Weak Variation of Gaussian Processes

Let X(t) (tR) be a real-valued centered Gaussian process with stationary increments. We assume that there exist positive constants ₀, C ₁, and c ₂ such that for any tR and hR with |h|₀ and for any 0r<min{|t|, ₀} where is regularly varying at zero of order (0 < < 1). Let be an inverse function of near zero such that (s)=(s) log log(1/s) is increasing near zero. We obtain exact estimates for the weak -variation of X(t) on [0,a].     

Newton polyhedra and the Bergman kernel

The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domain in ⁿ⁺¹ , the Bergman kernel B(z) of takes the form near a boundary point p: where (w,) is some polar coordinates on a nontangential cone with apex at p and means the distance from the boundary. Here admits some asymptotic expansion with respect to the variables ^{1/ m} and log(1/) as 0 on . The values of d _F >0, m _F ₊ and m are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of as 0 on is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel. Mathematics Subject Classification (2000):32A25, 32A36, 32T25, 14M25.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

A Simple Proof of the Borel Extension Theorem and Weak Compactness of Operators

Let T be a locally compact Hausdorff space and let C ₀(T) be the Banach space of all complex valued continuous functions vanishing at infinity in T, provided with the supremum norm. Let X be a quasicomplete locally convex Hausdorff space. A simple proof of the theorem on regular Borel extension of X-valued -additive Baire measures on T is given, which is more natural and direct than the existing ones. Using this result the integral representation and weak compactness of a continuous linear map u: C ₀(T) X when c ₀ X are obtained. The proof of the latter result is independent of the use of powerful results such as Theorem 6 of [6] or Theorem 3 (vii) of [13].     

An unconstrained convex programming view of linear programming

The major interest of this paper is to show that, at least in theory, a pair of primal and dual -optimal solutions to a general linear program in Karmarkar's standard form can be obtained by solving an unconstrained convex program. Hence unconstrained convex optimization methods are suggested to be carefully reviewed for this purpose.     

Invariance of closed convex sets and domination criteria for semigroups

Let a and b be two positive continuous and closed sesquilinear forms on the Hilbert space H=L ²(, ). Denote by T=T(t) _t0and S=S(t) _t0the semigroups generated by a and b on H. We give criteria in terms of a and b guaranteeing that the semigroup T is dominated by S, i.e. |T(t)f|S(t)|f| for all t0 and fH. The method proposed uses ideas on invariance of closed convex sets of H under semigroups. Applications to elliptic operators and concrete examples are given.     

Superficie cubiche di\mathbb{A}_\mathbb{C}^3 a curve sottoinsieme intersezione completa

Summary In this paper we classify the algebraic cubic surfaces of the affine space is the complex field, whose algebraic curves are set-theoretic complete intersections of ; in other words surfaces such that every prime ideal of height 1in the coordinate ring [] of is the radical of a principal ideal; if is non singular in codimension 1this means that [] is semifactorial. We give the equations of such surfaces within linear isomorphisms of providing also methods by which one can construct the equations of the surfaces cutting on its curves as set-theoretic complete intersections. Moreover for each of these surfaces we determine the minimum positive number such that every algebraic curve of with multiplicity of intersection , is complete intersection of itself with another surface § 8where the results are summarized). We tackle also the problem of such a classification over algebraically closed fields k different from .

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

On the existence of global first integrals in the plane

Summary A plane, autonomous, noncritical differential system x=f(x) of classC ^k is given. Under a suitable «discreteness» hypothesis on the behavior of orbits, the exeistence of a global first integral as regular as the field (provided k )is proved.     

Saturation on locally compact abelian groups: An extended theorem

Let G be a locally compact abelian group. The concern of the present note is to extend (for exponents p>2) the saturation theorem on G stated as Theorem 4 in [5]. The extension will be established for approximation processes (I_t)_t>0 acting on the submodule C^P(G), p]1,+[, of the convolutionM ¹(G)-module L^P(G) which consists of all functions fL^P(G) admitting as their Fourier transformsF _Gf (in the sense of the theory of quasimeasures) complex Radon measures not necessarily absolutely continuous with respect to any Haar measure on the dual group . Moreover, the relationship of the complex vector spaces C^P(G) to some other function spaces, in particular to the vector spaces B^P(G) introduced in [5], will be investigated.     

Primitive Illumination Systems for Families of Convex Bodies in the Plane

Let F= {C₁,C₂,...,C} be a family of ndisjoint convex bodies in the plane. We say that a set Vof exterior light sources illuminates F, if for every boundary point of any member of Fthere is a point in Vsuch that is visible from ,i.e. the open line segment joining and is disjoint from F. An illumination system Vis called primitive if no proper subset of Villuminates F. Let p_max(F) denote the maximum number of points forming a primitive illumination system for F, and letp_max(n) denote the minimum of F) taken over all families Fconsisting of ndisjoint convex bodies in the plane. The aim of this paper is to investigate the quantities p_max(F) and p_max(n).     

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A -symplectic structure on a complex manifold M of complex dimension2n is given by a smooth -closed (2, 0)-form such that ⁿ is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally can be represented as _i=1 ⁿ f _i f _{n + i} for appropriate smooth complex valuedfunctions f ₁, ..., f _2n . We also present a contact version of this theorem.     

The Theorem of Fáry and Milnor for Hadamard Manifolds

Suppose M is a complete noncompact surface C² immersed in E³ with K 0. It is proved that M is a cylinder if H 2 for some > 0. Moreover, the pinching constant 2 is optimal.     

