Die Verteilung der schlichten Funktionen in einem Funktionenraum

We consider the set of regular functions . We construct a Borel measure and a class of outer measures _h onH. With these and _h we show that: (HS)=0 and _h (HS)=0, (S is the set of normed univalent functions). From _h (HS)=0 follows—forh=t —that the Hausdorff—Billingsley-dimension ofHS is zero.     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Small Ball Probabilities Around Random Centers of Gaussian Measures and Applications to Quantization

Let be a centered Gaussian measure on a separable Hilbert space (E, ). We are concerned with the logarithmic small ball probabilities around a -distributed center X. It turns out that the asymptotic behavior of –log (B(X,)) is a.s. equivalent to that of a deterministic function _R (). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow.⁽⁸⁾     

Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs y=6y ²–x , which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1.     

Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus

Summary Two previous papers (in Vol. V) describe theory and some applications of the quotient-difference (=QD-) algorithm. Here we give an extension which allows the determination of the eigenvectors of a matrix. Letx ₍₀₎ ¹ , ...,x ₍₀₎ ⁿ be a coordinate system in whichA has Jacobi form (such a system may be constructed with methods ofC. Lanczos orW. Givens). Then the QD-algorithm allows the construction of a sequence of coordinate systemsx ₍₂₎ ¹ , ...,x ₍₂₎ ⁿ , (=0, 1, 2, ...) which converge for to the system of the eigenvectors ofA.     

Representations of integers as sums of powers with increasing exponents

Let(n) be the least integer such thatn may be represented in the formn=x ₁ ² +x ₂ ³ +...+x _(n) ⁽ⁿ⁾⁺¹ wherex ₁,x ₂, ...,x _(n) are natural numbers. We computed(n) forn 250 000 and found that(n) 5 for all thesen exceptn=56, 160 for which(n)=6. Also(n) 4 for 41542<n<=250 000.     

Distributional control and generalized analyticity

Let S be a strongly continuous, separation-preserving representation of a locally compact abelian group G in L^p(), where 1p<, and is an arbitrary measure. We show that S is uniformly bounded with respect to the L^p-and L-norms if and only if it satisfies a certain boundedness condition for distribution functions. These equivalent conditions facilitate the transference from L^p(G) to L^p() of the a.e. convergence for a wide class of sequences of convolution operators. The result unifies and generalizes various aspects of ergodic theory--in particular, the ergodic singular integral operators and ergodic Hardy spaces.     

Semicontinuità inferiore di un funzionale integrale dipendente da funzioni di classeL p a valori in uno spazio di Banach

Summary We study the sequential lower semicontinuity of an integral functional I: L ^s (, X) x L ^q (, Y) [– , ] where X and Y are separable B-spaces and is a finite positive regular complète measure. A necessary and sufficient condition is given on the integrand for the sequential lower semicontinuity of the integral with respect to the strong by weak topology on L ^s (, X)×L ^q (, Y).

---

---

Lavoro svolto nell'ambito del «Laboratorio per la matematica applicata» del C.N.R. presso l'Università di Genova.     

Small into isomorphisms onL ∞ spaces

IfT is an isomorphism ofL (A, ) intoL (B, ) which satisfies the condition T T ^–11+, where (A, ) is a -finite measure space, thenT/T is close to an isometry with an error less than 4.     

Surjectivity for Hamiltonian loop group spaces

Let G be a compact Lie group, and let LG denote the corresponding loop group. Let (X,) be a weakly symplectic Banach manifold. Consider a Hamiltonian action of LG on (X,), and assume that the moment map :XL ^* is proper. We consider the function ||²:X, and use a version of Morse theory to show that the inclusion map j:^-1(0)X induces a surjection j ^*:H _G ^*(X)H _G ^*(^-1(0)), in analogy with Kirwans surjectivity theorem in the finite-dimensional case. We also prove a version of this surjectivity theorem for quasi-Hamiltonian G-spaces.     

