DOCSLIB.ORG

K( *; Pi, P2) = 9( *; Pl)/G( ; P2)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
K( *; Pi, P2) = 9( *; Pl)/G( ; P2) VOL. 48, 1962 MATHEMATICS: I. S. GAL 775 is satisfied identically. The same is true of all equations of type (10) and the solution is completed. THE MARTIN BOUNDARY'* BY ISTVAN S. GkL UNIVERSITY OF MINNESOTA Communicated by Einar Hille, March 26, 1962 Here we shall study the Martin boundary and its known generalizations from the point of view of uniform structures. Our object is to obtain a topologically satisfactory characterization of these boundaries in terms of the associated uniform structures. We can then use the result to define the Martin boundary of arbitrary spaces possessing Green functions with respect to any given differential operator and an associated boundary-value condition. It is assumed that the reader is familiar with the concept of a uniform structure, its usefulness, and a few fundamental notions such as precompactness and inverse image. The necessary information is available in Kelley's General Topology or in the author's Topological Foundations of Analysis and Geometry. The significance of uniform structures in the definition of Martin boundaries was first pointed out by Brelot in reference 2 and the structure %'M to be defined below is mentioned there. However, Brelot made no attempt to prove that %"EM is identi- cal with the unique uniform structure 'CM associated with the Martin compactifica- tion. He proves only that the compactifications with respect to TEM and VM yield homeomorphic spaces, the question whether a uniform isomorphism indeed exists between these compactifications is left open. Our main result can be summarized as follows: The Martin structure "LM is the weakest uniform structure such that the quotient of any two Green functions K( *; pI, P2) = 9( *; pl)/g( ; P2) is uniformly continuous with respect to 'lM and the unique uniform structure compatible with the topology of the compactified real line [- , + o ]. This statement is both a theorem and a definition: In the classical case a domain D is compactified by the introduction of a set A of ideal points and by a proper topologization of the set OD = D U A. The Hausdorff space OI admits a unique uniform structure whose trace on D we denote by vM. Then the above proposition states that 'EM = %)M. The same is true in any other situation where a Martin- type compactification has been defined. This includes also the discrete case recently studied by Doob in reference 3. If instead of a domain D we consider a differentiable manifold admitting Green functions, then the above statement be- comes a definition. Let X be a nonvoid set, (1Y, 'V) a uniform space, and 0 a family of functions Downloaded by guest on September 25, 2021 776 MATHEMATICS: I. S. GAL PROC. N. A. S. (p: X -- Y. Then the weakest uniform structure on X which makes the functions <o uniformly continuous is cu = lub {y-1(V)t, where (p-l(l)), the inverse image of co under sp is determined by the structure base {s-l(V)} (V e V) consisting of the sets so-1(v) = { (xI, X2): (p(X1), p(X2)) e V}. If CU is precompact, so is every p-'(V) and thus 'U is a precompact structure. (See for example pp. 308-309 of ref. 4.) Since Y = [- c, + co] is compact, its unique uniform structure is precompact, and so our remark can be applied to the family + = I K( P; p2)}, and the corresponding uniform structure c11M* Hence, we obtain the following corollary to our definition: The Martin structure cUM is precompact. As soon as a uniform structure 'U is introduced on a set X, we immediately know the existence of a uniform space (X, 'U.) and an injection map f: X X such that f(X) is dense in X and 'U = f-'(qt). The triple (f, X, ca) is called the completion of X with respect to %U. Up to a uniform isomorphism, the completion is uniquely determined by X and 'U. The enveloping space X is compact if and only if 'U is precompact. Therefore, the foregoing corollary can be expressed also as follows: The completion of the underlying space X with respect to its Martin structure 'uM '/ields a compact space X, the Martin compactification of X. The set aX = X - X is the Martin boundary. We recall that in the classical case and also in any of its extensions a sequence m1, M2, ... of points mi e X converges