Parametrices for operators of tricomi type

Sunto Si costruisce una parametrice per il problema di Dirichlet relativo ad operatori P del tipo (1/i(/t))² + t_x, nella regione t0. Tenuto conto che per tali operatori è nota la parametrice del problema di Cauchy nella regione t 0, ciò consente di costruire una soluzione (modO ) dell'equazione Pu=f , t (–T, T), T>0.     

A graded scale of parametric families of distributions,and parameter estimates based on the sample mean

Summary Denote by _k a class of familiesP={P} of distributions on the line R¹ depending on a general scalar parameter , being an interval of R¹, and such that the moments µ₁()=xdP ,...,µ_2k ()=x ^2k dP are finite, ₁ (), ..., _k (), _k+1 () ..., _k () exist and are continuous, with ₁ () 0, and _j +1 ()= ₁ () _j () +[₂() -₁()²] _j ()/ ₁ (), J=2, ..., k. Let ₁=¯x=x ₁ + ... +x _n/n, ₂=x ₁ ² + ... +x _n ²/n, ..., _k =(x ₁ ^k + ... +x _n ^k/n denote the sample moments constructed for a sample x₁, ..., x_n from a population with distribution Pg. We prove that the estimator of the parameter by the method of moments determined from the equation ₁= ₁() and depending on the observations x₁, ..., x_n only via the sample mean ¯x is asymptotically admissible (and optimal) in the class _k of the estimators determined by the estimator equations of the form ₀ () + ₁ () ₁ + ... + _k () _k =0 if and only ifP _k .The asymptotic admissibility (respectively, optimality) means that the variance of the limit, as n (normal) distribution of an estimator normalized in a standard way is less than the same characteristic for any estimator in the class under consideration for at least one 9 (respectively, for every ).The scales arise of classes _{1 2}... of parametric families and of classes _{1 2} ... of estimators related so that the asymptotic admissibility of an estimator by the method of moments in the class _k is equivalent to the membership of the familyP in the class _k .The intersection consists only of the families of distributions with densities of the form h(x) exp {C₀() + C₁() x } when for the latter the problem of moments is definite, that is, there is no other family with the same moments ₁ (), ₂ (), ...Such scales in the problem of estimating the location parameter were predicted by Linnik about 20 years ago and were constructed by the author in [1] (see also [2, 3]) in exact, not asymptotic, formulation.Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei, pp. 41–47, 1981.     

On the representation type of one point extensions of tame concealed algebras

Letk be an algebraically closed field and a finite dimensionalk-algebra. Letq be the quadratic Tits form associated with . If is tame we show thatq is weakly semipositive. Let be a one-point extension of a tame concealed algebra, then is tame iffq is weakly semipositive.     

1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let be a graph with diameter d 2. Recall is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of the distance partition{{z V() | (z, y) = i, (x, z) = j} | 0 i, j d}is equitable and its parameters do not depend on the edge xy. Let be 1-homogeneous. Then is distance-regular and also locally strongly regular with parameters (v,k,,), where v = k, k = a ₁, (v – k – 1) = k(k – 1 – ) and c ₂ + 1, since a -graph is a regular graph with valency . If c ₂ = + 1 and c ₂ 1, then is a Terwilliger graph, i.e., all the -graphs of are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c ₂ 2 and obtained that there are only three such examples. In this article we consider the case c ₂ = + 2 3, i.e., the case when the -graphs of are the Cocktail Party graphs, and obtain that either = 0, = 2 or is one of the following graphs: (i) a Johnson graph J(2m, m) with m 2, (ii) a folded Johnson graph J¯(4m, 2m) with m 3, (iii) a halved m-cube with m 4, (iv) a folded halved (2m)-cube with m 5, (v) a Cocktail Party graph K _{m × 2} with m 3, (vi) the Schläfli graph, (vii) the Gosset graph.     

