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.     

Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties

We introduce new sufficient conditions for a finite algebraU to possess a finite basis of identities. The conditions are that the variety generated byU possess essentially only finitely many subdirectly irreducible algebras, and have definable principal congruences. Both conditions are satisfied if this variety is directly representable by a finite set of finite algebras. One task of the paper is to show that virtually no lattice varieties possess definable principal congruences. However, the main purpose of the paper is to apply the new criterion in proving that every para primal variety (congruence permutable variety generated by finitely many para primal algebras) is finitely axiomatizable. The paper also contains a completely new approach to the structure theory of para primal varieties which complements and extends somewhat the recent work of Clark and Krauss.     

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.     

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.     

Ein Approximationssatz für holomorphe Funktionen auf nicht-kompakten Riemannschen Flächen

Let R be a non-compact Riemann surface andO(R) the algebra of all holomorphic functions on R. A subalgebraA ?O(R) is calledfull (“voll”), if (F1) for every point ??R there is a function f∈A with a simple zero at ? and no other zeros; (F2) if f, g∈A and f/g has no poles, then f/∈A. In 1971 Ian RICHARDS set the problem whether full subalgebras are dense inO(R), with respect to the topology of compact convergence. We answer this question in the positive, using a lemma of I. RICHARDS and theorems of R. ARENS and the author. Does this approximation theorem remain true for Stein manifolds of dimension n>1, if one modifies condition (F1) in a natural way? A counterexample is provided by a domain of holomorphy G??² and a full, but not dense subalgebraA ?O(G).     

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.     

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.     

Some irreducibility theorems of parabolic induction on the metaplectic group via the Langlands-Shahidi method

Let $\overline {S{p_{2n}}({\rm{<Emphasis FontCategory="NonProportional">F</Emphasis>}})} $ be the metaplectic double cover of F where F is a local field of characteristic 0. We use the Uniqueness of Whittaker model to define a metaplectic analog to Shahidi local coefficients and we use these coefficients to define gamma factors. We show that these gamma factors are multiplicative and satisfy the crude global functional equation. Then, we compute these factors in various cases and obtain explicit formulas for Plancherel measures. These computations are then used to prove some irreducibility theorems for parabolic induction on the metaplectic group over p-adic fields. In particular, we show that all principal series representations induced from unitary characters are irreducible. We also prove that parabolic induction from unitary supercuspidal representation of the Siegel parabolic sub group is irreducible if and only if a certain parabolic induction on F is irreducible.     

Points of finite order on elliptic curves with complex multiplication

Let E be an elliptic curve defined overQ. The group ofQ- rational points of finite order on E is a finite group T(E). In this article T(E) is computed for all elliptic curves defined overQ admitting complex multiplication. The only possible values for the order t of T(E) are 1, 2, 3, 4, or 6 in these cases. A standard form for an affine equation describing an elliptic curve with a given j-invariant is obtained. This is used to show that if j ≠ 0, 2⁶ 3³, then the number ofQ- rational points of order 2 on E depends only on j. The results are summarized in the accompanying table.     

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.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

Nilpotent and solvable radicals in locally finite congruence modular varieties

The Loewy rank of a modular latticeL of finite height is defined as the leastn for which there exista ₀=0t, < ... r=1 inL such that each interval I[a_i, a_i+1] is a complemented lattice. In this paper, a generalized notion of Loewy rank is applied to obtain new results in the commutator theory of locally finite congruence modular varieties. LetV be a finitely generated congruence modular variety. We prove that every algebra inV has a largest nilpotent congruence and a largest solvable congruence. Moreover, there exist first order formulas which define these special congruences in every algebra ofV.     

On ray shooting in convex polytopes

LetP be a convex polytope withn facets in the Euclidean space of a (small) fixed dimensiond. We consider themembership problem forP (given a query point, decide whether it lies inP) and theray shooting problem inP (given a query ray originating insideP, determine the first facet ofP hit by it). It was shown in [AM2] that a data structure for the membership problem satisfying certain mild assumptions can also be used for the ray shooting problem, with a logarithmic overhead in query time. Here we show that some specific data structures for the membership problem can be used for ray shooting in a more direct way, reducing the overhead in the query time and eliminating the use of parametric search. We also describe an improved static solution for the membership problem, approaching the conjectured lower bounds more tightly.     

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.     

Eine explizite Einbettung für Möbius-Geometrien endlicher Dimension

A new and explicit embedding is given for geometries ?(R,L) of Möbius type (i.e.L any extension field ofR) and finite dimension n = (L:R). The image of a chain is an algebraic curve. Till now explicit embeddings of Möbius geometries have been known only in the case n = 2.     

Shintani lifting and real-valued characters

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over ${\mathbb{F}_q}$ , and ψ is an irreducible character of G( ${\mathbb{F}_{q^m}}$ ) which is the lift of an irreducible character χ of G( ${\mathbb{F}_q}$ ), we prove ψ is real-valued if and only if χ is real-valued. In the case m = 2, we show that if χ is invariant under the twisting operator of G( ${\mathbb{F}_{q^2}}$ ), and is a real-valued irreducible character in the image of lifting from G( ${\mathbb{F}_q}$ ), then χ must be an orthogonal character. We also study properties of the Frobenius–Schur indicator under Shintani lifting of regular, semisimple, and irreducible Deligne–Lusztig characters of finite reductive groups.     

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.