On orbit closure decompositions of tiling spaces by the generalized projection method
|
|
- Shannon Norris
- 5 years ago
- Views:
Transcription
1 Hiroshima Math. J. 30 (2000), 537±541 On orbit closure decompositions of tiling spaces by the generalized projection method (Received April 3, 2000) (Revised May 25, 2000) ABSTRACT. Let T E be the tiling space on a p-dimensional subspace E of R d with a xed lattice L by the generalized projection method. By using the dual of the lattice L we will construst explicitly a parameter family of the orbit closure decomposition of T E and characterize its dimension. As its application we obtain that the parameters of the orbit closure decomposition of T E correspond to the periods of T E?, provided that T E and T E? are given by the generalized projection method from an integral lattice L. 1. Introduction In 1981 de Bruijn introduced the multigrid and projection methods to construct aperiodic tilings such as Penrose tilings. The multigrid method was generalized by Kramer and Neri (1984). The projection method was generalized to a higher dimensional lattice Z d by Duneau and Katz (1985). GaÈhler and Rhyner (1986) extended the projection method to general lattices and showed that these generalized multigrid and projection methods are equivalent. First, we recall the de nitions of tilings and tiling spaces by the generalized projection method (cf. [3], [4]). Let L be a lattice in R d with a basis fb i ji ˆ 1; 2;...; dg. Let E be a p-dimensional subspace of R d, and E? be its orthogonal complement with respect to the standard innner product. Let p : R d! E and p? : R d! E? be the orthogonal projections. We put A ˆ f P d iˆ1 r ib i j0ur i U1g. For any x A E? we put x p? A ˆfx ujuap? A g, which is a compact set with a nonempty interior. For p vectors b ki A fb i g such that fp b ki g is linearly independent we put T k 1 ; k 2 ;...; k p ˆ f P p iˆ1 r ib ki j0ur i U1g. Taking such b ki we de ne F p L ˆfv T k 1 ; k 2 ;...; k p j v A L; k i such that fp b ki g is linearly independentg and T x ˆfp S js H x p? A E; S A F p L g. Note that T k 1 ; k 2 ;...; k p is a p-dimensional parallelotope. T x is a tiling on a p-dimensional subspace E of R d by the 2000 Mathematics Subject Classi cation. 05B45, 37B50, 52C20, 52C22, 52C23 Key words and phrases. tiling, tiling space, generalized projection method, orbit closure, period
2 538 generalized projection method. A tiling space T E ˆfu T x ju A E; x A E? g is de ned by a space of tilings consisting of all translates by E ˆ R p of the tilings T x for all x A E?. In the case L ˆ Z p the tiling space T E is de ned by any of three equivalent methods, the multigrid method, the projection method, or the oblique tiling method of Oguey, Duneau, and Katz (1988). Tiling spaces are topological dynamical systems with continuous R p translation action and with topology de ned by a tiling metric on tilings of R p (see for example [12]). In order to state theorems we remind of several de nitions. The dual lattice L is de ned by the set of vectors y A R d such that hy; xi AZ for all x A L, where h ; i denotes standard inner product. A lattice L is called integral if hx; yi AZ for all x; y A L. The standard lattice is both integral and self dual. Let Orb T x denote the orbit of T x in T E by the R p translation action and span A denote the R-linear span of a set A. The purpose of this paper is to show the following theorem: Theorem 1. Let T E be the tiling space on a p-dimensional subspace E of R d by the generalized projection method and p 0 : E?! span L V E? be the orthogonal projection. De ne p : L! span L V E? by p ˆ p 0 p? jl. We take a basis x 1 ;...; x k of any direct summand K such that L ˆ p 1 f0g l K. Then T E decomposes into a k parameter family of orbit closures Orb T t 1 x 1 t k x k for t 1 ;...; t k A R. In particular, we obtain that k is equal to rank L V E?. Note that any two orbit closures are either coincident or disjoint. A. Hof (1988) proved that E? V L ˆf0g if and only if T E ˆOrb T 0. Assume that L is integral. Then we see that rank L V E? ˆ rank L V E? because L H L and L =L is nite. In [7] we proved that the number of independent periods of the tiling space T E? is equal to rank L V E?. By Theorem 1 we immediately obtain the following theorem in the case that L is integral: Theorem 2. Let T E (resp. T E? ) be the tiling space on a p- dimensional subspace E (resp. d p -dimensional subspace E? ) of R d by the generalized projection method and assume that L is an integral lattice. Then T E decomposes into a k parameter family of orbit closures, where k is equal to the number of independent periods of the tiling space T E?. According to the unpublished thesis of C. Hillman (1988) the tiling space T E decomposes into a k parameter family of orbit closures, where k is equal to the number of independent periods of the tiling space T E? in the case that L ˆ Z d.
