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46Miller, Mirka
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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.

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.

**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:**false**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:**false**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.

Nonlinear transversality of collections of sets : dual space necessary characterizations

- Cuong, Nguyen, Kruger, Alexander

**Authors:**Cuong, Nguyen , Kruger, Alexander**Date:**2020**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 27, no. 1 (2020), p. 285-306**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**This paper continues the study of `good arrangements' of collections of sets in normed spaces near a point in their intersection. Our aim is to study general nonlinear transversality properties. We focus on dual space (subdifferential and normal cone) necessary characterizations of these properties. As an application, we provide dual necessary conditions for the nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.**Description:**The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

Orthogonality in locally convex spaces: Two nonlinear generalizations of Neumann's lemma

- Barbagallo, Annamaria, Ernst, Octavian-Emil, Théra, Michel

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

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

Polytopes close to being simple

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

**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**Full Text:****Reviewed:****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

**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**Full Text:****Reviewed:****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

Some new characterizations of intrinsic transversality in hilbert spaces

- Thao, Nguyen, Bui, Hoa, Cuong, Nguyen, Verhaegen, Michel

**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**Full Text:****Reviewed:****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).

**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**Full Text:****Reviewed:****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).

Spatial modelling of bacterial diversity over the selected regions in Bangladesh by next-generation sequencing : role of water temperature

- Akter, Nabila, Wahiduzzaman, Md, Yeasmin, Alea, Islam, Kazi, Luo, Jing-Jia

**Authors:**Akter, Nabila , Wahiduzzaman, Md , Yeasmin, Alea , Islam, Kazi , Luo, Jing-Jia**Date:**2020**Type:**Text , Journal article**Relation:**Applied Sciences (Switzerland) Vol. 10, no. 7 (2020), p.**Full Text:****Reviewed:****Description:**In this study, a spatial model has been developed to investigate the role of water temperature to the distribution of bacteria over the selected regions in the Bay of Bengal, located in the southern region of Bangladesh using next-generation sequencing. Bacterial concentration, quantitative polymerase chain reactions, and sequencing were performed on water samples and identified Acidobacteria, Actinobacteria, Bacteroidetes, Chlorobi, Chloroflexi, Cyanobacteria, Firmicutes, Nitrospirae, Planctomycetes, Proteobacteria, and Verrucomicrobia. The spatial model tessellated the parts of the Bay of Bengal with hexagons and analyzed the relationship between the distribution of bacteria and water temperature. A geographically weighted regression was used to observe whether water temperature contributed strongly or weakly to the distribution of bacteria. The residuals were examined to assess the model's fitness. The spatial model has the potential to predict the bacterial diversity in the selected regions of Bangladesh. © 2020 by the authors.

**Authors:**Akter, Nabila , Wahiduzzaman, Md , Yeasmin, Alea , Islam, Kazi , Luo, Jing-Jia**Date:**2020**Type:**Text , Journal article**Relation:**Applied Sciences (Switzerland) Vol. 10, no. 7 (2020), p.**Full Text:****Reviewed:****Description:**In this study, a spatial model has been developed to investigate the role of water temperature to the distribution of bacteria over the selected regions in the Bay of Bengal, located in the southern region of Bangladesh using next-generation sequencing. Bacterial concentration, quantitative polymerase chain reactions, and sequencing were performed on water samples and identified Acidobacteria, Actinobacteria, Bacteroidetes, Chlorobi, Chloroflexi, Cyanobacteria, Firmicutes, Nitrospirae, Planctomycetes, Proteobacteria, and Verrucomicrobia. The spatial model tessellated the parts of the Bay of Bengal with hexagons and analyzed the relationship between the distribution of bacteria and water temperature. A geographically weighted regression was used to observe whether water temperature contributed strongly or weakly to the distribution of bacteria. The residuals were examined to assess the model's fitness. The spatial model has the potential to predict the bacterial diversity in the selected regions of Bangladesh. © 2020 by the authors.

Stability analysis for parameterized variational systems with implicit constraints

- Benko, Matus, Gfrerer, Helmut, Outrata, Jiri

**Authors:**Benko, Matus , Gfrerer, Helmut , Outrata, Jiri**Date:**2020**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 167-193**Full Text:****Reviewed:****Description:**In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples. © 2019, The Author(s).

**Authors:**Benko, Matus , Gfrerer, Helmut , Outrata, Jiri**Date:**2020**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 28, no. 1 (2020), p. 167-193**Full Text:****Reviewed:****Description:**In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples. © 2019, The Author(s).

