On the value of a logarithmic-trigonometric integral

A closed expression is derived for the integral ₀ ^/2 log ⁿ cosxlog ^p sinxdx, wheren andp are non-negative integers. As already remarked by Nielsen in a monograph on the generalized polylogarithms published early in this century, this integral is equal to times a homogeneous polynomial in (q) (the Riemann zeta function for integer arguments) and log 2, with rational coefficients. Explicit expressions for the integral are given for 0<n4, 0p4, most of which have been found from the general formula by means of a computer.Part of this work was done during the author's leave of absence at the Institute of Theoretical Physics at McGill University, Montreal, Quebec, Canada. This work was supported in part by the National Research Council of Canada.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

Symmetry properties of higher spectral densities and nonparametric estimates of them

The symmetry properties of the spectral densitiesf ^(v)() of order v, equal to 4, 5, and 6 of a stationary stochastic process {(k), k Z} are studied. It is established that the fonctionf ^(v)(), R^v–1 is completely determined by its values on a set whose (v– 1)-dimensional volume is ⁴/6, ⁵/15, and ⁵/45 when v is equal to 4, 5, and 6, respectively. Results on the statistical estimation of the spectral densities of orders v equal to 3 and 4 with respect to samples of finite size are also given.Translated from Teoriya Sluchainykh Protsessov, No. 16, pp. 1–4, 1988.     

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     

On the unique finite (SI) decomposition of the Jordan operator ⊕ i=1 n S(θ) for certain inner function θ

In this paper, we have proven that for the Jordan blockS() withS() (SI), _i=1 ⁿ S() =S() ⁽ⁿ⁾ (n 1) has unique finite (SI) decomposition up to a similarity. As result, we obtain that ifV is a Volterra operator onH=L ²([0, 1]), thenV ⁽ⁿ⁾ has unique finite (SI) decomposition.This project was supported by National Natural Science Foundation of China.     

The spectral bound of Schrödinger operators

Let V: R ^N [0, ] be a measurable function, and >0 be a parameter. We consider the behaviour of the spectral bound of the operator 1/2–V as a function of . In particular, we give a formula for the limiting value as , in terms of the integrals of V over subsets of R ^N on which the Laplacian with Dirichlet boundary conditions has prescribed values. We also consider the question whether this limiting value is attained for finite .     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

Über nullstellenfreie Gebiete von ζ(itk) (s)

Let ^(itk) (s) denote thek-th derivative of the Riemann Zeta-function,s=+it, ,t real numbers,k1 rational integers. Using ideas fromT. C. Titchmarsh and from a paper ofR. Spira, lower bounds are derived for |^(itk)(s)|, |^(itk)(1-s) for >1 and some infinitely many, sufficiently large values oft. Further let be an algebraic number of degreen and heightH; then a lower bound for |^(itk)(its)|, dependent onn, H, k is established for alln,H1,k3, 2+7k/4 and all realt.     

Self-normalized central limit theorem for sums of weakly dependent random variables

Let {X _n,n1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX _n=,EX _n ² <,Var S _n /n² and the central limit theorem. This paper presents two estimators of ². Their weak and strong consistence as well as their rate of convergence are obtained for -mixing, -mixing and associated sequences.Supported by a NSF grant and a Taft travel grant. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025.Supported by a Taft Post-doctoral Fellowship at the University of Cincinnati and by the Fok Yingtung Education Foundation of China. Hangzhou University, Hangzhou, Zhejiang, P.R. China and Department of Mathematics, National University of Singapore, Singapore 0511.     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

Existence theorems for a class of nonconvex problems in the calculus of variations

We give some existence results of minima for a class of nonconvex functionals depending on the Laplacian. We minimize these functionals on the set of functionsu inW ^2,p () W ₀ ^1,p () such that u/n=0 on ,p>1, with either an annulus or the whole space ⁿ . Our approach allows us to deal with integrands without any regularity conditions. The results are obtained first by showing that the corresponding convexified problem has at least one radially symmetric solution via a rotation; then, by using a Liapunov's theorem on the range of a vector-valued measure, we construct a function that is a solution to our problem.The author wishes to thank Prof. A. Cellina for useful comments and Prof. G. Dal Maso for several helpful discussions during the preparation of this paper. He also is grateful to Prof. A. Salam and ICTP for the generous financial support that made this work possible.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Mappings that preserve cones in Lobachevskii space

Let ⁿ be n-dimensional Lobachevskii space, and {l_x:x ⁿ} be a family of lines, parallel to a linel ₀, 0ⁿ (in a given direction). Let {c_x:Xⁿ} be a family of circular cones in ⁿ of opening with axes l_X and vertex X. Then, iff:ⁿⁿ(n>2) is a bijective mapping andf(C_x)=C _f(x), it follows thatf is a motion in the space ⁿ.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 687–694, May, 1973.     

