Spherical 7-Designs in 2 n -Dimensional Euclidean Space

We consider a finite subgroup _n of the group O(N) of orthogonal matrices, where N = 2 ⁿ , n = 1, 2 .... This group was defined in [7]. We use it in this paper to construct spherical designs in 2 ⁿ -dimensional Euclidean space R ^N . We prove that representations of the group _n on spaces of harmonic polynomials of degrees 1, 2 and 3 are irreducible. This and the earlier results [1–3] imply that the orbit _n,2 x ^t of any initial point x on the sphere S _{N – 1} is a 7-design in the Euclidean space of dimension 2 ⁿ .     

Analysis of Henrici's Transformation for Singular Problems

Henrici's transformation is the underlying scheme that generates, by cycling, Steffensen's method for the approximation of the solution of a nonlinear equation in several variables. The aim of this paper is to analyze the asymptotic behavior of the obtained sequence (s _n ^* ) by applying Henrici's transformation when the initial sequence (s _n ) behaves sublinearly. We extend the work done in the regular case by Sadok [17] to vector sequences in the singular case. Under suitable conditions, we show that the slowest convergence rate of (s _n ^* ) is to be expected in a certain subspace N of R ^p . More precisely, if we write s _n ^* =s _n ^* ,N+s _n ^* ,N, the orthogonal decomposition into N and N , then the convergence is linear for (s _n ^* ,N) but ( _n ^* ,N) converges to the same limit faster than the initial one. In certain cases, we can have N=R ^p and the convergence is linear everywhere.     

A class of self-orthogonal 2-sequencings

Let 2K _ndenote the complete multigraph on n vertices in which each edge has multiplicity two. If 2K _ncan be partitioned into Hamiltonian paths such that any two distinct paths have exactly one edge in common, write 2K _n P _n. This paper considerably expands the set of known positive integers n such that 2K _n P _n. The solutions found have application to other similar problems. The basic idea is to consider an algebraic formulation of the problem in terms of 2-sequencings (terraces) with additional properties. Construction of these 2-sequencings gives a special type of solution for which very few examples have been known. The constructions detailed here hold eventually for certain classes of prime powers. For example, it is shown that there is a positive integer N such that if N < p ⁿ 5 (mod 8) and 3 is not a fourth power residue of GF[p ⁿ], then the additive group of GF[p ⁿ] has a 2-sequencing of the required type—a self-orthogonal 2-sequencing. Some of the solutions admit a 2-coloring which is important for applications. The method of construction appears to be much better than the theoretical bounds that are obtained. The general bounds are found by means of a character sum argument.     

A Family of Asymptotically e(n − 1) ! Polynomial Orders of Nn

Here, N is the set of nonnegative integers, while an order in N ⁿ is a bijective function : N ⁿ N. Two orders are equivalent if they differ only by a permutation of their arguments. Let s(x)=x₁+ ··· +x _n for 0 < n N and x =(x ₁, ···, x _n ) N ⁿ ; such an is a diagonal order if (x) < (y) whenever x,y N ⁿ , and s(x) < s(y). Lew composed Skolem"s diagonal polynomial orders to construct c _n inequivalent nondiagonal polynomial orders in N ⁿ . Afterwards, Morales and Lew did the same with respect to the Morales–Lew"s diagonal orders, obtaining additional d _n inequivalent nondiagonal polynomial orders. Moreover, they proved that d _n / c _n as n . Recently, Sanchez obtained a family of (n – 1) ! inequivalent diagonal orders in N ⁿ . In this paper, we compose the Sanchez diagonal polynomial orders to construct e _n inequivalent nondiagonal polynomial orders with e _n e(n – 1) !, where e is the base of natural logarithms. Furthermore, we prove that e _n / d _n as n .     

Method of moments on semigroups

If ( _j ) is a sequence of measures onR ^k having momentss _n ( _j ) of all ordersnN ₀ ^k and if for eachnN ₀ ^k the sequence (s _{n j} )) _jN converges to somet _n R then some subsequence of ( _j ) converges weakly to a measure with moments of all orders satisfyings _n ()=t _n for allnN^0/k . Thisindeterminate method of moments and the continuity theorems in probability theory suggest a common generalization, dealing with a commutative semigroupS, with involution and a neutral element, and measures on the dual semigroupS ^* ofcharacters on S—hermitian multiplicative complex functions not identically zero. In this setting, a continuity theorem holds for measures on the set of bounded characters,⁽²⁾ and an indeterminate method of moments whenS is finitely generated.⁽²⁾ The latter result is generalized in the present paper to the case of arbitraryS. This leads to a generalization of Haviland's criterion for theK-moment problem, and to a continuity theorem for the so-called perfect semigroups.     

