Perron Conditions for Uniform Exponential Expansiveness of Linear Skew-Product Flows

 We present necessary and sufficient conditions for uniform exponential expansiveness of discrete skew-product flows, in terms of uniform complete admissibility of the pair (c ₀(N, X), c ₀(N, X)). We give discrete and continuous characterizations for uniform exponential expansiveness of linear skew-product flows, using the uniform complete admissibility of the pairs (c ₀(N, X), c ₀(N, X)) and (C ₀(R ₊, X), C ₀(R ₊, X)), respectively. We generalize an expansiveness theorem due to Van Minh, R?biger and Schnaubelt, for the case of linear skew-product flows. Received August 10, 2001; in revised form June 25, 2002     

Limit theorems for bivariate Appell polynomials. Part II: Non-central limit theorems

Summary. Let (X _t ,t∈Z) be a linear sequence with non-Gaussian innovations and a spectral density which varies regularly at low frequencies. This includes situations, known as strong (or long-range) dependence, where the spectral density diverges at the origin. We study quadratic forms of bivariate Appell polynomials of the sequence (X _t ) and provide general conditions for these quadratic forms, adequately normalized, to converge to a non-Gaussian distribution. We consider, in particular, circumstances where strong and weak dependence interact. The limit is expressed in terms of multiple Wiener-It? integrals involving correlated Gaussian measures. Received: 22 August 1996 / In revised form: 30 August 1997     

On Existence and Weak Stability of Matrix Refinable Functions

We consider the existence of distributional (or L ₂ ) solutions of the matrix refinement equation where P is an r×r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P (0) has an eigenvalue of the form 2 ⁿ , . A characterization of the existence of L ₂ -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L ₂ -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. August 1, 1996. Date revised: July 28, 1997. Date accepted: August 12, 1997.     

The ranks of central unit groups of integral group rings of alternating groups

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r _n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A _n . We shall find all values n for r _n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r _n for n ≤ 600.     

On the existence of closed timelike geodesics in compact spacetimes. II

 The purpose of this paper is on the one hand to extend and generalize, in terms of Clifford translations, some results in a previous paper (Math. Z. 239 (2002), 277–291) concerning the existence of closed timelike geodesics in compact spacetimes, and on the other hand to prove that a compact flat spacetime (M, g) contains a closed timelike geodesic if and only if the fundamental group π₁(M) contains a non-trivial timelike translation. Received: 22 January 2002; in final form: 12 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 53C50, 53C22.     

Cobordism of immersions of surfaces¶in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. In particular, for a connected, non-orientable 3-manifold M, the group of cobordism clases of immersions of surfaces in M is isomorphic to a group structure on the set H ₂(M,Z/2Z)×H ₁(M,Z/2Z)×Z/2Z. Received: 8 June 2000 / Revised version: 2 October 2000     

On quantum mechanics and generalized Clifford algebras

In this note we give elementary examples of the naturalness of generalized Clifford algebras appearance, in some particular quantum mechanical models. First Weyl’s program [1] for quantum kinematics for the case of simplest Galois fieldsZ _n is realized in terms of generalized Clifford algebras. Dynamics might then be introduced, following the ideas of Hanney and Berry [2], as shown in [3]. Second the coherent state picture of the finite dimensional “Z _n — Quantum Mechanics” is presented. In the last part the known coherent states ofq-deformed quantum oscillators (q≡ω) are explicitly shown in the generalized Grassman algebras and the generalized Clifford algebras settings. Presented atThe Polish-Mexican Seminar, Kazimierz Dolny, August 1998 — Poland. 176     

Syzygies and Diagonal Resolutions for Dihedral Groups

Let G be a finite group with integral group ring Λ =Z[G]. The syzygies Ω_r(Z) are the stable classes of the intermediate modules in a free Λ-resolution of the trivial module. They are of significance in the cohomology theory of G via the “co-represention theorem” H^r(G, N) = Hom_𝒟er(Ω_r(Z), N). We describe the Ω_r(Z) explicitly for the dihedral groups D_4n+2, so allowing the construction of free resolutions whose differentials are diagonal matrices over Λ.     

