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15Kruger, Alexander
8Pineda-Villavicencio, Guillermo
7Yost, David
5Morris, Sidney
4Miller, Mirka
4Outrata, Jiri
3Kelarev, Andrei
3Mammadov, Musa
3Pierce, Robyn
3Rubinov, Alex
3Ugon, Julien
2Abawajy, Jemal
2Burachik, Regina
2Gabriyelyan, Saak
2Gao, David
2Henrion, René
2Hofmann, Karl
2Ivanov, Anatoli
2Nevo, Eran
2Plubtieng, Somyot

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120102 Applied Mathematics
5Mathematics
5Metric regularity
4Normal cone
4Slope
4Variational analysis
30802 Computation Theory and Mathematics
3Banach space
3Error bounds
3Metric subregularity
20906 Electrical and Electronic Engineering
2Alternating projections
2CAS
2Calmness
2Constraint qualification
2Intrinsic transversality
2Linear convergence
2Nonconvex optimization
2Perturbation

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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

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.

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.

About intrinsic transversality of pairs of sets

**Authors:**Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.

**Authors:**Kruger, Alexander**Date:**2018**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 26, no. 1 (2018), p. 111-142**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**The article continues the study of the ‘regular’ arrangement of a collection of sets near a point in their intersection. Such regular intersection or, in other words, transversality properties are crucial for the validity of qualification conditions in optimization as well as subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms. One of the main motivations for the development of the transversality theory of collections of sets comes from the convergence analysis of alternating projections for solving feasibility problems. This article targets infinite dimensional extensions of the intrinsic transversality property introduced recently by Drusvyatskiy, Ioffe and Lewis as a sufficient condition for local linear convergence of alternating projections. Several characterizations of this property are established involving new limiting objects defined for pairs of sets. Special attention is given to the convex case.

On modeling and complete solutions to general fixpoint problems in multi-scale systems with applications

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

**Authors:**Ruan, Ning , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Fixed Point Theory and Applications Vol. 2018, no. 1 (2018), p. 1-19**Full Text:****Reviewed:****Description:**This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e., the general fixed point problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on the objectivity principle in continuum physics; then the canonical duality theory is applied for solving this challenging problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.

On SPD method for solving canonical dual problem in post buckling of large deformed elastic beam

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

**Authors:**Ali, Elaf , Gao, David**Date:**2018**Type:**Text , Journal article**Relation:**Communications in Mathematical Sciences Vol. 16, no. 5 (2018), p. 1225-1240**Full Text:****Reviewed:****Description:**This paper presents a new methodology and algorithm for solving post buckling problems of a large deformed elastic beam. The total potential energy of this beam is a nonconvex functional, which can be used to model both pre- and post-buckling phenomena. By using a canonical dual finite element method, a new primal-dual semi-definite programming (PD-SDP) algorithm is presented, which can be used to obtain all possible post-buckled solutions. Applications are illustrated by several numerical examples with different boundary conditions. We find that the global minimum solution of the nonconvex potential leads to a stable configuration of the buckled beam, the local maximum solution leads to the unbuckled state, and both of these two solutions are numerically stable. However, the local minimum solution leads to an unstable buckled state, which is very sensitive to axial compressive forces, thickness of beam, numerical precision, and the size of finite elements. The method and algorithm proposed in this paper can be used for solving general nonconvex variational problems in engineering and sciences.

The excess degree of a polytope

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

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Discrete Mathematics Vol. 32, no. 3 (2018), p. 2011-2046**Full Text:****Reviewed:****Description:**We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d \not = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.

