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.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

Discrete Spectrum of Differential Operators in Spectral Gaps under Nonnegative Perturbations of High Order

Let A be a self-adjoint elliptic second-order differential operator, let (, ) be an inner gap in the spectrum of A, and let B(t) = A + tW ^* W, where W is a differential operator of higher order. Conditions are obtained under which the spectrum of the operator B(t) in the gap (, ) is either discrete, or does not accumulate to the right-hand boundary of the spectral gap, or is finite. The quantity N(, A, W, ), (, ), > 0 (the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to ) is considered. Estimates of N(, A, W, ) are obtained. For the perturbation W ^* W of a special form, the asymptotics of N(, A, W, ) as + is given. Bibliography: 5 titles.     

Applications of the zero–one law to problems connected with Cauchy's functional equation

Summary Marek Kuczma's book, entitled An Introduction To The Theory Of Functional Equations And Inequalities, mentions a certain setV ₀ in several places and presents references as to where this set is discussed in the literature. The main result of this paper is a proof of the fact that the setA _M (V ₀)={xV ₀ f(x)>M} is saturated non-measurable for each additive discontinuous functionf and each real numberM. Other results aboutV ₀ are also presented. Connections between measure and category are stressed. The main tool in our proofs is a certain so-called zero–one law and its topological analogue. In addition it is shown that the zero–one law is equivalent to Smital's lemma.     

Spectral properties of cosine operator functions

LetA be the generator of a cosine functionC _t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C _T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.     

Estimation of density functions by order statistics

The properties of the empirical density function,f _n(x) = k/n( _j ⁺ – _j-1 ⁺ ) if _j-1 ⁺ < x ⁺ where _j-1 ⁺ and _j ⁺ are sample elements and there are exactlyk – 1 sample elements between them, are studied in that practical point of view how to choose a suitablek for a good estimation. A bound is given for the expected value of the absolute value of difference between the empirical and theoretical density functions.     

Global Attractivity of the Equilibrium of a Nonlinear Difference Equation

The authors consider the nonlinear difference equation x_n+1=x_n+x_n-kf(x_n-k),n=0,1,(0.1) where (0,1), k 0,1, and f C¹[[0,), [0,)] with f(x) < 0.They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Waterman classes and triangular sums of double Fourier series

Let ^a ={nln^a (n+1)}, where a R. The following results are established: For every &fnof ^a BV ((- ]²), the triangular partial sums of its Fourier series are uniformly bounded if a = -1, and converge everywhere if a < -1.For every a>0, there exists &fnof ^a BV ((- ]²) such that the triangular partial sums of its Fourier series are unbounded at the point (0;0).     

On depth first search strees inm-out digraphs

We consider depth first search (DFS for short) trees in a class of random digraphs: am-out model. Let _i be thei ^th vertex encountered by DFS andL(i, m, n) be the height of _i in the corresponding DFS tree. We show that ifi/n asn, then there exists a constanta(,m), to be defined later, such thatL(i, m, n)/n converges in probability toa(,m) asn. We also obtain results concerning the number of vertices and the number of leaves in a DFS tree.     

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.     

Approximation under an arc length constraint

Let f C[a, b]. LetP be a subset ofC[a, b], L b – a be a given real number. We say thatp P is a best approximation tof fromP, with arc length constraintL, ifA[p] _b ^a [1 + (p(x)) ²]dx L andp – f q – f for allq P withA[q] L. represents an arbitrary norm onC[a, b]. The constraintA[p] L might be interpreted physically as a materials constraint.In this paper we consider the questions of existence, uniqueness and characterization of constrained best approximations. In addition a bound, independent of degree, is found for the arc length of a best unconstrained Chebyshev polynomial approximation.The work of L. L. Keener is supported by the National Research Council of Canada Grant A8755.     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

The disappearance of criticality in the theory of thermal ignition

Summary In the study of thermal ignition, when reactant consumption is neglected, criticality is usually taken as the point at which there is a large jump in the maximum steady temperature for a small change in the external parameters. If the equations are taken in their exact form, without the Frank-Kamenetskii approximation RT _a E0, criticality vanishes for large. In this paper a model is introduced for which the value of at which criticality vanishes can be determined precisely for geometries of an infinite slab and infinite circular cylinder.

