Narrowness implies uniformity

This paper is about varietiesV of universal algebras which satisfy the following numerical condition on the spectrum: there are only finitely many prime integersp such thatp is a divisor of the cardinality of some finite algebra inV. Such varieties are callednarrow. The variety (or equational class) generated by a classK of similar algebras is denoted by V(K)=HSPK. We define Pr (K) as the set of prime integers which divide the cardinality of a (some) finite member ofK. We callK narrow if Pr (K) is finite. The key result proved here states that for any finite setK of finite algebras of the same type, the following are equivalent: (1) SPK is a narrow class. (2) V(K) has uniform congruence relations. (3) SK has uniform congruences and (3) SK has permuting congruences. (4) Pr (V(K))= Pr(SK). A varietyV is calleddirectly representable if there is a finite setK of finite algebras such thatV= V(K) and such that all finite algebras inV belong to PK. An equivalent definition states thatV is finitely generated and, up to isomorphism,V has only finitely many finite directly indecomposable algebras. Directly representable varieties are narrow and hence congruence modular. The machinery of modular commutators is applied in this paper to derive the following results for any directly representable varietyV. Each finite, directly indecomposable algebra inV is either simple or abelian.V satisfies the commutator identity [x,y]=x·y·[1,1] holding for congruencesx andy over any member ofV. The problem of characterizing finite algebras which generate directly representable varieties is reduced to a problem of ring theory on which there exists an extensive literature: to characterize those finite ringsR with identity element for which the variety of all unitary leftR-modules is directly representable. (In the terminology of [7], the condition is thatR has finite representation type.) We show that the directly representable varieties of groups are precisely the finitely generated abelian varieties, and that a finite, subdirectly irreducible, ring generates a directly representable variety iff the ring is a field or a zero ring.     

Finite union theorem with restrictions

The aim of this paper is to prove the following extension of the Folkman-Rado-Sanders Finite Union Theorem: For every positive integersr andk there exists a familyL of sets having the following properties:

1. ifS ₁,S ₂, ...,S _{k + 1} are distinct pariwise disjoint elements ofL then there exists nonemptyI ? {1, 2, ...,k + 1} with ∪ _i∈I S _i ?L
2. ifL =L ₁ ?...?L _r is an arbitrary partition then there existsj ≤ r and pairwise disjoint setsS ₁,S ₂, ...,S _k ∈L _j , such thatL _i∈I S _i ∈L _j for every nonemptyI ? {1, 2, ...,k}.

     

Criteria for the injectivity of analytic sheaves

It is shown that a moduleL over the sheafO of germs of holomorphic functions on a domain G of Cⁿ is injective if and only if the following conditions are satisfied; a)L is flabby; b) for every closed set S ?G and every point z λ G, the stalk se _z of the sheafS ^L;U1→Γ S ^(U:L) is an injectiveO _z -module. It follows in particular that the sheaf of germs of hyperfunctions is injective over the sheaf of germs of analytic functions.     

Index formulae for subspaces of Kreîn spaces

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ^?[+]K ⁺ ofK, we prove the index formula $$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$ where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ^?)?dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.     

Two results for monocompact measures

For every finite measure space (Ω,A, P) whereA is K₁-generated we prove the equivalence of compactness and monocompactness for P . Moreover, we prove the existence of a perfect, not monocompaot probability, thus answering an open question in [6]. Let P be a charge on the algebraA andK ?A be a monocompact class. We show that P is o-additive ifK ^S P-approximatesK ^S, the family of finite unions inK , needs not to be monocompact.     

Directed graphs over topological spaces: Some set theoretical aspects

In this paper we shall consider problems of the following type. SupposeG is some set,U is some family of subsests ofG (e.g.G could be the Euclidean plane andU might be the family of all sets of Lebesgue measure zero), andG is any directed graph overG (i.e. any collection of ordered pairs of members ofG) such that for eachg∈G the set {h:<g,h>∈G} belongs to the familyU. How large a setSυG must there exist with the property that (S×S) ∩G=, i.e. such that it is totally disconnected? In many of the cases we shall consider (including the particular example above), the answer will turn out to be independent of the axioms of set theory and will remain so even after adjoining the negation of the continuum hypothesis.     

