The Embedding Theorem for Cantor Varieties

Let m and n be fixed integers, with 1 m < n. A Cantor variety C _m,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature ={g₁,...,g_m,f₁,...,f_n} by the identities f_ig₁x₁,...,x_n),...,g_mx₁,...,x_n) = x_i, _i=1,...,n, g_jf₁x₁,...,x_m),...,f_nx₁,...,x_m)) = x_j, _j=1,...,m. We prove the following: (a) every partial C _m,n-algebra A is isomorphically embeddable in the algebra G= A; S(A) of C _m,n; (b) for every finitely presented algebra G= A; S in C _m,n, the word problem is decidable; (c) for finitely presented algebras in _{C m}, the occurrence problem is decidable; (d) _{C m,n} has a hereditarily undecidable elementary theory.     

Complete unitary invariant for some subnormal operator

IfS is a subnormal operator with minimal normal extensionN satisfying the conditions that (i) \(\left[ {S^* ,S} \right]^{\frac{1}{2}} \in \mathcal{L}^1\) , (ii) sp (S) is the unit disk and (iii) sp (N)={N: |z|=1 orz=a ₁,...,a _k then $$tr\left( {\left[ {S^* ,(\lambda I - S)^{ - 1} } \right]\left[ {S^* ,(\mu I - S)^{ - 1} } \right]} \right) = \frac{n}{{\lambda ^2 \mu ^2 }} + \sum\limits_{i,j = 1}^k {\frac{{\gamma ij}}{{\lambda \mu (\lambda - a_i )(\mu - a_j )}}} $$ . wheren=index ( \(S^* - \bar zI\) ) forz∈sp (S)/sp (N) and (γij) is a real symmetric matrix. The set {n, γij,i,j = 1,...,k} is a complete unitary invariant for an operator in the class of all irreducible subnormal operators satisfying conditions (i), (ii), (iii) and that there is at least one positive simple eigenvalue of [S ^*,S].     

An Application of Polya's Enumeration Theorem to Partitions of Subsets of Positive Integers

Let S be a non-empty subset of positive integers. A partition of a positive integer n into S is a finite nondecreasing sequence of positive integers a ₁, a ₂,...,a _r in S with repetitions allowed such that . Here we apply Polya's enumeration theorem to find the number P(n; S) of partitions of n into S, and the number DP(n; S) of distinct partitions of n into S. We also present recursive formulas for computing P(n; S) and DP(n; S).     

Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) _{[ns] + 1} ^[nt] X _j ⁽ⁿ⁾ , where {X _j ⁽ⁿ⁾ } _{j = 1} ⁿ are the order statistics from independent random variables {X ₁,...,X _n } having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like _{j = 1} ^[nt] J(j/n)X _j ⁽ⁿ⁾ are also discussed.     

Some monotonicity properties of symmetric pólya densities and their exponential families

Summary In this note we observe that for independent symmetric random variables X and Y, when the pdf of X is PF, the conditional distributions of ¦Y¦ given S = X + Y form a MLR family. We then show that for a function : R ⁿR that is symmetric in each coordinate and increasing on (0, )ⁿ, E((S₁,...,S_n)¦S_n = s) is even and increasing in ¦s¦. Here S₁,...,S_n are partial sums with independent symmetric PF summands. Application is made to sequential tests that minimize the maximum expected sample size when the model is a one-parameter exponential family generated by a symmetric PF density.Work supported by NSF grants MPS 72-05082 AO2 and MCS 75-23344     

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.     

A generalization of the inversion formulas of systems of power series in systems of implicit functions

Using a multidimensional analog of the logarithmic residue, equations are derived expressing the coefficients of the power series of implicit functionsx _j =_j(w)=_j(w₁,...,w_m), j=1,...,n, defined by the system of equations f_j(w, x)=F_j (w₁,...,w_m:z₁,...,x _n )=0, j=1,...,n,f _j , (0, 0)=0, F_j(0, 0)/z_k=_jk in a neighborhood of the point (0, 0)C _(w,x) ^m+n , in terms of the coefficients of the power series of the functions F_j(w, z), j=1, ..., n. As a corollary, well-known formulas are obtained for the inversion of multiple power series.Translated from Matematicheskie Zametki, Vol. 23, No. 1, pp. 47–54, January, 1978.     

