Schwach irreduzible Markoff-Operatoren

For any weakly irreducible Markov operatorT on the space of continuous functions on a compact space and any continuous functionf we show that pointwise convergence of (T ⁿf) already implies uniform convergence. This improves a result ofJamison [2]. We also study convergence whenf is only measurable and give results related to the zero-two law. Finally two characterizations of uniform ergodicity for this class of operators are given. In particular a question ofLotz [4] is answered.     

Über die Wiener-Eigenschaft bei projektiven Limites lokalkompakter Gruppen

Leptin posed in [1] the problem to determine the class [W] of locally compact groups G characterized by the following property: Every proper closed two-sided idealJ in the Banach-*-algebraL ¹(G) is annihilated by some nondegenerate continuous *-representation ofL ¹(G) in a Hilbert space. Our main result: A locally compact group G, which is representable as a projective limit of a system of factor groups G/k, k compact normal subgroups, lies in [W] if and only if all the G/k are in [W].     

Duals of orbit spaces in groups with relatively compact inner automorphism groups are hypergroups

For a locally compact groupG and a groupB of topological automorphisms containing the inner automorphisms ofG and being relatively compact with respect to Birkhoff topology (that isG[FIA] _B,B I(G)) the spaceG _B of -orbits is a commutative hypergroup (=commutative convo inJewett's terminology) in a natural way asJewett has shown. Identifying the space of hypergroup characters ofG _B withE(G, B) (the extreme points ofB-invariant positive definite continuous functionsp withp (e)=1, endowed with the topology of compact convergence) we prove thatE(G, B) is a hypergroup, the hypergroup dual ofG _B.     

P-amenable locally compact groups

LetG denote a locally compact group andP(G) [P _c (G)] the topological semigroup of absolutely continuous [absolutely continuous and compactly supported] probability measures onG. We say thatG isP-amenable [P _c -amenable] if the topological semigroupP(G)[P _c (G)] is amenable. Some combinatorial properties of this class of groups are studied. The relationship between amenability andP-amenability ofG is investigated. It is shown that for a connected solvableP _c -amenable groupG, G has polynomial growth if, and only if,P _c (S)={fP _c (G)suppfS} is amenable for any open subsemigroupS ofG.This paper contains a part of the author's doctoral thesis at the State University of New York at Albany. The author wishes to thank ProfessorJoe Jenkins for his valuable suggestions and encouragement during the course of this work.     

Sull'esistenza delle soluzioni delle equazioni differenziali ordinarie in forma implicita negli spazi di Banach

Summary This paper deals with the existence of solutions for the implicit Cauchy problem F(t, x, x)=_B, x(t₀)=x₀ in a Banach space B. By using the Kuratowski and the Hausdorff measure of non compactness, we prove an existence theorem for the previous problem (Teorema 1.1) and its extension to non compact intervals (Teorema 2.1). These results generalize the previous ones by R.Conti [1] (in the case B=R), G.Pulvirenti [2] and T. Dominguez Benavides [3], [4] (in the general case). In particular, we relax a Lipschitz condition assumed by all of the abovementioned authors. Some applications of Teorema 2.1 are presented.

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Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.     

Some remarks on automorphisms of bounded displacement and bounded cocycles

Some time agoTits has considered the question of characterizingB(G), and more generally, of finding the automorphisms of bounded displacement ofG, particularly whenG is a connected real Lie group. Our purpose here is to extend these results to various other cases as well as to deal with the analogous questions for 1-cocycles. We concern ourselves, among other things, with the question of sufficient conditions forB(G)=Z(G), the center, or more generally, forG to have no non-trivial automorphisms of bounded displacement. The significance of such conditions can be seen in work of the author together withF. Greenleaf andL. Rothschild where, for example, the Selberg form of the Borel density theorem is considerably generalized. These conditions are therefore closely related to, but not identical with, sufficient conditions for the Zariski density of a closed subgroupH ofG withG/H having finite volume, see [11]. For this reason it is enlightening to compare these results with those of [11]. On the cocycle level, we give sufficient conditions for the points with bounded orbit under a linear representation to be fixed and more generally, for a bounded 1-cocycle to be identically zero. These conditions actually play a role in [11]; they are among the sufficient conditions necessary to establish Zariski density ofH inG. We also deal with certain converse questions and applications to homogeneous spaces of finite volume. For example, ifG/H has finite volume and is an automorphism ofG leavingH pointwise fixed, then has bounded displacement. If is a 1-cocycle and /H is trivial, then is itself bounded.Research partially supported by NSF Grant=MPS 75-08268.     

Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen

In this paper we study the class of all locally compact groupsG with the property that for each closed subgroupH ofG there exists a pair of homomorphisms into a compact group withH as coincidence set, and the class of all locally compact groupG with the property that finite dimensional unitary representations of subgroups ofG can be extended to finite dimensional representations ofG. It is shown that [MOORE]-groups (every irreducible unitary representation is finite dimensional) have these two properties. A solvable group in is a [MOORE]-group. Moreover, we prove a structure theorem for Lie groups in the class [MOORE], and show that compactly generated Lie groups in [MOORE] have faithful finite dimensional unitary representations.     

Locally compact groups acting on trees and propertyT

FollowingKazhdan, a separable locally compact groupG is said to have propertyT if the trivial representation is isolated in the dual space,, of equivalence classes of continuous irreducible unitary representations ofG. We generalize results ofMargulis—Tits by showing that groups which have propertyT can not be amalgams.Research supported by NSF.     

A generic existence theorem for convective motions in a viscous fluid

Summary Some years ago W.Velte [1]and A.Marino [2]have studied the problem of convective motions in a incompressible viscous fluid in dimensions two and three, respectively, focusing their interest in the case of nonuniqueness of the solution of the nonlinear system. They look at it as a bifurcation problem and prove the existence of solutions bifurcating from same line of trivial solutions under the hypothesis that the linearized operator has eigenvalues of odd multiplicities. They observe that the operators involved are not potential operators, thus variational tools can not be applied. In this paper, we prove that, in case of dimension two, for almost every domain , all the eigenvalues of the linearized operator are simple. Our procedure is related to that in [3];the fact that we deal with the system here, necessitates, however, some basic changes.Supported in part by C.N.R. (G.N.A.F.A.).     

Über Mittelwerte multiplikativer zahlentheoretischer Funktionen mehrerer Variablen

In [8] the author extended the concept of neighbouring functions (cp. [9]) to the case of several variables. Using these results it is shown that under some weak conditions a multiplicative functionf in two variables has a mean-value different from zero if and only if the two multiplicative functionsf ₁(n)=f(n, 1) andf ₂(n)=f(1,n) have mean-values different from zero. Applications to theorems ofDelange [3],Elliott [6] andDaboussi [1] are given.     

Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

On the row sum criterion and the convergence of some iterative processes

Summary The well known row sum criterion (in German Zeilensummen-kriterium) is reformulated for transformations on Banach spaces with cones, and some sufficient conditions guaranteeing the convergence ofJacobi, Gauss-Seidel and successive over-relaxation iterations are derived, similar to those given byWalter [12] andKulisch [3] for linear algebraic systems.     

Zum Nachweis arithmetischer Cohen-Macaulay Varietäten

There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example [4], [10], [11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner [3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner [7], p. 35 and, using our methods, we rather easily obtain the result of [7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay [5] (see also [6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.

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Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet     

Linearly independent zeros of quadratic forms over number-fields

LetK be an algebraic number-field of degree [K:Q] =n 1 and letO denote some fixed order ofK. Let, be a quadratic form which represents zero for some. For the special caseK =Q,O =Z, theorems ofCassels and ofDavenport provide estimates for the magnitude (in terms of the coefficients off(x)) of a zero and of a pair of linearly independent zeros off, respectively. Recently,Raghavan extendedCassels' result to arbitraryK. In this article, a new proof ofDavenport's theorem for a pair of linearly independent zeros is given which not only provides explicit constants in the estimates but also extends to generalK. A refinement of this proof leads to effectively computable bounds for rational representations of a numbern0 byf.     

A short proof of a theorem of Ruelle

Using the existence of conformal measures in the sense ofSullivan [4] andDenker, Urbaski [2] we give a simple proof of Ruelle's Theorem, that the Hausdorff dimension of a hyperbolic Julia set is the solution of the equationP(R,–tlog|R|)=0.     