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Zusammenfassung Wird beim Studium thermischer Explosionen der Reaktionsverbrauch vernachlässigt, so spricht man üblicherweise von einem kritischen Punkt, wenn dort bei einer kleinen Änderung der äußeren Parameter ein großer Sprung der stationären Maximaltemperatur auftritt. Bei den exakten Gleichungen (ohne die Frank-Kamenetskii'sche Approximation RT _a E0) gibt es für große keinen kritischen Punkt mehr. In dieser Arbeit wird ein Modell eingeführt, bei dem der Wert von , für welchen der kritische Punkt verschwindet, im Fall der unendlichen Platte und des unendlichen Zylinders genau bestimmt werden kann.

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Nomenclature A pre-exponential factor in Arrhenius law - d length scale (e.g. radius of cylinder and half-thickness of slab) - E Arrhenius activation energy - k thermal conductivity - Q exothermicity per unit mass - R gas constant - S surface of reacting material - T local absolute temperature - T _a surface temperature - V volume occupied by the reacting material - x dimensionless distance coordinate - =E(T-T _a )/RT _a ² dimensionless temperature - solution of Eqs. (6), (7) - maximum value of inV - =(1+)^–1 - maximum value of inV - density of material - =RT _a /E - ₀ smallest value of for which criticality vanishes - =QEAd ² exp (-E/RT _a )/kRT _a ² - _cr critical value of - ^* maximum value of for which positive solutions exist Supported in part by the Victoria Univeristy of Wellington Fellowship Committee and the European Research Office of the U.S. Army.     

Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

Symmetric balanced ternary designs with ϱ1 = 1 or 2

Summary We prove the following two non-existence theorems for symmetric balanced ternary designs. If ₁ = 1 and 0 (mod 4) then eitherV = + 1 or 4₂ – + 1 is a square and (4₂ – + 1) divides ² – 1. If ₁ = 2 thenV = ((m + 1)/2) ² + 2,K = (m ² + 7)/4 and = ((m – 1)/2)² + 1 wherem 3 (mod 4). An example belonging to the latter series withV = 18 is constructed.     

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.     

On the acceleration of limit periodic continued fractions

Summary A continued fraction (c.f.)K(a _n /1) is called limit periodic if . Fora anda(–,–1/4],a0, Thron-Waadeland (1980) examined a modification of a limit periodic c.f. for accelerating the convergence. This acceleration remains modest if thea _n converge only logarithmically. Thus it is proposed to apply an Euler summability method to the series equivalent to the c.f. Properties of the equivalent function are derived. These properties are used for choosing appropriate parameters for the summability method such that a considerable acceleration can be expected even if thea _n converge logarithmically.Dedicated to Prof. F.L. Bauer on the occasion of his 60th birthday     

Some extensions of the Hewitt-Savage zero-one law

Summary Let denote the class of infinite product probability measures = ₁× ₂× defined on an infinite product of replications of a given measurable space (X, A), and let denote the subset of for which (A) =0 or 1 for each permutation invariant event A. Previous works by Hewitt and Savage, Horn and Schach, Blum and Pathak, and Sendler (referenced in the paper) discuss very restrictive sufficient conditions under which a given member , of belongs to . In the present paper, the class is shown to possess several closure properties. E.g., if and _{0 n} for some n 1, then ₀× ₁× ₂×.... While the current results do not permit a complete characterization of they demonstrate conclusively that is a much larger subset of than previous results indicated. The interesting special case X={0,1} is discussed in detail.Research supported by the National Science Foundation under grant No. MCS75-07556     

Covering the hypercube with a bounded number of disjoint snakes

We present a construction of an induced cycle in then-dimensional hypercubeI[n] (n2), and a subgroup _n ofI[n] considered as the group ₂ ⁿ , such that | _n |16 and the induced cycle uses exactly one element of every coset of _n . This proves that for anyn2 the vertices ofI[n] can be covered using at most 16 vertex-disjoint induced cycles.