**Authors:**Pineda-Villavicencio, Guillermo , Ugon, Julien , Yost, David**Date:**2018**Type:**Text , Journal article**Relation:**SIAM Journal on Discrete Mathematics Vol. 32, no. 3 (2018), p. 2011-2046**Full Text:****Reviewed:****Description:**We define the excess degree \xi (P) of a d-polytope P as 2f1 - df0, where f0 and f1 denote the number of vertices and edges, respectively. This parameter measures how much P deviates from being simple. It turns out that the excess degree of a d-polytope does not take every natural number: the smallest possible values are 0 and d - 2, and the value d - 1 only occurs when d = 3 or 5. On the other hand, for fixed d, the number of values not taken by the excess degree is finite if d is odd, and the number of even values not taken by the excess degree is finite if d is even. The excess degree is then applied in three different settings. First, it is used to show that polytopes with small excess (i.e., \xi (P) < d) have a very particular structure: provided d \not = 5, either there is a unique nonsimple vertex, or every nonsimple vertex has degree d + 1. This implies that such polytopes behave in a similar manner to simple polytopes in terms of Minkowski decomposability: they are either decomposable or pyramidal, and their duals are always indecomposable. Second, we characterize completely the decomposable d-polytopes with 2d + 1 vertices (up to combinatorial equivalence). Third, all pairs (f0, f1), for which there exists a 5-polytope with f0 vertices and f1 edges, are determined.

About subtransversality of collections of sets

- Kruger, Alexander, Luke, Russell, Thao, Nguyen

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2017**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 25, no. 4 (2017), p. 701-729**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces.

**Authors:**Kruger, Alexander , Luke, Russell , Thao, Nguyen**Date:**2017**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 25, no. 4 (2017), p. 701-729**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**We provide dual sufficient conditions for subtransversality of collections of sets in an Asplund space setting. For the convex case, we formulate a necessary and sufficient dual criterion of subtransversality in general Banach spaces. Our more general results suggest an intermediate notion of subtransversality, what we call weak intrinsic subtransversality, which lies between intrinsic transversality and subtransversality in Asplund spaces.

Borwein–Preiss vector variational principle

- Kruger, Alexander, Plubtieng, Somyot, Seangwattana, Thidaporn

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2017**Type:**Text , Journal article**Relation:**Positivity Vol. 21, no. 4 (2017), p. 1273-1292**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:****Reviewed:****Description:**This article extends to the vector setting the results of our previous work Kruger et al. (J Math Anal Appl 435(2):1183–1193, 2016) which refined and slightly strengthened the metric space version of the Borwein–Preiss variational principle due to Li and Shi (J Math Anal Appl 246(1):308–319, 2000. doi:10.1006/jmaa.2000.6813). We introduce and characterize two seemingly new natural concepts of ε-minimality, one of them dependent on the chosen element in the ordering cone and the fixed “gauge-type” function. © 2017, Springer International Publishing.

Density character of subgroups of topological groups

- Leiderman, Arkady, Morris, Sidney, Tkachenko, Mikhail

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

**Authors:**Leiderman, Arkady , Morris, Sidney , Tkachenko, Mikhail**Date:**2017**Type:**Text , Journal article**Relation:**Transactions of the American Mathematical Society Vol. 369, no. 8 (2017), p. 5645-5664**Full Text:****Reviewed:****Description:**We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an omega-narrow topological group G: (i) G is homeomorphic to a subspace of a separable regular space; (ii) G is topologically isomorphic to a subgroup of a separable topological group; (iii) G is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A pro-Lie group is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality c of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group G which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that G is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to c is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight c. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.

Optimality conditions via weak subdifferentials in reflexive Banach spaces

- Hassani, Sara, Mammadov, Musa, Jamshidi, Mina

**Authors:**Hassani, Sara , Mammadov, Musa , Jamshidi, Mina**Date:**2017**Type:**Text , Journal article**Relation:**Turkish Journal of Mathematics Vol. 41, no. 1 (2017), p. 1-8**Full Text:****Reviewed:****Description:**In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. © Tübitak.

**Authors:**Hassani, Sara , Mammadov, Musa , Jamshidi, Mina**Date:**2017**Type:**Text , Journal article**Relation:**Turkish Journal of Mathematics Vol. 41, no. 1 (2017), p. 1-8**Full Text:****Reviewed:****Description:**In this paper the relation between the weak subdifferentials and the directional derivatives, as well as optimality conditions for nonconvex optimization problems in reflexive Banach spaces, are investigated. It partly generalizes several related results obtained for finite dimensional spaces. © Tübitak.

An open mapping theorem

- Gabriyelyan, Saak, Morris, Sidney

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2016**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 94, no. 1 (2016), p. 65-69**Full Text:****Reviewed:****Description:**It is proved that any surjective morphism f : Z(k) -> K onto a locally compact group K is open for every cardinal k. This answers a question posed by Hofmann and the second author.

