Divisibility Graph for Symmetric and Alternating Groups

Let X be a nonempty set of positive integers and X* = X?{1}. The divisibility graph D(X) has X* as the vertex set, and there is an edge connecting a and b with a, b ∈ X* whenever a divides b or b divides a. Let X = cs(G) be the set of conjugacy class sizes of a group G. In this case, we denote D(cs(G)) by D(G). In this paper, we will find the number of connected components of D(G) where G is the symmetric group S _n or is the alternating group A _n .     

On the law of the iterated logarithm for weighted sums of independent random variables in a Banach space

Assume that (X _n) are independent random variables in a Banach space, (b _n) is a sequence of real numbers, S_n= ₁ ⁿ b_iX_i, and B_n= ₁ ⁿ b _i ² . Under certain moment restrictions imposed on the variablesX _n, the conditions for the growth of the sequence (b_n) are established, which are sufficient for the almost sure boundedness and precompactness of the sequence (S_n/B _n ln ln B_n)^1/2).Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1225–1231, September, 1993.     

Extreme value limit laws in the nonidentically distributed case

LetX ₁, ...,X _n be independent random variables, letF _i be the distribution function ofX _i (1≦i≦n) and letX _1n ≦... ≦X _nn be the corresponding order statistics. We consider the statisticsX _kn, wherek=k(n),k/n → 1 andn−k → ∞. Under some additional restrictions concerning the behaviour of the sequences {a _n>0,b _n,k(n),F _n} we characterize the class of all distribution functionsH such that Prob{(X _kn −b _n )/a _n <x)}→H. Dedicated to the Memory of N. V. Smirnov (1900–1966)     

Automorphism Groups of 2-Valent Connected Cayley Digraphs on Regular <Emphasis Type="Italic">p</Emphasis>-Groups

 Let X=Cay(G,S) be a 2-valent connected Cayley digraph of a regular p-group G and let G _R be the right regular representation of G. It is proved that if G _R is not normal in Aut(X) then X≅[2K ₁ ] with n>1, Aut(X) ≅Z ₂ wrZ _2n , and either G=Z _2n+1 =<a> and S={a,a ²ⁿ⁺¹ }, or G=Z _2n ×Z ₂ =<a>×<b> and S={a,ab}. Received: May 26, 1999 Final version received: June 19, 2000     

A characterization of the exponential distribution

Summary A necessary and sufficient condition is given under which the minimum ofX _k /a _k ,k=1, 2, ...,n has the same distribution asX ₁, whereX's are i.i.d. positive random variables anda's are given positive constants. The Institute of Statistical Mathematics     

Only the first term of some series counts

Let X,X₁,X₂,… be i.i.d. random variables, and set S_n=X₁+?+X_n. We prove that for three important distributions of X, namely normal, exponential and geometric, series of the type ∑_n≥1a_nP(|S_n|≥xb_n) or ∑_n≥1a_nP(S_n≥xb_n) behave like their first term as x→∞.     

A note on asymptotics of linear combinations of iid random variables

Let X₁,X₂, ... be iid random variables, and let a _n = (a ₁,n, ..., a _n,n ) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination $ S_{a_n } $ S_{a_n } = a _1,n X ₁ + ... + a _n,n X _n under the natural negligibility condition lim _n→∞ max{|a _k,n |: k = 1, ..., n} = 0. We prove that if $ S_{a_n } $ S_{a_n } is asymptotically normal for a weight sequence a _n , in which the components are of the same magnitude, then the common distribution belongs to $ \mathbb{D} $ \mathbb{D} (2).     