Threshold phenomena in the transient behaviour of Markovian models of communication networks and databases

This paper accompanies a talk given at the Workshop on Mathematical Methods in Queueing Networks held at the Mathematical Sciences Institute at Cornell University in August 1988. In earlier work we had exhibited a threshold phenomenon in the transient behaviour of a closed network of ./M/1 nodes: When there areN customers circulating, and the initial state isx, letd _x ^N (t) denote the total variation distance between the distribution at timet and the stationary distribution. Let d^N(t) = max _x d _x ^N (t). We explicitly founda _N proportional toN such thatd ^N(ta_N)1 forevery t<1, andd ^N(ta_N)0 forevery t>1. Thus it appears that the network has not yet converged to stationarity uptoa _N , but has converged to stationarity aftera _N , soa _N can be naturally interpreted as the settling time of the network. Here we briefly deal with some other similar models — closed networks of ./M/m nodes, a well studied model for circuit switched networks, and a model of Mitra for studying concurrency control in databases. Similar threshold phenomena are established in the transient behaviour of these models.Research supported by the National Science Foundation, Grant No. NCR 8710840.     

Distributive multiplication rings

A ringR is said to be a left (right)n-distributive multiplication ring, n>1 a positive integer, if aa₁a₂...a_n=aa₁aa₂...aa_n (a₁a₂...a_na=a₁aa₂a...a_na) for all a, a₁,...,a_n R. It will be shown that the semi-primitive left (right)n-distributive rings are precisely the generalized boolean ringsA satisfying aⁿ=a for all a A. An arbitrary left (right)n-distributive multiplication ring will be seen to be an extension of a nilpotent ringN satisfyingN ⁿ⁺¹=0 by a generalized boolean ring described above. Under certain circumstances it will be shown that this extension splits.     

Extreme Value Distributions for Random Coupon Collector and Birthday Problems

Take n independent copies of a strictly positive random variable X and divide each copy with the sum of the copies, thus obtaining n random probabilities summing to one. These probabilities are used in independent multinomial trials with n outcomes. Let N _n(N ^* _n) be the number of trials needed until each (some) outcome has occurred at least c times. By embedding the sampling procedure in a Poisson point process the distributions of N _n and N ^* _n can be expressed using extremes of independent identically distributed random variables. Using this, asymptotic distributions as n are obtained from classical extreme value theory. The limits are determined by the behavior of the Laplace transform of X close to the origin or at infinity. Some examples are studied in detail.     

On extending backwards positive definite sequences

The two-sided Hamburger moment problem¹, also called the strong one [4], has been extensively studied in recent years in connection with rational approximation. We propose to consider the question of when a sequence, say {a _n } _n=0 can be extended backwards so that the resulting sequence {a _n } _n=–N has an integral representation of the Hamburger type. This was settled (without any proof) under different circumstances in [6]. Here we wish to discuss this completely, as well as the possibility of extending {a _n } _n=0 to {a _n } _n– .     

On Dirichlet Series for Sums of Squares

Hardy and Wright (An Introduction to the Theory of Numbers, 5th edn., Oxford, 1979) recorded elegant closed forms for the generating functions of the divisor functions _k (n) and _k ²(n) in the terms of Riemann Zeta function (s) only. In this paper, we explore other arithmetical functions enjoying this remarkable property. In Theorem 2.1 below, we are able to generalize the above result and prove that if f _i and g _i are completely multiplicative, then we have where L _f(s) := _{n = 1} f(n)n ^–s is the Dirichlet series corresponding to f. Let r _N(n) be the number of solutions of x ₁ ² + ··· + x _N ² = n and r _2,P (n) be the number of solutions of x ² + Py ² = n. One of the applications of Theorem 2.1 is to obtain closed forms, in terms of (s) and Dirichlet L-functions, for the generating functions of r _N(n), r _N ²(n), r _2,P (n) and r _2,P (n)² for certain N and P. We also use these generating functions to obtain asymptotic estimates of the average values for each function for which we obtain a Dirichlet series.     

Stability of Jackson Type Network Output

We consider an open Jackson type queueing network N with input epochs sequence I={T _n ⁽⁰⁾,n0}, T ₀ ⁽⁰⁾=0, assume another input ={ _n ⁽⁰⁾} and denote _k =| _k ⁽⁰⁾–T _k ⁽⁰⁾|, ₀=0, _n =max_{1kn k} , n1. Let {T _n } and { _n } be the output points in network N and in modified network, with input , accordingly. We study the long-run stability of the network output, establishing two-sided bounds for output perturbation via input perturbation. In particular, we obtain conditions that imply max _kn |T _k – _k |=o(n ^1/r ) with probability 1 as n for some r>0. This result is also extended to continuous time. We consider successively separate station (service node), tandem and feedforward networks. Then we extend stability analysis to general (feedback) networks and show that in our setting these networks can be reduced to feedforward ones. Similar stability results are also obtained in terms of the number of departures. Application to a tandem network with the overloaded stations is considered.     

