A chord-stretching map of a convex loop is an isometry

Let ⁿ be n-dimensional Euclidean space, and let : [0, L] ⁿ and : [0, L] ⁿ be closed rectifiable arcs in ⁿ of the same total length L which are parametrized via their arc length. is said to be a chord-stretched version of if for each 0s tL, |(t)–(s)| |(t)–(s)|. is said to be convex if is simple and if ([0, L]) is the frontier of some plane convex set. Individual work by Professors G. Choquet and G. T. Sallee demonstrated that if were simple then there existed a convex chord-stretched version of . This result led Professor Yang Lu to conjecture that if were convex and were a chord-stretched version of then and would be congruent, i.e. any chord-stretching map of a convex arc is an isometry. Professor Yang Lu has proved this conjecture in the case where and are C ² curves. In this paper we prove the conjecture in general.     

On Sums of Projections

In the paper, for all n, we describe the set _n of all real numbers admitting a collection of projections P ₁,...,P _n on a Hilbert space H such that _k=1 ⁿ P _k= I (I is the identity operator on H) and study the problem to find all collections of this kind for a given _n .     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

An inequality for a linear form in the logarithms of algebraic numbers

Let ln ₁, ..., ln _m–1 be the logarithms of fixed algebraic numbers which are linearly independent over the field of rational numbers, b₁, ..., b_m–1 rational integers, > 0. A bound from below is deduced for the height of the algebraic number _m under the condition that ¦b₁ ln ₁+...+_bm–1ln _m– ¦ < exp {–H},H=max ¦ b _k ¦ >0.Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 681–689, June, 1969.     

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.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

On the dispersion spectrum

A measure for the denseness of sequences (an) mod 1, irrational, is the dispersion constantD() introduced byH. Niederreiter. In this paper the smallest accumulation point ₁ of the set of theD() is determined and all those are explicitely given for whichD () < ₁ holds.     

A complete asymptotic expansion of the jacobi functions with error bounds

In this paper we give a complete asymptotic expansion of the Jacobi functions ^{(, )} (t) as + . The method we employed to get the complete expansion follows that of Olver in treating similar problems. By using a Gronwall-Bellman type inequality for an improper integral in which the integrand is an unbounded function and contains a parameter, we get an error bound of the asymptotic approximation which is different from that of Olver's.     

On the Heyde Theorem for Finite Abelian Groups

It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if _j are independent random variables, _j , _j are nonzero constants such that _i ± _j ^–1 _j 0 for all i j and the conditional distribution of L ₂=_{1 1} + ··· + _{n n} given L ₁=_{1 1} + ··· + _{n n} is symmetric, then all random variables _j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients _j , _j are automorphisms of X.     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

A class of non-Moufang Bol loops isomorphic to all their loop isotopes

For any two primes, , such that< and divides–1, it is shown that there exists a non-Moufang Bol loop of order ² which is isomorphic to each of its loop isotopes.     

Propriété de stabilité de la fonction extrémale relative

Nous donnons une caractérisation des domaines DX pour lesquels la fonction extrémale relative ^*(,E,D) a la propriété de stabilité pour tout ED, i.e. lim _k^*(,E,D _k )=^*(,E,D), ED. Ensuite, nous étudions la relation entre cette propriété et les enveloppes pluripolaires. Nous concluons par quelques remarques sur la propriété de stabilité lim _k^*(,E _k ,D)=^*(,E,D).     

Convergence of epigraphs and of sublevel sets

Let LSC(X) be the set of the proper lower semicontinuous extended real-valued functions defined on a metric spaceX. Given a sequence f _n in LSC(X) and a functionf LSC(X), we show that convergence of f _n tof in several variational convergence modes implies that for each , the sublevel set at height off is the limit, in the same variational sense, of an appropriately chosen sequence of sublevel sets of thef _n, at height _n approaching . The converse holds true whenever a form of stability of the sublevel sets of the limit function is verified. The results are obtained by regarding a hyperspace topology as the weakest topology for which each member of an appropriate family of excess functionals is upper semicontinuous, and each member of an appropriate family of gap functionals is lower semicontinuous. General facts about the representation of hyperspace topologies in this manner are given.     

Non polynomial splines and weakly singular two-point boundary value problems

In the present paper we shall consider an application of simple non-polynomial splines to a numerical solution of a weakly singular two-point boundary value problem:x ^– (x y)=f(x,y), (0<x1) subject toy(0)=0,y(1)=c ₁(1) ory(0)=c ₂,y(1)=c ₃(0<<1). Our collocation method gives a continuously differentiable approximation and isO(h ²)-convergent.     

On the weak convergence of stochastic processes without discontinuities of the second kind

Summary Subspaces D , > 0, of D[0, 1] are defined and given complete metrics d which are stronger than the Prokhorov metric. The spaces (D d ) are shown to be separable, and their pre-compact subsets are characterized. A condition which is known to guarantee weak pre-compactness of sets of probability measures over D[0, 1] is shown to also guarantee weak pre-compactness of probability measures over D for appropriate values of . Applications are made to the weak convergence of measures induced by stochastic processes, and some examples are included.     

On a Test Statistic for Linear Trend

Let {W(s)} _s 0 be a standard Wiener process. The supremum of the squared Euclidian norm Y (t)², of the R²-valued process Y(t)=(1/t W(t), {12/t ³ int_{0 t} s dW (s)– {3/t} W(t)), t [, 1], is the asymptotic, large sample distribution, of a test statistic for a change point detection problem, of appearance of linear trend. We determine the asymptotic behavior P {sup _t [, 1] Y(t)² > u as u , of this statistic, for a fixed (0,1), and for a moving = (u) 0 at a suitable rate as u . The statistical interest of our results lie in their use as approximate test levels.     

On a Darboux problem for a third-order hyperbolic equation with multiple characteristics

A Darboux type problem for a model hyperbolic equation of the third order with multiple characteristics is considered in the case of two independent variables. The Banach space, 0, is introduced where the problem under consideration is investigated. The real number ₀ is found such that for > ₀ the problem is solved uniquely and for < ₀ it is normally solvable in Hausdorff's sense. In the class of uniqueness an estimate of the solution of the problem is obtained which ensures stability of the solution.     

On a generalized Eulerian distribution

The distribution with probability function p _k(n, , ) = A _{n, k}(, )/(+ )^[p], k = 0, 1, 2, ..., n, where the parameters and are positive real numbers, A _{n, k} (, ) is the generalized Eulerian number and ( + )^[n] = ( + )( + +1) ... ( + +n – 1), introduced and discussed by Janardan (1988, Ann. Inst. Statist. Math., 40, 439–450), is further studied. The probability generating function of the generalized Eulerian distribution is expressed by a generalized Eulerian polynomial which, when expanded suitably, provides the factorial moments in closed form in terms of non-central Stirling numbers. Further, it is shown that the generalized Eulerian distribution is unimodal and asymptotically normal.     

On sequences of Fourier coefficients of functions of Hölder classes

The following theorem is proved. Let { _k(t)} be an arbitrary complete orthonormal system on [0, 1] and let 1/2<<1. Then anf(t) C exists for all< such that _k=1 · |c_k(f)|^p=, p=2/(l+2), where .Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 567–572, November, 1969.The authors wish to thank P. P. Zabreiko and P. L. Ul'yanov for helpful discussions and remarks.     

Repeated Angles inE 4

Let there be givenn points in four-dimensional euclidean spaceE ₄. We show that the number of occurrences of the angle iso(n ³) if is not a right angle and (n ³) otherwise.