to a boundary point m e aX if and only if (i) (i = 1, 2, ...) has no accumulation point in X and the numerical sequences (K(mi; Pi, P2)) (i = 1, 2, ...) are convergent for every pair (Pi, P2) e X X X - I. This fact is very important in various applications, e.g., in probability theory. Instead of sequences (mj) (i = 1, 2, . .), we may consider also nets (mi) (i e I) with values m, E X of filters ~F = {F } with sets F C X. This follows from the fact that all known Martin structures are metrizable and so for every net (in2) (i e I) there is a sequence (in,) (i = 1, 2, ... ) such that (mi) (i e I) and (mi) (i = 1, 2, . .) converge or diverge simultaneously and in the case of convergence they have the same limit. We are going to prove that if sequences are replaced by nets or filters then the proposition holds for any Martin structure 'aM. It is convenient to formulate it in terms of the Cauchy property in- stead of convergence: A net (mi) (i e I) is a Cauchy net with respect to the Martin structure 'aM if and only -if (K(m1; pi, P2)) (i I) is a Cauchy net in [-a, + o ] fo? every pair (pI, P2) e X X X - I. A similar proposition holds for filters ff instead of nets (mi) (i e I). The only difference is that the family IK(F; PI, p2)1 (F e l) is not the whole image filter K(5V; Pi, P2) but only a filter base. Since filters and nets exhibit the same adherence and convergence properties (see for instance refs. 1 and 5), it is sufficient to prove Downloaded by guest on September 25, 2021 VOL. 48, 1962 MATHEMATICS: I. S. GAL 7,7 the proposition for either one of these. In the present problem, it is easier to handle filters than nets. It turns out that the last proposition holds not only for the Martin structure EM but for any uniform structure 'U of the type 'U. = lub PROPOSITION 1. Let 4 = s4I be a family offunctions <o: X -* Y mapping the set X in the uniform space (Y, V)) and let cl = lub {so-lU(V)} Then a filter 1 in X is a Cauchy filter with respect to At if and only if <o(5) is a Cauchy filter in Y with respect to V) for every p e 4). Proof: Let ff be a filter in X such that so(ff) is a Cauchy filter for every (o 4). By the definition of cU for any given U E 'U, there exist functions (pi, ..., Sn e 4 and uniformities V1, ..., V. e VL such that (Pr-'(V1) n ... n Pn-'(Vn) C U. By hypothesis, there are suitable sets F1, . ., Fn in 9 such that (pi(Fi) X (p(Fi) C Vi (i = 1, ..., n). Therefore, Fj X Fj C spj-'(Vj) (i I1,.. n) and so F = F1 n ... nf Fn is an element of 3f which satisfies F X F C U. Since U e 'U is arbitrary, this shows that if is a Cauchy filter with respect to cU. In order to prove that so(5;) is a Cauchy filter for every (p E 4) when 5: is a Cauchy filter in X, it is sufficient to remark that each (p is uniformly continuous with respect to the structures 'U and 'C. For we have the following LEMMA 1. Let (X, 9U) and (Y, CL)) be uniform spaces and let <p: X -o Y be uni- formly continuous. Then so maps Cauchy filters into Cauchy filters. Proof: Let 9Y be a Cauchy filter with respect to 'l. A base for sp(F) consists of the sets p(F) (F e 5), so it is sufficient to prove that for each V in C) there is an F in 9: such that ip(F) X 4p(F) C V. Since (p is uniformly continuous V-'(u) < Ct and so U = '-'(V) e At. Thus, 9F being a Cauchy filter with respect to R, there is an F in 9F such that F X F C U. This set F satisfies the requirement. Next, we turn to the classical situations where the Martin boundary is already known and prove that the old and the new definitions coincide. The same reason- ing can be applied in the case of Euclidean domains, hyperbolic Riemann surfaces, and Green spaces and also for the so-called Martin exit boundary recently intro- duced by Doob in reference 3. The idea is very simple. First, we prove that a precompact uniform space (X, Ut) is uniquely determined by its Cauchy filters. This is the content of Proposition 3 below and it is proved by using a criterion for the possibility of a continuous extension of a map f: X -o Y from one uniform space (X, 'U) into another (Y, V)). As soon as the proposition is proved, we can deter- mine the Cauchy filters of the new uniform structure 91M by means of Proposition 1 and notice that these are identical with the Cauchy filters of the Martin compactifi- cation in the classical sense.