3 Orbit closure decompositions of tiling spaces Proof of Theorem 1. We de ne F : R d =L! T E by F xš ˆ T x for x A R d. Note that F is well-de ned by the construction of T x. Since T x u ˆu T x, we have F x uš ˆ u T x for any x A R d and u A E ˆ R p, where the tiling u T x is translation of the tiling T x by u A E ˆ R p. F is called a factor mapping (see for example [12]). Tiling spaces are compact due to [11, p. 357, Lemma 2]. So F satis es that F A ˆF A for any subset A H R d =L. Thus a parameter family of the orbit closure decomposition of T E is obtained from a parameter family of the orbit closure decomposition of R d =L by F. By a similar argument to [13, p. 52, Theorem 2.3] we can construct linear subspaces V; W H E? such that E? ˆ V l W, p? L V V is dense in V, p? L V W is a lattice in W and p? L ˆp? L V V p? L V W. In particular, V is given by V ˆ 7 r>0 span U r 0 V p? L, where span A denotes the R-linear span of a set A and U r v denotes the open ball of radius r and center v. Let Orb xš denote the orbit of xš in R d =L by the R p translation action. We take a basis x 1 ;...; x k of p? L V W. We have that the orbit closures of R d =L are the cosets of the closed connected subgroup Orb 0Š. By the properties of V and W we get that R d =L decomposes into a k parameter family of the orbit closures Orb t 1 x 1 t k x k Š for t 1 ;...; t k A R. Therefore T E also decomposes into a k parameter family of orbit closures Orb T t 1 x 1 t k x k for t 1 ;...; t k A R. We can easily see that k ˆ rank L V E?. In fact, we take a basis c 1 ;...; c k A p? L V W and a basis c k 1 ;...; c d p A V. By the de nition of L we get that hx; yi AZ for any x A p? L if and only if y A L V E? ˆ L V V? V E? for y A E?. Since p? L V V is dense in V, we obtain that L V E? is orthogonal to V. Hence, rank L V V? V E? U dim V? ˆ k. For any i with 1 U i U k we cantake a i A E? such that a i is orthogonal to fc j j 1 1 U j U d p; j 0 ig. Since x; ha i ; c i i a i A Z for any x A p? L, we see that 1 ha i ; c i i a i A E? V L 1. Now the ha i ; c i i a i are linearly independent in E? V L. Therefore we get k ˆ rank L V E?. We will construct explicitly the linear subspaces V; W H E? mentioned above from the dual of the lattice L. Let p 0 : E?! span L V E? be the orthogonal projection and de ne p : L! span L V E? by p ˆ p 0 p? jl. We take a direct summand K such that L ˆ p 1 f0g l K. By the de nition of p we have that span p? p 1 f0g is orthogonal to span L V E?. Because V H span p? p 1 f0g and k ˆ rank L V E?, we see that span p? p 1 f0g is equal to V given above. Then p? L V V is dense in V. We put W ˆ span p? K.
4 540 First, we will show that p K is discrete in span L V E?, which is enough to prove that p? L V W is a lattice in W. Suppose that fu i g is a sequence of elements of K such that f p u i g converges in L V E?. We may assume that f p u i g converges to 0. Write p? u i ˆv i w i with v i A V, w i A W for each i. By the hypothesis we see that fw i g converges to 0. Since p? K is dense in V we can choose y i A K such that jw i p? y i j < 1= 2i. Then jp? u i y i j ˆ jp? u i p? y i j ˆ jv i w i p? y i j U jv i j jw i p? y i j. So p? u i y i converges to 0. Thus for su½ciently large i, we have p? u i y i A V and u i y i A p 1 f0g. Then u i A p 1 f0g V K ˆ 0 and p u i ˆ0. This implies that p K is discrete in span L V E?. The rest of the proof is devoted to show that the two subspaces V; W H E? satisfy the following properties: E? ˆ V l W, p? L V W is a lattice in W and p? L ˆp? L V V p? L V W. Since p K spans span L V E? and p K is discrete, p K is a lattice of rank p K ˆdim E? dim V. Because the restriction pjk is injective, we obtain k ˆ rank K ˆ rank p K ˆ d p dim V. Since dim W U k ˆ d p dim V and E? ˆ W V we have dim W ˆ d p dim V and E? ˆ W l V. Since p K is discrete and p 0 is continuous, p? K is discrete. Because p? K has the same rank as K, we can see that K is isomorphic to p? K and p? K is a lattice in W. Since p? L ˆp? K p? p 1 f0g, we obtain p? K ˆp? L V W and p? p 1 f0g ˆ p? L V V. q.e.d. References [ 1 ] N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane. I, Kon. Nederl. Akad. Wetesch. Proc Ser. A. (ˆ Indag. Math.) (1981), 39±52. [ 2 ] N. G. de Bruijn, Algebraic theory of Penrose's non-periodic tilings of the plane. II, Kon. Nederl. Akad. Wetesch. Proc Ser. A. (ˆ Indag. Math.) (1981), 53±66. [ 3 ] M. Duneau and A. Katz, Quasiperiodic patterns, Phys Rev. Lett. 54 (1985), 2688±91. [ 4] F. GaÈhler and J. Rhyner, Equivalence of the generalized grid and projection methods for the construction of quasiperiodic tilings, J. Phys. A: Math. Gen. 19 (1986), 267±277. [ 5 ] C. Hillman, Sturmian dynamical systems, PhD. thesis, University of Washington, Seattle, [ 6 ] A. Hof, Uniform distribution and the projection method, in Quasicrystals and Discrete geometry, edited by Patera J., Fields Institute Monographs vol. 10, 1988, Amer. Math. Soc., 201±206. [ 7 ] K. Komatsu, Aperiodicity criterion for the tiling spaces given by the generalized projection method, Preprint. [ 8 ] K. Komatsu and K. Nakahira, Some remarks on tilings by the projection method, Mem. Fac. Sci. Kochi Univ. (Math.) 18 (1997), 117±122. [ 9 ] P. Kramer and R. Neri, On periodic and Non-periodic Space llings of E m obtained by projection, Acta Cryst. A40 (1984), 580±587. [10] C. Oguey, M. Duneau and A. Katz, A geometrical approach of quasiperiodic tilings, Comm. Math. Phys. 118 (1988), 99±118.