The martin boundary for harmonic functions on groups of automorphisms of a homogeneous tree

Consider a closed subgroup of the automorphism group of a homogeneous treeT, and assume that acts transitively on the vertex set. Suppose that is a probability measure on which has continuous density with respect to Haar measure and whose support is compact open and generates as a closed semigroup. It is shown that the Martin boundary of with respect to the random walk with law coincides with the space of ends ofT. This extends known results for free groups and applies, for example, to the affine group over a non archimedean local field.     

Tessellations generated by hyperplanes

Let be a locally finite system of hyperplanes in ^d with the property that the cells of the induced cell complex decomposition of ^d have uniformly bounded diameters. If is simple and the density of the vertices in exists, then the density of thek-cells in exists and can be given explicitly (k = 1, ...,d). Also, the mean number ofj-faces of thek-cells in exists and can be calculated. For certain nonsimple systems , corresponding inequalities are obtained.     

Additive functions

1<q<2 L:= _n=1 1/q ⁿ=1/q–1. [0,1] _n()=1, A _n:= _i=1 ^n–1 _i(x)/q^i+1/n x _n(x)=0, _n>. , = _{n=1 n}(x)/qⁿ. F: [0,L]R , F(x)= _{n=1 n}(x)a_n, _n=1 ¦a _n¦<. [0,L]. q(1,2), . , q(1, 2), . .     

Homothetic floating bodies

We show that a convex bodyK in ⁿ is homothetic to an ellipsoid if there is a sequence { _k }k converging to 0 so thatK is homothetic to its floating bodiesK _k.Supported by NSF grant DMS-9108003.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

L 2-approximation by the translates of a function and related attenuation factors

Summary Let be thek-dimensional subspace spanned by the translates (·–2j/k),j=0, 1, ...,k–1, of a continuous, piecewise smooth, complexvalued, 2-periodic function . For a given functionfL ²(–, ), its least squares approximantS _kf from can be expressed in terms of an orthonormal basis. Iff is continuous,S _kf can be computed via its discrete analogue by fast Fourier transform. The discrete least squares approximant is used to approximate Fourier coefficients, and this complements the works of Gautschi on attenuation factors. Examples of include the space of trigonometric polynomials where is the de la Valleé Poussin kernel, algebraic polynomial splines where is the periodic B-spline, and trigonometric polynomial splines where is the trigonometric B-spline.     

A variational principle for eigenvalues of pencils of Hermitian matrices

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number is an eigenvalue ofH ifdet(A–B)=0 (we include = ifdetB=0). For nonsingularH (i.e., for which some is not an eigenvalue), we show precisely which eigenvalues can be characterized as _k ⁺ =sup{inf{*A:*B=1,S},SS _k},S _k being the set of subspaces of C ^N of codimensionk–1.Dedicated to the memory of our friend and colleague Branko NajmanResearch supported by NSERC of Canada and the I.W.Killam FoundationProfessor Najman died suddenly while this work was at its final stage. His research was supported by the Ministry of Science of CroatiaResearch supported by NSERC of Canada     

On the smoothness of the moduli space of mathematical instanton bundles

In this paper we prove that the moduli spaces MI _2n+1(k) of mathematical instanton bundles on ²ⁿ⁺¹ with quantum number k are singular for n 2 and k 3 ,giving a positive answer to a conjecture made by Ancona and Ottaviani in 1993.     