Load more

Recommended publications
  • OBJ (Application/Pdf)
    ON METRIC SPACES AND UNIFORMITIES A THESIS SUBMITTED TO THE FACULTY OF ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY JOHN H. HARRIS DEPARTMENT OF MATHEMATICS ATLANTA, GEORGIA Ml 1966 RrU \ ~ 35 ACKNCMLEDGI*ENTS The most lucid presentation in English, to my knowledge, of the concept of a uniform space is found in John Kelly's classic, "General Topology". This text was trost influential in my choice and arrangement of materials for this thesis. I would also like to acknowledge and thank my instructors for their assistance and suggestions. A special thanks is extended to my parents without whose inspiration and encouragement this thesis would not have been possible. ii CONTENTS Page ACKNOWLEDGMENTS ii INTRODUCTION I Chapter I. Uniform Spaces « 3 1.1 Uniformities and The Uniform Topology 1.2 Uniform Continuity 1.3 The Metrization Theorem 1*JU Compactness II. Uniform Spaces and Topological Groups ... 23 III, Selected Problems ••..,.•.••...2? APPENDIX , . 31 INDEX OF SPECIAL SYMBOLS 3h BIBLIOGRAPHY 36 INTRODUCTION One of the major advantages of the theory of topology is that it lends itself readily to intuitive interpretation. This is especially true of metric spaces because of their close connection with the real number system and because of our intuitive concept of distance associated with that system. However, if we should attempt to generalize metric spaces, and we have, our intuition would often fail us for we would also have to generalize our already intuitive concept of distance. In this thesis, I shall attempt to present in a lucid and in an intuition appealing fashion, the generalized theory of metric spaces,“i.e., the theory of uniform spaces.
    [Show full text]
  • In Presenting the Dissertation As a Partial Fulfillment of • The
    thIne presentin requirementg sthe fo rdissertatio an advancen ads degrea partiae froml fulfillmen the Georgit oaf • Institute oshalf Technologyl make i,t availablI agreee thafort inspectiothe Librarn yan dof the materialcirculatiosn o fin thi accordancs type.e I witagrehe it stha regulationt permissios ngovernin to copgy bfromy th,e o rprofesso to publisr undeh fromr whos, ethis directio dissertation itn wa mas ywritten be grante, or,d sucin hhis copyin absenceg o,r bpublicatioy the Dean nis osolelf they foGraduatr scholarle Divisioy purposen whesn stooandd doe thast noanty involv copyineg potentia from, ol rfinancia publicatiol gainn of. ,I t thiiss under dis­­ bsertatioe allowen dwhic withouh involvet writtes npotentia permissionl financia. l gain will not 3/17/65 b I ! 'II L 'II LLill t 111 NETS WITH WELL-ORDERED DOMAINS A THESIS Presented to The Faculty of the Graduate Division by Gary Calvin Hamrick In Partial Fulfillment of the Requirements for the Degree Master of Science in Applied Mathematics Georgia Institute of Technology August 10, 1967 NETS WITH WELL-ORDERED DOMAINS Approved: j Date 'approved by Chairman: Z^tt/C7 11 ACKNOWLEDGMENTS I wish to thank ray thesis advisor, Dr. Roger D. Johnson, Jr., very much for his generous expenditure of time and effort in aiding me to pre­ pare this thesis. I further wish to thank Dr. William R. Smythe, Jr., and Dr. Peter B. Sherry for reading the manuscript. Also I am grateful to the National Science Foundation for supporting me financially with a Traineeship from January, 1966, until September, 1967. And for my wife, Jean, I hold much gratitude and affection for her forebearance of a thinly stocked cupboard while her husband was in graduate school.