5 Orbit closure decompositions of tiling spaces 541 [11] C. Radin and M. Wol, Space tilings and local isomorphism, Geometriae Dedicata 42 (1992), 355±360. [12] E. A. Robinson, The dynamical theory of tilings and quasicrystallography, in Ergodic theory of Z d -actions, London Math. Soc. Lecture Note Ser. 228, Cambridge University Press, 1996, 451±473. [13] M. Senechal, ``Quasicrystals and Geometry'', Cambridge University Press, Department of Mathematics, Faculty of Science, Kochi University, Kochi , Japan
A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey
A Highly Symmetric Four-Dimensional Quasicrystal * Veit Elser and N. J. A. Sloane AT&T Bell Laboratories Murray Hill, New Jersey 7974 Abstract A quasiperiodic pattern (or quasicrystal) is constructed in
More informationFACTOR MAPS BETWEEN TILING DYNAMICAL SYSTEMS KARL PETERSEN. systems which cannot be achieved by working within a nite window. By. 1.
FACTOR MAPS BETWEEN TILING DYNAMICAL SYSTEMS KARL PETERSEN Abstract. We show that there is no Curtis-Hedlund-Lyndon Theorem for factor maps between tiling dynamical systems: there are codes between such
More informationParameterizing orbits in flag varieties
Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.
More informationTopological groups with dense compactly generated subgroups
Applied General Topology c Universidad Politécnica de Valencia Volume 3, No. 1, 2002 pp. 85 89 Topological groups with dense compactly generated subgroups Hiroshi Fujita and Dmitri Shakhmatov Abstract.
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
More informationHomogenization of Penrose tilings
Homogenization of Penrose tilings A. Braides Dipartimento di Matematica, Università di Roma or Vergata via della ricerca scientifica, 0033 Roma, Italy G. Riey Dipartimento di Matematica, Università della
More informationWEAK INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO COMPARABLE CONTRA-PRECONTINUOUS (CONTRA-SEMI-CONTINUOUS) FUNCTIONS
MATHEMATICA MONTISNIGRI Vol XLI (2018) MATHEMATICS WEAK INSERTION OF A CONTRA-CONTINUOUS FUNCTION BETWEEN TWO COMPARABLE CONTRA-PRECONTINUOUS (CONTRA-SEMI-CONTINUOUS) FUNCTIONS Department of Mathematics,
More informationSome icosahedral patterns and their isometries
Some icosahedral patterns and their isometries Nicolae Cotfas 1 Faculty of Physics, University of Bucharest, PO Box 76-54, Postal Office 76, Bucharest, Romania Radu Slobodeanu 2 Faculty of Physics, University
More informationPoint Sets and Dynamical Systems in the Autocorrelation Topology
Point Sets and Dynamical Systems in the Autocorrelation Topology Robert V. Moody and Nicolae Strungaru Department of Mathematical and Statistical Sciences University of Alberta, Edmonton Canada, T6G 2G1
More informationAbelian topological groups and (A/k) k. 1. Compact-discrete duality
(December 21, 2010) Abelian topological groups and (A/k) k Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ 1. Compact-discrete duality 2. (A/k) k 3. Appendix: compact-open topology
More informationENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS. 1. Introduction
Trends in Mathematics Information Center for Mathematical Sciences Volume 6, Number 2, December, 2003, Pages 83 91 ENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS SEUNG-HYEOK KYE Abstract.