Absolutely convergent multiple fourier series and multiplicative Lipschitz classes of functions

We consider N-multiple trigonometric series whose complex coefficients c _j1,...,j _N , (j ₁,...,j _N ) ∈ ? ^N , form an absolutely convergent series. Then the series $$ \sum\limits_{(j_1 , \ldots ,j_N ) \in \mathbb{Z}^N } {c_{j_1 , \ldots j_N } } e^{i(j_1 x_1 + \ldots + j_N x_N )} = :f(x_1 , \ldots ,x_N ) $$ converges uniformly in Pringsheim’s sense, and consequently, it is the multiple Fourier series of its sum f, which is continuous on the N-dimensional torus $ \mathbb{T} $ ^N , $ \mathbb{T} $ := [?π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α₁,..., α _N ) and lip (α₁,..., α _N ) for some α₁,..., α _N > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients. Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular remaining sums of related N-multiple numerical series may be useful in other investigations, too.     

Distributions of Random Partitions and Their Applications

Assume that a random sample of size m is selected from a population containing a countable number of classes (subpopulations) of elements (individuals). A partition of the set of sample elements into (unordered) subsets, with each subset containing the elements that belong to same class, induces a random partition of the sample size m, with part sizes {Z ₁,Z ₂,...,Z _N } being positive integer-valued random variables. Alternatively, if N _j is the number of different classes that are represented in the sample by j elements, for j=1,2,...,m, then (N ₁,N ₂,...,N _m ) represents the same random partition. The joint and the marginal distributions of (N ₁,N ₂,...,N _m ), as well as the distribution of are of particular interest in statistical inference. From the inference point of view, it is desirable that all the information about the population is contained in (N ₁,N ₂,...,N _m ). This requires that no physical, genetical or other kind of significance is attached to the actual labels of the population classes. In the present paper, combinatorial, probabilistic and compound sampling models are reviewed. Also, sampling models with population classes of random weights (proportions), and in particular the Ewens and Pitman sampling models, on which many publications are devoted, are extensively presented.      

The Minimum Covering l pb -Hypersphere Problem

The minimu vering hypersphere problem is defined as to find a hypersphere of minimum radius enclosing a finite set of given points in ⁿ. A hypersphere is a set S(c,r)={x ⁿ : d(x,c) r}, where c is the center of S, r is the radius of S and d(x,c) is the Euclidean distance between x and c, i.e.,d(x,c)=l ₂ (x-c). We consider the extension of this problem when d(x, c) is given by any l _pb -norm, where 1<p and b=(b ₁,...,b _n ) with b _j >0, j=1,...,n, then S(c,r) is called an l _pb -hypersphere, in particular for p=2 and b _j =1, j=1,..., n, we obtain the l ₂-norm. We study some properties and propose some primal and dual algorithms for the extended problem , which are based on the feasible directions method and on the Wolfe duality theory. By computational experiments, we compare the proposed algorithms and show that they can be used to approximate the smallest l _pb -hypersphere enclosing a large set of points in ⁿ.     

Asymptotics for the Euclidean TSP with power weighted edges

Summary LetU ₁,...,U_n denote i.i.d. random variables with the uniform distribution on [0, 1]², and letT ²T²(U₁,...,U_n) denote the shortest tour throughU ₁,...,U_n with square-weighted edges. By drawing on the quasi-additive structure ofT ² and the boundary rooted dual process, it is shown that lim _n E T ²(U ₁,...,U_n)= for some finite constant .This work was supported in part by NSF Grant DMS-9200656, Swiss National Foundation Grant 21-298333.90, and the US Army Research Office through the Mathematical Sciences Institute of Cornell University, whose assistance is gratefully acknowledged     

Bounds for the Concentration of Asymptotically Normal Statistics

Let X₁,..., X _n be independent, not necessarily identically distributed random variables. An optimal bound is derived for the concentration function of an arbitrary real-valued statistic T = T (X ₁,...,X _n ) for which ET² < . Applications are given for Wilcoxon"s rank-sum statistic, U-statistics, Student"s statistic, the two-sample Student statistic and linear regression.     

Interacting Brownian particles and the Wigner law

Summary In this paper, we study interacting diffusing particles governed by the stochastic differential equationsdX _j (t)= _n dB _j (t) –D _jØ_n(X ₁,...,X _n)dt,j=1, 2,...,n. Here theB _jare independent Brownian motions in ^d , and Ø _n (X ₁,...,X _n)= _{n ij} V(X _iX _j) + _ni U(X ₁). The potentialV has a singularity at 0 strong enough to keep the particles apart, and the potentialU serves to keep the particles from escaping to infinity. Our interest is in the behaviour as the number of particles increases without limit, which we study through the empirical measure process. We prove tightness of these processes in the case ofd=1,V(x)=–log|x|,U(x)=x ²/2 where it is possible to prove uniqueness of the limiting evolution and deduce that a limiting measure-valued process exists. This process is deterministic, and converges to the Wigner law ast. Some information on the rates of convergence is derived, and the case of a Cauchy initial distribution is analysed completely.Supported by SERC grant number GR/H 00444     