    [Show full text]
  • Quasi-Arithmetic Filters for Topology Optimization
    Quasi-Arithmetic Filters for Topology Optimization Linus Hägg Licentiate thesis, 2016 Department of Computing Science This work is protected by the Swedish Copyright Legislation (Act 1960:729) ISBN: 978-91-7601-409-7 ISSN: 0348-0542 UMINF 16.04 Electronic version available at http://umu.diva-portal.org Printed by: Print & Media, Umeå University, 2016 Umeå, Sweden 2016 Acknowledgments I am grateful to my scientific advisors Martin Berggren and Eddie Wadbro for introducing me to the fascinating subject of topology optimization, for sharing their knowledge in countless discussions, and for helping me improve my scientific skills. Their patience in reading and commenting on the drafts of this thesis is deeply appreciated. I look forward to continue with our work. Without the loving support of my family this thesis would never have been finished. Especially, I am thankful to my wife Lovisa for constantly encouraging me, and covering for me at home when needed. I would also like to thank my colleges at the Department of Computing Science, UMIT research lab, and at SP Technical Research Institute of Sweden for providing a most pleasant working environment. Finally, I acknowledge financial support from the Swedish Research Council (grant number 621-3706), and the Swedish strategic research programme eSSENCE. Linus Hägg Skellefteå 2016 iii Abstract Topology optimization is a framework for finding the optimal layout of material within a given region of space. In material distribution topology optimization, a material indicator function determines the material state at each point within the design domain. It is well known that naive formulations of continuous material distribution topology optimization problems often lack solutions.
    [Show full text]
  • Arxiv:0810.1295V3 [Math.GR] 13 Jan 2011 Hc Sijciebtntsretv.Wt Hstriooya Hand, Sa at by Terminology Rephrased Set Be This a May with That Sets M Infinite Surjective
    EXPANSIVE ACTIONS ON UNIFORM SPACES AND SURJUNCTIVE MAPS TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT Abstract. We present a uniform version of a result of M. Gromov on the surjunctivity of maps commuting with expansive group ac- tions and discuss several applications. We prove in particular that for any group Γ and any field K, the space of Γ-marked groups G such that the group algebra K[G] is stably finite is compact. 1. Introduction A map f from a set X into itself is said to be surjunctive if it is surjective or not injective [Gro]. Thus, a non-surjunctive map is a map which is injective but not surjective. With this terminology at hand, Dedekind’s characterization of infinite sets may be rephrased by saying that a set X is infinite if and only if it admits a non-surjunctive map f : X → X. Similarly, elementary linear algebra tells us that a vector space is infinite-dimensional if and only if it admits a non-surjunctive endomorphism. In fact, it turns out that the absence of non-surjunctive endomorphisms for a given mathematical object X often reflects some “finiteness” property of X. The word surjunctive was created by W. Gottschalk [Got] who intro- duced the notion of a surjunctive group. Let G be a group. Given a set A, consider the set AG equipped with the prodiscrete topology, that is, with the product topology obtained by taking the discrete topology on each factor A of AG. There is a natural action of G on AG obtained by shifting coordinates via left multiplication in G (see Section 5).
    [Show full text]
  • Dynamics of Infinite-Dimensional Groups and Ramsey-Type Phenomena
    \rtp-impa" i i 2005/5/12 page 1 i i Dynamics of infinite-dimensional groups and Ramsey-type phenomena Vladimir Pestov May 12, 2005 i i i i \rtp-impa" i i 2005/5/12 page 2 i i i i i i \rtp-impa" i i 2005/5/12 page i i i Contents Preface iii 0 Introduction 1 1 The Ramsey{Dvoretzky{Milman phenomenon 17 1.1 Finite oscillation stability . 17 1.2 First example: the sphere S1 . 27 1.2.1 Finite oscillation stability of S1 . 27 1.2.2 Concentration of measure on spheres . 32 1.3 Second example: finite Ramsey theorem . 41 1.4 Counter-example: ordered pairs . 47 2 The fixed point on compacta property 49 2.1 Extremely amenable groups . 49 2.2 Three main examples . 55 2.2.1 Example: the unitary group . 55 2.2.2 Example: the group Aut (Q; ) . 60 ≤ 2.2.3 Counter-example: the infinite symmetric group 62 2.3 Equivariant compactification . 63 2.4 Essential sets and the concentration property . 66 2.5 Veech theorem . 71 2.6 Big sets . 76 2.7 Invariant means . 81 3 L´evy groups 95 3.1 Unitary group with the strong topology . 95 i i i i i \rtp-impa" i i 2005/5/12 page ii i i ii CONTENTS 3.2 Group of measurable maps . 98 3.2.1 Construction and example . 98 3.2.2 Martingales . 101 3.2.3 Unitary groups of operator algebras . 110 3.3 Groups of transformations of measure spaces . 115 3.3.1 Maurey's theorem .