More informationMath 413/513 Chapter 6 (from Friedberg, Insel, & Spence)
Math 413/513 Chapter 6 (from Friedberg, Insel, & Spence) David Glickenstein December 7, 2015 1 Inner product spaces In this chapter, we will only consider the elds R and C. De nition 1 Let V be a vector
More informationMaximal perpendicularity in certain Abelian groups
Acta Univ. Sapientiae, Mathematica, 9, 1 (2017) 235 247 DOI: 10.1515/ausm-2017-0016 Maximal perpendicularity in certain Abelian groups Mika Mattila Department of Mathematics, Tampere University of Technology,
More informationIntroduction to mathematical quasicrystals
Introduction to mathematical quasicrystals F S W Alan Haynes Topics to be covered Historical overview: aperiodic tilings of Euclidean space and quasicrystals Lattices, crystallographic point sets, and
More informationAssignment 3. Section 10.3: 6, 7ab, 8, 9, : 2, 3
Andrew van Herick Math 710 Dr. Alex Schuster Sept. 21, 2005 Assignment 3 Section 10.3: 6, 7ab, 8, 9, 10 10.4: 2, 3 10.3.6. Prove (3) : Let E X: Then x =2 E if and only if B r (x) \ E c 6= ; for all all
More informationTotally supra b continuous and slightly supra b continuous functions
Stud. Univ. Babeş-Bolyai Math. 57(2012), No. 1, 135 144 Totally supra b continuous and slightly supra b continuous functions Jamal M. Mustafa Abstract. In this paper, totally supra b-continuity and slightly
More informationG-frames in Hilbert Modules Over Pro-C*-algebras
Available online at http://ijim.srbiau.ac.ir/ Int. J. Industrial Mathematics (ISSN 2008-5621) Vol. 9, No. 4, 2017 Article ID IJIM-00744, 9 pages Research Article G-frames in Hilbert Modules Over Pro-C*-algebras
More informationAverage lattices for quasicrystals
Technische Fakultät Universität Bielefeld 23. May 2017 joint work with Alexey Garber, Brownsville, Texas Plan: 1. Lattices, Delone sets, basic notions 2. Dimension one 3. Aperiodic patterns 4. 5. Weighted
More informationg ωα-separation Axioms in Topological Spaces
Malaya J. Mat. 5(2)(2017) 449 455 g ωα-separation Axioms in Topological Spaces P. G. Patil, S. S. Benchalli and Pallavi S. Mirajakar Department of Mathematics, Karnatak University, Dharwad-580 003, Karnataka,
More informationCOMPACT ORTHOALGEBRAS
COMPACT ORTHOALGEBRAS ALEXANDER WILCE Abstract. We initiate a study of topological orthoalgebras (TOAs), concentrating on the compact case. Examples of TOAs include topological orthomodular lattices, and
More informationMetric Spaces. DEF. If (X; d) is a metric space and E is a nonempty subset, then (E; d) is also a metric space, called a subspace of X:
Metric Spaces DEF. A metric space X or (X; d) is a nonempty set X together with a function d : X X! [0; 1) such that for all x; y; and z in X : 1. d (x; y) 0 with equality i x = y 2. d (x; y) = d (y; x)
More information2 Garrett: `A Good Spectral Theorem' 1. von Neumann algebras, density theorem The commutant of a subring S of a ring R is S 0 = fr 2 R : rs = sr; 8s 2
1 A Good Spectral Theorem c1996, Paul Garrett, garrett@math.umn.edu version February 12, 1996 1 Measurable Hilbert bundles Measurable Banach bundles Direct integrals of Hilbert spaces Trivializing Hilbert
More informationAveraging Operators on the Unit Interval
Averaging Operators on the Unit Interval Mai Gehrke Carol Walker Elbert Walker New Mexico State University Las Cruces, New Mexico Abstract In working with negations and t-norms, it is not uncommon to call
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #17 11/05/2013 Throughout this lecture k denotes an algebraically closed field. 17.1 Tangent spaces and hypersurfaces For any polynomial f k[x
More informationGroup Gradings on Finite Dimensional Lie Algebras
Algebra Colloquium 20 : 4 (2013) 573 578 Algebra Colloquium c 2013 AMSS CAS & SUZHOU UNIV Group Gradings on Finite Dimensional Lie Algebras Dušan Pagon Faculty of Natural Sciences and Mathematics, University
More informationMATHÉMATIQUES PURES & APPLIQUÉES
MATHÉMATIQUES PURES & APPLIQUÉES Rashomon Pavages et rotations Charles RADIN (University of Texas, Austin) W e have all seen how illuminating it can be to view something, even something one knows very
More informationAvailable at: pocetna.html ON A GENERALIZATION OF NORMAL, ALMOST NORMAL AND MILDLY NORMAL SPACES II
Faculty of Sciences and Mathematics University of Niš Available at: www.pmf.ni.ac.yu/sajt/publikacije/publikacije pocetna.html Filomat 20:2 (2006), 67 80 ON A GENERALIZATION OF NORMAL, ALMOST NORMAL AND
More informationSung-Wook Park*, Hyuk Han**, and Se Won Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS Sung-Wook Park*, Hyuk Han**,
More informationA REPRESENTATION THEOREM FOR ORTHOGONALLY ADDITIVE POLYNOMIALS ON RIESZ SPACES
A REPRESENTATION THEOREM FOR ORTHOGONALLY ADDITIVE POLYNOMIALS ON RIESZ SPACES A. IBORT, P. LINARES AND J.G. LLAVONA Abstract. The aim of this article is to prove a representation theorem for orthogonally
More informationLECTURE 2: SYMPLECTIC VECTOR BUNDLES
LECTURE 2: SYMPLECTIC VECTOR BUNDLES WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic Vector Spaces Definition 1.1. A symplectic vector space is a pair (V, ω) where V is a finite dimensional vector space (over
More informationMath 5210, Definitions and Theorems on Metric Spaces
