Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

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.     

Spherical elastic-plastic waves

Zusammenfassung Es wird die Fortpflanzung elastisch-plastischer Spannungswellen in einem unendlichen Medium betrachtet, welches einer idealen Spannungs-Verformungs-Kurve folgt, Trescas Fliesskriterium unterworfen ist und einen sphärischen Hohlraum enthält, wobei an der Fläche des Hohlraumes ein Stoss angenommen wird. Ein rechnerisches Verfahren, basiert auf endliche Differenzen, wird entwickelt and ein Beispiel gegeben.

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Notation radial stress - tangential stress - K yield stress - _rr non-dimensional radial stress ( /K) - non-dimensional tangential stress ( /K) - , Lame's constants - K _b Bulk constant (=(3+2)/3) - v Poisson's constant - Material density - C Elastic wave velocity (=((+2)/)^1/2) - C _p Plastic wave velocity (=(K _b /)^1/2) - distance from center of cavity - r ₀ cavity radius - v non-dimensional radial co-ordinate (= /r ₀) - time - t non-dimensional time (=C /r ₀) - radial displacement - u non-dimensional radial displacement (=/r ₀) - particle velocity - v non-dimensional particle velocity (= /C) - pressure - P(t) non-dimensional pressure (= /K)     

Second-Order Subexponential Behavior of Subordinated Sequences

Suppose that , , and are three discrete probability distributions related by the equation (E): , where denotes the k-fold convolution of In this paper, we investigate the relation between the asymptotic behaviors of and . It turns out that, for wide classes of sequences and , relation (E) implies that , where is the mean of . The main object of this paper is to discuss the rate of convergence in this result. In our main results, we obtain O-estimates and exact asymptotic estimates for the difference .     

Some proximity and sensitivity results in quadratic integer programming

We show that for any optimal solution for a given separable quadratic integer programming problem there exist an optimal solution for its continuous relaxation such that wheren is the number of variables and(A) is the largest absolute subdeterminant of the integer constraint matrixA. Also for any feasible solutionz, which is not optimal for the separable quadratic integer programming problem, there exists a feasible solution having greater objective function value and with . We further prove, under some additional assumptions, that the distance between a pair of optimal solutions to an integer quadratic programming problem with right hand side vectorsb andb, respectively, depends linearly on b–b₁. Finally the validity of all the results for nonseparable mixed-integer quadratic programs is established. The proximity results obtained in this paper are extensions of some of the results described in Cook et al. (1986) for linear integer programming.This research was partially supported by Natural Sciences and Engineering Research Council of Canada Grant 5-83998.     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

Conditions of Local Asymptotic Normality for Gaussian Stationary Processes

Let {\bold x}[] be a stationary Gaussian process with zero mean and spectral density f, let be the -algebra induced by the random variables {\bold x}[], D(R¹), and let _t, t > 0, be the -algebra induced by the random variables x[],supp [-t,t]. Denote by (f) the Gaussian measure on generated by {\bold x}. Let _t(f) be the restriction of (f) to _t. Let f and g be nonnegative functions such that the measures _t(f) and _t(g) are absolutely continuous. Put

Win probabilities and simple majorities in probabilistic voting situations

This paper considers an election between candidatesA andB in which (1) voters may be uncertain about which candidate they will vote for, and (2) the winner is to be determined by a lottery betweenA andB that is based on their vote totals. This lottery is required to treat voters equally, to treat candidates equally, and to respond nonnegatively to increased support for a candidate. The set _n of all such lottery rules based on a total ofn voters is the convex hull of aboutn/2 basic lottery rules which include the simple majority rule. For odd values ofn 3 let , and for even values ofn 4 let . With the average of then voters probabilities of voting forA, it is shown that within _n the simple majority rule maximizes candidateA's overall win probability whenever , and that(n) is the smallest number for which this is true. Similarly, the simple majority rule maximizesB's overall win probability whenever (the average of the voters probabilities of voting forB) is as large as(n). This research was supported by the National Science Foundation, Grant SOC 75-00941.     