On the divisibility of homogeneous hypergraphs

We denote byK _k ,k, 2, the set of allk-uniform hypergraphsK which have the property that every element subset of the base ofK is a subset of one of the hyperedges ofK. So, the only element inK ₂ ² are the complete graphs. If is a subset ofK _k then there is exactly one homogeneous hypergraphH whose age is the set of all finite hypergraphs which do not embed any element of . We callH -free homogeneous graphsH _n have been shown to be indivisible, that is, for any partition ofH _n into two classes, oue of the classes embeds an isomorphic copy ofH _n . [5]. Here we will investigate this question of indivisibility in the more general context of-free homogeneous hypergraphs. We will derive a general necessary condition for a homogeneous structure to be indivisible and prove that all-free hypergraphs for K _k with 3 are indivisible. The-free hypergraphs with K _k ² satisfy a weaker form of indivisibility which was first shown by Henson [2] to hold forH _n . The general necessary condition for homogeneous structures to be indivisible will then be used to show that not all-free homogeneous hypergraphs are indivisible.This research has been supported by NSERC grant 69–1325.     

On a Generalization of Nikolskij's Extension Theorem in the Case of Two Variables

A modification of the Nikolskij extension theorem for functions from Sobolev spaces H ^k() is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary k however, it is restricted to the case of two-dimensional bounded domains.     

On certain imbedding and extension theorems for a weighted class of functions

n- , S n–1 m (0m<n), () S . () , ().     

Onp-Helson sets in R n

p- . E R ⁿ -, f () _p(R ⁿ)., ER ⁿ 2nq ₀, E— - q ₀(q ₀-1). : q₀>2 n1 E R ⁿ 2nq ₀, p- p<₀. , f-[-, ]ⁿ, f A _p(R ⁿ) , _p([-, ]ⁿ) (1 << ).     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

On the facial structure of the set covering polytope

Given a bipartite graphG = (V, U, E), a cover ofG is a subset with the property that each nodeu U is adjacent to at least one nodev D. If a positive weightc _v is associated with each nodev V, the covering problem (CP) is to find a cover ofG having minimum total weight.In this paper we study the properties of the polytopeQ(G) ^|V| , the convex hull of the incidence vectors of all the covers inG. After discussing some general properties ofQ(G) we introduce a large class of bipartite graphs with special structure and describe several types of rank facets of the associated polytopes.Furthermore we present two lifting procedures to derive valid inequalities and facets of the polytopeQ(G) from the facets of any polytopeQ(G) associated with a subgraphG ofG. An example of the application of the theory to a class of hard instances of the covering problem is also presented.     

Theory of Eisenstein series for the group SL(3,R) and its application to a binary problem

On the basis of arithmetic considerations, a Fourier expansion for the leading Eisenstein series is obtained for the principal homogeneous space of the group SL(3,), which is automorphic with respect to the discrete group SL(3,). The main result is Theorem 1 in which an explicit form of the Fourier expansion is presented which generalizes the well-known formula of Selberg and Chowla. From this, in particular, there follows a proof of the analytic continuation and the functional equations for this Eisentein series which is independent of the work of Langlands. The arithmetic coefficients in the Fourier expansion which generalize the number-theoretic functions _s(n)=_d¦n,d>od⁵ make it possible to relate the Eisenstein series considered to the problem of finding the asymptotics as of the sum _n3(n)₃(n+), where ₃(n) is the number of solutions of the equation d₁d₂d₃=n in natural numbers. Part II of the present work will be devoted to this binary problem. At the end of the paper properties of special functions used in Theorem 1 are discussed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 76, pp. 5–52, 1978.     

Wold decomposition,prediction and parameterization of stationary processes with infinite variance

Summary A discrete time stochastic process {_t} is said to be a p-stationary process (1<p2)if , for all integers n1, t ₁,...t _n,h and scalars b ₁,...b _n.The class of p-stationary processes includes the class of second-order weakly stationary stochastic processes, harmonizable stable processes of order (1<2), and p ^thorder strictly stationary processes. For any nondeterministic process in this class a finite Wold decomposition (moving average representation) and a finite predictive decomposition (autoregressive representation) are given without alluding to any notion of covariance or spectrum. These decompositions produce two unique (interrelated) sequences of scalar which are used as parameters of the process {_t}. It is shown that the finite Wold and predictive decomposition are all that one needs in developing a Kolmogorov-Wiener type prediction theory for such processes.