    [Show full text]
  • On Bitopological Spaces Marcus John Saegrove Iowa State University
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Digital Repository @ Iowa State University Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1971 On bitopological spaces Marcus John Saegrove Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mathematics Commons Recommended Citation Saegrove, Marcus John, "On bitopological spaces " (1971). Retrospective Theses and Dissertations. 4914. https://lib.dr.iastate.edu/rtd/4914 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. 71-26,888 SAEGROVE, Marcus John, 1944- ON BITOPOLOGICAL SPACES. Iowa State University, Ph.D., 1971 Mathematics University Microfilms, A XEROX Company, Ann Arbor, Michigan THIS DISSERTATION HAS BEOi MICROFILMED EXACTLY AS RECEIVED On bitopological spaces by Marcus John Saegrove A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of The Requirements for the Degree of DOCTOR OF PHILOSOPHY Major Subject: Mathematics Approved : Signature was redacted for privacy. In Charge oàMajor Work Signature was redacted for privacy. Signature was redacted for privacy. Iowa State University Of Science and Technology Ames, Iowa 1971 PLEASE NOTE: Some pages have indistinct print. Filmed as received. UNIVERSITY MICROFILMS. ii TABLE OF CONTENTS Page I. INTRODUCTION 1 II. SEPARATION PROPERTIES 4 III. AN INTERNAL CHARACTERIZATION OF PAIRWISE COMPLETE REGULARITY l8 IV. PRODUCTS AND BICOMPACTIFICATIONS 26 V.
    [Show full text]
  • Extremely Amenable Groups and Banach Representations
    Extremely Amenable Groups and Banach Representations A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Javier Ronquillo Rivera May 2018 © 2018 Javier Ronquillo Rivera. All Rights Reserved. 2 This dissertation titled Extremely Amenable Groups and Banach Representations by JAVIER RONQUILLO RIVERA has been approved for the Department of Mathematics and the College of Arts and Sciences by Vladimir Uspenskiy Professor of Mathematics Robert Frank Dean, College of Arts and Sciences 3 Abstract RONQUILLO RIVERA, JAVIER, Ph.D., May 2018, Mathematics Extremely Amenable Groups and Banach Representations (125 pp.) Director of Dissertation: Vladimir Uspenskiy A long-standing open problem in the theory of topological groups is as follows: [Glasner-Pestov problem] Let X be compact and Homeo(X) be endowed with the compact-open topology. If G ⊂ Homeo(X) is an abelian group, such that X has no G-fixed points, does G admit a non-trivial continuous character? In this dissertation we discuss some reformulations of this problem and its connections to other mathematical objects such as extremely amenable groups. When G is the closure of the group generated by a single map T ∈ Homeo(X) (with respect to the compact-open topology) and the action of G on X is minimal, the existence of non-trivial continuous characters of G is linked to the existence of equicontinuous factors of (X, T ). In this dissertation we present some connections between weakly mixing dynamical systems, continuous characters on groups, and the space of maximal chains of subcontinua of a given compact space.
