Finite Generation of Units in Alternative Loop Rings

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Suppose that L is finite and that any noncommutative division algebra appearing as a simple component in the Wedderburn decomposition of Q L is the classical Cayley–Dickson algebra over Q. Then the unit loop of the alternative loop ring Z L of L over the ring of rational integers is finitely generated.     

Jordan decompositions in alternative loop rings

It is observed that the additive as well as multiplicative Jordan decompositions hold in alternative loop algebras of finiteRA loops and theRA loops for which the additive Jordan decomposition holds in the integral loop ring are characterized. Multiplicative Jordan decomposition (MJD) inZL, whereL is a finiteRA loop with cyclic centre is analysed, besides settling MJD for integral loop rings of allRA loops of order ≤32. It is also shown that for any finiteRA loopL,U (ZL) is an almost splittable Moufang loop. Research of the second author is supported by CSIR.     

Families of finite sets with three intersections

LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n ²), and show that in some casesm(n, k, L)=O(n ^3/2), which is quite surprising.     

Galois modular representation of associated Jacobians in the tamely ramified cyclic case

Let L/K be an ℓ-cyclic extension with Galois group G of algebraic function fields over an algebraically closed field k of characteristic p ≠  ℓ. In this paper, the -module structure of the ℓ-torsion of the Jacobian associated to L is explicitly determined.     

On nonatomic submeasures on {\mathbb{N}}

A submeasure μ defined on the subsets of is nonatomic if for every ℓ ≥ 1 there exists a partition of into a finite number of parts on which μ is bounded from above by 1/ℓ. In this paper we answer several natural questions concerning nonatomic submeasures d _F that are determined (like the standard density) by a family F of finite subsets of . We first show that if the number of n-element sets in F grows at most exponentially with n, then d _F is nonatomic; but if this growth condition fails, then d _F need not be nonatomic in general. We next prove that, for a nonatomic submeasure d _F , the minimal number of sets in a 1/ℓ-small partition of can grow arbitrarily fast with ℓ. We also give a simple example of a nonatomic submeasure that is not equivalent to a submeasure of type d _F . The second author acknowledges a generous support of the Foundation for Polish Science.     

Denting and strongly extreme points in the unit ball of spaces of operators

For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(l^p). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L ¹ (μ), X) has denting points iffL ¹(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit ball of ℓ(X, Y) are strongly extreme points.     

Positive Semigroups of Kernel Operators

Extending results of Davies and of Keicher on ℓ^p we show that the peripheral point spectrum of the generator of a positive bounded C₀-semigroup of kernel operators on L^p is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators. Dedicated to the memory of H.H. Schaefer     

Dimension in algebraic frames

In an algebraic frame L the dimension, dim(L), is defined, as in classical ideal theory, to be the maximum of the lengths n of chains of primes p ₀ < p ₁ < ... < p _n , if such a maximum exists, and ∞ otherwise. A notion of “dominance” is then defined among the compact elements of L, which affords one a primefree way to compute dimension. Various subordinate dimensions are considered on a number of frame quotients of L, including the frames dL and zL of d-elements and z-elements, respectively. The more concrete illustrations regarding the frame convex ℓ-subgroups of a lattice-ordered group and its various natural frame quotients occupy the second half of this exposition. For example, it is shown that if A is a commutative semiprime f-ring with finite ℓ-dimension then A must be hyperarchimedean. The d-dimension of an ℓ-group is invariant under formation of direct products, whereas ℓ-dimension is not. r-dimension of a commutative semiprime f-ring is either 0 or infinite, but this fails if nilpotent elements are present. sp-dimension coincides with classical Krull dimension in commutative semiprime f-rings with bounded inversion.     

