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.     

On covering designs with block size 5 and index 5

LetV be a finite set of order . A (, , ) covering design of index and block size is a collection of -element subsets, called blocks, such that every 2-subset ofV occurs in at least blocks. The covering problem is to determine the minimum number of blocks, (, , ), in a covering design. It is well known that , where [x] is the smallest integer satisfyingx[X]. It is shown here that (, 5, 5)=(, 5, 5) for all positive integers 5 with the possible exception of =24, 28, 56, 104, 124, 144, 164, 184.     

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     

Blocking Sets in (υ,{2, 4}, 1)-Designs

The existence of blocking sets in (, {2, 4}, 1)-designs is examined. We show that for 0, 3, 5, 6, 7, 8, 9, 11 (mod 12>), blocking sets cannot exist. We prove that for each 1, 2, 4 (mod 12) there is a (, {2, 4}, 1)-design with a blocking set with three possible exceptions. The case 10 (mod 12) is still open; we consider the first four values of in this situation.     

A finitely based theory of a non-trivial language,with exactly ℵ0 subcovers

LetL=f, g be the language with two unary operation symbols. I prove that the finitely based equational theory =[f₀=₀] ofL covers exactly ₀ others.Presented by S. Burris.Dedicated to George McNulty, my mentor in equational logic.     

An evolution problem for plastic antiplanar shear

We study the parabolic problemu/t=div_{x p}() in the function space L²(0, T; BV()). Here is the linear growth functional arising from the study of plastic antiplanar shear deformations.     

Signed Total Domination Nnumber of a Graph

The signed total domination number of a graph is a certain variant of the domination number. If is a vertex of a graph G, then N() is its oper neighbourhood, i.e. the set of all vertices adjacent to in G. A mapping f: V(G)-1, 1, where V(G) is the vertex set of G, is called a signed total dominating function (STDF) on G, if for each V(G). The minimum of values , taken over all STDF's of G, is called the signed total domination number of G and denoted by _st(G). A theorem stating lower bounds for _st(G) is stated for the case of regular graphs. The values of this number are found for complete graphs, circuits, complete bipartite graphs and graphs on n-side prisms. At the end it is proved that _st(G) is not bounded from below in general.     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.     

An inequality linking packing and covering densities of plane convex bodies

For every convex body K in R ², let (K) denote the packing density of K, i.e. the density of the tightest packing of congruent copies of K in R ², and let (K) denote the covering density of K, i.e. the density of the thinnest covering of R ² with congruent copies of K. It is shown here that 4(K)3(K) for every convex body K in R ². This inequality is the strongest possible, since if E is an ellipse, then the equality 4(E)=3(E) holds. Two corollaries are presented, and a summary of known bounds for packing and covering densities is given.     

Local controllability of observables in finite and infinite dimensional nonlinear control systems

LetA, be evolution operators (possibly nonlinear) which act within a Banach spaceB andu(·) a measurable, real valued, control function. We study control systems of the form _t /t=A _t +u(t) _t , ₀= B. An observation of this system is defined to be a continuous linear mapg:B ^k . Our main result gives a computable sufficient condition to assure that fort > 0 and sufficiently small, the observation of the reference solution (which corresponds tou(t)0) at timet is interior to the set of observations of all solutions at timet. An example to illustrate the theory is the local controllability, via tension, of various observations of a vibrating string.This research was supported by NSF Grant MCS 76-04419 A01.     

Large Sets of Mutually Almost Disjoint Steiner Triple Systems Not From Steiner Quadruple Systems

We construct what we believe to be the first examples of large sets of MAD STS( ) which are not obtained from Steiner quadruple systems     

Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Bounded isometries and homogeneous Riemannian quotient manifolds

Let M be a Riemannian manifold that admits a transitive semisimple group G of isometries, G of noncompact type. Then every bounded isometry of M centralizes G and so is a Clifford translation (constant displacement). Thus a Riemannian quotient M is homogeneous if and only if consists of Clifford translations of M. The technique of proof also leads to a determination of the group of all isometries of M.IMAF, Córdoba, Argentina. Partially supported by Conicet, Argentina, and by IMPA, Rio de Janeiro, Brazil.IMAF, Córdoba, Argentina. Partially supported by Conicet, Argentina.University of California at Berkely, U.S.A. Partially supported by National Science Foundation Grant DMS-8200235.     

Induktive Definitionen und Dilatoren

Summary In this paper we give a new and comparatively simple proof of the following theorem by Girard [1]:If x y (x,y) (where the relation is arithmetic and positive in Kleene's ), then there exists a recursive DilatorD such that x ^<y ^{(x, y).The essential feature of our proof is its very direct definition of the dilatorD. Within a certain infinitary cutfree system of inductive logic (which in fact is a modification of Girard's system in [1]) we construct in a uniform way for each ordinal a derivation T of the formula x <y (x, y), and then defineD immediately from the family (T)_On. Especially we set D():=Kleene-Brouwer length of (T).}     

Four-dimensional compact projective planes with doubly transitive action on the fixed pencil

We consider a four-dimensional compact projective plane =( , ) whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag pW, acts transitively on _p \{W}, and fixes no point in the set W{p}. We study the actions of and N on and on the pencil _p \{W}, in the case that does not contain a three-dimensional elation group. In the special situation that acts doubly transitively on _p {W}, we will determine all possible planes . There are exactly two series of such planes.     