Specialization of a generic hyperplane section through a rational point of an algebraic variety

Summary Let V/k be an irreducible algebraic variety of dimension ≥3, defined over a field k of characteristic O, passing through the origin (O) of the ambient affine n-space. Let p be the prime ideal of V in the polynomial ring k[X₁, ..., X_n]. If V/k is k-normal at (O), then for almost all hyperplanes H_a: a₁X₁+...+a_nX_n=0 with a₁, ..., a_n ε k the ideal (p, a₁X₁+...+a_nX_n) is a prime ideal. If the generic hyperplane section of V through (O) is normal over the ground field of the generic section of V, then the section V ∩ H_a is k-normal at (O) for almost all H_a. Entrata in Redazione il 1 ottobre 1971.     

Method of solution to fuzzy relation equations in a complete Brouwerian lattice

Up to now, how to solve a fuzzy relation equation in a complete Brouwerian lattice is still an open problem as Di Nola et al. point out. To this problem, the key problem is whether there exists a minimal element in the solution set when a fuzzy relation equation is solvable. In this paper, we first show that there is a minimal element in the solution set of a fuzzy relation equation AX=b (where A=(a₁,a₂,…,a_n) and b are known, and X=(x₁,x₂,…,x_n)^T is unknown) when its solution set is nonempty, and b has an irredundant finite join-decomposition. Further, we give the method to solve AX=b in a complete Brouwerian lattice under the same conditions. Finally, a method to solve a more general fuzzy relation equation in a complete Brouwerian lattice when its solution set is nonempty is also given under similar conditions.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

The occurrence of large values in stationary sequences

Summary A direct proof is given of the Tanny (1974) result that for certain non-decreasing sequences a _n , it is true that lim supa _n ^–1 X _n = 0 or + with probability one for all ergodic stationary sequences X _n . The condition on a _n is shown to be necessary. For all non-decreasing a _n and stationary X _n , lim sup a _n ^–1 X _n= lim sup a _n ^–1 X _–n a.s. Similar continuous-time theorems are also given.This research supported in part by the Natural Sciences and Engineering Research Council of Canada     

Solution Methodologies for the Smallest Enclosing Circle Problem

Given a set of circles C = {c ₁, ..., c _n} on the Euclidean plane with centers {(a ₁, b ₁), ..., (a _n, b _n)} and radii {r ₁, ..., r _n}, the smallest enclosing circle (of fixed circles) problem is to find the circle of minimum radius that encloses all circles in C. We survey four known approaches for this problem, including a second order cone reformulation, a subgradient approach, a quadratic programming scheme, and a randomized incremental algorithm. For the last algorithm we also give some implementation details. It turns out the quadratic programming scheme outperforms the other three in our computational experiment.     

Ordering distributions by scaled order statistics

Motivated by applications in reliability theory, we define a preordering (X ₁, ...,X _n) (Y ₁ ...,Y _n) of nonnegative random vectors by requiring thek-th order statistic ofa ₁ X ₁,..., a _n X _n to be stochastically smaller than thek-th order statistic ofa ₁ Y ₁, ...,a _n Y _n for all choices ofa _i >0,i=1, 2, ...,n. We identify a class of functionsM _{k, n} such that if and only ifE(X)E(Y) for allM _k,n. Some preservation results related to the ordering are obtained. Some applications of the results in reliability theory are given.Supported by the Air Force Office of Scientific Research, U.S.A.F., under Grant AFOSR-84-0205.     

Co-monotone allocations,Bickel-Lehmann dispersion and the Arrow-Pratt measure of risk aversion

For every integrable allocation (X ₁,X ₂, ...,X _n ) of a random endowmentY= _i ^=1/n X _i amongn agents, there is another allocation (X ₁*,X ₂*, ...,X _n *) such that for every 1in,X _i * is a nondecreasing function ofY (or, (X ₁*,X ₂*, ...,X _n *) areco-monotone) andX _i * dominatesX _i by Second Degree Dominance.If (X ₁*,X ₂*, ...,X _n *) is a co-monotone allocation ofY= _i ^=1/n X _i *, then for every 1in, Y is more dispersed thanX _i * in the sense of the Bickel and Lehmann stochastic order.To illustrate the potential use of this concept in economics, consider insurance markets. It follows that unless the uninsured position is Bickel and Lehmann more dispersed than the insured position, the existing contract can be improved so as to raise the expected utility of both parties, regardless of their (concave) utility functions.     