Additive patterns in a multiplicative group in a finite field

 Let F _q be a field with q elements, let d>1 be a divisor of q−1 and U _d be the subgroup of F _q ^× of index d. Under some growth conditions, we show that the distribution of s-tuples of elements of U _d which follow a given additive pattern approaches a Poissonian distribution. Received: 28 August 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 11T99     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

MECHANISMS OF THE ICOSAHEDRAL COMPOUND OF TEN TETRAHEDRA

Ten regular tetrahedra can be arranged in such a way that their vertices are coincident with the vertices of a regular dodecahedron and that two tetrahedra meet at each vertex of the dodecahedron. If the resultant structure is considered as a bar-and-joint structure, there will be 60 bars, lying along the edges of the tetrahedra, and 20 joints at the vertices of the dodecahedra; six bars meet at each joint. Although the structure more than satisfies Maxwell's rule, it is known to admit finite mechanisms. Recently, a new method for detecting symmetric finite mechanisms in symmetric bar-and-node structures has been developed. The method only requires a count of the number of bars, and the number of nodes, that are left unmoved by each of the symmetry operations allowable for the structure. This paper will describe the application of this method to the structure described above. The structure has icosahedral symmetry, I _h , and the analysis confirms the existence of the mechanisms with C _3v and C _5v symmetry that have previously been detected using ad-hoc methods. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the Steinberg module of Chevalley groups

For a simple Chevalley group G an explicit version of the Solomon-Tits theorem is proved by describing the generators of the kernel of the map Z[G(K)]S_K, where K is any field and where S_K is the Steinberg module of the group G(K). As a corollary it is shown that if is a Euclidean domain whose fraction field is K, then S_K is cyclic as a G() module when G is either a classical group or an exceptional group of type E₆, or E₇.Acknowledgement I would like to thank Matt Emerton, Özlem Imamoglu, Paul Gunnells for several helpful comments.     

CirculantGH(p 2; Z p ) exist for all primesp

The only known circulant ordinary Hadamard matrix is developed from the initial row-1, 1, 1, 1. Letp be a prime, and letZ _p denote the cyclic group of orderp. In this paper, we construct circulantGH(p ²;Z _p ) for all primesp. Whenp is odd, this result also extends the earlier result that there exist circulantGH(p;Z _p ) for all odd primesp. Other families ofGH-matrices which are developed modulo a group are discussed.     

The Central Limit Theorem on SO0(p, q)/SO(p)×SO(q)

We obtain a central limit theorem for the space SO ₀(p, q)/SO(p)×SO(q). To achieve this, we derive a Taylor expansion of the spherical function on the group SO ₀(p, q).     

Group Chromatic Number of Graphs without K5-Minors

 Let G be a graph with a fixed orientation and let A be a group. Let F(G,A) denote the set of all functions f: E(G) ↦A. The graph G is A -colorable if for any function f∈F(G,A), there is a function c: V(G) ↦A such that for every directed e=u v∈E(G), c(u)−c(v)≠f(e). The group chromatic numberχ₁(G) of a graph G is the minimum m such that G is A-colorable for any group A of order at least m under a given orientation D. In [J. Combin. Theory Ser. B, 56 (1992), 165–182], Jaeger et al. proved that if G is a simple planar graph, then χ₁(G)≤6. We prove in this paper that if G is a simple graph without a K ₅-minor, then χ₁(G)≤5. Received: August 18, 1999 Final version received: December 12, 2000     

On the Grigorchuk–Kurchanov conjecture

 Let denote the free group of rank 2g. An automorphism φ? Aut(F _{2 g} ) is generating if N _a φ (N _b ) = F _{2 g} , where N _a is the normal closure of and N _b is defined analogously. We present a characterization of generating automorphisms in Aut(F ₂) and observe that there exists a unique (up to equivalence) epimorphism F ₂→Z×Z: this is a particular case of the Grigorchuk–Kurchanov conjecture. This leads to further investigations for splitting homomorphisms for the pairs (F _{2 g} , F _g) and (G _g, F _g) where G _g denotes the fundamental group of a closed orientable surface of genus g and a reformulation of the Poincaré and Grigorchuk–Kurchanov conjectures is derived. Received: 1 October 2001     

Fitting ideals and the Gorenstein property

Let p be a prime number and G be a finite commutative group such that p ² does not divide the order of G. In this note we prove that for every finite module M over the group ring Z _p [G], the inequality #M  ￡  #Z_p[G]/Fit_Zp[G](M){\#M\,\leq\,\#{\bf Z}_{p}[G]/{{\rm Fit}}_{{\bf Z}_{p}[G]}(M)} holds. Here, Fit_Zp[G](M){\rm Fit}_{{\bf Z}_{p}[G]}(M) is the Z _p [G]-Fitting ideal of M.     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.