Showing papers in "Pacific Journal of Mathematics in 1988"

Representing polynomials by positive linear functions on compact convex polyhedra.

Abstract: If K is a compact polyhedron in Euclidean έ/-space, defined by linear inequalities, βt > 0, and if / is a polynomial in d variables that is strictly positive on AT, then / can be expressed as a positive linear combination of products of members of {/?,}. In proving this and subsidiary results, we construct an ordered ring that is a complete AGL(d, R)-invariant for K, and discuss some of its properties. For example, the ordered ring associated to K admits the Riesz interpolation property if and only if it is AGL(d, R)-equivalent to a product of simplices. This is exploited to show that certain polynomials are not in the positive cone generated by the set {/?,}. Let L be a subfield of the real numbers, and let j8, = y ai4+\ (Ϊ = 1,2, 3,..., J) be linear polynomials ("linear forms") in the d variables {Xj}9 with coefficients from L. Suppose the convex polyhedron in R^ defined by K = f){βi)~~ι{[0,oo)) is compact and has interior. Let / be a polynomial in the d variables with entries from L, such that the restriction, f\K, is strictly positive. Then our first result (1.3) asserts that / may be represented as a combination with coefficients from L n R+ (that is, positive numbers in L) of terms that are products of the original set of /Γs that determine K. If / vanishes at only a vertex of K (and is strictly positive elsewhere), this decomposition does not hold in general (§ΠI). Our second principal result concerns the Riesz decomposition property in an ordered ring naturally associated to K, and leads to some interesting geometric characterizations of those polytopes that are affinely homeomorphic to products of simplices. With K defined as above, define a monomial (in the βfs) to be a polynomial in the JΓs that can be expressed as a product of the form βw = βw(\)βw(2) m m , βw(s)

Integrated semigroups and their applications to the abstract cauchy problem

Abstract: This paper is concerned with characterizations of those linear, closed, but not necessarily densely defined operators A on a Banach space E with nonempty resolvent set for which the abstract Cauchy problem u'(t) = Au(t), u(0) = x has unique, exponentially bounded solutions for every initial value x e D(An). Investigating these operators we are led to the class of "integrated semigroups". Among others, this class contains the classes of strongly continuous semigroups and cosine families and the class of exponentially bounded distribution semigroups. The given characterizations of the generators of these integrated semigroups unify and generalize the classical characterizations of generators of strongly continuous semigroups, cosine families or exponentially bounded distribution semigroups. We indicate how integrated semigroups can be used studying second order Cauchy problems u"{t) — A\u'{t) - Aiu(t) = 0, operator valued equations U'(t) = A{ U(t) + U(t)A2 and nonautonomous equations u'{t) = A(t)u(t).

Lipschitz convergence of Riemannian manifolds.

Abstract: Soit #7B-C≡#7B-C(n,Λ,So,Vo) l'ensemble de toutes les varietes de Riemann a n dimensions C ∞ compactes connexes de /courbure sectionnelle/ Vo. On montre que cette classe #7B-C a certaines proprietes de precompacite

Uniform rank over differential operator rings and Poincaré-Birkhoff-Witt extensions

Abstract: This paper is principally concerned with the question of when a generalized differential operator ring T over a ring R must have the same uniform rank (Goldie dimension) or reduced rank as R, and with the corresponding questions for induced modules. In particular, when R is either a right and left noetherian Q-algebra, or a right noetherian right fully bounded Q-algebra, it is proved that Tτ and RR have the same uniform rank. For any right noetherian ring R, it is proved that Tτ and RR have the same reduced rank. The type of generalized differential operator ring considered is any ring extension T D R generated by a finite set of elements satisfying a suitable version of the PoincareBirkhoff-Witt Theorem.

On matricially normed spaces.

Abstract: Arveson and Wittstock have proved a "non-commutati ve HahnBanach Theorem" for completely hounded operator-valued maps on spaces of operators. In this paper it is shown that if T is a linear map from the dual of an operator space into a C*-algebra, then the usual operator norm of T coincides with the completely bounded norm. This is used to prove that the Arveson-Wittstock theorem does not generalize to "matricially normed spaces". An elementary proof of the Arveson-Wittstock result is presented. Finally a simple bimodule interpretation is given for the "Haagerup" and "matricial" tensor products of matricially normed spaces. 1. Introduction. A. function space V on a set X is a linear subspace of the bounded complex functions on X. With the uniform norm, this is a normed vector space. Conversely, any (complex) normed vector space V may be realized as a function space on the closed unit ball X of the dual space V*. Thus one may regard a normed vector space as simply an abstract function space. An operator space V on a Hubert space H is a linear subspace of the bounded operators on H. For each «GN, the operator norm associated with Hn determines a distinguished norm on the n x n matrices over V. The second author recently gave an abstract characterization for the operator spaces by taking into consideration these systems of matrix norms. The operator spaces V are characterized among the "matricially normed spaces" (see §2), by the "L°°-property": given matrices v = [v/7], w = [wkl] with vzy, wkι e V, ||vθH>||=max{||v||,|M|}.