Classification of gradient space of dimension 8 by a reducible <Emphasis Type="Italic">s</Emphasis>ℓ(2, C) action

This paper deals with a reducible sℓ(2,C) action on the formal power series ring. The purpose of this paper is to confirm a special case of the Yau conjecture: Suppose that sℓ(2,C) acts on the formal power series ring via (1.1). Then I(f) = (ℓ _i1) ⊕ (ℓ _i2) ⊕... ⊕ (ℓ _is ) modulo some one dimensional sℓ(2,C) representations where (ℓ _i ) is an irreducible sℓ(2,C) representation of ℓ _i dimension and {ℓ _i1 ℓ _i2,...,ℓ _is } ⊆ {ℓ ₁ , ℓ ₂...,ℓ _r }. Unlike classical invariant theory which deals only with irreducible action and 1-dimensional representations, we treat the reducible action and higher dimensional representations successively.     

Hypergraphs in which all disjoint pairs have distinct unions

Let ℓ be a set-system ofr-element subsets on ann-element set,r≧3. It is proved that if |ℓ|>3.5 then ℓ contains four distinct membersA, B, C, D such thatA∪B=C∪D andA∩B=C∩D=0.     

The Radon-Nikodym Property for Tensor Products of Banach Lattices II

Let φ be an Orlicz function that has a complementary function φ* and let ℓ_φ be an Orlicz sequence space. We prove two results in this paper. Result 1: , the Fremlin projective tensor product of ℓ_φ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓ_φ and X have the Radon-Nikodym property. Result 2: , the Wittstock injective tensor product of ℓ_φ with a Banach lattice X, has the Radon-Nikodym property if and only if both ℓ_φ and X have the Radon-Nikodym property and each positive continuous linear operator from h_φ* to X is compact. We dedicate this paper to the memory of H. H. Schaefer The first-named author gratefully acknowledges support from the Faculty Research Program of the University of Mississippi in summer 2004.     

Weak* Hypercyclicity and Supercyclicity of Shifts on ℓ∞

We study hypercyclicity and supercyclicity of weighted shifts on ℓ^∞, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c₀ or ℓ^p (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ^∞ is norm nowhere dense. Finally, we show that ℓ^∞ supports an isometry that is weak * sequentially supercyclic.     

Operator-semistable,operator semi-selfdecomposable probability measures and related nested classes on <Emphasis Type="Italic">p</Emphasis>-adic vector spaces

Let V be a finite dimensional p-adic vector space and let τ be an operator in GL(V). A probability measure μ on V is called τ-decomposable or if μ = τ(μ)* ρ for some probability measure ρ on V. Moreover, when τ is contracting, if ρ is infinitely divisible, so is μ, and if ρ is embeddable, so is μ. These two subclasses of are denoted by L ₀(τ) and L ₀ ^#(τ) respectively. When μ is infinitely divisible τ-decomposable for a contracting τ and has no idempotent factors, then it is τ-semi-selfdecomposable or operator semi-selfdecomposable. In this paper, sequences of decreasing subclasses of the above mentioned three classes, , are introduced and several properties and characterizations are studied. The results obtained here are p-adic vector space versions of those given for probability measures on Euclidean spaces.     

Large derivatives,backward contraction and invariant densities for interval maps

In this paper, we study the dynamics of a smooth multimodal interval map f with non-flat critical points and all periodic points hyperbolic repelling. Assuming that |Dfⁿ(f(c))|→∞ as n→∞ holds for all critical points c, we show that f satisfies the so-called backward contracting property with an arbitrarily large constant, and that f has an invariant probability μ which is absolutely continuous with respect to Lebesgue measure and the density of μ belongs to L^p for all p<ℓ_max/(ℓ_max-1), where ℓ_max denotes the maximal critical order of f. In the appendix, we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction.     

Intersection theorems with geometric consequences

In this paper we prove that ifℱ is a family ofk-subsets of ann-set, μ₀, μ₁, ..., μ_s are distinct residues modp (p is a prime) such thatk ≡ μ₀ (modp) and forF ≠ F′ ≠ℱ we have |F ∩F′| ≡ μ_i (modp) for somei, 1 ≦i≦s, then |ℱ|≦( _s ⁿ ). As a consequence we show that ifR ⁿ is covered bym sets withm<(1+o(1)) (1.2) ⁿ then there is one set within which all the distances are realised. It is left open whether the same conclusion holds for compositep.     

Restricted summability of Fourier transforms and local Hardy spaces

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms. Under some conditions on θ, it is proved that the maximal operator of the θ-means defined in a cone is bounded from the amalgam Hardy space W（hp, e∞） to W（Lp,e∞）. This implies the almost everywhere convergence of the θ-means in a cone for all f ∈ W（L1, e∞） velong to L1.     

Semidirect Product of Loops and Fibrations

Starting from two loops (H, +) and (K, ·), a new loop L can be defined by means of a suitable map Θ : K → Sym H (cf. [3]). Such a loop is called semidirect product of H and K with respect to Θ and denoted by H ×_Θ K =: L. Here we consider the class of those semidirect products in which Θ : K → Aut(H, +) is a homomorphism, this situation being quite akin to the group case. Some relevant algebraic properties of the loop L (Bol condition, Moufang etc.) can be inherited from H and K. In the case that K is a group we investigate the possibility of describing L by a partition (or fibration). In this way we propose a generalization of [8] for the non associative case. Received: September 20, 2007. Revised: November 8, 2007.     

Relations entre isopérimétrie et trou spectral pour les chaînes de Markov finies

Let G be a finite and connected graph, we will note by l(G) the maximum length of an injective path in G. We will show (by two dictinct proofs, one using sub-trees of G and the other based on multiflows of paths) that sup _{(P,μ)∈?(G)} I(P, μ)/λ(P, μ) = l(G), where the supremum is taken over all Markovian kernels P reversible with respect to a probability μ and whose allowed transitions are given by the edges of G, and where I(P, μ) (respectively λ(P, μ)) is the isoperimetric constant (resp. the spectral gap) associated to the couple (P, μ). Then we will study more precisely the same supremum, but where the probability μ is also fixed. These results give optimal minorations of the spectral gap which are linear with respect to the isoperimetric constant (and not quadratic, as in the Cheeger inequality), and we will give several examples on infinite graphs. Re?u: 12 ao?t 1997 / Version révisée: 9 novembre 1998     

Lattice-embedding scales ofL p spaces into orlicz spaces

We study the setP _X of scalarsp such thatL ^p is lattice-isomorphically embedded into a given rearrangement invariant (r.i.) function spaceX[0, 1]. Given 0<α≤β<∞, we construct a family of Orlicz function spacesX=L ^F [0, 1], with Boyd indicesα andβ, whose associated setsP _X are the closed intervals [γ, β], for everyγ withα≤γ≤β. In particular forα>2, this proves the existence of separable 2-convex r.i. function spaces on [0,1] containing isomorphically scales ofL ^p -spaces for different values ofp. We also show that, in general, the associated setP _X is not closed. Similar questions in the setting of Banach spaces with uncountable symmetric basis are also considered. Thus, we construct a family of Orlicz spaces ℓ ^F (I), with symmetric basis and indices fixed in advance, containing ℓ ^p (Γ-subspaces for differentp’s and uncountable Λ⊂I. In contrast with the behavior in the countable case (Lindenstrauss and Tzafriri [L-T₁]), we show that the set of scalarsp for which ℓ ^p (Γ) is isomorphic to a subspace of a given Orlicz space ℓ ^F (I) is not in general closed. Supported in part by DGICYT grant PB 94-0243.     

Convex Solid Subsets of <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>(<Emphasis Type="Italic">X</Emphasis>, μ)

If A⊂ L₀(X, μ) is a convex solid subset of L₀(X, μ), then there exist disjoint X₀ and X₁ with X = X₀∪ X₁ such that A_{| X_0} is dense in L₀(X₀, μ) and A_{|X_1} is bounded in measure in L₀(X₁, μ).