    [Show full text]
  • ON UNIFORM TOPOLOGY and ITS APPLICATIONS 1. Introduction The
    TWMS J. Pure Appl. Math., V.6, N.2, 2015, pp.165-179 ON UNIFORM TOPOLOGY AND ITS APPLICATIONS ALTAY A. BORUBAEV1, ASYLBEK A. CHEKEEV2 Abstract. This paper is a review and an extended version of the report made by A.A. Borubaev at the V World Congress of the Turkic world. It contains results on some classes of uniform spaces and uniformly continuous mappings and absolutes, generalizations of metrics, normed, uniform unitary spaces, topological and uniform groups, its completions, its spectral characterizations. Keywords: uniform space, uniformly continuous mapping, multiscalar product, multimetrics, multinorm, topological group, uniform group, completion. AMS Subject Classi¯cation: 54C05, 54C10, 54C30, 54C35, 54C45, 54C50. 1. Introduction The main notions relevant to the uniform topology gradually revealed in the theory of Real Analysis. Historically the ¯rst notions in the theory of uniform spaces clearly can be consid- ered the notion of what subsequently was called as "Cauchy sequence" (1827) and the notion of uniformly continuous function which appeared in the last half of the XIX's century. French mathematician M. Frechet devised [20] (1906) a notion of "metric space" which is a special kind of "uniform space". The theory of metric spaces was deeply developed by the German mathematician F. Hausdor® [24] (1914) and especially due to the fundamental papers of the Polish mathematician S. Banach [6] (1920). With the notion of nonmetrizable spaces appeared idea of creating some natural structure expressing the idea of uniformity and in the ¯rst turn the notion of completeness and uniformly continuous function and constructing research instru- ment to generalize the metric approach.
    [Show full text]
  • Ultrafilters and Tychonoff's Theorem
    Ultrafilters and Tychonoff's Theorem (October, 2015 version) G. Eric Moorhouse, University of Wyoming Tychonoff's Theorem, often cited as one of the cornerstone results in general topol- ogy, states that an arbitrary product of compact topological spaces is compact. In the 1930's, Tychonoff published a proof of this result in the special case of [0; 1]A, also stating that the general case could be proved by similar methods. The first proof in the gen- eral case was published by Cechˇ in 1937. All proofs rely on the Axiom of Choice in one of its equivalent forms (such as Zorn's Lemma or the Well-Ordering Principle). In fact there are several proofs available of the equiv- alence of Tychonoff's Theorem and the Axiom of Choice|from either one we can obtain the other. Here we provide one of the most popular proofs available for Ty- chonoff's Theorem, using ultrafilters. We see this approach as an opportunity to learn also about the larger role of ultra- filters in topology and in the foundations of mathematics, including non-standard analysis. As an appendix, we also include an outline of an alternative proof of Tychonoff's Theorem using a transfinite induction, using the two-factor case X × Y (previously done in class) for the inductive step. This approach is found in the exercises of the Munkres textbook. The proof of existence of nonprincipal ultrafilters was first published by Tarski in 1930; but the concept of ultrafilters is attributed to H. Cartan. 1. Compactness and the Finite Intersection Property Before proceeding, let us recall that a collection of sets S has the finite intersection property if S1 \···\ Sn 6= ? for all S1;:::;Sn 2 S, n > 0.
    [Show full text]
  • Expansive Actions with Specification on Uniform Spaces, Topological Entropy, and the Myhill Property
    EXPANSIVE ACTIONS WITH SPECIFICATION ON UNIFORM SPACES, TOPOLOGICAL ENTROPY, AND THE MYHILL PROPERTY TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT Abstract. We prove that every expansive continuous action with the weak specification property of an amenable group G on a compact Hausdorff space X has the Myhill property, i.e., every pre-injective continuous self-mapping of X commuting with the action of G on X is surjective. This extends a result previously obtained by Hanfeng Li in the case when X is metrizable. Contents 1. Introduction 2 2. Background material 4 2.1. General notation 4 2.2. Subsets of X × X 4 2.3. Uniform spaces 5 2.4. Ultrauniform spaces 6 2.5. Actions 7 2.6. Covers 7 2.7. Shift spaces 8 2.8. Amenable groups 8 3. Expansiveness 9 4. Topological entropy 10 4.1. Topological entropy 10 4.2. (F, U)-separated subsets, (F, U)-spanning subsets, and (F, U)-covers 12 4.3. Uniformapproachestotopologicalentropy 14 4.4. Topological entropy and expansiveness 17 5. Homoclinicity 19 6. Weak specification 20 arXiv:1905.02740v3 [math.DS] 6 Mar 2020 7. Proof of the main result 25 References 31 Date: March 9, 2020. 2010 Mathematics Subject Classification. 54E15, 37B05, 37B10, 37B15, 37B40, 43A07, 68Q80. Key words and phrases. uniform space, uniformly continuous action, expansiveness, specification, amenable group, entropy, homoclinicity, strongly irreducible subshift, cellular automaton. 1 2 TULLIO CECCHERINI-SILBERSTEIN AND MICHEL COORNAERT 1. Introduction A topological dynamical system is a pair (X,G), where X is a topological space and G is a group acting continuously on X.