A New Critical Behavior for Nonlinear Wave Equations

We study the inhomogeneous semilinear wave equations on with initial values and ,where is a noncompact, complete manifold. We founda new critical behavior in the following sense. There exists ap* > 0. When 1 < p p*, the above problem hasno global solution for any nonnegative not identicallyzero and for any and ; when the problem has a global solution for some and some and . If , which is equipped with the Euclideanmetric, then . If we show that belongs to the blow upcase. Although homogeneous semilinear wave equations are known to exhibit acritical behavior for a long time, this seems to be the first result oninhomogeneous ones.     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

The elements of K2(ℤs)

Let S={p₁,...,p_s} be a set of rational primes, . One has K₂(_s)K₂(_su{2} and we want to assume 2 S. It is snown that every element of K₂(_S) is a Dehnis-Stein-symbol <a,b>, 1+ab being a unit of _S.Here b can be determined concretely, depending only on S, and we obtain a normal form of the elements of K₂(Q) as Steinberg-symbols, which is unique in some way and expresses the quadratic reciprocity law.     

Finitely generated relatively universal varieties of Heyting algebras

Given distinct varieties and of the same type, we say that is relatively -universal if there exists an embedding :K from a universal categoryK such that for every pairA, B ofK-objects, a homomorphismf:A B has the formf=_g for someK-morphismg:A B if and only if Im(f) . Finitely generated relatively -universal varieties of Heyting algebras are described for the variety of Boolean algebras, the variety generated by a three element chain, and for the variety generated by the four element Boolean algebra with an added greatest element.Dedicated to the memory of Alan DayPresented by J. Sichler.The support of the NSERC is gratefully acknowledged.     

Strong Positivity in Right-Invariant Order on Braid Groups and Quasipositivity

Dehornoy constructed a right invariant order on the braid group B _n uniquely defined by the condition 1{\text{ if }}\beta _0 ,\beta _1$$ " align="middle" border="0"> are words in . A braid is called strongly positive if 1$$ " align="middle" border="0"> for any . In the present paper it is proved that the braid is strongly positive if the word does not contain . We also provide a geometric proof of the result by Burckel and Laver that the standard generators of a braid group are strongly positive. Finally, we discuss relations between the right invariant order and quasipositivity.     

A structure theorem for weak buildings of spherical type

Following earlier work of Tits [8], this paper deals with the structure of buildings which are not necessarily thick; that is, possessing panels (faces of codimension 1) which are contained in two chambers, only. To every building , there is canonically associated a thick building whose Weyl group W( ) can be considered as a reflection subgroup of the Weyl group W() of . One can reconstruct from together with the embedding W( ) W(). Conversely, if is any thick building and W any reflection group containing W( ) as a reflection subgroup, there exists a weak building with Weyl group W and associated thick building .     

A Delocalization Property for Assembly Maps and an Application in Algebraic K-Theory of Group Rings

Let be a group, F the free -module on the set of finite order elements in , with acting by conjugation, and the ring extension of by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeaacaGaaiaabeqaamaabaabaaGcbaWaaiWaaeaada% WcaaqaaiaaigdaaeaatCvAUfKttLearyGqLXgBG0evaGqbciab-5ga% UbaaieaacaGFLbGaaGOmaiaabc8acqWFPbqAcaqGVaGae8NBa42aaq% qaaeaacqGHdicjcqaHZoWzcqGHiiIZcqqHtoWrcaqGGaGaae4Baiaa% bAgacaqGGaGaae4BaiaabkhacaqGKbGaaeyzaiaabkhacaqGGaGae8% NBa4gacaGLhWoaaiaawUhacaGL9baaaaa!563E!\[\left\{ {\frac{1}{n}e2{\text{\pi }}i{\text{/}}n\left| {\exists \gamma \in \Gamma {\text{ of order }}n} \right.} \right\}\]. For a ring R with , we build an injective assembly map , detected by the Dennis trace map. This is proved by establishing a delocalization property for the assembly map in Hochschild homology, namely providing a gluing of simpler assembly maps (i.e. localized at the identity of ) to build , and by delocalizing a known assembly map in K-theory to define . We also prove the delocalization property in cyclic homology and in related theories.     

