Existence of compatible families of proper regular conditional probabilities

Summary Let (, , ) be a perfect probability space with countably generated, and let IB be a family of sub--fields of . Under a countability condition on the family IB, I show that there exists a family {}_IB of regular conditional probabilities which are everywhere compatible. Under a more stringent condition on IB, I show that the can furthermore be chosen to be everywhere proper. It follows that in the Dobrushin-Lanford-Ruelle formulation of the statistical mechanics of classical lattice systems, every (perfect) probability measure is a Gibbs measure for some specification.Research supported in part by NSF PHY-78-23952NSF Predoctoral Fellow (1976–79) and Danforth Fellow (1979–81).     

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.     

Eine Verallgemeinerung des Satzes vom abgeschlossenen Wertebereich in lokalkonvexen Räumen

This paper extends Kato's proof [5] of Banach's closed range theorem to locally convex spaces. Thus we consider a locally convex space (E,) and pairs (M,N) of closed subspaces. We call such a pair -open, if and only if there exists a directed, total system of seminorms generating the topology induced by a on M+N, such that the minimal gap _p(M,N)>O for each p. Our main result is a generalisation of the closed range theorem and it consists of statements on relationships between the following properties: (a) M+N -closed, (b) M+N (E,E)-closed, (c) M+N (E,E)-closed, (d) (M,N) -open, (e) (M,N) (E,E)-open, (f) (M,N) (E,E)-open, (g) (M,N) (E,E)-open, (h) M+N=(MN), (i) M+N=(MN).By specialising the space (E,) and the subspaces M,N, our generalisation includes the closed range theorems of Dieudonné and Schwartz [4], Browder [1] and Mochizuki [12]. It is shown that these theorems not only hold for closed linear operators but even for closed linear relations. We are therefore able to obtain closed domain theorems which extend Brown's examinations in Banach-spaces [2] to locally convex spaces.

---

---

Herrn Gottfried Köthe zum 70. Geburtstag am 25.12.1975 gewidmet     

Cutting disjoint disks by straight lines

Fork>0 letf(k) denote the minimum integerf such that, for any family ofk pairwise disjoint congruent disks in the plane, there is a direction such that any line having direction intersects at mostf of the disks. We determine the exact asymptotic behavior off(k) by proving that there are two positive constantsd ₁,d ₂ such thatd ₁k logkf(k)d ₂k logk. This result has been motivated by problems dealing with the separation of convex sets by straight lines.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

Analysis of a class of fractional programming problems

We propose a solution strategy for fractional programming problems of the form max _xx g(x)/ (u(x)), where the function satisfies certain convexity conditions. It is shown that subject to these conditions optimal solutions to this problem can be obtained from the solution of the problem max _xx g(x) + u(x), where is an exogenous parameter. The proposed strategy combines fractional programming andc-programming techniques. A maximal mean-standard deviation ratio problem is solved to illustrate the strategy in action.     

Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H₀(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

Extremes of a Gaussian Process and the Constant H α

Consider a triangular array of standard Gaussian random variables {_n,i, i 0, n 1} such that {_n,i, i 0} is a stationary normal sequence for each n 1. Let _n,k = corr(_n,i,_n,i+k). If (1-_n,k)log n _k (0,) as n for some k, then the locations where the extreme values occur cluster and the limiting distribution of the maxima is still the Gumbel distribution as in the stationary or i.i.d. case, but shifted by a parameter measuring the clustering. Triangular arrays of Gaussian sequences are used to approximate a continuous Gaussian process X(t), t 0. The cluster behavior of the random sequence refers to the behavior of the extremes values of the continuous process. The relation is analyzed. It reveals a new definition of the constants H used for the limiting distribution of maxima of continuous Gaussian processes and provides further understanding of the limit result for these extremes.     

Efficient Regular Polygon Dissections

We study the minimum number g(m,n) (respectively, p(m,n)) of pieces needed to dissect a regular m-gon into a regular n-gon of the same area using glass-cuts (respectively, polygonal cuts). First we study regular polygon-square dissections and show that n/2 -2 g(4,n) (n/2) + o(n) and n/4 g(n,4) (n/2) + o(n) hold for sufficiently large n. We also consider polygonal cuts, i.e., the minimum number p(4,n) of pieces needed to dissect a square into a regular n-gon of the same area using polygonal cuts and show that n/4 p(4,n) (n/2) + o(n) holds for sufficiently large n. We also consider regular polygon-polygon dissections and obtain similar bounds for g(m,n) and p(m,n).     

Eulerian numbers with fractional order parameters

Summary The aim of this paper is to generalize the well-known Eulerian numbers, defined by the recursion relationE(n, k) = (k + 1)E(n – 1, k) + (n – k)E(n – 1, k – 1), to the case thatn is replaced by . It is shown that these Eulerian functionsE(, k), which can also be defined in terms of a generating function, can be represented as a certain sum, as a determinant, or as a fractional Weyl integral. TheE(, k) satisfy recursion formulae, they are monotone ink and, as functions of , are arbitrarily often differentiable. Further, connections with the fractional Stirling numbers of second kind, theS(, k), > 0, introduced by the authors (1989), are discussed. Finally, a certain counterpart of the famous Worpitzky formula is given; it is essentially an approximation ofx in terms of a sum involving theE(, k) and a hypergeometric function.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

Asymptotic behavior of the dwell time distribution for a random walk on a positive semi-axis

Let ₁, ₂, ... be a sequence of independent identically distributed random variables with zero means. We consider the functional _n = _k=o ⁿ (S _k ) where S₁=0, S_k= _i=1 ^k _i (k1) and(x)=1 for x0,(x) = 0 for x<0. It is readily seen that _n is the time spent by the random walk S_n, n0, on the positive semi-axis after n steps. For the simplest walk the asymptotics of the distribution P (_n = k) for n and k, as well as for k = O(n) and k/n<1, was studied in [1]. In this paper we obtain the asymptotic expansions in powers of n^–1 of the probabilities P(h_n = nx) and P(nx_{1 n} nx₂) for 0<₁, x = k/n ₂<1, 0<₁x₁2₂<1.Translated from Matematicheskie Zametki, Vol. 15, No. 4, pp. 613–620, April, 1974.The author wishes to thank B. A. Rogozin for valuable discussions in the course of his work.     

