic copy of the free topological vector space for every finite-dimensional cube , thereby answering an open question in the literature. We show that this result cannot be extended from the closed unit interval to general metrisable spaces. Indeed, we prove that the free topological vector space does not even have a vector subspace isomorphic as a topological vector space to , where is a Cook continuum, which is a one-dimensional compact metric space. This is also shown to be the case for a rigid Bernstein set, which is a zero-dimensional subspace of the real line. © 2019 Australian Mathematical Publishing Association Inc..

Energy sector development : system dynamics analysis

- Laimon, Mohamd, Mai, Thanh, Goh, Steven, Yusaf, Talal

**Authors:**Laimon, Mohamd , Mai, Thanh , Goh, Steven , Yusaf, Talal**Date:**2020**Type:**Text , Journal article**Relation:**Applied Sciences-Basel Vol. 10, no. 1 (Jan 2020), p. 19**Full Text:****Reviewed:****Description:**The development of a complex and dynamic system such as the energy sector requires a comprehensive understanding of its constituent components and their interactions, and thus requires approaches that can adapt to the dynamic complexity in systems. Previous efforts mainly used reductionist approaches, which examine the components of the system in isolation, neglecting their interdependent nature. Such approaches reduce our ability to understand the system and/or mitigate undesirable outcomes. We adopt a system dynamics approach to construct an integrated model for analysing the behaviour of the energy sector. Although the Australian energy sector is used as a case study, the model can be applied in other context elsewhere around the world The results indicate that the current trajectory of the Australian energy sector is unsustainable and growth is not being controlled. Limits to growth are fast approaching due to excessive fossil fuel extraction, high emissions and high energy dependency. With the current growth, Australia's global CO2 emissions footprint will increase to unprecedented levels reaching 12% by 2030 (9.5% for exports and 2.5% for domestic). Oil dependency will account for 43% and 47% of total consumption by 2030 and 2050. By 2032, coal will be the only fossil fuel resource available in Australia. Expansion of investment in coal and gas production is a large risk.

**Authors:**Laimon, Mohamd , Mai, Thanh , Goh, Steven , Yusaf, Talal**Date:**2020**Type:**Text , Journal article**Relation:**Applied Sciences-Basel Vol. 10, no. 1 (Jan 2020), p. 19**Full Text:****Reviewed:****Description:**The development of a complex and dynamic system such as the energy sector requires a comprehensive understanding of its constituent components and their interactions, and thus requires approaches that can adapt to the dynamic complexity in systems. Previous efforts mainly used reductionist approaches, which examine the components of the system in isolation, neglecting their interdependent nature. Such approaches reduce our ability to understand the system and/or mitigate undesirable outcomes. We adopt a system dynamics approach to construct an integrated model for analysing the behaviour of the energy sector. Although the Australian energy sector is used as a case study, the model can be applied in other context elsewhere around the world The results indicate that the current trajectory of the Australian energy sector is unsustainable and growth is not being controlled. Limits to growth are fast approaching due to excessive fossil fuel extraction, high emissions and high energy dependency. With the current growth, Australia's global CO2 emissions footprint will increase to unprecedented levels reaching 12% by 2030 (9.5% for exports and 2.5% for domestic). Oil dependency will account for 43% and 47% of total consumption by 2030 and 2050. By 2032, coal will be the only fossil fuel resource available in Australia. Expansion of investment in coal and gas production is a large risk.

- Crouzeix, Jean-Pierre, Sukhorukova, Nadezda, Ugon, Julien

**Authors:**Crouzeix, Jean-Pierre , Sukhorukova, Nadezda , Ugon, Julien**Date:**2020**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 123-147. http://purl.org/au-research/grants/arc/DP180100602**Full Text:**false**Reviewed:****Description:**One of the purposes in this paper is to provide a better understanding of the alternance property which occurs in Chebyshev polynomial approximation and continuous piecewise polynomial approximation problems. In the first part of this paper, we prove that alternating sequences of any continuous function are finite in any given segment and then propose an original approach to obtain new proofs of the well known necessary and sufficient optimality conditions. There are two main advantages of this approach. First of all, the proofs are intuitive and easy to understand. Second, these proofs are constructive and therefore they lead to new alternation-based algorithms. In the second part of this paper, we develop new local optimality conditions for free knot polynomial spline approximation. The proofs for free knot approximation are relying on the techniques developed in the first part of this paper. The piecewise polynomials are required to be continuous on the approximation segment. © 2020, Springer Nature B.V.

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