Math 5210, Definitions and Theorems on Metric Spaces Let (X, d) be a metric space. We will use the following definitions (see Rudin, chap 2, particularly 2.18) 1. Let p X and r R, r > 0, The ball of radius
More informationRepresentations of Sp(6,R) and SU(3) carried by homogeneous polynomials
Representations of Sp(6,R) and SU(3) carried by homogeneous polynomials Govindan Rangarajan a) Department of Mathematics and Centre for Theoretical Studies, Indian Institute of Science, Bangalore 560 012,
More informationCHAPTER 6. Representations of compact groups
CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive
More informationHomotopy and homology groups of the n-dimensional Hawaiian earring
F U N D A M E N T A MATHEMATICAE 165 (2000) Homotopy and homology groups of the n-dimensional Hawaiian earring by Katsuya E d a (Tokyo) and Kazuhiro K a w a m u r a (Tsukuba) Abstract. For the n-dimensional
More informationModeling Ergodic, Measure Preserving Actions on Z d Shifts of Finite Type
Monatsh. Math. 132, 237±253 (2001) Modeling Ergodic, Measure Preserving Actions on Z d Shifts of Finite Type By E. Arthur Robinson, Jr. 1 AysËe A.SËahin 2 1 George Washington University, Washington DC,
More informationDUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS
MATH. SCAND. 86 (2000), 130^142 DUALITY FOR POSITIVE LINEAR MAPS IN MATRIX ALGEBRAS MYOUNG-HOE EOM AND SEUNG-HYEOK KYE Abstract We characterize extreme rays of the dual cone of the cone consisting of all
More informationINTRODUCTION TO LIE ALGEBRAS. LECTURE 2.
INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined
More informationRepresentations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III
Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group
More informationg -Pre Regular and g -Pre Normal Spaces
International Mathematical Forum, 4, 2009, no. 48, 2399-2408 g -Pre Regular and g -Pre Normal Spaces S. S. Benchalli Department of Mathematics Karnatak University, Dharwad-580 003 Karnataka State, India.
More informationThe Residual Spectrum and the Continuous Spectrum of Upper Triangular Operator Matrices
Filomat 28:1 (2014, 65 71 DOI 10.2298/FIL1401065H Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat The Residual Spectrum and the
More informationAlvaro Rodrigues-Neto Research School of Economics, Australian National University. ANU Working Papers in Economics and Econometrics # 587
Cycles of length two in monotonic models José Alvaro Rodrigues-Neto Research School of Economics, Australian National University ANU Working Papers in Economics and Econometrics # 587 October 20122 JEL:
More informationGROUPS DEFINABLE IN O-MINIMAL STRUCTURES
GROUPS DEFINABLE IN O-MINIMAL STRUCTURES PANTELIS E. ELEFTHERIOU Abstract. In this series of lectures, we will a) introduce the basics of o- minimality, b) describe the manifold topology of groups definable
More informationMath Linear Algebra
Math 220 - Linear Algebra (Summer 208) Solutions to Homework #7 Exercise 6..20 (a) TRUE. u v v u = 0 is equivalent to u v = v u. The latter identity is true due to the commutative property of the inner
More informationDETERMINING THE HURWITZ ORBIT OF THE STANDARD GENERATORS OF A BRAID GROUP
Yaguchi, Y. Osaka J. Math. 52 (2015), 59 70 DETERMINING THE HURWITZ ORBIT OF THE STANDARD GENERATORS OF A BRAID GROUP YOSHIRO YAGUCHI (Received January 16, 2012, revised June 18, 2013) Abstract The Hurwitz
More informationhal , version 1-22 Jun 2010
HYPERCYCLIC ABELIAN SEMIGROUP OF MATRICES ON C n AND R n AND k-transitivity (k 2) ADLENE AYADI Abstract. We prove that the minimal number of matrices on C n required to form a hypercyclic abelian semigroup
More informationCorrelation dimension for self-similar Cantor sets with overlaps
F U N D A M E N T A MATHEMATICAE 155 (1998) Correlation dimension for self-similar Cantor sets with overlaps by Károly S i m o n (Miskolc) and Boris S o l o m y a k (Seattle, Wash.) Abstract. We consider
More informationWeek 5 Lectures 13-15
Week 5 Lectures 13-15 Lecture 13 Definition 29 Let Y be a subset X. A subset A Y is open in Y if there exists an open set U in X such that A = U Y. It is not difficult to show that the collection of all
More informationCanonical systems of basic invariants for unitary reflection groups
Canonical systems of basic invariants for unitary reflection groups Norihiro Nakashima, Hiroaki Terao and Shuhei Tsujie Abstract It has been known that there exists a canonical system for every finite
More informationSome results on g-regular and g-normal spaces
SCIENTIA Series A: Mathematical Sciences, Vol. 23 (2012), 67 73 Universidad Técnica Federico Santa María Valparaíso, Chile ISSN 0716-8446 c Universidad Técnica Federico Santa María 2012 Some results on
More informationCONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP
CONSIDERATION OF COMPACT MINIMAL SURFACES IN 4-DIMENSIONAL FLAT TORI IN TERMS OF DEGENERATE GAUSS MAP TOSHIHIRO SHODA Abstract. In this paper, we study a compact minimal surface in a 4-dimensional flat
More informationIntroduction to Topology
Introduction to Topology Randall R. Holmes Auburn University Typeset by AMS-TEX Chapter 1. Metric Spaces 1. Definition and Examples. As the course progresses we will need to review some basic notions about
More informationTHE LARGEST INTERSECTION LATTICE OF A CHRISTOS A. ATHANASIADIS. Abstract. We prove a conjecture of Bayer and Brandt [J. Alg. Combin.
