The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

The order of approximation to functions of the Zα. Class by means of positive linear operators

Let C_n (, ) be the upper bound for deviations of periodic functions which form the Zygmund class Z,0 0<<2 from a class of positive linear operators. A study is made of the conditions under which there exists a limit nC_n(, )=C(, ). An explicit expression is given for the functions C(,).Translated from Matematicheskie Zametki, Vol. 4, No. 2, pp. 201–210, August, 1968.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

On the Measure of a Nonreciprocal Algebraic Number

Let M() be the Mahler measure of an algebraic number and let G() be the modulus of the product of logarithms of absolute values of its conjugates. We prove that if is a nonreciprocal algebraic number of degree d 2 then M()² G()^1/d 1/2d. This estimate is sharp up to a constant. As a main tool for the proof we develop an idea of Cassels on an estimate for the resultant of and 1/. We give a number of immediate corollaries, e.g., some versions of Smyth's inequality for the Mahler measure of a nonreciprocal algebraic integer from below.     

On the positive definiteness of n ↦ epnα for α close to 2

For >2, let Q ⁺() be the infimum of those q>0 for which the function n e^pn is positive definite on N ₀ for every pq. We shall prove that Q ⁺()0 as 2.     

Finding the αn-th largest element

We describe an algorithm for selecting the n-th largest element (where 0<<1), from a totally ordered set ofn elements, using at most (1+(1+o(1))H())·n comparisons whereH() is the binary entropy function and theo(1) stands for a function that tends to 0 as tends to 0. For small values of this is almost the best possible as there is a lower bound of about (1+H())·n comparisons. The algorithm obtained beats the global 3n upper bound of Schönhage, Paterson and Pippenger for <1/3.     

Inequalities for moments of tails of random variables,with a queueing application

Summary For a non-negative random variable X and 1 such that EX<, E(X-Y) ⁺ /{E(X-y) ₊}⁺ is monotonic-decreasing in y, and hence no smaller than EX . Inequalities for E(X-Y) ₊ E(, 1, y, z0) are also given. This relation enables an inequality of Kingman for the mean waiting time in a stationary GI/G/1 queue to be sharpened.Work done as Visiting Fellow, Department of Statistics, University of Melbourne     

OnC 2-diffeomorphisms of the circle which are of type III1

For every irrational number [0, 1) which is not of constant type we construct aC ²-diffeomorphism of the circle with rotation number which is of type III₁. This diffeomorphism can be chosen arbitrarily close to the rotationR . Our methods also allow us to construct, for every Liouville number [0, 1), aC -diffeomorphism of the circle with rotation number which is of type III₁.     

On a hypothesis on Poincaré series

Let F(x₁,..., x_m) (m1) be a polynomial with integral p-adic coefficients, and let N, be the number of solutions of the congruence F(x₁,..., X_m)=0 mod A proof is given that the Poincaré series (t) = ₀ N t is rational for a class of isometrically-equivalent polynomials of m variables (m2) containing a form of degree n2 of two variables.Translated from Matematicheskie Zametki, Vol. 14, No. 3, pp. 453–463, September, 1973.The author wishes to thank N. G. Chudakov for discussing this paper and for his helpful advice.     

Quadratic-Cycloidal Curves

Interesting curves can be represented in the space Q=span{1,t,t ²,cost,sint}, t[0,] (0<<2). In this paper we introduce quadratic-cycloidal B-splines associated to equally spaced knots and the properties of the generated curves for 0<<. It is proved that, when 0, the limit of a QC-B-spline curve approaches a B-spline curve of degree 4.     