Subdifferential of the supremum via compactification of the index set

- Correa, Rafael, Hantoute, Abderrahim, López, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2020**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 48, no. 3 (2020), p. 569-588, http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.**Description:**Funding details: Fondo Nacional de Desarrollo CientÃfico, TecnolÃ³gico y de InnovaciÃ³n TecnolÃ³gica, FONDECYT, PIA AFB-170001, 1190110, 1190012 Funding details: Universidad de Alicante, BEA- GAL 18/00205, PGC2018-097960-B-C21 Funding details: Australian Research Council, ARC, DP 180100602 Funding details: ComisiÃ³n Nacional de InvestigaciÃ³n CientÃfica y TecnolÃ³gica, CONICYT Funding details: Ministerio de Ciencia e InnovaciÃ³n, MICINN Funding text 1: Research supported by CONICYT (Fondecyt 1190012 and 1190110), Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Grant Beatriz Galindo BEA- GAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC - Discovery Projects DP 180100602

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2020**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 48, no. 3 (2020), p. 569-588, http://purl.org/au-research/grants/arc/DP180100602**Full Text:****Reviewed:****Description:**We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming. © 2020, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.**Description:**Funding details: Fondo Nacional de Desarrollo CientÃfico, TecnolÃ³gico y de InnovaciÃ³n TecnolÃ³gica, FONDECYT, PIA AFB-170001, 1190110, 1190012 Funding details: Universidad de Alicante, BEA- GAL 18/00205, PGC2018-097960-B-C21 Funding details: Australian Research Council, ARC, DP 180100602 Funding details: ComisiÃ³n Nacional de InvestigaciÃ³n CientÃfica y TecnolÃ³gica, CONICYT Funding details: Ministerio de Ciencia e InnovaciÃ³n, MICINN Funding text 1: Research supported by CONICYT (Fondecyt 1190012 and 1190110), Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Grant Beatriz Galindo BEA- GAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC - Discovery Projects DP 180100602

Subdifferentials and stability analysis of feasible set and pareto front mappings in linear multiobjective optimization

- Cánovas, Maria, López, Marco, Mordukhovich, Boris, Parra, Juan

**Authors:**Cánovas, Maria , López, Marco , Mordukhovich, Boris , Parra, Juan**Date:**2020**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 315-334**Full Text:****Reviewed:****Description:**The paper concerns multiobjective linear optimization problems in**Description:**Funding details: European Commission, EC Funding details: European Regional Development Fund, FEDER Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP180100602 Funding details: Australian Research Council, ARC, DP-190100555 Funding details: Air Force Office of Scientific Research, AFOSR, 15RT04 Funding details: DMS-1512846, DMS-1808978 Funding text 1: This research has been partially supported by grants MTM2014-59179-C2-(1,2)-P and PGC2018-097960-B-C2(1,2) from MINECO/MICINN, Spain, and ERDF, “A way to make Europe”, European Union. Funding text 2: Research of the second author is also partially supported by the Australian Research Council (ARC) Discovery Grants Scheme (Project Grant # DP180100602). Funding text 3: Research of third author was partially supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research grant #15RT04, and by Australian Research Council under grant DP-190100555.

**Authors:**Cánovas, Maria , López, Marco , Mordukhovich, Boris , Parra, Juan**Date:**2020**Type:**Text , Journal article**Relation:**Vietnam Journal of Mathematics Vol. 48, no. 2 (2020), p. 315-334**Full Text:****Reviewed:****Description:**The paper concerns multiobjective linear optimization problems in**Description:**Funding details: European Commission, EC Funding details: European Regional Development Fund, FEDER Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP180100602 Funding details: Australian Research Council, ARC, DP-190100555 Funding details: Air Force Office of Scientific Research, AFOSR, 15RT04 Funding details: DMS-1512846, DMS-1808978 Funding text 1: This research has been partially supported by grants MTM2014-59179-C2-(1,2)-P and PGC2018-097960-B-C2(1,2) from MINECO/MICINN, Spain, and ERDF, “A way to make Europe”, European Union. Funding text 2: Research of the second author is also partially supported by the Australian Research Council (ARC) Discovery Grants Scheme (Project Grant # DP180100602). Funding text 3: Research of third author was partially supported by the USA National Science Foundation under grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research grant #15RT04, and by Australian Research Council under grant DP-190100555.

The non-smooth and bi-objective team orienteering problem with soft constraints

- Estrada-Moreno, Alejandro, Ferrer, Albert, Juan, Angel, Panadero, Javier, Bagirov, Adil

**Authors:**Estrada-Moreno, Alejandro , Ferrer, Albert , Juan, Angel , Panadero, Javier , Bagirov, Adil**Date:**2020**Type:**Text , Journal article**Relation:**Mathematics Vol. 8, no. 9 (2020), p.**Full Text:****Reviewed:****Description:**In the classical team orienteering problem (TOP), a fixed fleet of vehicles is employed, each of them with a limited driving range. The manager has to decide about the subset of customers to visit, as well as the visiting order (routes). Each customer offers a different reward, which is gathered the first time that it is visited. The goal is then to maximize the total reward collected without exceeding the driving range constraint. This paper analyzes a more realistic version of the TOP in which the driving range limitation is considered as a soft constraint: every time that this range is exceeded, a penalty cost is triggered. This cost is modeled as a piece-wise function, which depends on factors such as the distance of the vehicle to the destination depot. As a result, the traditional reward-maximization objective becomes a non-smooth function. In addition, a second objective, regarding the design of balanced routing plans, is considered as well. A mathematical model for this non-smooth and bi-objective TOP is provided, and a biased-randomized algorithm is proposed as a solving approach. © 2020 by the authors.**Description:**This work has been partially supported by the Spanish Ministry of Economy and Competitiveness & FEDER (SEV-2015-0563), the Spanish Ministry of Science (PID2019-111100RB-C21, RED2018-102642-T), and the Erasmus+ Program (2019-I-ES01-KA103-062602).