Conditioned limit theorems for random walks with negative drift

Summary In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S _n is a random walk with negative mean and finite variance then there is a constant so that (S _[n.]/n ^1/2¦N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES ₁=–a<0, ES ₁ ² <, and there is a slowly varying function L so that P(S ₁>x)x ^–q L(x) as x then (S _[n.]/n¦S _n >0) and (S _[n.]/n¦N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1–(x/a)^–q )⁺) and are otherwise linear with slope –a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second.The research for this paper was started while the author was visiting W. Vervaat at the Katholieke Universiteit in Nijmegen, Holland, and was completed while the author was at UCLA being supported by funds from NSF grant MCS 77-02121     

On Probability and Moment Inequalities for Supermartingales and Martingales

The probability inequality for sum S _n = _j=1 ⁿ X _j is proved under the assumption that the sequence S _k , k= , forms a supermartingale. This inequality is stated in terms of the tail probabilities P(X _j >y) and conditional variances of the random variables X _j , j= . The well-known Burkholder moment inequality is deduced as a simple consequence.     

Itération de pliages de quadrilatères

Iteration of quadrilateral foldings. Starting with a quadrilateral q₀=(A₁,A₂,A₃,A₄) of ², one constructs a sequence of quadrilaterals q_n=(A_4n+1,...,A_4n+4) by iteration of foldings: q_n=₄°₃°₂°₁(q_n-1) where the folding _j replaces the vertex number j by its symmetric with respect to the opposite diagonal (see Fig. 1).We study the dynamical behavior of this sequence. In particular, we prove that:– The drift exists.– When none of the q_n is isometric to q₀, then the drift v is zero if and only if one has maxa_j+mina_j1/2a_j where a₁,...,a₄ are the sidelengths of q₀.– For Lebesgue almost all q₀ the sequence (q_n-nv)_n1 is dense on a bounded analytic curve with a center or an axis of symmetry. However, for Baire generic q₀, the sequence (q_n-nv)_n1 is unbounded (see Figs. 2 to 7).      

Circle grids and bipartite graphs of distances

Fort fixed,n+t pointsA ₁,A ₂,...,A _n andB ₁,B ₂,...,B _t are constructed in the plane withO(n) distinct distancesd(A _i B _j ) As a by-product we show that the graph of thek largest distances can contain a complete subgraphK _{t, n} withn=(k ²), which settles a problem of Erds, Lovász and Vesztergombi.Research partially supported by the Hungarian National Science Fund (OTKA) # 2117.     

Elementary symmetric polynomials of increasing order

Summary The asymptotic behaviour of elementary symmetric polynomials S _n ^(k) of order k, based on n independent and identically distributed random variables X ₁,..., X _n,is investigated for the case that both k and n get large. If , then the distribution function of a suitably normalised S _n ^(k) is shown to converge to a standard normal limit. The speed of this convergence to normality is of order , provided and certain natural moment assumptions are imposed. This order bound is sharp, and cannot be inferred from one of the existing Berry-Esseen bounds for U-statistics. If k at the rate n ^1/2 then a non-normal weak limit appears, provided the X _i's are positive and S _n ^(k) is standardised appropriately. On the other hand, if k at a rate faster than n ^1/2 then it is shown that for positive X _j'sthere exists no linear norming which causes S _n ^(k) to converge weakly to a nondegenerate weak limit.     

The k-orbit reconstruction and the orbit algebra

Let (G, W) be a permutation group on a finite set W = {w ₁,..., w _n}. We consider the natural action of G on the set of all subsets of W. Let h ₀, h ₁,..., h _N be the orbits of this action. For each i, 1 i N, there exists k, 1 k n, such that h _i is a set of k-element subsets of W. In this case h _i is called a symmetrized k-orbit of the group (G, W) or simply a k-orbit. With a k-orbit h _i we associate a multiset H(h _i ) = % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyykJeoaaa!3690!\[\langle \]h _i ⁽¹⁾, h _i ⁽²⁾,..., h _i ^(k)% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGGipm0dc9vqaqpepu0xbbG8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOkJepaaa!36A1!\[\rangle \] of its (k – 1)-suborbits. Orbits h _i and h _j are called equivalent if H(h _i ) = H(h _j ). An orbit is reconstructible if it is equivalent to itself only. The paper concerns the k-orbit reconstruction problem and its connections with different problems in combinatorics. The technique developed is based on the notion of orbit and co-orbit algebras associated with a given permutation group (G, W).     

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.