Bollettino di Matematica pura e applicata

Transcription

1 BmPa Bollettino di Matematica pura e applicata II

2 The Dipartimento di Metodi e Modelli Matematici of the Università di Palermo is operative from It was created thanks to the will of the all the professors of the mathematical area of the Facoltà di Ingegneria. Both the scientific and didactic activity of the Dipartimento di Metodi e Modelli Matematici can be found in the web-site From its constitution it has been involved in many collaborations with Italian and foreign researchers. The aim of this Bollettino, with annual periodicity, is that of covering the areas of mathematics existing at the Dipartimento di Metodi e Modelli Matematici. The papers emphasize the advances of knowledge in mathematics problems and new applications. The Bollettino di Matematica pura e applicata comprehend the following sectors: Analysis, Geometry, Mathematical Physics, Probability... and it is also open to the contribution of the other Italian or stranger researchers.

3 Editors Tommaso Brugarino, Maria Stella Mongiovì. Editorial Committee Pietro Aiena (Palermo) S. Twareque Ali (Montreal) Maria Letizia Bertotti (Bolzano) Luis Funar (Grenoble) Renata Grimaldi (Palermo) David Jou (Barcelona) Valentin Poenaru (Paris-Sud) Karl Strambach (Erlangen)

4 Bollettino di Matematica pura e applicata Volume II Editors Tommaso Brugarino Maria Stella Mongiovì

5 Copyright MMIX ARACNE editrice S.r.l. via Raffaele Garofalo, 133/A-B Roma (06) ISBN I diritti di traduzione, di memorizzazione elettronica, di riproduzione e di adattamento anche parziale, con qualsiasi mezzo, sono riservati per tutti i Paesi. Non sono assolutamente consentite le fotocopie senza il permesso scritto dell Editore. I edizione: dicembre 009

6 Indice Some additive (v, 4, λ) designs Andrea Caggegi On the fixed ends of hyperbolic translations of infinite graphs Marco Pavone Pseudo-bosons arising from Riesz bases Fabio Bagarello, Fabio Calabrese Slight extensions of positive linear functionals: two concrete realizations Giorgia Bellomonte On the spectral properties of some classes of operators Pietro Aiena Far From Equilibrium Constitutive Relations in a Nonlinear Model of Superfluid Turbulence Lucia Ardizzone, Giuseppa Gaeta Gabor-type Frames from Generalized Weyl-Heisenberg Groups G. Honnouvo, S. Twareque Ali Polaroid type operators under quasi-affinities Pietro Aiena, Muneo Chō and Manuel González A note on semifinite von Neumann algebras Salvatore Triolo

7 Stability in the plane Couette flow of superfluid helium Maria Stella Mongiovì, Michele Sciacca Flow of turbulent superfluid helium inside a porous medium L. Ardizzone, G. Gaeta, M.S. Mongiovì A short note on O -algebras and quantum dynamics Fabio Bagarello, Francesco Tschinke Precessioni non uniformi di un solido in campo gravitazionale Francesco Rigoli A note on stochastic model of malaria with periodic coefficients Elisabetta Tornatore, Stefania Maria Buccellato (v, 3, r) designs and the equation x+y +z = 0 in finite abelian groups Marco Pavone Weyl type theorems for left and right polaroid operators Pietro Aiena, Elvis Aponte and Edixon Balzan A short survey on isoperimetric problem on noncompact Riemannian surfaces Renata Grimaldi, Stefano Nardulli Exact solutions of the Zakharov equations Tommaso Brugarino, Michele Sciacca

