A representation theorem for weak automorphisms of a universal algebra

Given a group G and a descending chainG ₀,G ₁,...,G _n, of normal subgroups ofG, we prove that there exists a universal algebra , such that the chain ...W_n( )...W₁( }) W₀( )W( ) is isomorphic to the chain ...G _n ...G ₁G ₀G, where W( ) is the group of weak automorphisms of , and W_n( ) is the group of weak automorphisms of that leaves alln-ary operations fixed.We also prove that there are an infinite number of non-isomorphic algebras that satisfy the above.These results are a generalization of those proved by J. Sichler, in the special case when G=G₀, and G₁=G₂=...=G_n=....Presented by J. Mycielski.This paper comprises part of the author's doctoral dissertation at the University of Notre Dame in 1983. The author wishes to express her deep gratitude to Professor Abraham Goetz for suggesting this problem, for being extremely generous with his time and experience, and for giving her his constant encouragement. The author also thanks the reviewer for his helpful comments.     

Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas

Direct finite interpolation formulas are developed for the Paley–Wiener function spaces and , where contains all bivariate entire functions whose Fourier spectrum is supported by the set = Cl{(u, v) |u| + |v| < ], while in the Fourier spectrum support set of its d-variate entire elements is [–, ] ^d . The multidimensional Kotel'nikov–Shannon sampling formula remains valid when only finitely many sampling knots are deviated from the uniform spacing. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases.According to the Sun–Zhou extension of the Kadets -theorem, the magnitudes of deviations are limited coordinatewise to . To avoid this inconvenience, we introduce weighted Kotel'nikov–Shannon sampling sums. For , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Divisibility of Certain Partition Functions by Powers of Primes

Let be the prime factorization of a positive integer k and let b _k (n) denote the number of partitions of a non-negative integer n into parts none of which are multiples of k. If M is a positive integer, let S _k (N; M) be the number of positive integers N for which b _k(n ) 0(mod M). If we prove that, for every positive integer j In other words for every positive integer j, b _k(n) is a multiple of for almost every non-negative integer n. In the special case when k=p is prime, then in representation-theoretic terms this means that the number ofp -modular irreducible representations of almost every symmetric groupS _n is a multiple of p ^j. We also examine the behavior of b _k(n) (mod ) where the non-negative integers n belong to an arithmetic progression. Although almost every non-negative integer n (mod t) satisfies b _k(n) 0 (mod ), we show that there are infinitely many non-negative integers n r (mod t) for which b _k(n) 0 (mod ) provided that there is at least one such n. Moreover the smallest such n (if there are any) is less than 2 .     

A classification of martingale Hardy spaces associated with rearrangement-invariant function spaces

Let X be a rearrangement-invariant Banach function space over a complete probability space , and denote by the Hardy space consisting of all martingales such that . We prove that implies for any filtration if and only if Doobs inequality holds in X, where denotes the martingale defined by , n = 0, 1, 2, ..., and a.s.Received: 1 August 2000     

The property of having independent basis in semigroup varieties

There exist independently based semigroup varieties and , , such that has no cover in the interval [ ; ].Translated from Algebra i Logika, Vol. 44, No. 1, pp. 81–96, January–February, 2005.     

Feynman-Kac propagators and viscosity solutions

The aim of this paper is to study viscosity solutions to the following terminal value problem on [0, t] × E:

Un théorème de Choquet-Deny pour les groupes moyennables

Résumé SoitG un groupe moyennable connexe, locallement compact, à base dénombrable. Soit une mesure positive sur les boréliens deG. Nous étudions les fonctions boréliennes positivesh vérifiant: g G, . Sous de bonnes hypothèses sur , nous obtenons, pour ces fonctions, une représentation intégrale à l'aide d'exponentielles.

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Summary LetG be a connected locally compact separable amenable group. Let be a positive measure on the Borel -field ofG. We study the positive Borel functionsh onG which satisfy: g G, . Under smooth assumptions on , we establish an integral representation of these functions in term of exponentials.

     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations

Summary In this paper we establish a large deviations principle for the invariant measure of the non-Gaussian stochastic partial differential equation (SPDE) _t v =v +f(x,v )+(x,v ) . Here is a strongly-elliptic second-order operator with constant coefficients, h:=DH _xx-h, and the space variablex takes values on the unit circleS ¹. The functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic PDE, and in particular we require that 0<mM wherem andM are some finite positive constants. The perturbationW is a Brownian sheet. It is well-known that under some simple assumptions, the solutionv ² is aC ^k (S ¹)-valued Markov process for each 0<1/2, whereC (S ¹) is the Banach space of real-valued continuous functions onS ¹ which are Hölder-continuous of exponent . We prove, under some further natural assumptions onf and which imply that the zero element ofC (S ¹) is a globally exponentially stable critical point of the unperturbed equation _t ⁰ = ⁰ +f(x,⁰), that has a unique stationary distributionv ^K, on (C (S ¹), (C ^K (S ¹))) when the perturbation parameter is small enough. Some further calculations show that as tends to zero,v ^K, tends tov ^K,0, the point mass centered on the zero element ofC (S ¹). The main goal of this paper is to show that in factv ^K, is governed by a large deviations principle (LDP). Our starting point in establishing the LDP forv ^K, is the LDP for the process , which has been shown in an earlier paper. Our methods of deriving the LDP forv ^K, based on the LDP for are slightly non-standard compared to the corresponding proofs for finite-dimensional stochastic differential equations, since the state spaceC (S ¹) is inherently infinite-dimensional.This work was performed while the author was with the Department of Mathematics, University of Maryland, College Park, MD 20742, USA     