    [Show full text]
  • Topology Optimization for FDM Parts Considering the Hybrid Deposition Path Pattern
    micromachines Article Topology Optimization for FDM Parts Considering the Hybrid Deposition Path Pattern Shuzhi Xu 1, Jiaqi Huang 2, Jikai Liu 2 and Yongsheng Ma 1,* 1 Department of Mechanical Engineering, University of Alberta, Edmonton, AB T2G 2G8, Canada; [email protected] 2 Center for Advanced Jet Engineering Technologies (CaJET), Key Laboratory of High Efficiency and Clean Mechanical Manufacture (Ministry of Education), School of Mechanical Engineering, Shandong University, Jinan 250100, China; [email protected] (J.H.); [email protected] (J.L.) * Correspondence: [email protected] Received: 11 June 2020; Accepted: 15 July 2020; Published: 22 July 2020 Abstract: Based on a solid orthotropic material with penalization (SOMP) and a double smoothing and projection (DSP) approach, this work proposes a methodology to find an optimal structure design which takes the hybrid deposition path (HDP) pattern and the anisotropic material properties into consideration. The optimized structure consists of a boundary layer and a substrate. The substrate domain is assumed to be filled with unidirectional zig-zag deposition paths and customized infill patterns, while the boundary is made by the contour offset deposition paths. This HDP is the most commonly employed path pattern for the fused deposition modeling (FDM) process. A critical derivative of the sensitivity analysis is presented in this paper, which ensures the optimality of the final design solutions. The effectiveness of the proposed method is validated through several 2D numerical examples. Keywords: solid orthotropic material with penalization; hybrid deposition paths; double smoothing and projection; fused deposition modeling 1. Introduction Additive manufacturing (AM) has gained fast development in research and industrial applications.
    [Show full text]
  • Compliance–Stress Constrained Mass Minimization for Topology Optimization on Anisotropic Meshes
    Research Article Compliance–stress constrained mass minimization for topology optimization on anisotropic meshes Nicola Ferro1 · Stefano Micheletti1 · Simona Perotto1 Received: 7 January 2020 / Accepted: 22 May 2020 / Published online: 11 June 2020 © Springer Nature Switzerland AG 2020 Abstract In this paper, we generalize the SIMPATY algorithm, which combines the SIMP method with anisotropic mesh adapta- tion to solve the minimum compliance problem with a mass constraint. In particular, the mass of the fnal layout is now minimized and both a maximum compliance and a maximum stress can be enforced as either mono- or multi-constraints. The new algorithm, named MSC-SIMPATY, is able to sharply detect the material–void interface, thanks to the anisotropic mesh adaptation. The presented test cases deal with three diferent scenarios, with a focus on the efect of the constraints on the fnal layouts and on the performance of the algorithm. Keywords Topology optimization · Stress constraint · Anisotropic mesh adaptation 1 Introduction in the literature driving topology optimization. Among these, we mention the density-based approaches [8, 10, Topology optimization is of utmost interest in diferent 46], the level-set methods [5, 53], topological derivative branches of industrial design, such as biomedical, space, procedures [49], phase feld techniques [12, 20], evolution- automotive, mechanical, architecture (see, e.g., [2, 13, ary approaches [54], homogenization [4, 9], performance- 19, 23, 57]). Similar formulations can also be adopted for based optimization [39]. We focus on the frst class and, the optimization of structures in diferent contexts, from in particular, on the SIMP (Solid Isotropic Material with fuid–structure interaction to the tailored design of mag- Penalization) method [9, 10, 46] where the material dis- netic or auxetic metamaterials [26, 36, 47, 56].
    [Show full text]