### 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.

### Embedding into free topological vector spaces on compact metrizable spaces

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2018**Type:**Text , Journal article**Relation:**Topology and its Applications Vol. 233, no. (2018), p. 33-43**Full Text:**false**Reviewed:****Description:**For a Tychonoff space X, let V(X) be the free topological vector space over X. Denote by I, G, Q and Sk the closed unit interval, the Cantor space, the Hilbert cube Q=IN and the k-dimensional unit sphere for k

### 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:**false**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.

### Subspaces of the free topological vector space on the unit interval

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2018**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 97, no. 1 (2018), p. 110-118**Full Text:**false**Reviewed:****Description:**For a Tychonoff space X, let V(X) be the free topological vector space over X, A(X) the free abelian topological group over X and I the unit interval with its usual topology. It is proved here that if X is a subspace of I, then the following are equivalent: V(X) can be embedded in V(I) as a topological vector subspace; A(X) can be embedded in A(I) as a topological subgroup; X is locally compact. © 2017 Australian Mathematical Publishing Association Inc..

### The excess degree of a polytope

**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.

### Valadier-like formulas for the supremum function I

**Authors:**Correa, Rafael , Hantoute, Abderrahim , López, Marco**Date:**2018**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 25, no. 4 (2018), p. 1253-1278**Relation:**http://purl.org/au-research/grants/arc/DP160100854**Full Text:**false**Reviewed:****Description:**We generalize and improve the original characterization given by Valadier [19, Theorem 1] of the subdifferential of the pointwise supremum of convex functions, involving the subdifferentials of the data functions at nearby points. We remove the continuity assumption made in that work and obtain a general formula for such a subdifferential. In particular, when the supremum is continuous at some point of its domain, but not necessarily at the reference point, we get a simpler version which gives rise to the Valadier formula. Our starting result is the characterization given in [11, Theorem 4], which uses the e-subdifferential at the reference point.

### About subtransversality of collections of sets

**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

**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

**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.

### Free topological vector spaces

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2017**Type:**Text , Journal article**Relation:**Topology and its Applications Vol. 223, no. (2017), p. 30-49**Full Text:**false**Reviewed:****Description:**In this paper the free topological vector space V(X) over a Tychonoff space X is defined and studied. It is proved that V(X) is a kω-space if and only if X is a kω-space. If X is infinite, then V(X) contains a closed vector subspace which is topologically isomorphic to V(N). It is proved that for X a k-space, the free topological vector space V(X) is locally convex if and only if X is discrete and countable. The free topological vector space V(X) is shown to be metrizable if and only if X is finite if and only if V(X) is locally compact. Further, V(X) is a cosmic space if and only if X is a cosmic space if and only if the free locally convex space L(X) on X is a cosmic space. If a sequential (for example, metrizable) space Y is such that the free locally convex space L(Y) embeds as a subspace of V(X), then Y is a discrete space. It is proved that V(X) is a barreled topological vector space if and only if X is discrete. This result is applied to free locally convex spaces L(X) over a Tychonoff space X by showing that: (1) L(X) is quasibarreled if and only if L(X) is barreled if and only if X is discrete, and (2) L(X) is a Baire space if and only if X is finite. © 2017 Elsevier B.V.