The Lattice of Interpretability Types of Cantor Varieties

For integers 1 m < n, a Cantor variety with m basic n-ary operations _i and n basic m-ary operations _k is a variety of algebras defined by identities _k(₁( ), ... , _m( )) = _k and i(₁( ), ... ,_n( )) = y _i, where = (x ₁., ... , x _n) and = (y ₁, ... , y _m). We prove that interpretability types of Cantor varieties form a distributive lattice, , which is dual to the direct product ₁ × ₂ of a lattice, ₁, of positive integers respecting the natural linear ordering and a lattice, ₂, of positive integers with divisibility. The lattice is an upper subsemilattice of the lattice of all interpretability types of varieties of algebras.     

Determinants in Triangulated Categories

For a small triangulated category , Bass's K ₁ group is described, and the theorem of the heart is proved. We define the determinant map from to Neeman's , and we compute this map when is the derived category of an Abelian category .     

An infinite order operator on the lattice of varieties of completely regular semigroups

Let be a variety of completely regular semigroups. Define C ^* to be the class of all completely regular semigroupsS whose least full and self-conjugate subsemigroupC ^*(S) belongs to . ThenC ^* is an operator on the lattice of varieties of completely regular semigroups. In this note we show that the order ofC ^* is infinite. This fact yields that the Mal'cev project is not associative on . We describe (C ^*)¹, andi 0, in terms of -invariant normal subgroups of the free group over a countably infinite set. The lattice theoretic properties ofC ^* are also studied.Presented by W. Taylor.     

Complete Subobjects of Fuzzy Sets Over MV-Algebras

A subobjects structure of the category -FSet of -fuzzy sets over a complete MV-algebra is investigated, where an -fuzzy set is a pair A = (A, ) such that A is a set and : A × A is a special map. Special subobjects (called complete) of an -fuzzy set A which can be identified with some characteristic morphisms A ^* = (L × L, ) are then investigated. It is proved that some truth-valued morphisms are characteristic morphisms of complete subobjects.     

On a Problem of the Theory of Multiply Local Formations

We describe the -closed n-multiply local formations such that the lattice of all -closed n-multiply local formations between and is Boolean.     

Disintegration with respect to L p-density functions and singular measures

Summary Let A be a real or complex commutative ordered algebra with identity and involution. Let denote the set of positive multiplicative linear functionals on A. Equip with the topology of simple convergence. For a fixed non-negative probability measure on the set _p of linear functionals f on A which admit an integral representation of the form with FL _p () (1p) is biuniquely identified with L _p () via the map tfF. The norm on _p under which this map becomes an isometry is characterized and a formula for approximating F is derived. The linear functionals which admit representation of the form with are also characterized and appropriately normed. The theory is applied to solve abstract versions of trigonometric and n-dimensional moment problems as well as provide an alternate point of view to the theory of L _p-spaces. New proofs of classical theorems are offered.Research for this paper was sponsored in part by the Danish Natural Science Research Council (Grant No.511-10302) and in part by the National Science Foundation (Grant No. MCS78-03397)The results contained herein include the proofs of theorems announced in [15]     

Comparison of Sampling Schemes with and without Replacement

An urn contains colored balls, ~balls of each of different colors. The balls are drawn sequentially and equiprobably, one ball at a time, and then each drawn ball drawn is either returned to the urn (sampling with replacement) or left outside the urn (sampling without replacement). The drawing continues until some colors are drawn at least ~times each. Observable statistics are the numbers , , of colors that have appeared precisely ~times each by the stopping time. The asymptotic behavior as of these values for each of the two sampling models is studied; the possibility of their use for identifying the model is discussed.     

Spectral-Null Codes and Null Spaces of Hadamard Submatrices

Codes of length 2 ^m over {1, -1} are defined as null spaces of certain submatrices of Hadamard matrices. It is shown that the codewords of all have an rth order spectral null at zero frequency. Establishing the connection between and the parity-check matrix of Reed-Muller codes, the minimum distance of is obtained along with upper bounds on the redundancy of . An efficient algorithm is presented for encoding unconstrained binary sequences into .     

Estimates of the Kolmogorov Widths of Classes of Analytic Functions Representable by Cauchy-Type Integrals. I

In the Banach space of functions analytic in a Jordan domain , we establish order estimates for the Kolmogorov widths of certain classes of functions that can be represented in by Cauchy-type integrals along the rectifiable curve = and can be analytically continued to or to .