Convolution Operators with Fractional Measures Associated to Holomorphic Functions

Let be an open set in the complex plane and let be a holomorphic function on . Let K be a compact subset of with nonempty interior such that 0 K. Let be the Borel measure of R ⁴ C ² given by(E = _K E(z, (z))|z|^–2 d(z)where 0 < 2 and d(x ₁ + ix ₂) = dx ₁ dx ₂ denotes the Lebesgue measure on C. Let T be the convolution operator T f = * f. In this paper we characterize the type set E associated to T .     

On the Monotone Error Rule for Parameter Choice in Iterative and Continuous Regularization Methods

We consider in Hilbert spaces linear ill-posed problems Ax = y with noisy data y satisfying y –y. Regularized approximations x _r to the minimum-norm solution x of Ax = y are constructed by continuous regularization methods or by iterative methods. For the choice of the regularization parameter r (the stopping index n in iterative methods) the following monotone error rule (ME rule) is used: we choose r = r _ME (n = n _ME) as the largest r-value with the guaranteed monotonical decrease of the error x _r – x for r [0, r _ME] (x _n – x <#60; x _n–1 – x for n = 1, 2, ..., n _ME). Main attention is paid to iterative methods of gradient type and to nonstationary implicit iteration methods. As shown, the ME rule leads for many methods to order optimal error bounds. Comparisons with other rules for the choice of the stopping index are made and numerical examples are given.This revised version was published online in October 2005 with corrections to the Cover Date.     

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

On the boundedness of Weyl sums

Leta be irrational and letf:[0,1] be Riemann-integrable with integral zero. Letf _n (x) denote the Weyl sumf _n (x):= _k=0 ^n–1 f({x k>}),x/[0,1[,n. We prove criteria for the boundedness of the sequence (f _n ) _n1 and discuss the relation of this question to irregularities of the distribution of sequences.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

Independence in hypergraphs

Suppose an integral function (|A|)q₁ defined on the subsets of edges of a hypergraph (X,u,) satisfies the following two conditions: 1) any set W u such that |A|(|A|) for any AW is matroidally independent; 2) if W is an independent set, then there exists a unique partitionW=T₁+ T₂+...+T_v such that |T ⁱ |=(|T ⁱ |),i1:v, and for any AW, |A|(|A|) there exists a T_i such that AT_i. The form of such a function is found, in terms of parameters of generalized connected components, hypercycles, and hypertrees.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 196–204, 1982.     

Algebraic polynomial bases of space LP

Let (_n) be a system, close to the orthonormal complete system (x _n). An estimate is obtained for the deviation of the system {f_n}, obtained from {_n} by Schmidt's method, from the system {x_n}. This estimate is used to show that, in any L_P(–1,1), withp (1,4/3] [4,), and for any >e¦4 = i,13..., there exists an orthogonal algebraic system (P _n (x)) _n=0 , forming a basis in L_P and such that _n = degP _n (x) n for n>n_o(p,).Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 223–230, February, 1978.     

Some analytical properties of γ-convex functions on the real line

This paper deals with the analytical properties of -convex functions, which are defined as those functions satisfying the inequalityf(x ₁ )+f(x ₂ )f(x ₁)+f(x ₂), forx _i [x ₁,x ₂], |x _i –x _i |=, i=1,2, whenever |x ₁–x ₂|>, for some given positive . This class contains all convex functions and all periodic functions with period . In general, -convex functions do not have ideal properties as convex functions. For instance, there exist -convex functions which are totally discontinuous or not locally bounded. But -convex functions possess so-called conservation properties, meaning good properties which remain true on every bounded interval or even on the entire domain, if only they hold true on an arbitrary closed interval with length . It is shown that boundedness, bounded variation, integrability, continuity, and differentiability almost everywhere are conservation properties of -convex functions on the real line. However, -convex functions have also infection properties, meaning bad properties which propagate to other points, once they appear somewhere (for example, discontinuity). Some equivalent properties of -convexity are given. Ways for generating and representing -convex functions are described.This research was supported by the Deutsche Forschungsgemeinschaft. The first author thanks Prof. Dr. E. Zeidler and Prof. Dr. H. G. Bock for their hospitality and valuable support.     

Infinitesimal obstructions to weakly mixing

Let ^t be the flow (parametrized with respect to arc length) of a smooth unit vector field v on a closed Riemannian manifold M ⁿ , whose orbits are geodesics. Then the (n-1)-plane field normal to v, v, is invariant under d ^t and, for each x M, we define a smooth real function _x (t) : (1 + _i (t)), where the _i(t) are the eigenvalues of AA ^T, A being the matrix (with respect to orthonormal bases) of the non-singular linear map d^2t , restricted to v at the point _x ^-t M ⁿ.Among other things, we prove the Theorem (Theorem II, below). Assume v is also volume preserving and that _x ^' (t) 0 for all x M and real t; then, if _x ^t : M M is weakly missng for some t, it is necessary that vx 0 at all x M.     