8 Boll. di mat. pura ed appl. Vol. II (009) Some additive (v, 4, λ) designs Andrea Caggegi Dipartimento di Metodi e Modelli Matematici, Università di Palermo Viale delle Scienze Ed. 8, I-9018 Palermo (Italy) caggegi@unipa.it Abstract Given a finite abelian (additive) group G and an integer k with 3 k < G, denote by D k (G) the simple incidence structure whose point-set is G and whose blocks are the k-subsets C = {c 1,..., c k } of G such that c c k = 0. From [] we know that D k (G) is a -design, if G is an elementary abelian p-group with p a divisor of k. It is also known (see [3]) that D 3(G) is a -design if and only if G is an elementary abelian 3-group. Here we shall prove that G is necessarily an elementary abelian -group, if D 4(G) is a -design. AMS MSC 05B05, 05B5 Let v, k, t, λ be positive integers with v > k > t. By a t-design with parameters v, k, λ (or shortly: a t (v, k, λ) design) one understands a pair D = (P, B) where P is a finite set with v elements (called points) and B is a collection of distinguished subsets of P called blocks such that each block contains k points and any t distinct points are contained in exactly λ common blocks (cfr. [1], [4]). We say that a t (v, k, λ) design D = (P, B) is an additive design, if there are a finite abelian (additive) group G and an injective mapping φ : P G with the property that φ(c 1 ) + φ(c ) + + φ(c k ) = 0 whenever C = {c 1,..., c k } B is a block of D = (P, B) (cfr. [3]). Note that each -design of the form D k (G) is automatically an additive -design. Having said this, we prove now the following Theorem. Suppose G is a finite abelian (additive) group of order n > 4 and let D 4 (G) be the simple incidence structure whose point-set is G and whose blocks are the 4-subsets C = {c 1, c, c 3, c 4 } of G such that c 1 + c + c 3 + c 4 = 0. If D 4 (G) is a (n, 4, λ) design, then G is an elementary abelian -group. Proof: Suppose D = (P, B) is a (n, 4, λ) design. Then we must have: 1) if n is an odd integer and if y G is not zero, there are exactly n 3 blocks of D 4 (G) through {y, y}, because every z G \ {0, y, y} lies in just one block of D 4 (G) through {y, y}; 1 Andrea Caggegi: Some additive (v, 4, λ) designs (pp. 1-3)

9 Boll. di mat. pura ed appl. Vol. II (009) ) if n is an odd integer and y G has the property that 3y 0, there are precisely n 5 blocks of D 4 (G)through {0, y}, since any z G \ {0, y, y, y, 1 y} belongs to a unique block of D 4 (G) containing {0, y}; { x 0 y 3) if 3G = 0 and if x, y G are such that, there are exactly x y 0 x + y n 5 blocks of D 4 (G) through {x, y}, because each z G\{x, y, x y, y x, x+y} is contained in just one block of D 4 (G) through {x, y}. Using 1), ), 3) we see that the hypothesis n is an odd integer gives n 3 = λ = n 5, a contradiction. Therefore (n is an even integer and) the elementary abelian -group G = {g G g = 0} has order m. Suppose G G. Then the following statements hold: i) if y G \ G, then there are exactly λ = n (m +) blocks of D 4 (G) through {y, y}, since every z G \ G with y z y lies in a unique block of D 4 (G) through {y, y}; { x + y / G ii) if x, y are elements of G such that, there are precisely λ = x + y / G n 4 blocks of D 4 (G) through {x, y}, because any z G \ {x, y, x y, x y} belongs to just one block of D 4 (G) through {x, y}; { x + y / G iii) if x, y are elements of G such that, there are exactly x + y G n blocks of D 4 (G) through {x, y}, since each z G \ {x, y} is contained in just one block of D 4 (G) through {x, y}. Using i), ii), iii) we obtain j) m = 1 and hence G = {0, ω} for a suitable (unique) ω G such that ω 0 = ω; jj) if g G, then the equation z = g has exactly two solutions (namely z = g and z = g + ω); jjj) λ = n 4 is the number of blocks of D 4 (G) through any two given points (elements of G). Using j) and jj) we deduce that, if x G is such that x 0 3x, then there are exactly n 6 blocks of D 4 (G) through {0, x}, because every z G \ {0, x, x, x} such that z x lies in just one block of D 4 (G) through {0, x}. This and jjj) give n 4 = λ = n 6, a contradiction. Thus we may assume that x = 0 or 3x = 0 whenever x G. It follows from j) that one of the following holds: Andrea Caggegi: Some additive (v, 4, λ) designs (pp. 1-3)

