Fractal penrose tiles II: Tiles with fractal boundary as duals of penrose triangles

Summary Suppose given a quasi-periodic tiling of some Euclidean space E _u which is self-similar under the linear expansiong: E_μ→E_μ. It is known that there is an embedding of E_μ into some higher-dimensional space ℝ ^N and a linear automorphism with integer coefficients such that E _u ⊂ ℝ ^N is invariant under andg is the restriction of to E _u . Let E _s be the -invariant complement of E _u , and . If certain conditions are fulfilled (e.g. must be a lattice automorphism,g ^* is an expansion), we construct a self-similar tiling of E _s whose expansion isg ^*, using the information contained in the original tiling of E_μ. The term “Galois duality” of tilings is motivated by the fact that the eigenvalues ofg ^* are Galois conjugates of those ofg. Our method can be applied to find the Galois duals which are given by Thurston, obtained in a somewhat other way for the case that dim E_μ=1, andg is the multiplication by a cubic Pisot unit. Bandt and Gummelt have found fractally shaped tilings which can be considered as strictly self-similar modifications of the kites-and-darts, and the rhombi tilings of Penrose. As one of the examples, we show that these fractal versions can be constructed by dualizing tilings by Penrose triangles.     

Group algebras with almost maximal lie nilpotency index

LetK G be a non-commutative Lie nilpotent group algebra of a groupG over a fieldK. It is known that the Lie nilpotency index ofKG is at most |G′|+1, where |G′| is the order of the commutator subgroup ofG. In [4] the groupsG for which this index is maximal were determined. Here we list theG’s for which it assumes the next highest possible value. The present paper is a part of the PhD dissertation of the author.     

Automorfismi spezzanti di ordine 2

LetG be a group and α an automorphism ofG; α is calledn-splitting if for allg∈G. In this note we study the structure of finite groups admitting an-splitting automorphism of order 2.

     

Dilatation estimates for quasiconformal extensions

LetD andD′ be ring domains inB ⁿ , withS ⁿ⁻¹ as one boundary component, and let be a homeomorphism which isK-quasiconformal inD and withf(S ⁿ⁻¹)=S ⁿ⁻¹. According to a result of Gehringf÷S ⁿ⁻¹ admits an extension which is quasiconformal inB ⁿ . We find here an upper bound for the dilatation ofg in terms ofn, K, and modD. This work was started during a visit to Université de Paris, financed by a cultural exchange program between France and Finland.     

The frattini subgroup of the absolute Galois group of a local field

LetK be a local field,T the maximal tamely ramified extension ofK, F the fixed field inK _sof the Frattini subgroup ofG(K), andJ the compositum of all minimal Galois extensions ofK containingT. The main result of the paper is thatF=J. IfK is a global field andK _solv is the maximal prosolvable extension ofK, then the Frattini group of % MathType!End!2!1!(K _solv/K) is trivial. Partially supported by a grant from the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Representations of double coset hypergroups and induced representations

The principal goal of this paper is to investigate the representation theory of double coset hypergroups. IfK=G//H is a double coset hypergroup, representations ofK can canonically be obtained from those ofG. However, not every representation ofK originates from this construction in general, i.e., extends to a representation ofG. Properties of this construction are discussed, and as the main result it turns out that extending representations ofK is compatible with the inducing process (as introduced in [7]). It follows that a representation weakly contained in the left-regular representation ofK always admits an extension toG. Furthermore, we realize the Gelfand pair (where are a local field andR its ring of integers) as a polynomial hypergroup on ℕ₀ and characterize the (proper) subset of its dual consisting of extensible representations.     

Gruppi Policiclici Dotati di un Automorfismo Uniforme di Ordinepq

An automorphismϕ of a groupG is said to be uniform il for everyg ∈G there exists anh ∈G such thatG=h ⁻¹ h ^ρ . It is a well-known fact that ifG is finite, an automorphism ofG is uniform if and only if it is fixed-point-free. In [7] Zappa proved that if a polycyclic groupG admits an uniform automorphism of prime orderp thenG is a finite (nilpotent)p′-group. In this paper we continue Zappa’s work considering uniform automorphism of orderpg (p andq distinct prime numbers). In particular we prove that there exists a constantμ (depending only onp andq) such that every torsion-free polycyclic groupG admitting an uniform automorphism of orderpq is nilpotent of class at mostμ. As a consequence we prove that if a polycyclic groupG admits an uniform automorphism of orderpq thenZ _μ (G) has finite index inG.

Al professore Guido Zappa per il suo 90⁰ compleanno     

Limit distributions of expanding translates of certain orbits on homogeneous spaces

LetL be a Lie group and λ a lattice inL. SupposeG is a non-compact simple Lie group realized as a Lie subgroup ofL and . LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U⁺ ={gεG:a ⁻ⁿ gaⁿ} →e as n → ∞. Let Ω be a non-empty open subset ofU ⁺ andn _i → ∞ be any sequence. It is showed that . A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeG-equivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

Automorphisms of linear semigroups over division rings

Given an-dimensional right vector spaceV over a division ring we denote byS the semigroup of the endomorphisms ofV and designate this semigroup as alinear semigroup. First we prove that every automorphism ofS can be written asT→fTf ⁻¹, wheref∶V→V is a semilinear homeomorphism. Furthermore, we show that every isomorphism between maximal compact subsemigroups ofS is also of this type.     

