Further Results on the Existence of Splitting BIBDs and Application to Authentication Codes

Let (v,u×c,λ)-splitting BIBD denote a (v,u×c,λ)-splitting balanced incomplete block design of order v with block size u×c and index λ. Necessary conditions for the existence of a (v,u×c,λ)-splitting BIBD are v≥uc, λ(v−1)≡0 (mod c(u−1)) and λ v(v−1)≡0 (mod (c ² u(u−1))). We show in this paper that the necessary conditions for the existence of a (v,3×3,λ)-splitting BIBD are also sufficient with possible exceptions when (1) (v,λ)∈{(55,1),(39,9k):k=1,2,…}, (2) λ≡0 (mod 54) and v≡0 (mod 2). We also show that there exists a (v,3×4,1)-splitting BIBD when v≡1 (mod 96). As its application, we obtain a new infinite class of optimal 4-splitting authentication codes.     

The spectrum for large sets of pure directed triple systems

An LPDTS(ν) is a collection of 3(ν-2) disjoint pure directed triple systems on the same set ofνelements. It is showed in Tian's doctoral thesis that there exists an LPDTS(ν) forν=0,4 (mod 6),ν≥4. In this paper, we establish the existence of an LPDTS(ν) forν= 1,3 (mod 6),ν> 3. Thus the spectrum for LPDTS(ν) is completely determined to be the set {ν:ν= 0, 1 (mod 3),ν≥4}.     

On the vanishing viscosity limit in a disk

We say that a solution of the Navier–Stokes equations converges in the vanishing viscosity limit to a solution of the Euler equations if their velocities converge in the energy (L ²) norm uniformly in time as the viscosity ν vanishes. We show that a necessary and sufficient condition for the vanishing viscosity limit to hold in a disk is that the space–time energy density of the solution to the Navier–Stokes equations in a boundary layer of width proportional to ν vanish with ν, and that one need only consider spatial variations whose frequencies in the radial or tangential direction lie in a band centered around 1/ν. The author was supported in part by NSF grant DMS-0705586 during the period of this work.     

UNA generalizzazione del problema di burnside relativo all'esponente 4

An automorphism σ of order (a divisor of)n of the groupG is calledn-splitting if for everyg∈G. In this paper we prove that a 2-group admitting a 4-splitting automorphism, is locally finite.     

Tricyclic Steiner Triple Systems

A Steiner triple system of order ν, denoted STS(ν), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this paper we give necessary and sufficient conditions for the existence of a tricyclic STS(ν) for several cases. We also pose conjectures concerning their existence in two remaining cases.     

Automorfismi spezzanti di ordine primo

LetG be a group andα εAut(G);α is calledn-splitting if gg^α...g^{α n-1}=1 ∀g εG. In this note we study the structure of finite groups admitting an-splitting automorphism of orderp (p an odd prime number).

     

Mixed weak continuity on generalized topological spaces

We introduce the notion of mixed weak (μ,ν₁ν₂)-continuity between a generalized topology μ and two generalized topologies ν₁, ν₂. We characterize such continuity in terms of mixed generalized open sets: (ν₁,ν₂)′-semiopen sets, (ν₁,ν₂)′-preopen sets, (ν₁,ν₂)-preopen sets [2], (ν₁,ν₂)′-β′-open sets and θ(ν₁,ν₂)-open sets [3]. In particular, we show that for a given mixed weakly (μ,ν₁ν₂)-continuous function, if the codomain of the given function is mixed regular (=(ν₁,ν₂)-regular), then the function is also (μ,ν₁)-continuous.     

Anomalous behaviour for the random corrections to the cumulants of random walks in fluctuating random media

The Central Limit Theorem for a model of discrete-time random walks on the lattice ℤ^ν in a fluctuating random environment was proved for almost-all realizations of the space-time nvironment, for all ν > 1 in [BMP1] and for all ν≥ 1 in [BBMP]. In [BMP1] it was proved that the random correction to the average of the random walk for ν≥ 3 is finite. In the present paper we consider the cases ν = 1,2 and prove the Central Limit Theorem as T→∞ for the random correction to the first two cumulants. The rescaling factor for theaverage is for ν = 1 and (ln T), for ν=2; for the covariance it is , ν = 1,2. Received: 25 November 1999 / Revised version: 7 June 2000 / Published online: 15 February 2001     

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.

     

Stochastic comparison and preservation of positive correlations for Lévy-type processes

The stochastic comparison and preservation of positive correlations for Levy-type processes on R^d are studied under the condition that Levy measure v satisfies f{0〈｜z｜≤1）｜z｜｜v（x, dz） - v（x, d（-z））｜ 〈 ∞, x∈ R^d, while the sufficient conditions and necessary ones for them are obtained. In some cases the conditions for stochastic comparison are not only sufficient but also necessary.     