10 Boll. di mat. pura ed appl. Vol. II (009) a) G is an abelian -group in which any element g G \ G has order 4; b) G is an elementary abelian 3-group. Hypothesis a) implies that (G is a cyclic group of order 4 and hence) there is no block of D 4 (G) through {0,{ ω}, a contradiction. Therefore b) holds. Then for any given x 0 y x, y G such that there are exactly x y 0 x + y n 6 blocks of D 4 (G) through {x, y}, since any z G \ {x, y, x y, y x, x + y, x + y + ω} belongs to just one block of D 4 (G) through {x, y}. This and jjj) give n 6 = λ = n 4, again a contradiction, which guarantees that x+y = 0 whenever x, y G \{0}, with x y. So G = 3 and G = G G is an abelian (cyclic) group of order n = 6. Therefore λ = n 4 { = 1 and the number b of blocks of D 4 (G) satisfies the condition ( n ) ( λ = 4 n = 6 ) b, where. Hence we have 15 = 6b, a contradiction. Such a final λ = 1 contradiction shows that G = G is necessary an elementary abelian -group. Remark. Suppose G is an elementary abelian -group of order n = ν > 4. If x 1, x, x 3 are three distinct elements of G, then x 4 = x 1 + x + x 3 / {x 1, x, x 3 }, hence {x 1, x, x 3, x 4 } is a block of D 4 (G). Thus D 4 (G) is (also) a 3 (n, 4, 1) design with exactly b = 4( 1 n ) 3 = n(n 1)(n ) 4 blocks. Moreover, from ( ( n ) λ = 4 ) b it follows that λ = n : hence D 4(G) is a (n, 4, n ) design. Acknowledgments: Supported by M.I.U.R., Università di Palermo. References [1] T. Beth, D. Jungnickel, H. Lenz, Design theory, nd ed., Cambridge University Press (1999). [] A. Caggegi, A. Di Bartolo, G. Falcone, Boolean designs and the embedding of a design in a group, submitted (arxiv v). [3] A. Caggegi, G. Falcone, M. Pavone, Additivity of affine designs and of Steiner triple systems, submitted. [4] C. J. Colbourn, J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton (1996). 3 Andrea Caggegi: Some additive (v, 4, λ) designs (pp. 1-3)

11 Boll. di mat. pura ed appl. Vol. II (009) On the fixed ends of hyperbolic translations of infinite graphs Marco Pavone Dipartimento di Metodi e Modelli Matematici, Università di Palermo Viale delle Scienze Ed. 8, I-9018 Palermo (Italy) pavone@unipa.it Abstract Let X be an infinite, connected, locally finite and vertex-transitive graph with infinitely many ends and let G be a subgroup of Aut(X) which acts transitively on X. In this note we provide a necessary and sufficient condition for the existence of a hyperbolic translation g G with fixed ends in two prescribed open subsets of the space of ends ΩX. We also give an explicit combinatorial construction of the hyperbolic translation g in the special case where X is a (right) Cayley graph of a (non-abelian) free group of finite type G. AMS MSC 010: 05C5, 0F05, 54D35. Contents 1 Introduction 5 Ends and automorphisms 6 3 Hyperbolic translations 7 4 Cayley graphs of finitely generated groups 11 References 1 1 Introduction We assume throughout this paper that X is an infinite graph which is connected, locally finite and vertex-transitive. We exclude multiple edges and oriented edges, 5 Marco Pavone: On the fixed ends of hyperbolic translations of infinite graphs (pp. 5-14)