On singular perturbation of a semilinear hyperbolic equation

We consider a hyperbolic version of Eells-Sampson's equation: . This equation is semilinear with respect to space derivative and time derivative. Letu (x) be the solution with initial data u(0) and (0), and putv (t,x)=u (t,x). We show that when the resistance ,V (t,x) converges to a solution of the original parabolic Eells-Sampson's equation: . Note thatv _t(0)= (0) diverges when . We show that this phenomena occurs in more general situations.This article was processed by the author using the Springer-Verlag Pjourlg macro package.     

Zeros of the densities of infinitely divisible measures on R n

Let be an infinitely divisible probability measure onR ⁿ without Gaussian component and let be its Lévy measure. Suppose that is absolutely continuous with respect to the Lebesgue measure . We investigate the structure of the set ⁿ of admissible translates of . This yields a unified presentation of previously known results. We also show that if(S)>0 then is equivalent to , under the assumption that supp =R ⁿ , whereS is the closure of the semigroup generated by the support of .The research of this author is supported by KBN Grant.The research of this author is supported by AFSOR Grant No. 90-0168, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

An algebraic test forA 0-stability

The stability of a large class of numerical methods to solve initial value problems of ordinary differential equations is governed by a two-variable polynomial (,) when the method is applied toy'=qy. Here=hq, whereh is the stepsize. This class of methods includes Runge-Kutta methods, linear multistep methods, predictor-corrector methods, composite multistep methods and linear multistep-multiderivative methods. An algebraic test is given to determineA ₀-stability of such methods in a finite number of operations (additions, subtractions, multiplications and divisions). It is shown that the number of multiplications and divisions is of order 1/8²(⁴ +O(³)), where is the degree of (,) in the variable and the degree in the variable. The test has been implemented for multistep-multiderivative methods in a symbol manipulation language. For Enright's second derivativek-step methods it is proved that the methods areA ₀-stable if and only ifk<8.Supported by the Swiss National Foundation Grant No. 82.524.077. On leave from Institute of Mathematics, Ruhr-University Bochum, D-463 Germany.     

Convolutions of random measures on a compact topological semigroup

We investigate the asymptotic behavior of a sequence of convolutionsv ⁽ⁿ ₁() ** _n (), where { _n } _n=1 is some random process taking values in a semigroupM ¹(S) of probability Borel measures on a compact topological semigroupS.     

Interpolation of spectra

By the M.Riesz Convexity Theorem, an operator T on the space of simple integrable functions into the measurable functions (on some measure space) which has continuous extensions to L^p() and L^q() , where 1 p q , also has continuous exten — sions to all L^r () , p r q . It is shown that, whenever (T_p) and (T_q) are o-dimensional (in particular, countable) then the spectra (T_r) (p r q) are pairwise identical. For q = , only w^*-continuous extensions are considered. An example due to Dayanithy shows that the conclusion fails in general.     

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 .     

Conditioned super-Brownian motion

Summary We investigate classes of conditioned super-Brownian motions, namely H-transformsP ^H with non-negative finitely-based space-time harmonic functionsH(t, ). We prove thatH ^H is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching Brownian motions. We identify the limit behaviour of H-transforms with functionsH(t, )=h(t, (1)) depending only on the total mass (1). Using the Palm measures of the super-Brownian motion we describe for an additive spacetime harmonic functionH(t, )=h(t, x) (dx) theH-transformP ^H as a conditioned super-Brownian motion in which an immortal particle moves like an h-transform of Brownian motion.     

Numerical studies of the Möbius power series

We define the Möbius power series throughf(z)= _n-1 z ⁿ ,g(z)= _n=1 (n)z ⁿ /n where (n) is the usual Möbius function. This paper presents some heuristic estimates describing the behavior off(z) andg(z) when |z| is close to 1 together with representations in terms of elementary functions for real values ofz. Function tables are also given together with zeros and a few other special values.     

Covering a convex body by smaller homothetic copies

Let a convex bodyAE ⁿ be covered bys smaller homothetic copies with coefficients ₁, ..., _s , respectively. It is conjectured that ₁ + ...+ _s n. This conjecture is confirmed in two cases:n is arbitrary ands=n+1;s is arbitrary andn=2.     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.