The product of finitely based varieties of lattice-ordered groups

The question as to whether a product of two finitely based varieties of lattice-ordered groups is finitely based is considered. It is proved that varieties and are finitely based; here is a variety of lattice-ordered groups defined by identities [x ⁿ,y ⁿ] =e and [[x,y] z, [x ₁,y ₁] z ₁] =e; is a variety of lattice-ordered nilpotent groups of class s, defined by an identity [x ₁,x ₂,...,x _(s+1)] =e; V is an arbitrary finitely based variety of lattice-ordered groups. Translated fromAlgebra i Logika, Vol. 33, No. 3, pp. 255–263, May–June, 1994.Supported by the Russian Foundation for Fundamental Research, grant No. 93-011-1524.     

Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2-periodic functions x with bounded variation of the derivative x, namely

On the asymptotic behaviour of first passage times for transient random walk

Summary Let _x denote the time at which a random walk with finite positive mean first passes into (x, ), wherex0. This paper establishes the asymptotic behaviour of Pr { _x >n} asn for fixedx in two cases. In the first case the left hand tail of the step-distribution is regularly varying, and in the second the step-distribution satisfies a one-sided Cramér type condition. As a corollary, it follows that in the first case Pr { _x >n}/Pr{ ₀ >n} coincides with the limit of the same quantity for recurrent random walk satisfying Spitzer's condition, but in the second case the limit is more complicated.     

Steepest Descent with Curvature Dynamical System

Let H be a real Hilbert space and let <..,.> denote the corresponding scalar product. Given a function that is bounded from below, we consider the following dynamical system:

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is 0 such that the essential spectral radius of is smaller than || (for example, if G R ³ is a domain with a piecewise smooth boundary and the restriction of the Newtonian potential on G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition for which (G) = 0.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

The Most Visited Sites of Certain Lévy Processes

Let X be a symmetric Lévy process with

On the essential approximate point spectrum of operators

If belongs to the essential approximate point spectrum of a Banach space operatorTB(X) and is a sequence of positive numbers with lim ^j a _j =0, then there existsxX such that for every polynomialp. This result is the best possible — if for some constantc>0 thenT has already a non-trivial invariant subspace, which is not true in general.     

Renewal Theorems for Singular Differential Operators

Let * be the convolution on M( ₊) associated with a second order singular differential operator L on ]0, +[. If is a probability measure on ₊ with suitable moment conditions, we study how to normalize the measures * ⁿ ; n } (resp. ) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.     

Integral averages and oscillation of second order sublinear differential equations

New oscillation criteria are given for the second order sublinear differential equation

On the Normal Order of ϕk+1(n)/ϕk(n), where ϕk is the k-Fold Iterate of Euler's Function

Let be the j-fold iterated function of . Let and > 0 be fixed, Q be a prime, and let N _k(Q|x) denote the number of those nx for which Q . We give the asymptotics of N _k(Q|x) in the range .     

On Interpolation Sequences of One Class of Functions, Analytic in the Half-Plane

A new criterion of solvability of the interpolation problem f( _n )=b_n in the class of functions f, analytic in the right half-plane and such that there exists c ₁(0;+) such that |f(z)|c ₁exp((c₁|z|)) for all z , where is a positive increasing continuous differentiable function on [0;+), for which (t)+ as t+ and there exists c ₂(0;+) such that

Estimates for the Distributions of the Sums of Subexponential Random Variables

Let be a random walk with independent identically distributed increments . We study the ratios of the probabilities P(S _n >x) / P(₁ > x) for all n and x. For some subclasses of subexponential distributions we find upper estimates uniform in x for the ratios which improve the available estimates for the whole class of subexponential distributions. We give some conditions sufficient for the asymptotic equivalence P(S > x) E P(₁ > x) as x . Here is a positive integer-valued random variable independent of . The estimates obtained are also used to find the asymptotics of the tail distribution of the maximum of a random walk modulated by a regenerative process.     