Semigroups generated by certain operators on varieties of completely regular semigroups

Abstract: The operators C,K,L,T,T( and Tr on the lattice Se(^0t) of varieties of completely regular semigroups have played an important role in recent studies of S^(^9t\ Although each of these operators is idempotent, when applied in various combinations to the trivial variety they yield varieties for which the only upper bound is ^^. The semigroups generated by various subsets of (C, K, L, T, Tr, T{) are determined here in terms of generators and relations. 1. Introduction and summary. Completely regular semigroups (unions of groups) may be regarded as algebras with the operations of (binary) multiplication and (unary) inversion. As such they form a variety <&{% defined by the identities (1) (ab)c = (ab)c, a = aa~ιa, aa'1 = a~ιa, (a~ι)~ = a. The lattice ££{^01) of all subvarieties of <£!% turns out to be amenable to a thorough analysis both globally and locally. The former includes various (complete) congruences that emerge naturally in the study either of the varieties themselves or of the corresponding fully invariant congruences on a free completely regular semigroup FΉffl on a countably infinite set. Local studies of the lattice SP^Si) usually amount to rather complete descriptions of relatively small intervals in JS?(^^) modulo «S?(^), the lattice of group varieties, starting from the bottom of the lattice. In the local approach, a number of operators make their appearance in the description of certain varieties in terms of some of their proper subvarieties. But these operators may be defined on all of «£?(^^) thereby providing a certain amount of information for varieties scattered throughout S^{^0i) and hence may be used for a global study of this lattice. Another source of operators on ^(Ήέft) are the kernel and trace relations on the lattice of fully invariant congruences on FΉffl now translated into relations on JS?(^^). Of the considerable literature on varieties of completely regular semigroups, we mention only the following ones because they are directly related to our object of study. We thus cite Jones [6], [7], Kadourek [8],

Systems of nonlinear wave equations with nonlinear viscosity.

Abstract: On etablit l'existence et la regularite des solutions pour le probleme de Cauchy associe a l'equation hyperbolique non lineaire avec viscosite non lineaire: u-Σ i=1 n ∂[∂W(p)/∂Pi]/∂x i -Σ i=1 n ∂[∂V(q)/∂qi]/∂x i =f ou p=⊇u, q=⊇u˙

Topological entropy and recurrence of countable chains.

Abstract: On considere un systeme dynamique (X, σ) sur un espace etat denombrable. On introduit une sorte d'entropie topologique pour de tels systemes, h*, qui coincide avec l'entropie topologique usuelle quand X est compact. On utilise une approche figurative pour classer un graphe Γ (ou une chaine) comme transitoire, recurrent nul ou recurrent positif. On montre qu'etant donnes 0≤α≤β≤∞, il existe une chaine dont l'entropie h* est β et ou l'entropie de Gurevic est α

On divisors of sums of integers V

Abstract: 1. Let P{ή) and p(n) denote the greatest and smallest prime factor of n, respectively. Recently in several papers, Balog, Erdόs, Maier, Sarkozy, and Stewart have studied problems of the following type: if A\,...,Ak are "dense" sets of positive integers, then what can be said about the arithmetical properties of the sums a\ H h % with a,\ G A\,...,ak G Akl In particular, Balog and Sarkόzy proved that there is a sum a\+ci2 (#i G A\9 aι G A2) for which P(a\+aι) is "small", i.e., all the prime factors of a\ +aι are small. On the other hand, Balog and Sarkόzy and Sarkόzy and Stewart studied the existence of a sum CL\ Λ h % for which P(a\ H h ak) is large. In this paper we study p(a\ H \ak). Our goal is to show that if A\,..., Ak are sets of positive integers then there exists a sum a\ + h cik with a\ G A\,..., ak € Ak that is divisible by a "small" prime. In the most interesting special case, namely A\ = = Ak, there are sums CL\ Λ Vak divisible by /c, so that p(a\ H h ak) 2) by studying the case min, \At\ > N /+. Here Gallagher's larger sieve will be used. The results in §§3 and 4 do not give especially good results when the sets A\,...,Ak are very "dense". In §5, we will give an essentially best possible result for the small prime factors of the sums d\ Λ h ak in the case when (\AX\ \Ak\) / > Nexp(-clogklogN/loglogN)

Szegő’s conjecture on Lebesgue constants for Legendre series

Abstract: On etablit la conjecture de Szego (1926) selon laquelle les constantes de Lebesgue des series de Legendre forment une suite monotone croissante