### Nonseparable closed vector subspaces of separable topological vector spaces

**Authors:**Kakol, Jerzy , Leiderman, Arkady , Morris, Sidney**Date:**2017**Type:**Text , Journal article**Relation:**Monatshefte Fur Mathematik Vol. 182, no. 1 (2017), p. 39-47**Full Text:**false**Reviewed:****Description:**In 1983 P. Domanski investigated the question: For which separable topological vector spaces E, does the separable space have a nonseparable closed vector subspace, where is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line the space of all continuous real-valued functions on endowed with the pointwise convergence topology, contains a nonseparable closed vector subspace while is separable.**Description:**In 1983 P. DomaA"ski investigated the question: For which separable topological vector spaces E, does the separable space have a nonseparable closed vector subspace, where is the cardinality of the continuum? He provided a partial answer, proving that every separable topological vector space whose completion is not q-minimal (in particular, every separable infinite-dimensional Banach space) E has this property. Using a result of S.A. Saxon, we show that for a separable locally convex space (lcs) E, the product space has a nonseparable closed vector subspace if and only if E does not have the weak topology. On the other hand, we prove that every metrizable vector subspace of the product of any number of separable Hausdorff lcs is separable. We show however that for the classical Michael line the space of all continuous real-valued functions on endowed with the pointwise convergence topology, contains a nonseparable closed vector subspace while is separable.

### On varieties of Abelian topological groups with coproducts

**Authors:**Gabriyelyan, Saak , Morris, Sidney**Date:**2017**Type:**Text , Journal article**Relation:**Bulletin of the Australian Mathematical Society Vol. 95, no. 1 (2017), p. 54-65**Full Text:**false**Reviewed:****Description:**A class of abelian topological groups was previously defined to be a variety of topological groups with coproducts if it is closed under forming subgroups, quotients, products and coproducts in the category of all abelian topological groups and continuous homomorphisms. This extended research on varieties of topological groups initiated by the second author. The key to describing varieties of topological groups generated by various classes was proving that all topological groups in the variety are a quotient of a subgroup of a product of groups in the generating class. This paper analyses generating varieties of topological groups with coproducts. It focuses on the interplay between forming products and coproducts. It is proved that the variety of topological groups with coproducts generated by all discrete groups contains topological groups which cannot be expressed as a quotient of a subgroup of a product of a coproduct of discrete groups. It is proved that the variety of topological groups with coproducts generated by any infinite-dimensional Hilbert space contains all infinite-dimensional Hilbert spaces, answering an open question. This contrasts with the result that a variety of topological groups generated by a topological group does not contain any infinite-dimensional Hilbert space of greater cardinality. © 2016 Australian Mathematical Publishing Association Inc..

### Optimality conditions via weak subdifferentials in reflexive Banach spaces

**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

**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

**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

**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

**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 Cournot-Nash-Walras equilibria and their computation

**Authors:**Outrata, Jiri , Ferris, Michael , Červinka, Michal , Outrata, Michal**Date:**2016**Type:**Text , Journal article**Relation:**Set-Valued and Variational Analysis Vol. 24, no. 3 (2016), p. 387-402**Full Text:**false**Reviewed:****Description:**This paper concerns a model of Cournot-Nash-Walras (CNW) equilibrium where the Cournot-Nash concept is used to capture equilibrium of an oligopolistic market with non-cooperative players/firms who share a certain amount of a so-called rare resource needed for their production, and the Walras equilibrium determines the price of that rare resource. We prove the existence of CNW equilibria under reasonable conditions and examine their local stability with respect to small perturbations of problem data. In this way we show the uniqueness of CNW equilibria under mild additional requirements. Finally, we suggest some efficient numerical approaches and compute several instances of an illustrative test example. © 2016, Springer Science+Business Media Dordrecht.

### On the maximum order of graphs embedded in surfaces

**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:**false**Reviewed:****Description:**The maximum number of vertices in a graph of maximum degree

### Regularity of collections of sets and convergence of inexact alternating projections

**Authors:**Kruger, Alexander , Thao, Nguyen**Date:**2016**Type:**Text , Journal article**Relation:**Journal of Convex Analysis Vol. 23, no. 3 (2016), p. 823-847**Relation:**http://purl.org/au-research/grants/arc/DP110102011**Full Text:**false**Reviewed:****Description:**We study the usage of regularity properties of collections of sets in convergence analysis of alternating projection methods for solving feasibility problems. Several equivalent characterizations of these properties are provided. Two settings of inexact alternating projections are considered and the corresponding convergence estimates are established and discussed.