On the stochastic independence properties of hard-core distributions

A probability measurep on the set of matchings in a graph (or, more generally 2-bounded hypergraph) ishard-core if for some : [0,), the probabilityp(M) ofM is proportional to . We show that such distributions enjoy substantial approximate stochastic independence properties. This is based on showing that, withM chosen according to the hard-core distributionp, MP () the matching polytope of , and >0, if the vector ofmarginals, (Pr(AM):A an edge of ), is in (1–) MP (), then the weights (A) are bounded by someA(). This eventually implies, for example, that under the same assumption, with fixed, as the distance betweenA, B tends to infinity.Thought to be of independent interest, our results have already been applied in the resolutions of several questions involving asymptotic behaviour of graphs and hypergraphs (see [14, 16], [11]–[13]).Supported in part by NSFThis work forms part of the author's doctoral dissertation [16]; see also [17]. The author gratefully acknowledges NSERC for partial support in the form of a 1967 Science and Engineering Scholarship.     

On Fourier multipliers between Besov spaces with 0 <p o ≤ min{1,p 1}

, 0<0 min{1,p1}.     

The Selberg trace formula for SL(3,Z)

In this article the first step toward the generalization of the Selberg trace formula to the case of a rank 2 symmetric space S and a discrete group for which the fundamental region \S goes to infinity nontrivially appears. For S we use the space SL(3,)/SO(3) and for we use SL(3,). The fundamental results are Theorems 9 and 10, in which is calculated the contribution to the matrix trace of the operator K which appears in the right side of the trace formula of the expression h()d^c(), where ^c() is the continuous part of the spectral measure of the quasiregular representation on the space IL₂(\S).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 63, pp. 8–66, 1976.     

Random time change and an integral representation for marked stopping times

Summary Consider the set of all possible distributions of triples (, , ), such that is a finite stopping time with associated mark in some fixed Polish space, while is the compensator random measure of (, ). We prove that is convex, and that the extreme points of are the distributions obtained when the underlying filtration is the one induced by (, ). Moreover, every element of has a corresponding unique integral representation. The proof is based on the peculiar fact that EV _, =0 for every predictable processV which satisfies a certain moment condition. From this it also follows thatT _, isU(0, 1) wheneverT is a predictable mapping into [0, 1] such that the image of , a suitably discounted version of , is a.s. bounded by Lebesgue measure. Iterating this, one gets a time change reduction of any simple point process to Poisson, without the usual condition of quasileftcontinuity. The paper also contains a very general version of the Knight-Meyer multivariate time change theorem.Research supported by NSF grant DMS-8703804     

Parameter-dependent renewal theorems with applications

Let ₁, ₂, ... be a sequence of i.i.d. random variables with positive mean and finite variance and letr(b), b0, be real numbers tending to 0 asb . Definings _n=₁+...+_n andS _n=S_n(b)=s_n+r(b)n, the stopping time =(b)=inf {n>/1:S_n >b} whereb=b(b) , will be considered with special regard to the excess over the boundaryR _b=s+r(b)–b. It turns out that the limiting distribution ofR _b is the same as in the caser(b)0 for allb. Proving this, Blackwell's renewal theorem and its integral version have to be established first in the above stated situation. Finally, an expansion ofE to vanishing terms asb will be provided and applied to some examples arising in economics.

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Zusammenfassung Seien ₁, ₂, ... unabhängige identisch verteilte Zufallsgrößen mit positivem Erwartungswert und endlicher Varianz sowier(b), b0, reelle Zahlen mitr(b)0 für b. Sei ferners ₁, s₂, ... der zugehörige Summenprozeß,S _n= S_n(b)=s_n+r(b)n fürn1 und =(b)=inf {n1: S_n>b, wobeib=b(b) fürb . Es wird gezeigt, daß die asymptotische Verteilung des ExzessesR _b=s +r(b) –b mit der im Fallr(·)0 übereinstimmt. Dazu werden sowohl das Blackwellsche Erneuerungstheorem als auch seine Integralversion in der vorher beschriebenen parameterabhängigen Situation geeignet formuliert und bewiesen. Als Folgerung ergibt sich dann eine asymptotische Entwicklung vonE(b) fürb bis zu Termen o(1). Anh- and einiger Beispiele aus dem ökonomischen Bereich wird schließlich noch aufgezeigt, wo Approximationen fürE(b) von Interesse sein können.