THE LARGEST INTERSECTION LATTICE OF A DISCRIMINANTAL ARRANGEMENT CHRISTOS A. ATHANASIADIS Abstract. We prove a conjecture of Bayer and Brandt [J. Alg. Combin. 6 (1997), 229{246] about the \largest" intersection
More informationON µ-compact SETS IN µ-spaces
Questions and Answers in General Topology 31 (2013), pp. 49 57 ON µ-compact SETS IN µ-spaces MOHAMMAD S. SARSAK (Communicated by Yasunao Hattori) Abstract. The primary purpose of this paper is to introduce
More informationBEST APPROXIMATIONS AND ORTHOGONALITIES IN 2k-INNER PRODUCT SPACES. Seong Sik Kim* and Mircea Crâşmăreanu. 1. Introduction
Bull Korean Math Soc 43 (2006), No 2, pp 377 387 BEST APPROXIMATIONS AND ORTHOGONALITIES IN -INNER PRODUCT SPACES Seong Sik Kim* and Mircea Crâşmăreanu Abstract In this paper, some characterizations of
More informationTakashi Noiri, Ahmad Al-Omari, Mohd. Salmi Md. Noorani WEAK FORMS OF OPEN AND CLOSED FUNCTIONS
DEMONSTRATIO MATHEMATICA Vol. XLII No 1 2009 Takashi Noiri, Ahmad Al-Omari, Mohd. Salmi Md. Noorani WEAK FORMS OF OPEN AND CLOSED FUNCTIONS VIA b-θ-open SETS Abstract. In this paper, we introduce and study
More informationLINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO
ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS N. 39 2018 (757 763) 757 LINEAR MAPS ON M n (C) PRESERVING INNER LOCAL SPECTRAL RADIUS ZERO Hassane Benbouziane Mustapha Ech-Chérif Elkettani Ahmedou Mohamed
More informationTROPICAL BRILL-NOETHER THEORY
TROPICAL BRILL-NOETHER THEORY 11. Berkovich Analytification and Skeletons of Curves We discuss the Berkovich analytification of an algebraic curve and its skeletons, which have the structure of metric
More informationProof: The coding of T (x) is the left shift of the coding of x. φ(t x) n = L if T n+1 (x) L
Lecture 24: Defn: Topological conjugacy: Given Z + d (resp, Zd ), actions T, S a topological conjugacy from T to S is a homeomorphism φ : M N s.t. φ T = S φ i.e., φ T n = S n φ for all n Z + d (resp, Zd
More informationRESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES RIMS A Note on an Anabelian Open Basis for a Smooth Variety. Yuichiro HOSHI.