Sets with fixed congruence in a steiner system

In this paper we study particular sets of a Steiner systemS. More precisely, we study the setsA such that ¦A ¦ d modh for all lines ofS, withd andh integers satisfyingd 0,h 2.Dedicated to Professor M. Scafati Tallini on the occasion of her sixtyfifth birthday     

On Fraenkel’s <Emphasis Type="Italic">N</Emphasis>-Heap Wythoff’s Conjectures

The N-heap Wythoffs game is a two-player impartial game with N piles of tokens of sizes Players take turns removing any number of tokens from a single pile, or removing (a₁,..., a_N) from all piles - a_i tokens from the i-th pile, providing that where is the nim addition. The first player that cannot make a move loses. Denote all the P-positions (i.e., losing positions) by Two conjectures were proposed on the game by Fraenkel [7]. When are fixed, i) there exists an integer N₁ such that when . ii) there exist integers N₂ and _2 such that when , the golden section.In this paper, we provide a sufficient condition for the conjectures to hold, and subsequently prove them for the three-heap Wythoffs game with the first piles having up to 10 tokens.AMS Subject Classification: 91A46, 68R05.     

The quality of the Diophantine approximations found by the Jacobi-Perron algorithm and related algorithms

Given a vector of real numbers=(₁,... _d ) ^d , the Jacobi-Perron algorithm and related algorithms, such as Brun's algorithm and Selmer's algorithm, produce a sequence of (d+1)×(d+1) convergent matrices {C⁽ⁿ⁾():n1} whose rows provide Diophantine approximations to . Such algorithms are specified by two mapsT:[0, 1] ^d [0, 1] ^d and A:[0,1] ^d GL(d+1,), which compute convergent matrices C⁽ⁿ⁾())...A(T())A(). The quality of the Diophantine approximations these algorithms find can be measured in two ways. The best approximation exponent is the upper bound of those values of for which there is some row of the convergent matrices such that for infinitely many values ofn that row of C⁽ⁿ⁾() has . The uniform approximation exponent is the upper bound of those values of such that for all sufficiently large values ofn and all rows of C⁽ⁿ⁾⁽⁾ one has . The paper applies Oseledec's multiplicative ergodic theorem to show that for a large class of such algorithms and take constant values and on a set of Lebesgue measure one. It establishes the formula where are the two largest Lyapunov exponents attached by Oseledec's multiplicative ergodic theorem to the skew-product (T, A,d), whered is aT-invariant measure, absolutely continuous with respect to Lebesgue measure. We conjecture that holds for a large class of such algorithms. These results apply to thed-dimensional Jacobi-Perron algorithm and Selmer's algorithm. We show that; experimental evidence of Baldwin (1992) indicates (nonrigorously) that. We conjecture that holds for alld2.     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

The Busy Period of the M/GI/∞ Queue

An equation for the distribution Z() of the duration T of the busy period in a stationary M/GI/ service system is constructed from first principles. Two scenarios are examined, being distinguished by the half-plane Re()>₀ for some ₀0 in which the generic service time random variable S, always assumed to have a finite mean E(S), has an analytic Laplace–Stieltjes transform E(e^–S ). If ₀<0 then E(e^–T ) is analytic in a half-plane (₁,), where ₀₁<0 and ₁ is determined by the distribution of S; then for any 0<s<|₁|.When ₀=0, E(e^–T ) is analytic in (0,), and now more is known about T. Inequalities on the tail () are used to show that for any 1, E(T ) is finite if and only if E(S ) is finite. It follows that the point process consisting of the starting epochs of busy periods is long range dependent if and only if E(S ²)=, in which case it has Hurst index equal to [frac12](3–), where is the moment index of S.If also the tail (x)=Pr{Sx} of the service time distribution satisfies the subexponential density condition ₀ ^x (x–u) (u)du/ (x)2E(S) as x, then (x)/ (x)e^E(S), where is the arrival rate.