The Maslov index revisited

LetD be a Hermitian symmetric space of tube type,S its Shilov boundary andG the neutral component of the group of bi-holomorphic diffeomorphisms ofD. In the model situationD is the Siegel disc,S is the manifold of Lagrangian subspaces andG is the symplectic group. We introduce a notion of transversality for pairs of elements inS, and then study the action ofG on the set of triples of mutually transversal points inS. We show that there is a finite number ofG-orbits, and to each orbit we associate an integer, thus generalizing theMaslov index. Using the scalar automorphy kernel ofD, we construct a ^*,G-invariant kernel onD×D×D. Taking a specific determination of its argument and studying its limit when approaching the Shilov boundary, we are able to define a -valued,G-invariant kernel for triples of mutually transversal points inS. It is shown to coincide with the Maslov index. Symmetry properties and cocycle properties of the Maslov index are then easily obtained.Both authors acknowledge partial support from the European Commission (European TMR Network Harmonic Analysis 1998–2001, Contract ERBFMRX-CT97-0159).     

Spectra of ergodic group actions

LetG be a locally compact second countable abelian group, (X, μ) aσ-finite Lebesgue space, and (g, x) →gx a non-singular, properly ergodic action ofG on (X, μ). Let furthermore Γ be the character group ofG and let Sp(G, X) ⊂ Γ denote theL ^∞-spectrum ofG on (X, μ). It has been shown in [5] that Sp(G, X) is a Borel subgroup of Γ and thatσ (Sp(G, X))<1 for every probability measureσ on Γ with lim sup_g→∞Re (g)<1, where is the Fourier transform ofσ. In this note we prove the following converse: ifσ is a probability measure on Γ with lim sup_g→∞Re (g)<1 (g)=1 then there exists a non-singular, properly ergodic action ofG on (X, μ) withσ(Sp(G, X))=1.     

Una nota sui gruppi policiclici che ammettono un automorfismo con numero di Reidemeister finito

LetG be a group and ϕ an automorphism ofG. Two elementsx, y ∈ G are called ϕ-conjugate if there existsg ∈ G such thatx=g ⁻¹ yg ^θ. It is easily verified that the ϕ-conjugation is an equivalence relation; the numberR(ϕ) of ϕ-classes ofG is called the Reidemeister number of the automorphism ϕ. In this paper we prove that if a polycyclic groupsG admits an automorphism ϕ of ordern such thatR(ϕ)<∞, thenG contains a subgroup of finite index with derived length at most 2 ⁿ⁻¹ .

     

On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.     

Isometries of JB-algebras

A bijective linear mapping between two JB-algebrasA andB is an isometry if and only if it commutes with the Jordan triple products ofA andB. Other algebraic characterizations of isometries between JB-algebras are given. Derivations on a JB-algebraA are those bounded linear operators onA with zero numerical range. For JB-algebras of selfadjoint operators we have: IfH andK are left Hilbert spaces of dimension ≥3 over the same fieldF (the real, complex, or quaternion numbers), then every surjective real linear isometryf fromS(H) ontoS(K) is of the formf(x)=UoxoU ⁻¹ forx inS(H), whereτ is a real-linear automorphism ofF andU is a real linear isometry fromH ontoK withU(λh)=τ(λ)U(h) for λ inF andh inH. Supported by Acción Integrada Hispano-Alemana HA 94 066 B     

On conditional edge-connectivity of graphs

1. IntroductionIn this paper, a graph G ~ (V,E) always means a simple graph (without loops andmultiple edges) with the vertex-set V and the edge-set E. We follow [1] for graph-theoreticalterllilnology and notation not defined here.It is well known that when the underlying topology of a computer interconnectionnetwork is modeled by a graph G, the edge-connectivity A(G) of G is an important measurefor fault-tolerance of the network. However, it has many deficiencies (see [2]). MotiVatedby t…     

Path-closed sets

Given a digraphG = (V, E), call a node setT ⊆V path-closed ifv, v′ εT andw εV is on a path fromv tov′ impliesw εT. IfG is the comparability graph of a posetP, the path-closed sets ofG are the convex sets ofP. We characterize the convex hull of (the incidence vectors of) all path-closed sets ofG and its antiblocking polyhedron inR ^v , using lattice polyhedra, and give a minmax theorem on partitioning a given subset ofV into path-closed sets. We then derive good algorithms for the linear programs associated to the convex hull, solving the problem of finding a path-closed set of maximum weight sum, and prove another min-max result closely resembling Dilworth’s theorem.     

Regular and normal closure operators and categorical compactness for groups

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF F, so thatA = {x G |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A F. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, ₂(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.     

Division by a holomorphic matrix in strictly pseudoconvex Domains

LetF andG be respectively a vector- and a matrix-function in a bounded strictly pseudoconvex domainD, with entries holomorphic inD and continuous in . We prove that ifF can be divided locally byG with holomorphic factors in a neighborhood of a given pointw inD, and the rank ofG is maximal at all points of , then the division ofF byG holds globally, with some factors which are holomorphic inD and continuous in . This method applies also to other function algebras in pseudoconvex domains.     

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.