Type,infratype and the Elton-Pajor theorem

Summary Consider a sequence(x _i ) _in of norm one vectors in a Banach space. For a subsetJ of {1,...,n} consider the equivalence constant (J) between(x _i ) _iJ and the ₁ basis, and consider (k)=min{(J); cardJ=k}. We give a near optimal relationship between the rate of decay of (k) and the averageE of over all choices of signs. In particular, we show that one can choosek such that, for some universal constantK, kE ² /Kn and . This is optimal within the logarithmic term. We also prove, that forp<2, the notions of type and infratype coincide.Oblatum 27-III-1991Work partially supported by an NSF grant     

Prime ideal decomposition inF({}^m\sqrt \mu )

LetF be an algebraic number field and F such thatx ^m– is irreducible, wherem is an integer. Let be a prime ideal inF with . The prime decomposition of in is explicitly obtained in the following cases. Case 1: , (a,m) = 1 (where means , 0 ). Case 2:m lt, wherel is a prime andl 0 . Case 3:m 0 and every prime that dividesm also dividespf–1. It is not assumed that thev ^th roots of unity are inF for anyv 2.     

On the asymptotic behaviour of the free gas and its fluctuations in the hydrodynamical limit

We consider the time evolved states of the free motion _t (q, v)=(q+tv,v),q,v ^d , starting in some non-equilibrium state and look at the associated processX _t of fluctuations of the actual number _t/ () of particles of the realization in with velocities inB at timet/ around its mean as 0 (i.e., in the hydrodynamic limit). It is shown that under natural conditions on the initial state , especially a mixing condition in the space variables, for eacht the laws of the fluctuations become Gaussian in the hydrodynamic limit in the following sense: as 0, where denotes weak convergence and is a centered Gaussian state, which is translation invariant in the space variables. Furthermore the time evolution is also given by the free motion in the sense that On the other hand we shall see that ast, whereP _z× is the Poisson process with intensity measurez·×, i.e., the equilibrium state for the free motion with particle densityz and velocity distribution . In the hydrodynamic limit this behaviour corresponds to the ergodic theorem for the fluctuation process: ast. Here is a centered Gaussian state describing the equilibrium fluctuations, i.e., the fluctuations ofP _z× . Thus we prove the central limit theorem for the ideal gas: fluctuations are Gaussian even in non-equilibrium. The proofs rest on an adaption of the method of moments for sequences of generalized fields.     

Zur Interpolation und Integration differenzierbarer periodischer Funktionen

Summary Letx ₀<x ₁<...<x _n–1<x ₀+2 be nodes having multiplicitiesv ₀,...,v _n–1, 1v _k r (0k<n). We approximate the evaluation functional ,x fixed, and the integral respectively by linear functionals of the form and determine optimal weights for the Favard classesW _r C ₂. In the even case of optimal interpolation these weights are unique except forr=1,x(x _k +x _k–1)/2 mod 2. Moreover we get periodic polynomial splinesw _{k, j} (0k<n, 0j<v _k ) of orderr such that are the optimal weights. Certain optimal quadrature formulas are shown to be of interpolatory type with respect to these splines. For the odd case of optimal interpolation we merely have obtained a partial solution.

Bojanov hat in [4, 5] ähnliche Resultate wie wir erzielt. Um Wiederholungen zu vermeiden, werden Resultate, deren Beweise man bereits in [4, 5] findet, nur zitiert     

An approach to the constructivization of Cantor's set theory

A new approach is proposed for the construction of constructive analogs of set theory in hyperarithmetic languages , where is a scale of constructive ordinals. For every ordinal in the language , a special relation of equality = is defined for codes of one-parameter formulas (conditions) of the level in a constructive hyperarithmetic hierarchy corresponding to the scale . The membership relation, (also expressible in the language ), is defined by the conditionx y=z(z= x&z y), where the relation is obtained by suitable refinement of the traditional representations of the constructive relation of membership. This results in a hierarchy of constructive analogsM of the theory of sets (in which the sets are represented by codes of conditions of level , identified modulo the relation =, and is taken as the relation of membership). Some properties of this hierarchy are introduced which show that for the limits ,M is sufficiently rich from the traditional set theoretic standpoint.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 68, pp. 38–49, 1977.     

On Rounding Error Sums Related to the Circle Problem

For a large real parameter t and 0 a b we consider sums where is the rounding error function, i.e. (z) = z - [z] - 1/2. We generalize Huxley's well known estimate by showing that holds uniformly in 0 a b . Fruther, we investigate an analogous question related to the divisor problem and show that the inequality , which (due to Huxley) holds uniformly in 0 a b , and which is in general not true for 1 a b t, is true uniformly in 0 a b .     

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati