The strong closure of $\sigma$-complete Boolean algebras of projections

A result of T. A. Gillespie implies that the strong operator closure of any abstractly s\sigma -complete Boolean algebra of projections in a Banach space X which does not contain a copy of c₀ is Bade complete. It is shown that the same conclusion is valid for another (extensive) class of Banach spaces X, namely those which are weakly compactly generated. As a consequence, it follows that a Boolean algebra of projections in a separable Banach space is abstractly s\sigma -complete iff it is abstractly complete. It is also shown that a Banach space X has the property that the strong closure of every abstractly complete Boolean algebra of projections in X is Bade complete iff X does not contain a copy of l^￥\ell ^\infty \!.     

Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis

Let X be a rearrangement invariant function space on [0,1]. We consider the subspace Radi X of X which consists of all functions of the form , where xk are arbitrary independent functions from X and rk are usual Rademacher functions independent of {xk}. We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X^′ possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [0,∞). This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)].     

Uniform convergence and sequence of maps on a compact metric space with some chaotic properties

Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f : X → X of a sequence of continuous topologically transitive (in strongly successive way) functions fn : X → X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney’s definition is lost on the limit function.     

On isomorphic factorizations of circulant graphs

We investigate the conjecture that every circulant graph X admits a k‐isofactorization for every k dividing |E(X)|. We obtain partial results with an emphasis on small values of k. © 2006 Wiley Periodicals, Inc. J Combin Designs 14: 406–414, 2006     

A Note on Boolean Algebras with Few Partitions Modulo some Filter

We show that for every uncountable regular κ and every κ-complete Boolean algebra B of density ≤ κ there is a filter F ? B such that the number of partitions of length < modulo κF is ≤2^<κ. We apply this to Boolean algebras of the form P(X)/I, where I is a κ-complete κ-dense ideal on X. Mathematics Subject Classification: 06E05, 03C20.     

On the Schwartz space of the basic affine space

Let G be the group of points of a split reductive algebraic group G over a local field k and let X = G / U where U is the group of k-points of a maximal unipotent subgroup of G. In this paper we construct a certain canonical G-invariant space (called the Schwartz space of X) of functions on X, which is an extension of the space of smooth compactly supported functions on X. We show that the space of all elements of , which are invariant under the Iwahori subgroup I of G, coincides with the space generated by the elements of the so called periodic Lusztig basis, introduced recently by G. Lusztig (cf. [10] and [11]). We also give an interpretation of this space in terms of a certain equivariant K-group (this was also done by G. Lusztig — cf. [12]). Finally we present a global analogue of , which allows us to give a somewhat non-traditional treatment of the theory of the principal Eisenstein series.     

Multi-secant lemma

We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations [8, 10, 1, 2, 4, 7]. Let X be an equidimensional projective variety of dimension d. For a given k ≤ d + 1, we are interested in the study of the variety of k-secants. The classical trisecant lemma just considers the case where k = 3 while in [10] the case k = d + 2 is considered. Secants of order from 4 to d + 1 provide service for our main result. In this paper, we prove that if the variety of k-secants (k ≤ d +1) satisfies the following three conditions: (i) through every point in X, there passes at least one k-secant, (ii) the variety of k-secants satisfies a strong connectivity property that we define in the sequel, (iii) every k-secant is also a (k +1)-secant; then the variety X can be embedded into ℙ ^d+1. The new assumption, introduced here, that we call strong connectivity, is essential because a naive generalization that does not incorporate this assumption fails, as we show in an example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption.     

Rigidity for pseudo pretheories

Assume F is a homotopy invariant pseudo pretheory with torsion values and X is a smooth scheme of finite type over a field k. We show for certain field extensions k⊆K the map F(X)→F(X _K ) is an isomorphism. Mathematics Subject Classification (2000) 14A99     

Principal bundles whose restriction to curves are trivial

Let k be an algebraically closed field and X a smooth projective variety defined over k. Let E_G be a principal G–bundle over X, where G is an algebraic group defined over k, with the property that for every smooth curve C in X the restriction of E_G to C is the trivial G–bundle. We prove that the principal G–bundle E_G over X is trivial. We also give examples of nontrivial principal bundle over a quasi-projective variety Y whose restriction to every smooth curve in Y is trivial.     

Supplementary Note on 'Rational Invariants of Certain Orthogonal and Unitary Groups'

Given a field k and a finite group G acting on the rationalfunction field k(X₁, ..., X_n) as a group of k-automorphisms,an important Noether's problem asks whether the invariant subfield [forumal] is purely transcendental over k. 1991 Mathematics Subject Classification12F20, 20G40.     

Pullbacks of Arithmetical Rings

We give necessary and sufficient conditions that the pullback of a conductor square be a chain ring (i.e., a ring whose ideals are totally ordered by inclusion). We also give necessary and sufficient conditions that the pullback of a conductor square be an arithmetical ring (i.e., a ring which is locally a chain ring at every maximal ideal). For any integral domain D with field of fractions K, we characterize all Prüfer domains R between D[X] and K[X] such that the conductor C of K[X] into R is nonzero. As an application, we show that for n ≥ 2, such a ring R has the n-generator property (every finitely generated ideal can be generated by n elements) if and only if R/C has the same property.     

Min-Wise Independent Families with Respect to any Linear Order

A set of permutations 𝒮 on a finite linearly ordered set Ω is said to be k-min-wise independent, k-MWI for short, if Pr (min (π(X)) = π(x)) = 1/|X| for every X ? Ω such that |X| ≤ k and for every x ∈ X. (Here π(x) and π(X) denote the image of the element x or subset X of Ω under the permutation π, and Pr refers to a probability distribution on 𝒮, which we take to be the uniform distribution.) We are concerned with sets of permutations which are k-MWI families for any linear order. Indeed, we characterize such families in a way that does not involve the underlying order. As an application of this result, and using the Classification of Finite Simple Groups, we deduce a complete classification of the k-MWI families that are groups, for k ≥ 3.     

Ergodicity of canonical Gibbs measures with respect to the diffeomorphism group

For general potentials we prove that every canonical Gibbs measure on configurations over a manifold X is quasi‐invariant w.r.t. the group of diffeomorphisms on X. We show that this quasi‐invariance property also characterizes the class of canonical Gibbs measures. From this we conclude that the extremal canonical Gibbs measures are just the ergodic ones w.r.t. the diffeomorphism group. Thus we provide a whole class of different irreducible representations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Note on independent sets in steiner systems

A partial Steiner (n, k, l)-system or briefly (n, k, l)-system is a pair (V, S), where V is an n-set and S is a collection of k-subsets of V, such that every l-subset of V is contained in at most one k-subset of S. A subset X ? V is called independent if [X]^k ∩ S = 0. The size of the largest independent set in S is denoted by α(S). Define The purpose of this note is to prove that for every k, l, k > l holds, where c, d are positive constants depending on k and l only.     

Weakly almost periodic functionals on certain Banach algebras

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common fixed point property.     

Toughness and the existence of k-factors

A connected graph G is called t-tough if t · w(G - S) ? |S| for any subset S of V(G) with w(G - S) > 1, where w(G - S) is the number of connected components of G - S. We prove that every k-tough graph has a k-factor if k|G| is even and |G| ? k + 1. This result, first conjectured by Chvátal, is sharp in the following sense: For any positive integer k and for any positive real number ε, there exists a (k - ε)-tough graph G with k|G| even and |G| ? k + 1 which has no k-factor.     

Generalized Derivations with Invertible or Nilpotent Values on Multilinear Polynomials

Let R be a ring with unity, g a generalized derivation on R and f(X ₁,…,X _k ) a multilinear polynomial. In this article we describe the structure of R provided that g(f(x ₁,…,x _k )) is either invertible or nilpotent for every x ₁,…,x _k in some nonzero ideal of R.     

Generation of k-jets on toric varieties

The notion of a k-convex -support function for a toric variety is introduced. A criterion for a line bundle L to generate k-jets on X is given in terms of the k-convexity of the -support function . Equivalently L is proved to be k-jet ample if and only if the restriction to each invariant curve has degree at least k. Received October 22, 1997; in final form January 12, 1998     

Diameter and diametrical pairs of points in ultrametric spaces

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function τ: F(X) → ℝ there is an ultrametric on X such that τ(A) = diamA for every A ∈ F(X). For finite nondegenerate ultrametric spaces (X, d) it is shown that X together with the subset of diametrical pairs of points of X forms a complete k-partite graph, k ⩾ 2, and, conversely, every finite complete k-partite graph with k ⩾ 2 can be obtained by this way. We use this result to characterize the finite ultrametric spaces (X, d) having the minimal card{(x, y): d(x, y) = diamX, x, y ∈ X} for given card X.