12 Boll. di mat. pura ed appl. Vol. II (009) while loops are permitted. We use the letter X to denote the graph itself as well as its vertex set, and denote by d the geodesic distance on X. If X has more than one end, then the hyperbolic limit set of X is non-empty and dense in the space of ends ΩX (see [W, Corollary.] and [W3, Theorem and Corollary 3]). In other words, for any non-empty open subset A of ΩX, there exists a hyperbolic translation in Aut(X) with a fixed end in A. The proof in [W] resorts to direct graph-theoretical arguments, whereas the approach in [W3] relies on a natural convergence property of the action of Aut(X) on X ΩX, called contractivity. In [P1] we show that if X has infinitely many ends and G is a closed non-amenable subgroup of Aut(X) which acts transitively on X, then H(G), the hyperbolic limit set of G, is bilaterally dense in ΩX : if A and B are two non-empty disjoint open subsets of ΩX, then there exists a hyperbolic translation g G with one fixed end in A and one fixed end in B [P1, Proposition 8]. Our proof is non-constructive and relies on Woess notion of contractivity, in the topological spirit of [W3]. It is then natural to ask whether the bilateral denseness of H(G) in ΩX can be proved by combinatorial or graph-theoretical methods. Given G, A and B as above, and given an automorphism g G, the purpose of this note is to provide sufficient graph-theoretical conditions on g so that g is a hyperbolic translation with one fixed end in A and one fixed end in B. Such conditions become particularly simple in the case where X is the Cayley graph of a finitely generated group G with infinitely many ends. In the case where G is a free group of finite type, we describe an explicit construction of the hyperbolic translation g. We wish to point out that the dynamics of kleinian groups acting on their boundaries have been much studied in recent years. In fact, the essential features were axiomatized by Gehring and Martin [GM], giving rise to what are called convergence groups. These ideas, which have since been investigated by several authors, carry through to word-hyperbolic groups (cf. [TU] and [FRE]). Ends and automorphisms We briefly recall the definition of the end space of X. See [F], [H1], [H] and [J1] for fundamental facts concerning ends, and [W1] for a more detailed graph-theoretical description close to the present setting. Let F denote the set of all finite subsets of X. If F F, then, by the local finiteness of X, X \F decomposes into finitely many components (i.e. maximal connected subgraphs). A ray (also called infinite path or half line) in X is a sequence {x n } n=0 of successively adjacent vertices without repetitions. Two rays are equivalent if, for every F F, they lie eventually in the same component of X \F. An end of X is an equivalence class of rays [H1], and the set of ends is denoted by ΩX. Roughly speaking, the ends describe how the graph branches, and each end can be thought of as a way of going to infinity. It is known that an infinite connected vertex-transitive graph X has one, two or infinitely many ends [H, Theorem 10]. The number of ends of X is the supremum over all F F of the number of infinite 6 Marco Pavone: On the fixed ends of hyperbolic translations of infinite graphs (pp. 5-14)