On the existence of one-sided designs

A collection of subsets (called blocks) of a fixed vertex set (possibly with repetition) is called a (t _n , t _n –1, ..., t ₁; a _m , a _m –1, ..., a ₁)-design if it satisfies certain regularity conditions on the number of blocks which contain subsets of the vertex set of certain size, and other regularity conditions on the size of the intersections of certain numbers of the blocks. (For example, a BIBD (or (b, v, r, k, )-configuration) is a (1, 2; 1)-design, and a t-design is a (t, t–1, ..., 1; 1)-design.) A design has design-type (t _n , ..., t ₁; a _m , ..., a ₁) if it satisfies only those conditions. A one-sided design is a design with design-type (t _n , ..., t ₁;) or (;a _m , ..., a ₁). In this paper we show, by construction, that any one-sided design-type is possible.     

Extremes of independent Gaussian processes

For every n ∈ ℕ, let X _1n ,..., X _nn be independent copies of a zero-mean Gaussian process X _n  = {X _n (t), t ∈ T}. We describe all processes which can be obtained as limits, as n→ ∞, of the process a _n (M _n  − b _n ), where M _n (t) =  max _{i = 1,...,n} X _in (t), and a _n , b _n are normalizing constants. We also provide an analogous characterization for the limits of the process a _n L _n , where L _n (t) =  min _{i = 1,...,n} |X _in (t)|.     

The strong approximation of extremal processes

Summary If X ₁, X ₂, ..., are i.i.d. random variables and Y _n =Max(X ₁, ..., X _n ); if for some sequences A _n , B_n, n=1, 2, ..., E _n (t)=A_nY_[nt]+B_n is such that E _n (1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E ⁽ⁿ⁾(t) on the same probability space, such that d(E _n (.), E ⁽ⁿ⁾(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations.     

Large-deviation probability and the local dimension of sets

Let X ₁ , X ₂ , ..., X_n be n independent identically distributed real random variables and S_n = Σ _n=1 ⁿ X_i. We obtain precise asymptotics forP (S_n ∈ nA) for rather arbitrary Borel sets A₁ in terms of the density of the dominating points in A. Our result extends classical theorems in the field of large deviations for independent samples. We also obtain asymptotics forP (S_n ∈ γ_nA), with γ_n/n → ∞. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

Quadrature based collocation methods for integral equations of the first kind

Problem of solving integral equations of the first kind, ò_a^b k(s, t) x(t) dt = y(s), s ? [a, b]\int_a^b k(s, t) x(t)\, dt = y(s),\, s\in [a, b] arises in many of the inverse problems that appears in applications. The above problem is a prototype of an ill-posed problem. Thus, for obtaining stable approximate solutions based on noisy data, one has to rely on regularization methods. In practice, the noisy data may be based on a finite number of observations of y, say y(τ ₁), y(τ ₂), ..., y(τ _n ) for some τ ₁, ..., τ _n in [a, b]. In this paper, we consider approximations based on a collocation method when the nodes τ ₁, ..., τ _n are associated with a convergent quadrature rule. We shall also consider further regularization of the procedure and show that the derived error estimates are of the same order as in the case of Tikhonov regularization when there is no approximation of the integral operator is involved.     

On P.I. ring and standard identities

An example is given of a ringR (with 1) satisfying the standard identityS ₆[x ₁, ...,x ₆] butM ₂(R), the 2 × 2 matrix ring overR, does not satisfyS ₁₂[x ₁, ...,x ₁₂]. This is in contrast to the caseR=M _n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS _2n [x ₁, ...,x _2n] andM ₂(R) satisfiesS _4n [x ₁, ...,x _n]. Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.