RIMS-1898 A Note on an Anabelian Open Basis for a Smooth Variety By Yuichiro HOSHI January 2019 RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES KYOTO UNIVERSITY, Kyoto, Japan A Note on an Anabelian Open Basis
More informationLinear Algebra. Linear Algebra. Chih-Wei Yi. Dept. of Computer Science National Chiao Tung University. November 12, 2008
Linear Algebra Chih-Wei Yi Dept. of Computer Science National Chiao Tung University November, 008 Section De nition and Examples Section De nition and Examples Section De nition and Examples De nition
More informationAN EFFECTIVE METRIC ON C(H, K) WITH NORMAL STRUCTURE. Mona Nabiei (Received 23 June, 2015)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 46 (2016), 53-64 AN EFFECTIVE METRIC ON C(H, K) WITH NORMAL STRUCTURE Mona Nabiei (Received 23 June, 2015) Abstract. This study first defines a new metric with
More informationHalf twists of Hodge structures of CM-type
J. Math. Soc. Japan Vol. 53, No. 4, 2001 Half twists of Hodge structures of CM-type By Bert van Geemen (Received Sept. 6, 1999) (Revised Apr. 21, 2000) Abstract. To a Hodge structure V of weight k with
More informationTHE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES
THE GEOMETRY OF CONVEX TRANSITIVE BANACH SPACES JULIO BECERRA GUERRERO AND ANGEL RODRIGUEZ PALACIOS 1. Introduction Throughout this paper, X will denote a Banach space, S S(X) and B B(X) will be the unit
More informationarxiv: v1 [math.rt] 14 Nov 2007
arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof
More informationA DECOMPOSITION OF E 0 -SEMIGROUPS
A DECOMPOSITION OF E 0 -SEMIGROUPS Remus Floricel Abstract. Any E 0 -semigroup of a von Neumann algebra can be uniquely decomposed as the direct sum of an inner E 0 -semigroup and a properly outer E 0
More informationDAVIS WIELANDT SHELLS OF NORMAL OPERATORS
DAVIS WIELANDT SHELLS OF NORMAL OPERATORS CHI-KWONG LI AND YIU-TUNG POON Dedicated to Professor Hans Schneider for his 80th birthday. Abstract. For a finite-dimensional operator A with spectrum σ(a), the
More informationModel sets, Meyer sets and quasicrystals
Model sets, Meyer sets and quasicrystals Technische Fakultät Universität Bielefeld University of the Philippines Manila 27. Jan. 2014 Model sets, Meyer sets and quasicrystals Quasicrystals, cut-and-project
More informationCHAOTIC UNIMODAL AND BIMODAL MAPS
CHAOTIC UNIMODAL AND BIMODAL MAPS FRED SHULTZ Abstract. We describe up to conjugacy all unimodal and bimodal maps that are chaotic, by giving necessary and sufficient conditions for unimodal and bimodal
More informationGenericity: A Measure Theoretic Analysis
Genericity: A Measure Theoretic Analysis Massimo Marinacci Draft of December 1994 y 1 Introduction In many results of Mathematical Economics the notion of smallness plays a crucial role. Indeed, it often
More informationTHE CLASSIFICATION OF TILING SPACE FLOWS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 THE CLASSIFICATION OF TILING SPACE FLOWS by Alex Clark Abstract. We consider the conjugacy of the natural flows on one-dimensional tiling
More informationAuerbach bases and minimal volume sufficient enlargements
Auerbach bases and minimal volume sufficient enlargements M. I. Ostrovskii January, 2009 Abstract. Let B Y denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in
More informationSEMI-FREDHOLM THEORY: HYPERCYCLIC AND SUPERCYCLIC SUBSPACES
SEMI-FREDHOLM THEORY: HYPERCYCLIC AND SUPERCYCLIC SUBSPACES MANUEL GONZAÂ LEZ, FERNANDO LEOÂ N-SAAVEDRA and ALFONSO MONTES-RODRIÂGUEZ [Received 27 April 1998; revised 28 June 1999] 1. Introduction Let
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 124, Number 7, July 1996 ON COMPLEMENTED SUBSPACES OF SUMS AND PRODUCTS OF BANACH SPACES M. I. OSTROVSKII (Communicated by Dale Alspach) Abstract.
More information1 Selected Homework Solutions
Selected Homework Solutions Mathematics 4600 A. Bathi Kasturiarachi September 2006. Selected Solutions to HW # HW #: (.) 5, 7, 8, 0; (.2):, 2 ; (.4): ; (.5): 3 (.): #0 For each of the following subsets
More informationIntroduction to Group Theory
Introduction to Group Theory Ling-Fong Li (Institute) Group 1 / 6 INTRODUCTION Group theory : framework for studying symmetry. The representation theory of the group simpli es the physical solutions. For
More informationShift Invariant Spaces and Shift Generated Dual Frames for Local Fields
Communications in Mathematics and Applications Volume 3 (2012), Number 3, pp. 205 214 RGN Publications http://www.rgnpublications.com Shift Invariant Spaces and Shift Generated Dual Frames for Local Fields
More informationExtensions of Lipschitz functions and Grothendieck s bounded approximation property
North-Western European Journal of Mathematics Extensions of Lipschitz functions and Grothendieck s bounded approximation property Gilles Godefroy 1 Received: January 29, 2015/Accepted: March 6, 2015/Online:
More informationSOME ABSOLUTELY CONTINUOUS REPRESENTATIONS OF FUNCTION ALGEBRAS
Surveys in Mathematics and its Applications ISSN 1842-6298 Volume 1 (2006), 51 60 SOME ABSOLUTELY CONTINUOUS REPRESENTATIONS OF FUNCTION ALGEBRAS Adina Juratoni Abstract. In this paper we study some absolutely
More informationMore on sg-compact spaces
arxiv:math/9809068v1 [math.gn] 12 Sep 1998 More on sg-compact spaces Julian Dontchev Department of Mathematics University of Helsinki PL 4, Yliopistonkatu 15 00014 Helsinki 10 Finland Abstract Maximilian
More informationMATH Linear Algebra
MATH 304 - Linear Algebra In the previous note we learned an important algorithm to produce orthogonal sequences of vectors called the Gramm-Schmidt orthogonalization process. Gramm-Schmidt orthogonalization
More informationFACIAL STRUCTURES FOR SEPARABLE STATES
J. Korean Math. Soc. 49 (202), No. 3, pp. 623 639 http://dx.doi.org/0.434/jkms.202.49.3.623 FACIAL STRUCTURES FOR SEPARABLE STATES Hyun-Suk Choi and Seung-Hyeok Kye Abstract. The convex cone V generated
More informationExercises on chapter 0
Exercises on chapter 0 1. A partially ordered set (poset) is a set X together with a relation such that (a) x x for all x X; (b) x y and y x implies that x = y for all x, y X; (c) x y and y z implies that
More informationON REGULARITY OF FINITE REFLECTION GROUPS. School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
ON REGULARITY OF FINITE REFLECTION GROUPS R. B. Howlett and Jian-yi Shi School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia Department of Mathematics, East China Normal University,
More informationContra-Pre-Semi-Continuous Functions
BULLETIN of the Malaysian Mathematical Sciences Society http://math.usm.my/bulletin Bull. Malays. Math. Sci. Soc. (2) 28(1) (2005), 67 71 Contra-Pre-Semi-Continuous Functions M.K.R.S. Veera Kumar P.G.