Super-simple (ν, 5, 5) Designs

Super-simple designs are useful in constructing codes and designs such as superimposed codes and perfect hash families. In this article, we investigate the existence of a super-simple (ν, 5, 5) balanced incomplete block design and show that such a design exists if and only if ν ≡ 1 (mod 4) and ν ≥ 17 except possibly when ν = 21. Applications of the results to optical orthogonal codes are also mentioned. Research supported by NSERC grant 239135-01.     

On Principal Hook Length Partitions and Durfee Sizes in Skew Characters

We construct for a given arbitrary skew diagram A{\mathcal A} all partitions ν with maximal principal hook lengths among all partitions with [ν] appearing in [A{\mathcal A}]. Furthermore, we show that these are also partitions with minimal Durfee size. We use this to give the maximal Durfee size for [ν] appearing in [A{\mathcal A}] for the cases when A{\mathcal A} decays into two partitions and for some special cases of A{\mathcal A}. We also deduce necessary conditions for two skew diagrams to represent the same skew character.     

Gruppi che ammettono un automorfismo 4-spezzante

LetG be a group admitting a 4-splitting automorphism (i.e. an automorphism σ such that for everyg∈G). In this paper we prove that ifG≠1 is solvable with derived lengthd thenG′ is nilpotent of class not greater than (4 ^d−1−1)/3.     

A note on quantum products of Schubert classes in a Grassmannian

Given two Schubert classes σ_λ and σ_μ in the quantum cohomology of a Grassmannian, we construct a partition ν, depending on λ and μ, such that σ_ν appears with coefficient 1 in the lowest (or highest) degree part of the quantum product σ_λ⋆σ_μ. To do this, we show that for any two partitions λ and μ, contained in a k × (n − k) rectangle and such that the 180^∘-rotation of one does not overlap the other, there is a third partition ν, also contained in the rectangle, such that the Littlewood-Richardson number c _λμ ^ν is 1.     

Weierstrass factorizations in compact Riemann surfaces

Letν′ be the complementary in a compact Riemann surface ν of a point (or a finite set). In this paper are characterized the subfields, of the field of meromorphic functions inν′, containing sufficient functions to verify a factorization property, similar to that of the classical Weierstrass theorem. It is also seen that the field generated by the Baker functions is not of this type, and the problem is solved of determining the divisors, inν′, of the holomorphic functions admiting Weierstrass factorizations with Baker functions as factors. As an application, a theorem is obtained characterizing the infinite products, of meromorphic functions in ν with bounded degree, which converge normally inν′. The first-named author is partially supported by PB91-0188.     

On infinite products of functions in compact riemann surfaces

Letν′ be the complementary of a point ∞ in a compact Riemann surfaceν. The normal convergence in compact subsets ofν′ of an infinite product of meromorphic functions (with polynomic exponential singularities at ∞ of bounded degree) is shown in this paper to be equivalent to a certain type of convergence of the double series of Newton sums of the divisors of its factors. This applies, for instance, to products of Baker functions inν′ and to products of meromorphic functions inν. The result for this last case is also generalized to complementaries of arbitrary nonvoid finite subsets ofν. Research supported by SA30/00B.     

Some balanced complete block designs

Letq=6t±1, ν=2q+2. The (ν/3) tri ples on ν marks may be partitioned intoq sets, each forming a BIBD of parameters (ν,3,2). Related results, some of them known, are also discussed briefly.     

Computable numberings of families of low sets and Turing jumps in the Ershov hierarchy

If ν and μ are some Δ₂⁰-computable numberings of families of sets of the naturals then P(x,y) ⇔ ν(x)′ ≠ μ(y) is a Σ₂⁰-predicate. Deriving corollaries from this result, we obtain a sufficient condition for existence of a Δ₂⁰-computable numbering of the subfamily of all sets in a given family with the Turing jumps belonging to a fixed level of the Ershov hierarchy, and we deduce existence of a Σ_ω⁻¹-computable numbering of the family of all superlow sets.     

Propriétés diophantiennes du développement en cotangente continue de Lehmer

Diophantine Properties of Lehmer’s Continued Cotangent Developments. This article deals with an algorithm devised by Lehmer which enables us to write any real positive number as the sum of an alternating series of cotangents of integers n _ν, ν ≥ 0, in a unique way. We continue the work begun by Lehmer and continued by Shallit: amongst other things, we give explicitly the link between the rational approximations of a given real number coming from this algorithm and the usual convergents of the same real number and we produce a quasi-optimal bound for the growth of the sequence (n _ν)_{ν ≥ 0} associated to an algebraic number. We also determine the regular continued fractions of an exceptional class of continued cotangent developments, which enables us to produce optimal irrationality measures of these expansions.