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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**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**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.

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**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.

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.

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).

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

Calculus for directional limiting normal cones and subdifferentials

- Benko, Matúš, Gfrerer, Helmut, Outrata, Jiri

**Authors:**Benko, Matúš , Gfrerer, Helmut , Outrata, Jiri**Date:**2019**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 27, no. 3 (2019), p. 713-745**Full Text:****Reviewed:****Description:**The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

**Authors:**Benko, Matúš , Gfrerer, Helmut , Outrata, Jiri**Date:**2019**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 27, no. 3 (2019), p. 713-745**Full Text:****Reviewed:****Description:**The paper is devoted to the development of a comprehensive calculus for directional limiting normal cones, subdifferentials and coderivatives in finite dimensions. This calculus encompasses the whole range of the standard generalized differential calculus for (non-directional) limiting notions and relies on very weak (non-restrictive) qualification conditions having also a directional character. The derived rules facilitate the application of tools exploiting the directional limiting notions to difficult problems of variational analysis including, for instance, various stability and sensitivity issues. This is illustrated by some selected applications in the last part of the paper.

Connectivity of cubical polytopes

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

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.

**Authors:**Bui, Hoa , Pineda-Villavicencio, Guillermo , Ugon, Julien**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Combinatorial Theory Series A Vol. 169, no. (Jan 2019), p. 21**Full Text:****Reviewed:****Description:**A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any d >= 3, the graph of a cubical d-polytope with minimum degree 5 is min{delta, 2d - 2}-connected. Second, we show, for any d >= 4, that every minimum separator of cardinality at most 2d - 3 in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself. (C) 2019 Elsevier Inc. All rights reserved.

Embedding of the free abelian topological group A (X ⊕ X) into A (X)

- Krupski, Mikolaj, Leiderman, Arkady, Morris, Sidney

**Authors:**Krupski, Mikolaj , Leiderman, Arkady , Morris, Sidney**Date:**2019**Type:**Text , Journal article**Relation:**Mathematika Vol. 65, no. 3 (2019), p. 708-718**Full Text:**false**Reviewed:****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

Holder error bounds and holder calmness with applications to convex semi-infinite optimization

- Kruger, Alexander, Lopez, Marco, Yang, Xiaoqi, Zhu, Jiangxing

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

Lower bound theorems for general polytopes

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

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**European Journal of Combinatorics Vol. 79, no. (2019), p. 27-45**Full Text:****Reviewed:****Description:**For a d-dimensional polytope with v vertices, d + 1 <= v <= 2d, we calculate precisely the minimum possible number of m-dimensional faces, when m = 1 or m >= 0.62d. This confirms a conjecture of Grunbaum, for these values of m. For v = 2d + 1, we solve the same problem when m = 1 or d - 2; the solution was already known for m = d - 1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**European Journal of Combinatorics Vol. 79, no. (2019), p. 27-45**Full Text:****Reviewed:****Description:**For a d-dimensional polytope with v vertices, d + 1 <= v <= 2d, we calculate precisely the minimum possible number of m-dimensional faces, when m = 1 or m >= 0.62d. This confirms a conjecture of Grunbaum, for these values of m. For v = 2d + 1, we solve the same problem when m = 1 or d - 2; the solution was already known for m = d - 1. In all these cases, we give a characterisation of the minimising polytopes. We also show that there are many gaps in the possible number of m-faces: for example, there is no polytope with 80 edges in dimension 10, and a polytope with 407 edges can have dimension at most 23.

- Cibulka, Radek, Fabian, Marian, Kruger, Alexander

**Authors:**Cibulka, Radek , Fabian, Marian , Kruger, Alexander**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 473, no. 2 (2019), p. 811-836**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**There are two basic ways of weakening the definition of the well-known metric regularity property by fixing one of the points involved in the definition. The first resulting property is called metric subregularity and has attracted a lot of attention during the last decades. On the other hand, the latter property which we call semiregularity can be found under several names and the corresponding results are scattered in the literature. We provide a self-contained material gathering and extending the existing theory on the topic. We demonstrate a clear relationship with other regularity properties, for example, the equivalence with the so-called openness with a linear rate at the reference point is shown. In particular cases, we derive necessary and/or sufficient conditions of both primal and dual type. We illustrate the importance of semiregularity in the convergence analysis of an inexact Newton-type scheme for generalized equations with not necessarily differentiable single-valued part. © 2019 Elsevier Inc.

On the reconstruction of polytopes

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

**Authors:**Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302**Full Text:****Reviewed:****Description:**Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

**Authors:**Doolittle, Joseph , Nevo, Eran , Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2019**Type:**Text , Journal article**Relation:**Discrete and Computational Geometry Vol. 61, no. 2 (2019), p. 285-302**Full Text:****Reviewed:****Description:**Blind and Mani, and later Kalai, showed that the face lattice of a simple polytope is determined by its graph, namely its 1-skeleton. Call a vertex of a d-polytope nonsimple if the number of edges incident to it is more than d. We show that (1) the face lattice of any d-polytope with at most two nonsimple vertices is determined by its 1-skeleton; (2) the face lattice of any d-polytope with at most d- 2 nonsimple vertices is determined by its 2-skeleton; and (3) for any d> 3 there are two d-polytopes with d- 1 nonsimple vertices, isomorphic (d- 3) -skeleta and nonisomorphic face lattices. In particular, the result (1) is best possible for 4-polytopes. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

Valadier-like Formulas for the Supremum Function II: The Compactly Indexed Case

- Correa, Rafael, Hantoute, Abderrahim, Lopez, Marco

**Authors:**Correa, Rafael , Hantoute, Abderrahim , Lopez, Marco**Date:**2019**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 26, no. 1 (2019), p. 299-324**Full Text:**false**Reviewed:****Description:**Continuing with the work on the subdifferential of the pointwise supremum of convex functions, started in part I of this paper [R. Correa, A. Hantoute, M. A. Lopez, Valadier-like formulas for the supremum function I, J. Convex Analysis 25 (2018) 1253-1278], we focus now on the compactly indexed case. We assume that the index set is compact and that the data functions are upper semicontinuous with respect to the index variable (actually, this assumption will only affect the set of epsilon-active indices at the reference point). As in the previous work, we do not require any continuity assumption with respect to the decision variable. The current compact setting gives rise to more explicit formulas, which only involve subdifferentials at the reference point of active data functions. Other formulas are derived under weak continuity assumptions. These formulas reduce to the characterization given by M. Valadier [Sous-differentiels d'une borne superieure et d'une somme continue de fonctions convexes, C. R. Acad. Sci. Paris Ser. A-B Math. 268 (1969) 39-42, Theorem 2], when the supremum function is continuous.

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