**Authors:**Estrada-Moreno, Alejandro , Ferrer, Albert , Juan, Angel , Panadero, Javier , Bagirov, Adil**Date:**2020**Type:**Text , Journal article**Relation:**Mathematics Vol. 8, no. 9 (2020), p.**Full Text:****Reviewed:****Description:**In the classical team orienteering problem (TOP), a fixed fleet of vehicles is employed, each of them with a limited driving range. The manager has to decide about the subset of customers to visit, as well as the visiting order (routes). Each customer offers a different reward, which is gathered the first time that it is visited. The goal is then to maximize the total reward collected without exceeding the driving range constraint. This paper analyzes a more realistic version of the TOP in which the driving range limitation is considered as a soft constraint: every time that this range is exceeded, a penalty cost is triggered. This cost is modeled as a piece-wise function, which depends on factors such as the distance of the vehicle to the destination depot. As a result, the traditional reward-maximization objective becomes a non-smooth function. In addition, a second objective, regarding the design of balanced routing plans, is considered as well. A mathematical model for this non-smooth and bi-objective TOP is provided, and a biased-randomized algorithm is proposed as a solving approach. © 2020 by the authors.**Description:**This work has been partially supported by the Spanish Ministry of Economy and Competitiveness & FEDER (SEV-2015-0563), the Spanish Ministry of Science (PID2019-111100RB-C21, RED2018-102642-T), and the Erasmus+ Program (2019-I-ES01-KA103-062602).

The radius of metric subregularity

- Dontchev, Asen, Gfrerer, Helmut, Kruger, Alexander, Outrata, Jiri

**Authors:**Dontchev, Asen , Gfrerer, Helmut , Kruger, Alexander , Outrata, Jiri**Date:**2020**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 28, no. 3 (2020), p. 451-473, http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings. © 2019, Springer Nature B.V.**Description:**Funding details: Austrian Science Fund, FWF, P26132-N25, P26640-N25, P29190-N32 Funding details: National Science Foundation, NSF Funding details: Australian Research Council, ARC Funding details: Australian Research Council, ARC, DP160100854 Funding details: Austrian Science Fund, FWF Funding details: Universiteit Stellenbosch, US, P26640-N25 P26132-N25, BodyRef/PDF/11228_2019_Article_523.pdf Funding details: Grantová Agentura

A remark on the separable quotient problem for topological groups

**Authors:**Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 100, no. 3 (Dec 2019), p. 453-457**Full Text:**false**Reviewed:****Description:**The Banach-Mazur separable quotient problem asks whether every infinite-dimensional Banach space B has a quotient space that is an infinite-dimensional separable Banach space. The question has remained open for over 80 years, although an affirmative answer is known in special cases such as when B is reflexive or even a dual of a Banach space. Very recently, it has been shown to be true for dual-like spaces. An analogous problem for topological groups is: Does every infinite-dimensional (in the topological sense) connected (Hausdorff) topological group G have a quotient topological group that is infinite dimensional and metrisable? While this is known to be true if G is the underlying topological group of an infinite-dimensional Banach space, it is shown here to be false even if G is the underlying topological group of an infinite-dimensional locally convex space. Indeed, it is shown that the free topological vector space on any countably infinite k(omega)-space is an infinite-dimensional toplogical vector space which does not have any quotient topological group that is infinite dimensional and metrisable. By contrast, the Graev free abelian topological group and the Graev free topological group on any infinite connected Tychonoff space, both of which are connected topological groups, are shown here to have the tubby torus T-omega, which is an infinite-dimensional metrisable group, as a quotient group.

A topological group observation on the Banach-Mazur separable quotient problem

- Gabriyelyan, Saak, Morris, Sidney

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Topology and Its Applications Vol. 259, no. (2019), p. 283-286**Full Text:****Reviewed:****Description:**The Separable Quotient Problem of Banach and Mazur asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient Banach space. It has remained unsolved for 85 years but has been answered in the affirmative for special cases such as reflexive Banach spaces. An affirmative answer to the Separable Quotient Problem would obviously imply that every infinite-dimensional Banach space has a quotient topological group which is separable, metrizable, and infinite-dimensional in the sense of topology. In this paper it is proved that every infinite-dimensional Banach space has as a quotient group the separable metrizable infinite-dimensional topological group, T

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