Backward Stability of Clenshaw's Algorithm

We study numerical properties of Clenshaw's algorithm for summing the series w = _{n = 0} ^N b _n p _n where p _n satisfy the linear three-term recurrence relation. We prove that under natural assumptions Clenshaw's algorithm is backward stable with respect to the data b _n, n = 0,N.     

Limit Theorems for the Maximum Terms of a Sequence of Random Variables with Marginal Geometric Distributions

Let X _n1 ^* , ... X _nn ^* be a sequence of n independent random variables which have a geometric distribution with the parameter p _n = 1/n, and M _n ^* = \max\{X _n1 ^* , ... X _nn ^* }. Let Z ₁, Z₂, Z₃, ... be a sequence of independent random variables with the uniform distribution over the set N _n = {1, 2, ... n}. For each j N _n let us denote X _nj = min{k : Z_k = j}, M _n = max{X_n1, ... X_nn}, and let S _n be the 2nd largest among X _n1, X_n2, ... X_nn. Using the methodology of verifying D(u_n) and D'(u_n) mixing conditions we prove herein that the maximum M _n has the same type I limiting distribution as the maximum M _n ^* and estimate the rate of convergence. The limiting bivariate distribution of (S_n, M_n) is also obtained. Let _n, _n N_n, , and T _n = min{M(A_n), M(B_n)}. We determine herein the limiting distribution of random variable T _n in the case _n , _n/_n > 0, as n .     

The k-record processes are i.i.d.

Let X ₁, X ₂, ... be i.i.d. positive random variables, and let _n be the initial rank of X _n (that is, the rank of X _n among X ₁, ..., X _n). Those observations whose initial rank is k are collected into a point process N ^k on ⁺, called the k-record process. The fact that {itN^k; k=1, 2, ... are independent and identically distributed point processes is the main result of the paper. The proof, based on martingales, is very rapid. We also show that given N ¹, ..., N ^k, the lifetimes in rank k of all observations of initial rank at most k are independent geometric random variables.These results are generalised to continuous time, where the analogue of the i.i.d. sequence is a time-space Poisson process. Initially, we think of this Poisson process as having values in ⁺, but subsequently we extend to Poisson processes with values in more general Polish spaces (for example, Brownian excursion space) where ranking is performed using real-valued attributes.     

Linear widths of Hölder-Nikol'skii classes of periodic functions of several variables

We consider the linear widths _N (W _p ^r (Tⁿ), L_q) and _N (H _p ^r (Tⁿ), L_q) of the classesW _p ^r (Tⁿ) andH _p ^r (Tⁿ) of periodic functions of one or several variables in the spaceL _q. For the Sobolev classesW _p ^r (Tⁿ) of functions of one or several variables, we state some well-known results without proof; for the Hölder-Nikol'skii classesH _p ^r (Tⁿ), we state some well-known results, prove some new results, and present some previously unpublished proofs.Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 189–199, February, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

Trace formulas for a class of subnormal tuples of operators

Trace formulas are established for the product of commutators related to subnormal tuple of operators (S ₁,...,S _n ) with minimal normal extension (N ₁,...,N _n ) satisfying conditions that sp(S _j )/sp(N _j ) is simply-connected with smooth boundary Jordan curve sp(N _i ) and [S _j ^* ,S _j ]^1/2 L ¹,j=1, 2,...,n.Some complete unitary invariants related to the trace formulas are found.This work is supported in part by NSF Grant no. DMS-9101268.     

Approximating the ball by a minkowski sum of segments with equal length

It is proved that forn 2 the Euclidean ballB _n can be approximated up to (in the Hausdorff distance) by a zonotope havingN summands of equal length withN c(n)( ^–2|log|)^{(n–1)/(n+2)}.Research supported in part by the U.S.-Israeli Binational Science Foundation. [Please see the Editors' note on the first page of the preceding paper.]     

A criterion for <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript>-rectifiability of sets

The following result is proved: Let be a n-dimensional C¹-submanifold of R^N which is domain of a given _nR^N-valued map of class C¹. Then the set of all points P such that (P) is non-zero, simple and enveloped by T_P is C²-rectifiable. As a corollary we get a criterion for the C²-rectifiability of a rectifiable set based on the rectifiability of some generalized Gauss lift to the Grassmanian bundle R^N×G(N,n). Mathematics Subject Classification (2000) Primary 49Q15, 53A07; Secondary 49Q20, 49N60