Simple algebras of Weyl type

Over a fieldF of arbitrary characteristic, we define the associative and the Lie algebras of Weyl type on the same vector spaceA[D] =A?F[D] from any pair of a commutative associative algebra,A with an identity element and the polynomial algebraF[D] of a commutative derivation subalgebraD ofA We prove thatA[D], as a Lie algebra (modulo its center) or as an associative algebra, is simple if and only ifA isD-simple andA[D] acts faithfully onA. Thus we obtain a lot of simple algebras.     

When is the radical associated with a module a torsion?

For an arbitrary R-module M we consider the radical (in the sense of Maranda)G _M, namely, the largest radical among all radicalsG, such thatG(M). We determine necessary and sufficient on M in order for the radicalG(M) to be a torsion. In particular,G(M) is a torsion if and only if M is a pseudo-injective module.     

Expansion of open sets and decomposition of continuous mappings

The notion of expansionA of open sets is introduced. ThenA-expansion continuous mappingf:X→Y is defined. The main result of this note is that a mappingf is continuous if and only if it is bothA-expansion continuous andB-expansion continuous, whereA-expansion,B-expansion are two mutually dual expansions.     

Multifractal analysis of the divergence of Fourier series: The extreme cases

We study the size, in terms of the Hausdorff dimension, of the subsets of T such that the Fourier series of a generic function in L ¹(T), L ^p (T), or C(T) may behave badly. Genericity is related to the Baire Category Theorem or the notion of prevalence. This paper is a continuation of [3].     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

On the subfields of cyclotomic function fields

Let K = F(T) be the rational function field over a finite field of q elements. For any polynomial f(T) ∈ F [T] with positive degree, denote by Λ _f the torsion points of the Carlitz module for the polynomial ring F[T]. In this short paper, we will determine an explicit formula for the analytic class number for the unique subfield M of the cyclotomic function field K(Λ _P ) of degree k over F(T), where P ∈ F[T] is an irreducible polynomial of positive degree and k > 1 is a positive divisor of q ? 1. A formula for the analytic class number for the maximal real subfield M ⁺ of M is also presented. Futhermore, a relative class number formula for ideal class group of M will be given in terms of Artin L-function in this paper.     

Quasi-factors in ergodic theory

Motivated by the notion of quasi-factor in topological dynamics, we introduce an analogous notion in the context of ergodic theory. For two processes,X andY , we haveX?Y if and only ifY has a factor which is isomorphic to a quasi-factor ofX. On the other hand, weakly mixing processes can have nontrivial quasifactors which are not w.m. We characterize those ergodic processes which admit only trivial continuous ergodic quasi-factors, and use this characterization to conclude that a process with minimal selfjoinings is of this type. From this we derive the fact that for every suchX and any ergodicY eitherX ⊥Y orY extends some symmetric product ofX.     

Straight line arrangements in the real projective plane

Let A be an arrangement of n pseudolines in the real projective plane and let p ₃(A) be the number of triangles of A. Grünbaum has proposed the following question. Are there infinitely many simple arrangements of straight lines with p ₃(A)=1/3n(n?1)? In this paper we answer this question affirmatively.     

Asymptotic behaviour of the minima to a class of optimization problems for differential inclusions

Denoting byS _k (ξ _k ) the set of solutions of the Cauchy problem $\dot x \in F_k (t,x),x(0) = \xi _k $ , fork∈N∪{∞}, we prove that, under appropriate assumptions, the sequence {S _k (ξ _k )} _{k ∈ N} converges toS _∞(∈_∞) in the Kuratowski sense as well as in the Mosco sense. This result together with some facts from Γ-convergence theory are used to prove a result concerning the existence and the asymptotic behavior of the minima to the optimization problem $$\min \int_0^T {[g_k (t,x(t)) + h_k (t,\dot x(t))]} dt + \psi _k (\xi ),x \in S_k (\xi ),\xi \in K$$ withK a compact subset ofR ⁿ .     

Using RngStreams for parallel random number generation in C++ and R

The RngStreams software package provides one viable solution to the problem of creating independent random number streams for simulations in parallel processing environments. Techniques are presented for effectively using RngStreams with C++ programs that are parallelized via OpenMP or MPI. Ways to access the backbone generator from RngStreams in R through the parallel and rstream packages are also described. The ideas in the paper are illustrated with both a simple running example and a Monte Carlo integration application.     

Sulle foliazioni misurate arazionali parte II

In this sequel to our previous paper [2] we show that on a surfacesS of genusg>1, the arational measured foliations are generic inM F. We also show that ergodic measured foliations are necessarily non-rational (i.e. without closed leaves).