A relationship between the Mahler measure and the discriminant of algebraic numbers

In this note we show that in the well-known Dobrowolski estimate lnM() (ln lnd/ lnd)3,d , where is a nonzero algebraic number of degreed that is not a root of unity andM() is its Mahler measure, the parameterd can be replaced by the quantity=d/() ^1/d, where () is the modulus of the discriminant of. To this end, must satisfy the condition deg ^p=deg for any primep.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 415–420, March, 1996.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

Sur la convergence presque sure,au sens de Cesaro d'ordre α, 0<α<1, de variables aléatoires indepéndantes et identiquement distribuées

Summary We prove the following theorem: «Given 0<1, the (C, )-means of a sequence of i.i.d. random variables X _n converge a.s. iff E|X _n|^1/<.» For 1/2<1 and 0<<1/2 this result is essentially known. We give here a proof of the case =1/2; an important tool is a theorem of Hsu and Robbins [5].     

Fredholm integro-differential equation

We consider numerical solution of an integro-differential equation with nonsmooth initspaial values. Unique solvability in Sobolev spaceW ₂ (0, 1), =1,2, is proved. We establish the rate of convergence of the approximate solution to the exact solution in fractional spacesW ₂ ⁺¹ , 01, with approximation order O(h ^++1/2 ) for 01/2 andO(h ⁺¹ |ln h|1/2, for 1/2 #x2264;1.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 8–16, 1988.     

Net subgroups of Chevalley groups. II. Gauss decomposition

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup () of the Chevalley group G(,R) of type over a commutative ring R that corresponds to a net , i.e., to a set =(),, of ideals of R such that ₊ whenever ,,+ . It is proved that if the ring R is semilocal, then () coincides with the group ₀ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of () into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 62–76, 1982.In conclusion, the authors would like to thank Z. I. Borevich for his interest in this paper.     

On Some Interpolation Rules for Lattice Ordered Groups

Let be an infinite cardinal. In this paper we define an interpolation rule IR() for lattice ordered groups. We denote by C() the class of all lattice ordered groups satisfying IR(), and prove that C() is a radical class.     

Existence and uniqueness of solutions to two point boundary value problems for ordinary differential equations

In this paper, using the optimal control method, we deal with the following boundary value problemy + f(x, y)=0,y(0)=c,y(1)=d, under new nonresonance conditions of the form –A f _y ^/ (x, y) (x) B, where A > 0. We obtain the existence and uniqueness of solutions of the BVP (1).     

A method to obtain some similarity solutions to the generalized Newtonian fluid

It is shown that a similarity solution to the Rayieigh flow problem for a generalized Newtonian fluid exists if the fluid boundary (y=0) is given a velocityU(t) t ^1/2. Furthermore, if the fluid is of the power-law type then similarity solutions exist ifU(t) t (orU(t) e ^t), where is any real number.

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Zusammenfassung Eine Methode wird aufgezeigt, wonach eine Ähnlichkeitslösung für das Rayleigh Strömungsproblem für eine allgemeine Newtonische Flüssigkeit besteht, falls der Flüssigkeitsgrenze (y=0) eine GeschwindigkeitU (t) t ^1/2 zugeordnet wird. Für Flüssigkeiten von Potenz-Gesetz-Charakter gibt es außerdem Ähnlichkeitslösungen, fallsU(t) t (orU(t) e ^t) gilt, wobei eine beliebige reelle Zahl darstellt.

     

Cyclic Just-In-Time Sequences Are Optimal

Consider the Product Rate Variation problem. Given n products 1,...,i,...,n, and n positive integer demands d ₁,..., d_i,...,d_n. Find a sequence =₁,...,_T, T = _i=1 ⁿ d _i, of the products, where product i occurs exactly d _i times that always keeps the actual production level, equal the number of product i occurrences in the prefix ₁,..., _t, t=1,...,T, and the desired production level, equal r _i t, where r _i=d_i/T, of each product i as close to each other as possible. The problem is one of the most fundamental problems in sequencing flexible just-in-time production systems. We show that if is an optimal sequence for d ₁,...,d_i,...,d_n, then concatenation ^m of m copies of is an optimal sequence for md ₁,..., md_i,...,md_n.     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.