13 Boll. di mat. pura ed appl. Vol. II (009) components of X\F (see e.g. [C, Proposition.14]). One may put the discrete topology on X and think of the ends as a boundary at infinity of X. Then ΩX compactifies X according to the following definition of end topology, which can be traced back to papers of Hopf and Freudenthal in the thirties and fourties (see e.g. [F]). Let X = X ΩX and let F F. If C is (the set of vertices of) a component of X\F, then we denote by C the subset of X obtained by adding to C all the ends which have a representative ray with all vertices in C. Varying F in F and C among the components of X\F, the family of all sets C forms the basis of a topology which makes X a totally disconnected compact Hausdorff space, called the end compactification of X (see e.g. [J1]). One may actually suppose that F varies only among the finite connected subgraphs of X. It follows from the definition above that X is open dense in X. If C is a component of X \F for some F in F, then the set C defined above is open and closed, and is precisely the closure of C in X. Finally, if {x n } n=0 is a representative ray of an end ξ Ω, then n ξ in X. x n An automorphism of X is a bijection of (the vertex set of) X onto itself which preserves the adjacency relation (or, equivalently, a self-isometry of (X, d)). By Aut(X) we denote the group of all automorphisms of X. Any g Aut(X) extends to a homeomorphism of X in an obvious way. In fact, X is a contractive Aut(X)-compactification in the sense of Woess [W3]. Finally, we endow Aut(X) with the topology of pointwise convergence. 3 Hyperbolic translations We say that g Aut(X) is a translation if there exists no non-empty finite subset of X which is fixed by g. A translation g is called hyperbolic (or proper) if g fixes precisely two ends of X. The set H(X) of all the fixed ends of the hyperbolic translations in Aut(X) is called hyperbolic limit set of X (in [M1] the set H(X) is called set of directions with finite thickness and is denoted by FEX). Halin [H, Theorem 9] showed that if g is a hyperbolic translation of X, then there exist two distinct ends g +, g in ΩX which are fixed by g and are such that g n x n g + and g n x n g for all x in X. (1) We say that g Aut(X) is a shift if there exists F in F and a (necessarily infinite) component C of X \F such that g(f C) C. () It is known that h Aut(X) is a hyperbolic translation if and only if h has a power which is a shift (this is implicit in [H, Theorem 9]. See also [J3, Theorem.7]). Moreover, if a shift g satisfies () above, then g + C and g X \C. (3) 7 Marco Pavone: On the fixed ends of hyperbolic translations of infinite graphs (pp. 5-14)

14 Boll. di mat. pura ed appl. Vol. II (009) The question of the existence of hyperbolic translations in Aut(X) is settled by the following result, which is essentially due to Jung [J, Theorem 1] (see also [SOW, Lemma 5] for a short proof. For a slightly different rephrasement, see [M1, Theorem ] and [M, Theorem 1] ). Lemma 1. If F is a finite subset of X, and if C is an infinite component of X \F with infinite complement, then there exists a shift g Aut(X) such that g(f C) C. Suppose that X has more than one end. If A is a non-empty open subset of ΩX, then, by definition of the end topology, there exist F F and an infinite component C of X \F with infinite complement such that C ΩX A. (4) In view of (3) and (4) above, by Lemma 1 there exists a hyperbolic translation g such that g + A. One may then state the following result (see [W, Corollary.] and [W3, Theorem ]). Theorem. If X has more than one end, then the hyperbolic limit set H(X) is non-empty and dense in ΩX. Let X have more than one end, and let A and B be two non-empty disjoint open subsets of ΩX. By Theorem there exist two hyperbolic translations g, h such that g + A and h B. One may ask whether there exists just one hyperbolic translation g such that g + A and g B. By Lemma 1 and (3) above, this is obvious if X has two ends. Hence we will only consider the case where X has infinitely many ends. By definition of the end topology, the question is whether, given a finite connected subgraph F of X and two distinct infinite components C 1, C of X \F, there exists a hyperbolic translation g such that g + C 1 and g C. As shown by the following result, the answer is affirmative if and only if Aut(X) is non-amenable (as in the case where X is a homogeneous tree). Note that there exist examples of infinite graphs X with infinitely many ends which are connected, locally finite and vertex-transitive, and are such that Aut(X) is amenable. See e.g. [SOW, 4, Example ]. Theorem 3. Suppose that X has infinitely many ends. Let G be a closed subgroup of Aut(X) which acts transitively on X. Then the following two conditions are equivalent: (I) The set of all pairs (g +, g ) of fixed ends of the hyperbolic translations g G is dense in ΩX ΩX. (II) G is non-amenable. Proof. If G is non-amenable, then condition (I) is satisfied by [P1, Proposition 8]. Conversely, suppose that condition (I) holds. Since ΩX is an infinite Hausdorff space, G contains two hyperbolic translations with no common fixed ends. Hence there exists no end of X which is fixed by G. Therefore G is non-amenable by [SOW, Proposition ]. 8 Marco Pavone: On the fixed ends of hyperbolic translations of infinite graphs (pp. 5-14)