Linear equations depending differentiably on a parameter

We investigate operator functionsT(x) inBanach spaces, depending differentiably (meaning of classC orC ) on a parameterx and enjoying a certain regularity property. Iff is a given differentiable function such that the equationT(x)e=f(x) is solvable for eachx then the existence of a functione is proved which belongs to the same differentiability class asf andT, solving the equationT(x)e(x)f(x) identically inx. As an application we extend a result ofJ. Leiterer [9] and give a comprehensive answer to a question posed byJ.L. Taylor in [15] concerning the exactness of certain cochain complexes.     

Convergence of theQR algorithm

TheQR algorithm ofJ. G. F. Francis is used in computing matrix eigenvalues. The convergence proof given here is an analogue ofRutishauser's proof of the convergence of theLR algorithm, but our proof covers the case of disorder of the eigenvalues.The work presented in this paper was supported by the AEC Computing and Applied Mathematics Center, Courant Institute of Mathematical Sciences, New York University, under contract AT(30-1)-1480 with the U.S.Atomic Energy Commission.     

A condition for absolute continuity of infinitely divisible distribution functions

Summary Throughout this paper the symbols r.v., d.f., ch.f., and i.d. will stand, respectively, for random variable, distribution function, characteristic function, and infinitely divisible.Let F(x) be an i.d.d.f. Hartman and Wintner [5] and Blum and Rosenblatt [1] have given a condition, necessary and sufficient, for F(x) to be a continuous d.f. In this note a sufficient condition for F(x) to be an absolutely continuous d.f. is given.Research supported by ONR Contract No. NONR-285(46).Research supported in part by a National Science Foundation fellowship.     

A sequence has almost nowhere small discrepancy

LetD _n (x) be the discrepancy function of a sequence in [0,1) as defined in the first lines of the introduction. A well known result ofW. M. Schmidt states that limsup _n sup _x |D _n (x)|/logn1/100 for every sequence. In this paper it is proved that for every sequence sup _x |D _n (x)|/logn1/100 for almost all values ofn. A more refined assertion is given in Theorem 1, a more general one in Theorem 2, while it is proved in Theorem 3 that these results are essentially the best possible. Another result ofSchmidt is that for every sequence limsup _n|D _n (x)|/loglogn1/2000 for almost allx in [0, 1). In this paper it is shown that for every sequence limsup _n|D _n (x)|/logn1/400 for almost allx in [0, 1). Theorem 4 contains a more refined and Theorem 5 a more general statement. Finally Theorem 6 implies that also these results are essentially the best possible.     

Regularity properties of functional equations and inequalities

Summary By a well-known theorem of Lebesgue and Fréchet every measurable additive real function is continuous. This result was improved by Ostrowski who showed that a (Jensen-) convex real function must be continuous if it is bounded above on a set of positive Lebesgue measure. Recently, R. Trautner provided a short and elegant proof of the Lebesgue—Fréchet theorem based on a representation theorem for sequences on the real line.We consider here a locally compact topological groupX with some Haar measure. Then the following generalizes Trautner's theorem: Theorem.Let M be a measurable subset of X of positive finite Haar measure. Then there is a neighbourhood W of the identity e such that for each sequence (z _n )in W there is a subsequence (z _nk )and points y and x _k in M with z _nk =x _k ·y ^–1 for k . Using this theorem we obtain the following extensions of the theorems of Lebesgue and Fréchet and of Ostrowski. Theorem.Let R and T be topological spaces. Suppose that R has a countable base and that X is metrizable. If g: X R and H: R × X T are mappings where g is measurable on a set M of positive finite Haar measure and H is continuous in its first variable, then any solution f: X T of f(x · y) = H(g)(x), y) for x, yX is continuous. Theorem.Let G: X × X be a mapping. If there is a subset M of X of positive finite Haar measure such that for each yX the mapping x G(x, y) is bounded above on M, then any solution f: x of f(x · y) G(x, y) for x, yX is locally bounded above. We also prove category analogues of the above results and obtain similar results for general binary mappings in place of the group operation in the argument off.