Superrigidity of lattices in solvable Lie groups

Summary Let be a closed, cocompact subgroup of a simply connected, solvable Lie groupG, such that Ad _G has the same Zariski closure as AdG. If : GL _n () is any finite-dimensional representation of , we show that virtually extends to a representation ofG. (By combining this with work of Margulis on lattices in semisimple groups, we obtain a similar result for lattices in many groups that are neither solvable nor semisimple.) Furthermore, we show that if is isomorphic to a closed, cocompact subgroup of another simply connected, solvable Lie groupG, then any isomorphism from to extends to a crossed isomorphism fromG toG. In the same vein, we prove a more concrete form of Mostow's theorem that compact solvmanifolds with isomorphic fundamental groups are diffeomorphic.Oblatum 5-VII-1994 & 15-IV-1995     

K-theory for finite covariant systems

LetA be a real or complex Banach algebra and assume that is an action of a finite groupG onA by means of continuous automorphisms. To such a finite covariant system (A, G, ), we associate an Abelian groupK(A, G, ). We obtain some classical exact sequences for an algebraA and a closed invariant idealI. We also compute the group in a few important special cases. Doing so, we relate our new invariant to the classicalK ₀ andK ₁ of a Banach algebra and to theK-theory of ₂-graded Banach algebras. Finally, we obtain a result that gives a close relationship of our groupK(A, G, ) with theK-theory of the crossed productA G. In particular, we prove a six-term exact sequence involving our groupK(A, G, ) and theK-groups ofA G. In this way, we hope to contribute to the well-known problem of finding theK-theory of the crossed productA G in the case of an action of a finite group.     

A remark on generalized iterates

Letf be an invertible function on the real lineR, and letZ denote the set of integers. For eachx Z, letf ^|n| denote then'th iterate off. Clearlyf ^|m|(f ^|n|(x))=f ^|m+n|(x) for allm,nZ and allxR. LetG be any group of orderc, the cardinality of the continuum, which contains (an isomorphic copy of)Z as a normal subgroup. If for eachxR, the iteration trajectory (orbit) ofx is non-periodic, then there exists a set of invertible functionsF={F ^||:G} on the real line with the properties (i)F ^||(F ^||(x))=F ^|+| (x) for allxR and (ii)F ^|n|(x)=f ^|n|(x) for allnZ andxR. That is,f can be embedded in a set ofG-generalized iterates. In particular,f can be embedded in a set of complex generalized iterates.Dedicated to Professor Janos Aczél on his 60th birthday     

Symmetriegerechte Basisvektoren für Permutationsdarstellungen aller endlichen Punktgruppen der Dimensionen 3 und 2

Zusammenfassung Es seiG eine endliche Untergruppe der orthogonalen Gruppe (det=±1) des ^k mitk=2 oder 3 undN eine endliche Menge von Punkten des ^k , welche unterG invariant ist. Dies gibt Anlass zu einer Permutationsdarstellung vonG im Vektorraum der komplexen Funktionen aufN.In Abschn. 3 wird für eine symmetriegerechte Basis angegeben. Dabei sind die Funktionswerte jeweils exakt tabelliert.

---

Let G be a finite subgroup of the orthogonal group (det=±1) of ^k wherek=2 or 3 and letN be a finite set of points of ^k , which is invariant underG. In this way one gets a permutation representation ofG in the vector space of the complex functions onN.In Section 3, a symmetry adapted basis is given for , where the function values are tabulated exactly.

---

---

Im Buch [1] wurden lediglich die Diedergruppen behandelt (in Abschn. 3.1).     

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

Nonlinear eigenvalue approximation

Summary For each in some domainD in the complex plane, letF() be a linear, compact operator on a Banach spaceX and letF be holomorphic in . Assuming that there is a so thatI–F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I–F(),F()] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I–F(),I] with algebraic multiplicitym. For fixed leth() denote the arithmetic mean of them eigenvalues of the pencil [I–F(),I] which are closest to 0. Thenh is holomorphic in a neighborhood of andh()=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.     

Interpolation problems and Toeplitz operators on multiply connected domains

LetR be a bounded domain in the complex plane bounded by n + 1 nonintersecting analytic Jordan curves, letE, F, andG be flat unitary vector bundles (in the sense of Abrahamse and Douglas) and let :F G and :E G be bounded analytic bundle maps. A condition is given for the existence of a bounded analytic map D:E F such that D = , together with an estimate for D. An interesting special case is the case whereE = G and = I _E , for which the condition involves a uniform lower bound for a class of Toeplitz operators overR, all of which are induced (formally) by the bundle map (N = rankE). When interpreted for a finite column of analytic scalar functions, this special case gives quantitative information on the corona theorem forR. The main tool is the Sz.Nagy-Foias commutant lifting theorem for regionsR recently obtained by the author.Research supported by National Science Foundation Grant No. MCS 77-00966.     

On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

Compact abelian groups of automorphisms of simpleC *-algebras

If is a continuous representation of a compact abelian groupG as *-automorphisms of a prime or simpleC ^*-algebraA we study how algebraic properties of the fixed-point algebraA ₀ are reflected in properties of the spectrum of and outerness of ₁,tG. For example, ifA is prime, ()=Sp() if and only ifA ₀ is prime, and the same with simple (instead of prime) whenG is finite. In the latter case the relative commutant ofA ₀ inA is trivial if and only if all ₁l are outer, and ifG is cyclic of prime order all the conditions above are equivalent. This paper replaces the earlier versions [12] and [9].     

Choosing the maximum from a sequence with a discount function

Choosing the maximum value from a sequence ofN independent values is a well known problem often called the candidate problem or secretary problem. This paper treats the above problem with a discount penalty (0<<1) for each additional observation taken. It is shown that asN increases indefinitely, the optimal stopping policy is bounded although the maximum expected payoff goes to zero, and that there exists a sequence 0= ₀<₁<₂<<1, such that the asymptotic optimal stopping rule is the same for all _i–1<_i.     

On the positive definiteness of n ↦ epnα for α close to 2

For >2, let Q ⁺() be the infimum of those q>0 for which the function n e^pn is positive definite on N ₀ for every pq. We shall prove that Q ⁺()0 as 2.     

Ouverts d"harmonicité pour les fonctions séparément harmoniques

Let D ^N , G ^M be two open sets, E D and F G two compact sets which satisfy the condition (H) (that is a harmonic condition similar to Leja"s condition). We find an open set ^N+M such that each separately harmonic function f : X : = (D× F) (E × G) (i.e.: for all x in E, f(x,.) is harmonic on G; for all y in F, f(., y) is harmonic on D) extends to a harmonic function on .     

Reguläre Hjelmslev-Homomorphismen

Zusammenfassung Es werden Homomorphismen von Hjelmslev-Gruppen betrachtet und der Begriff des regulären Hjelmslev-Homomorphismus eingeführt. Ein Hjelmslev-Homomorphismus : (G, S)(G, S) heißt regulär, wenn für jede Drehung in (G, S) gilt: F()={C}F()={C}. Dabei bedeutet F()={C}, daß C der einzige Fixpunkt von ist. Reguläre Hjelmslev-Homomorphismen lassen sich unter Voraussetzung des Gitteraxioms durch homogene Überdeckungen aus vollständigen Flecken charakterisieren. Wichtige Spezialfälle sind die zu Drehungen konstruierten Hjelmslev-Homomorphismen. Wenn die Großgeometrie von (G, S) existiert, bilden die Kerne der regulären Hjelmslev-Homomorphismen, für die das Zentrum der geraden Bewegungen in der Bildgruppe gleich Eins ist, einen vollständigen Verband mit Z(S^ger) als kleinstem und der Gruppe der Nachbarbewegungen als größtem Element. Auf der analytischen Seite entsprechen diesen Hjelmslev-Homomorphismen die durch echte Ideale des Koordinatenringes induzierten Homomorphismen. Reguläre Hjelmslev-Homomorphismen eignen sich dazu, das Axiom von der Eindeutigkeit der Verbindungsgeraden innerhalb einer allgemeineren Theorie zu studieren.Friedrich Bachmann zum siebzigsten Geburtstag     

Weight functions on the Kneser graph and the solution of an intersection problem of Sali

LetX, Y be finite sets and suppose thatF is a collection of pairs of sets (F, G),FX,GY satisfying |FF|s, |GG|t and |FF|+|GG|s+t+1 for all (F, G),F, GF. Extending a result of Sali, we determine the maximum ofF.     

On recurrence

Summary LetT be a non-singular ergodic automorphism of a Lebesgue space (X,L,) and letf: X be a measurable function. We define the notion of recurrence of such a functionf and introduce the recurrence setR(f)={:f– is recurrent}. If , then R()={0}, but in general recurrence sets can be very complicated. We prove various conditions for a number to lie in R(f) and, more generally, forR(f) to be non-empty. The results in this paper have applications to the theory of random walks with stationary increments.     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

Invariant equations defining coincident root loci

Let F(x) = xⁿ+₁ x^n-1+₂ x^n-2+ ··· +_n be a polynomial with complex coefficients, and suppose we are given a partition (₁,...,_r) of n. It is a classical problem to determine explicit algebraic conditions on the _i so that F may have roots with multiplicities ₁,...,_r. We give an invariant theoretic solution to this problem, to wit, we exhibit a set of covariants of F whose vanishing is a necessary and sufficient condition. The construction of such covariants is combinatorial, and involves associating a set of graphs on n vertices (called decisive graphs) to each .Received: 28 September 2003     

Mouvement brownien,cônes et processus stables

Summary LetW=(W _t, t0) denote a two-dimensional Brownian motion starting at 0 and, for 0<<, letC be a wedge in ² with vertex 0 and angle 2. We consider the set of timest's such that the path ofW, up to timet, stays inside the translated wedgeW _t-C. It follows from recent results of Burdzy and Shimura that this set, which we denote byH , contains nonzero times if, and only if, >/4. Here we construct a measure, a local time, supported onH . For /4W, time-changed by the inverse of this local time, is shown to be a two-dimensional stable process with index 2-/2. This results extends Spitzer's construction of the Cauchy process, which is recovered by taking =/2. A formula which describes the behaviour ofW before a timetH is established and applied to the proof of a conjecture of Burdzy. We also obtain a two-dimensional version of the famous theorem of Lévy concerning the maximum process of linear Brownian motion. Precisely, for 0<S _t denote the vertex of the smallest wedge of the typez-C which contains the path ofW up to timet. The processS _t-W_t is shown to be a reflected Brownian motion in the wedgeC , with oblique reflection on the sides. Finally, we investigate various extensions of the previous results to Brownian motion inR ^d, d3. LetC be the cone associated with an open subset of the sphereS ^d-1, and letH be defined asH above. Sufficient conditions are given forH to contain nonzero times, in terms of the first eigenvalue of the Dirichlet Laplacian on .