On meromorphic solutions of algebraic differential equations in angular domains

We obtain asymptotic estimates of meromorphic solutions to the differential equationP _n (z, , )=P _n–1 (z, , ,..., ^(m) ) in the angular domain P={z: arg z · }. Here P_n(z, w, w) is a polynomial in all variables, and of degree n with respect to w and w; P_n–1(z, w, w, ..., w^(m)) is a polynomial in all variables, and of degree n –1 with respect to w, w, ..., w^(m) In the particular case, when the solutions are entire functions, these estimates are more precise than the known estimates that are obtained by using the method of Wiman-Valiron, which cannot be applied to meromorphic solutions in the domain P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 514–523, April, 1992.     

Darstellung einer Ordnung von Maßen durch Stoppzeiten

Summary Given a stochastic matrixP on the state spaceI an ordering for measures inI can be defined in the following way: iff(f)(f) for allf in a sufficiently rich subcone of the cone of positiveP-subharmonic functions. It is shown that, if, are probability measures with , then in theP-process (X _n)_n0 having as initial distribution there exists a stopping time such thatX is distributed according to. In addition, can be chosen in such a way, that for every positive subharmonicf with(f)< the submartingale (f(X _n))_n0 is uniformly integrable.     

Proof of two conjectures of H. Brezis and L.A. Peletier

In this paper, we consider the problem: –u=N(N–2)u ^p– , u>0 on ; u=0 on , where is a smooth and bounded domain inR ^N, N3, p= , and >0. We prove a conjecture of H. Brezis and L.A. Peletier about the asymptotic behaviour of solutions of this problem which are minimizing for the Sobolev inequality as goes to zero. We give similar results concerning the related problem: –u=N(N–2)u^p+u, u>0 on ; u=0 on , for N is larger than 4.     

Improvement of the Nash-Tognoli theorem

Let be a smooth closed manifold in ⁿ. The Nash-Tognoli theorem says that M can be arbitrarily well approximated (in the C^r-topology with r < ) in ⁿ by a nonsingular real algebraic set under the condition that dim <(n-1)/2 There is a familiar conjecture, going back at least to Nash, that the restriction on dim in the Nash-Tognoli theorem is unnecessary. However, up to now in unstable dimensions [i.e., for dim (n-1)/2 ] the possibility of approximating was known only for orientable of codimension (in ⁿ) 1 or 2. The goal of the paper is to prove the following theorem, relaxing the restriction on dim in the Nash-Tognoli theorem to dim M<(2n-1)/3. If is a smooth closed manifold in IK and dim M<(2n–1)/3, then can be arbitrarily well approximated in ⁿ by a nonsingular real algebraic set.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 66–71, 1982.     

The MoravaK-theory of algebraicK-theory spectra

Ifn2 the MoravaK-theoryK(n) _* of an algebraicK-theory spectrumKX vanishes for any ring or schemeX. This is proved using thev _n -complexes of Hopkins and Smith, together with the following theorem. The natural mapf:Q ₀S⁰BGL⁺ factors through the space ImJ. In particularf _*: _* ^s K _* annihilates CokerJ. These results are closely related to the Lichtenbaum-Quillen conjectures.Partially supported by an NSF grant.     

A joint central limit theorem for the sample mean and regenerative variance estimator

Let {V(k) :K1} be a sequence of independent, identically distributed random vectors in ^d with mean vector . The mappingg is a twice differentiable mapping from ^d to ¹. Setr=g(). A bivariate central limit theorem is proved involving a point estimator forr and the asymptotic variance of this point estimate. This result can be applied immediately to the ratio estimation problem that arises in regenerative simulation. Numerical examples show that the variance of the regenerative variance estimator is not necessarily minimized by using the return state with the smallest expected cycle length.This research was supported by Army Research Office Contract DAAG29-84-K-0030. The first author was also supported by National Science Foundation Grant ECS-8404809 and the second author by National Science Foundation Grant MCS-8203483.