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2016**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 94, no. 1 (2016), p. 65-69**Full Text:****Reviewed:****Description:**It is proved that any surjective morphism f : Z(k) -> K onto a locally compact group K is open for every cardinal k. This answers a question posed by Hofmann and the second author.

Borwein-Preiss variational principle revisited

- Kruger, Alexander, Plubtieng, Somyot, Seangwattana, Thidaporn

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.

**Authors:**Kruger, Alexander , Plubtieng, Somyot , Seangwattana, Thidaporn**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Mathematical Analysis and Applications Vol. 435, no. 2 (2016), p. 1183-1193**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:****Reviewed:****Description:**In this article, we refine and slightly strengthen the metric space version of the Borwein-Preiss variational principle due to Li and Shi (2000) [12], clarify the assumptions and conclusions of their Theorem 1 as well as Theorem 2.5.2 in Borwein and Zhu (2005) [4] and streamline the proofs. Our main result, Theorem 3 is formulated in the metric space setting. When reduced to Banach spaces (Corollary 9), it extends and strengthens the smooth variational principle established in Borwein and Preiss (1987) [3] along several directions. (C) 2015 Elsevier Inc. All rights reserved.

Comparative analysis of numerical solution of optimal control problems

- Shangareeva, Gulnaz, Grigoryev, Igor, Mustafina, Svetlana

**Authors:**Shangareeva, Gulnaz , Grigoryev, Igor , Mustafina, Svetlana**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Pure and Applied Mathematics Vol. 110, no. 4 (2016), p. 645-649**Full Text:****Reviewed:****Description:**In this article step by step algorithms were developed for solving optimal control problems based on the method of successive approximations and the method of variations in the space of controls. The algorithm of the method of successive approximations requires details of the problem to the boundary problem of the maximum principle. In turn, the algorithm of the variations is more versatile because it is based on iterating state variables and control in the phase space. A numerical study and comparative analysis of the developed algorithms performed at different values of accuracy. © 2016 Academic Publications, Ltd.

**Authors:**Shangareeva, Gulnaz , Grigoryev, Igor , Mustafina, Svetlana**Date:**2016**Type:**Text , Journal article**Relation:**International Journal of Pure and Applied Mathematics Vol. 110, no. 4 (2016), p. 645-649**Full Text:****Reviewed:****Description:**In this article step by step algorithms were developed for solving optimal control problems based on the method of successive approximations and the method of variations in the space of controls. The algorithm of the method of successive approximations requires details of the problem to the boundary problem of the maximum principle. In turn, the algorithm of the variations is more versatile because it is based on iterating state variables and control in the phase space. A numerical study and comparative analysis of the developed algorithms performed at different values of accuracy. © 2016 Academic Publications, Ltd.

Nonmeasurable subgroups of compact groups

- Hernández, Salvador, Hofmann, Karl, Morris, Sidney

**Authors:**Hernández, Salvador , Hofmann, Karl , Morris, Sidney**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Group Theory Vol. 19, no. 1 (2016), p. 179-189**Full Text:****Reviewed:****Description:**In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. © 2016 by De Gruyter 2016 Generalitat Valenciana PROMETEO/2014/062 We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [Pacific J. Math. 116 (1985), 217-241] shortened the proof of Theorem 4.3 considerably.

**Authors:**Hernández, Salvador , Hofmann, Karl , Morris, Sidney**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Group Theory Vol. 19, no. 1 (2016), p. 179-189**Full Text:****Reviewed:****Description:**In 1985 S. Saeki and K. Stromberg published the following question: Does every infinite compact group have a subgroup which is not Haar measurable? An affirmative answer is given for all compact groups with the exception of some metric profinite groups which are almost perfect and strongly complete. In this spirit it is also shown that every compact group contains a non-Borel subgroup. © 2016 by De Gruyter 2016 Generalitat Valenciana PROMETEO/2014/062 We are grateful for our referee's useful comments. In particular, the suggestion that originally we had overlooked [Pacific J. Math. 116 (1985), 217-241] shortened the proof of Theorem 4.3 considerably.