On the convergence of the kadanoff transformation towards trivial fixed points

Summary We consider the Kadanoff transformation T (depending on a positive parameter p) acting on probability measures on the space {+1, –}^d. A measure is called a non-trivial fixed point of T, if it is extremal in the set of T-invariant measures but is not a product measure. We describe the set of trivial fixed points and show that non-trivial fixed points exist provided that d2 and p large enough. A strong mixing condition on implies convergence of T ⁿ towards a trivial fixed point. In particular this applies to the two-dimensional Ising model except at the critical point. What happens at the critical point still remains unknown.Research supported by the Deutsche Forschungsgemeinschaft (Sonderforschungsbereich 123)     

Symmetric Harmonic Sheaves Possessing Bipotentials

Let be the family of sheaves H of continuous functions on a Brelot harmonic space with a countable base such that locally the Dirichlet problem with respect to H is solvable, H satisfies Harnack inequalities and also H has a symmetry property. Defining the notions of H-biharmonic functions, H-biharmonic Green functions, H-bipotentials, H-biharmonic extensions, etc. we study the interrelation between them and exhibit various classifications of the family of sheaves.     

Bernstein functions, complete hyperexpansivity and subnormality—I

Special classes of functions on the classical semigroupN of non-negative integers, as defined using the classical backward and forward difference operators, get associated in a natural way with special classes of bounded linear operators on Hilbert spaces. In particular, the class of completely monotone functions, which is a subclass of the class of positive definite functions ofN, gets associated with subnormal operators, and the class of completely alternating functions, which is a subclass of the class of negative definite functions onN, with completely hyper-expansive operators. The interplay between the theories of completely monotone and completely alternating functions has previously been exploited to unravel some interesting connections between subnormals and completely hyperexpansive operators. For example, it is known that a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {1/_n}(n0). The present paper discovers some new connections between the two classes of operators by building upon some well-known results in the literature that relate positive and negative definite functions on cartesian products of arbitrary sets using Bernstein functions. In particular, it is observed that the weight sequence of a completely hyperexpansive weighted shift with the weight sequence {_n}(n0) (of positive reals) gives rise to a subnormal weighted shift whose weight sequence is {_n+1/_n}(n0). It is also established that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. Further, the connection of Bernstein functions with Stieltjes functions and generalizations thereof is exploited to link certain classes of subnormal weighted shifts to completely hyperexpansive ones.     

Convergence theorems for set-valued amarts and uniform amarts

( ) , — , - . .     

Foliated control theory, II

The main result is a control theorem for the structure space of E with control near the leaves F in M, where : E M is a fiber bundle over the Riemannian manifold M having a compact closed manifold for fiber and F is a smooth foliation of M, each leaf of which inherits a flat Riemannian geometry from M. A similar result has been proved by the authors under the assumption that each leaf of F is one-dimensional and the fiber of : E M is homotopy stable.Both authors were supported in part by the National Science Foundations.     

Powers of Sign Portraits of Real Matrices

The sign portrait S of a real n× n matrix is a matrix over the semiring with elements 0, 1, -1, and , where symbolizes indeterminateness. It is proved that if k is the least positive integer such that all the entries of S ^k are equal to , then k 2n ² – 3n + 2, and this bound is sharp. Bibliography: 6 titles.     

On some formalized conservation results in arithmetic

I _n andB _n are well known fragments of first-order arithmetic with induction and collection for _n formulas respectively;I _n ⁰ andB _n ⁰ are their second-order counterparts. RCA₀ is the well known fragment of second-order arithmetic with recursive comprehension;WKL ₀ isRCA ₀ plus weak König's lemma. We first strengthen Harrington's conservation result by showing thatWKL ₀ +B _n ⁰ is ₁ ¹ -conservative overRCA ₀ +B _n ⁰ . Then we develop some model theory inWKL ₀ and illustrate the use of formalized model theory by giving a relatively simple proof of the fact thatI ₁ provesB _n+1 to be _n+2-conservative overI _n . Finally, we present a proof-theoretic proof of the stronger fact that the _n+2 conservation result is provable already inI ₀ + superexp. ThusI _n+1 proves 1-Con (B _n+1) andI ₀ +superexp proves Con (I _n )Con(B _n+1).The first author was partially supported by NSF Grant #DCR-860615     

In the square of graphs,hamiltonicity and pancyclicity,Hamiltonian connectedness and panconnectedness are equivalent concepts

The squareG ² of a graphG has the same point set asG, and two points ofG ² are adjacent inG ² if and only if their distance inG is at most two. The result thatG ² is Hamiltonian ifG is two-connected, has been established early in 1971. A conjecture (ofA. Bondy) followed immediately: SupposeG ² to have a Hamiltonian cycle; is it true that for anyvV(G), there exist cyclesC _j containingv and having arbitrary lengthj, 3j|V(G)|. The proof of this conjecture is one of the two main results of this paper. The other main result states that ifG ² contains a Hamiltonian pathP(v, w) joining the pointsv andw, thenG ² contains for anyj withd _G ² (v, w)j|V(G)|–1 a pathP _j (v, w) of lengthj joiningv andw. By this, a conjecture ofF. J. Faudree andR. H. Schelp is proved and generalized for the square of graphs.However, to prove these two results extensive preliminary work is necessary in order to make the proof of the main results transparent (Theorem 1 through 5); and Theorem 3 plays a central role for the main results. As can be seen from the statement of Theorem 3, the following known results follow in a stronger form: (a) IfG is two-connected, thenG ² is Hamiltonian-connected; (b) IfG is two-connected, thenG ² is 1-Hamiltonian.Dedicated to Prof. Dr. E. Hlawka on the occasion of his 60th birthday