Randomly Weighted Sums of Subexponential Random Variables with Application to Ruin Theory

Let {X _k , 1 k n} be n independent and real-valued random variables with common subexponential distribution function, and let {k, 1 k n} be other n random variables independent of {X _k , 1 k n} and satisfying a _k b for some 0 < a b < for all 1 k n. This paper proves that the asymptotic relations P (max_{1 m n k=1} ^m _k X _k > x) P (sum _k=1 ⁿ _k X _k > x) sum _k=1 ⁿ P ( _k X _k > x) hold as x . In doing so, no any assumption is made on the dependence structure of the sequence { _k , 1 k n}. An application to ruin theory is proposed.     

Mixing and some integro-functional equations

Summary It is proved that the operatorP: L ¹ (0, ) L ¹(0, ), given byPg(z) = _z/c [g(x)/cx]dx, is completely mixing, i.e.,P ⁿ g ₁ 0 forg L ¹(0, ) with g dx = 0. This implies that, forc (0, 1), each continuous and bounded solution of the equationf(x)= ₀ ^cx f(t)dt/(cx) (x (0, 1]) is constant.     

Über Tuttes Cages

An undirected graph of valencyd and girth is called a (d, )-cage if its automorphism group acts transitively on the set of alls-paths in ands(+1)/2. We discuss an elementary construction of two known families of cages which allows us to prove easily some facts about their automorphism groups. We give, for example, a new proof of the fact that the automorphism group ofSp ₄(2 ⁿ ) contains elements which are not induced by Sp ₄(2 ⁿ ).     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

Ein neues Surjektivitätskriterium im Hilbertraum

LetA, B andC be linearm-accretive operators in a Hilbert space. Suppose further thatC is bounded, thatb:=inf {Re (C y, y)| y=1}>0, thatA ^–1 exists as a bounded operator and that Re (B ^* x, A ^–1 x)+a x²0 holds for allxD (B ^*) and a constanta with 0a<b. ThenCA+B is surjective, (CA+B)^–1 exists and C Ax+Bx (b–a) A x holds for allxD (A) D (B). This criterion can be applied to evolution equations of the formdu/dt+C(t)A(t)u=f(t) whereB:=d/dt.     

Estimate of a random field based on the trajectory of a point moving in it

We consider the stochastic differential equationd _t =( _t )dt+ _t ( _t )dw _t in Euclidean space, where (x) is a Gaussian random field andw _t is a standard Wiener process. Let f _t ={ _s ,st}. Equations are obtained for the conditional meansm _t (x)=f _t } andB _t (x, y)=M{(x)(y)|f _t }.Translated fromTeariya Sluchaínykh Protsessov, Vol. 14, pp. 7–9, 1986.     

On properties of the probabilistic constrained linear programming problem and its dual

In this paper, the two problems inf{inf{cx:x R ⁿ,A ₁ xy,A ₂ xb}:y suppF R ^m,F(y)p} and sup{inf{uy:y suppF R ^m,F(y)p}+vb:uA ₁+vA ₂=c, (u,v0} are investigated, whereA ₁,A ₂,b,c are given matrices and vectors of finite dimension,F is the joint probability distribution of the random variables ₁,..., _m, and 0<p<1. The first problem was introduced as the deterministic equivalent and the second problem was introduced as the dual of the probabilistic constrained linear programming problem inf{cx:P(A ₁ x)p,A ₂ xb}.b}. Properties of the sets and the functions involved in the two problems and regularity conditions of optimality are discussed.     

Extended Classical Asymptotic Expansions in the Case of Gaussian Limit Distribution

Let X, X ₁, X ₂,... be a sequence of independent and identically distributed random variables with common distribution function F. Denote by F _n the distribution function of centered and normed sum S _n . Let F belong to the domain of attraction of the standard normal law , that is, lim F _n (x)= (x), as n , uniformly in x . We obtain extended asymptotic expansions for the particular case where the distribution function F has the density p(x) = cx ^––1 ln(x), x > r, where 2, , c > 0, and r > 1. We write the classical asymptotic expansion (in powers of n ^–1/2) and then add new terms of orders n ^–/2 ln n, n ^–/2 ln^-1 n, etc., where 0.     

Flocks of a quadratic cone in PG(3, q), q ≤ 8

If (O) is a quadratic cone in PG(3,q), with vertex x, then a flock of (O) is a partition of (O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q ² and a generalized quadrangle of order (q ², q). Here we determine all flocks of (O) for q 8.     

An extension of the fenchel theorem

We give an extension of the Fenchel-Borsuk result by introducing the total absolute torsion-curvature KT() for regular curves whose tangent indicatrix is a piecewise regular curve (-closed curves). We prove that KT() is low bounded by 2 and we give a geometric characterization for the -closed curves whose KT() is minimal.     

Singularities of the Green function of a random walk on a discrete group

LetX be a countable discrete group and let be an irreducible probability onX. The radius of convergence of the Green function is finite, and independent ofx. Let 0} \right\}$$ " align="middle" border="0"> be the period of . We show that for eachxX the singularities of the analytic functionzG(x; z) on the circle {z:|z|=} are precisely the points e ^2ik/d k=0, ...,d–1. In particular, is the only singularity on the circle in the aperiodic cased=1 (which occurs, for example, when (e)>0). This affirms a conjecture ofLalley [5]. When is symmetric, i.e., (x ^–1)=(x) for allxX, d is either 1 or 2. As another particular case of our result, we see that- is then a singularity ofzG (x; z) if and only ifd=2, in which caseX is bicolored. This answers a question ofde la Harpe, Robertson andValette [2].