An infinite dimensional analogue of the Albeverio-Röckner's theorem on Fukushima's conjecture

LetX be the solution of the SDE:dX _t = (X _t)dB _t +b(X _t)dt, with andb C _b (R) such that >0 for some constant , andB a real Brownian motion. Let be the law ofX onE=C([0, 1],R) andk E* – {0}, whereE* is the topological dual space ofE. Consider the classical form: _k (u, v)=u / kv / kd, whereu andv are smooth functions onE. We prove that, if _k is closable for anyk in a dense subset ofE* and if the smooth functions are contained in the domain of the generator of the closure of _k , must be a constant function.     

Minimality of root vectors of operator functions analytic in an angle

We study the minimality of elementsx _h,j,k of canonical systems of root vectors. These systems correspond to the characteristic numbers _k of operator functionsL() analytic in an angle; we assume that operators act in a Hilbert space . In particular, we consider the case whereL()=I+T()c, >0,I is an identity operator,C is a completely continuous operator, (I- C)^–1c for ¦arg¦, 0<<, the operator functionT() is analytic, and T()c for ¦arg¦<. It is proved that, in this case, there exists >0 such that the system of vectorsC ^v x _h,j,k is minimal in for arbitrary positive <1+, provided that ¦_k¦>.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 545–566, May, 1994.This research was partially supported by the Ukrainian State Committee of Science and Technology.     

Quasiclassical asymptotic behavior of the scattering cross section for asymptotically homogeneous potentials

One considers the total scattering cross section on the potential gV(x), x^m, m3, for large values of the coupling constant g and of the wave number k. One assumes that V(x)(x/|1x|)|x|^–, 2>m+1, as ¦x¦. It is shown that for gk^–1 , g^3–ak^2(a–2) the scattering cross section is equal asymptotically to _a(gk^–1), x=(m–1)(–1)^–1. Here the coefficient _a is determined only by the function and the number . Under the additional conditions >0, V>0, the indicated asymptotic behavior holds in the large domain gk^–1 , gk^a–z c(gk^–1), >0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 152, pp. 105–136, 1986.     

Criteria of algebraic independence of complex numbers over a field of finite transcendence type (II)

In the present paper we establish a criterion of algebraic independence of complex numbers ₁, ..., _n over a field of finite transcendence type using a sequence of nonzero polynomials in several variables with integral coefficients, which satisfy simultaneously certain upper and lower estimates in different orders of magnitude at the point ( ₁, ..., _q , ₁, ..., _n ), where { ₁, ..., _q } is a transcendence basis of over .The Project Supported by the National Natural Science Foundation of China     

On optimality of spline interpolation for recovery of differentiable functions

W _p ^r (1p<) , . - r -1. =1 r-2.     

(q, δ)-Numeration Systems with Missing Digits

We consider the (q, ) numeration system, with basis q2 and the set of digits {, +1,,q+–1} where –(q–1)0. We study properties of numbers where some digits do not occur. This is analogous to the Cantor set {0.a₁a₂a_i{0,2}}. We compute an asymptotic equivalent of the nth moment of the Cantor (q, D)-distribution which can be described as the numbers 0. w₁w₂ with w_iD{,,q+–1}, and each such letter can occur with the same probability 1/CardD. Furthermore, we consider n random strings according to the distribution and the expected minimum of them. We find a recursion which we solve asymptotically.This author was supported by the CNRS/NRF-project no 10959. Part of this work was done during the first authors visit to the John Knopfmacher Centre for Applicable Analysis and Number Theory at the University of the Witwatersrand, Johannesburg, South Africa.This author was supported by the CNRS/NRF-project no 10959.     

Recurrence and transience for random walks with stationary increments

Summary Let S _n = ₁+...+ _n , n1, be the partial sums of stationary, dependent random variables in ^m . The probability space can be partitioned into I _t I _r , where I _t = {S _n} and I _r ={each S _n is limit point of (S _n)_n1}. This result follows from the inclusion{S _n > for n>0}I _t a.s., which is obtained by using Kac's inequality.     

Improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set

We will establish the following improved Krasnosel'skii theorems for the dimension of the kernel of a starshaped set: For each k and d, 0 k d, define f(d,k) = d+1 if k = 0 and f(d,k) = max{d+1,2d–2k+2} if 1 k d.Theorem 1. Let S be a compact, connected, locally starshaped set in R^d, S not convex. Then for a k with 0 k d, dim ker S k if and only if every f(d, k) lnc points of S are clearly visible from a common k-dimensional subset of S.Theorem 2. Let S be a nonempty compact set in R^d. Then for a k with 0 k d, dim ker S k if and only if every f (d, k) boundary points of S are clearly visible from a common k-dimensional subset of S. In each case, the number f(d, k) is best possible for every d and k.