On the maximum order of graphs embedded in surfaces

- Nevo, Eran, Pineda-Villavicencio, Guillermo, Wood, David

**Authors:**Nevo, Eran , Pineda-Villavicencio, Guillermo , Wood, David**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Combinatorial Theory. Series B Vol. 119, no. (2016), p. 28-41**Full Text:****Reviewed:****Description:**The maximum number of vertices in a graph of maximum degree

**Authors:**Nevo, Eran , Pineda-Villavicencio, Guillermo , Wood, David**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Combinatorial Theory. Series B Vol. 119, no. (2016), p. 28-41**Full Text:****Reviewed:****Description:**The maximum number of vertices in a graph of maximum degree

An additive subfamily of enlargements of a maximally monotone operator

- Burachik, Regina, Martinez-Legaz, Juan, Rezaie, Mahboubeh, Thera, Michel

**Authors:**Burachik, Regina , Martinez-Legaz, Juan , Rezaie, Mahboubeh , Thera, Michel**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 643-665**Full Text:****Reviewed:****Description:**We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical epsilon-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the epsilon-subdifferential enlargement.

**Authors:**Burachik, Regina , Martinez-Legaz, Juan , Rezaie, Mahboubeh , Thera, Michel**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 643-665**Full Text:****Reviewed:****Description:**We introduce a subfamily of additive enlargements of a maximally monotone operator. Our definition is inspired by the early work of Simon Fitzpatrick. These enlargements constitute a subfamily of the family of enlargements introduced by Svaiter. When the operator under consideration is the subdifferential of a convex lower semicontinuous proper function, we prove that some members of the subfamily are smaller than the classical epsilon-subdifferential enlargement widely used in convex analysis. We also recover the epsilon-subdifferential within the subfamily. Since they are all additive, the enlargements in our subfamily can be seen as structurally closer to the epsilon-subdifferential enlargement.

Continuants and some decompositions into squares

- Delorme, Charles, Pineda-Villavicencio, Guillermo

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2015**Type:**Text , Journal article**Relation:**Integers Vol. 15, no. (2015), p. 1**Full Text:****Reviewed:****Description:**In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number

**Authors:**Delorme, Charles , Pineda-Villavicencio, Guillermo**Date:**2015**Type:**Text , Journal article**Relation:**Integers Vol. 15, no. (2015), p. 1**Full Text:****Reviewed:****Description:**In 1855 H. J. S. Smith proved Fermat's two-square using the notion of palindromic continuants. In his paper, Smith constructed a proper representation of a prime number

Error bounds and Hölder metric subregularity

**Authors:**Kruger, Alexander**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 705-736**Full Text:****Reviewed:****Description:**The Holder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Holder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

**Authors:**Kruger, Alexander**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 705-736**Full Text:****Reviewed:****Description:**The Holder setting of the metric subregularity property of set-valued mappings between general metric or Banach/Asplund spaces is investigated in the framework of the theory of error bounds for extended real-valued functions of two variables. A classification scheme for the general Holder metric subregularity criteria is presented. The criteria are formulated in terms of several kinds of primal and subdifferential slopes.

Graphical derivatives and stability analysis for parameterized equilibria with conic constraints

- Mordukhovich, Boris, Outrata, Jiri, Ramirez, Hector

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Ramirez, Hector**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 687-704**Full Text:****Reviewed:****Description:**The paper concerns parameterized equilibria governed by generalized equations whose multivalued parts are modeled via regular normals to nonconvex conic constraints. Our main goal is to derive a precise pointwise second-order formula for calculating the graphical derivative of the solution maps to such generalized equations that involves Lagrange multipliers of the corresponding KKT systems and critical cone directions. Then we apply the obtained formula to characterizing a Lipschitzian stability notion for the solution maps that is known as isolated calmness.

**Authors:**Mordukhovich, Boris , Outrata, Jiri , Ramirez, Hector**Date:**2015**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 23, no. 4 (2015), p. 687-704**Full Text:****Reviewed:****Description:**The paper concerns parameterized equilibria governed by generalized equations whose multivalued parts are modeled via regular normals to nonconvex conic constraints. Our main goal is to derive a precise pointwise second-order formula for calculating the graphical derivative of the solution maps to such generalized equations that involves Lagrange multipliers of the corresponding KKT systems and critical cone directions. Then we apply the obtained formula to characterizing a Lipschitzian stability notion for the solution maps that is known as isolated calmness.

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