More informationON THE PRODUCT OF SEPARABLE METRIC SPACES
Georgian Mathematical Journal Volume 8 (2001), Number 4, 785 790 ON THE PRODUCT OF SEPARABLE METRIC SPACES D. KIGHURADZE Abstract. Some properties of the dimension function dim on the class of separable
More informationSystems of Linear Equations
LECTURE 6 Systems of Linear Equations You may recall that in Math 303, matrices were first introduced as a means of encapsulating the essential data underlying a system of linear equations; that is to
More informationSlightly γ-continuous Functions. Key words: clopen, γ-open, γ-continuity, slightly continuity, slightly γ-continuity. Contents
Bol. Soc. Paran. Mat. (3s.) v. 22 2 (2004): 63 74. c SPM ISNN-00378712 Slightly γ-continuous Functions Erdal Ekici and Miguel Caldas abstract: The purpose of this paper is to give a new weak form of some
More informationThe Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria
ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Common Homoclinic Points of Commuting Toral Automorphisms Anthony Manning Klaus Schmidt
More informationTHE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE. In memory of A. Pe lczyński
THE DUAL FORM OF THE APPROXIMATION PROPERTY FOR A BANACH SPACE AND A SUBSPACE T. FIGIEL AND W. B. JOHNSON Abstract. Given a Banach space X and a subspace Y, the pair (X, Y ) is said to have the approximation
More informationTOPOLOGICAL GROUPS MATH 519
TOPOLOGICAL GROUPS MATH 519 The purpose of these notes is to give a mostly self-contained topological background for the study of the representations of locally compact totally disconnected groups, as
More informationContra Pre Generalized b - Continuous Functions in Topological Spaces
Mathematica Aeterna, Vol. 7, 2017, no. 1, 57-67 Contra Pre Generalized b - Continuous Functions in Topological Spaces S. Sekar Department of Mathematics, Government Arts College (Autonomous), Salem 636
More informationOn bτ-closed sets. Maximilian Ganster and Markus Steiner
On bτ-closed sets Maximilian Ganster and Markus Steiner Abstract. This paper is closely related to the work of Cao, Greenwood and Reilly in [10] as it expands and completes their fundamental diagram by
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationLECTURE 12 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT
MARCH 29, 26 LECTURE 2 UNIT ROOT, WEAK CONVERGENCE, FUNCTIONAL CLT (Davidson (2), Chapter 4; Phillips Lectures on Unit Roots, Cointegration and Nonstationarity; White (999), Chapter 7) Unit root processes
More informationLecture 1. Toric Varieties: Basics
Lecture 1. Toric Varieties: Basics Taras Panov Lomonosov Moscow State University Summer School Current Developments in Geometry Novosibirsk, 27 August1 September 2018 Taras Panov (Moscow University) Lecture
More informationarxiv:math/ v1 [math.co] 3 Sep 2000
arxiv:math/0009026v1 [math.co] 3 Sep 2000 Max Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department San Francisco State University San Francisco, CA 94132 sergei@sfsu.edu
More informationThe symmetric group action on rank-selected posets of injective words
The symmetric group action on rank-selected posets of injective words Christos A. Athanasiadis Department of Mathematics University of Athens Athens 15784, Hellas (Greece) caath@math.uoa.gr October 28,
More informationOn Preclosed Sets and Their Generalizations
On Preclosed Sets and Their Generalizations Jiling Cao Maximilian Ganster Chariklia Konstadilaki Ivan L. Reilly Abstract This paper continues the study of preclosed sets and of generalized preclosed sets
More information