The embedding of linearly ordered sets

It is shown that if a linearly ordered set B does not contain as subsets sets of order type and ^* then B can be embedded in 2 . We construct an example of a set satisfying the above conditions which cannot be embedded in any 2 if < . Simultaneously we show that for any ordinal, 2 ⁺¹ cannot be embedded in 2 and that there exists at least ₊₁ distinct dense order types of cardinality 2 .Translated from Matematicheskie Zametki, Vol. 11, No. 1, pp. 83–88, January, 1972.In conclusion, I wish to take the opportunity to thank Yu. L. Ershov for kindness and assistance in this work.     

Stability properties of collocation methods

Lower bounds for are given for which equidistant s-point collocation methods areA()-stable for arbitrarys.     

On the complexity of following the central path of linear programs by linear extrapolation II

A class of algorithms is proposed for solving linear programming problems (withm inequality constraints) by following the central path using linear extrapolation with a special adaptive choice of steplengths. The latter is based on explicit results concerning the convergence behaviour of Newton's method to compute points on the central pathx(r), r>0, and this allows to estimate the complexity, i.e. the total numberN = N(R, ) of steps needed to go from an initial pointx(R) to a final pointx(), R>>0, by an integral of the local weighted curvature of the (primal—dual) path. Here, the central curve is parametrized with the logarithmic penalty parameterr0. It is shown that for large classes of problems the complexity integral, i.e. the number of stepsN, is not greater than constm log(R/), where < 1/2 e.g. = 1/4 or = 3/8 (note that = 1/2 gives the complexity of zero order methods). We also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1/3.As a byproduct, many analytical and structural properties of the primal—dual central path are obtained: there are, for instance, close relations between the weighted curvature and the logarithmic derivatives of the slack variables; the dependence of these quantities on the parameterr is described. Also, related results hold for a family of weighted trajectories, into which the central path can be embedded.On leave from the Institute of Mathematics, Eötvös University Budapest, H-1080 Budapest, Hungary.     

Ordered sets with no chains of ideals of a given type

We study the possible order types of chains of ideals in an ordered set. Our main result is this. Given an indecomposable countable order type , there is a finite listA ₁ , ...,A _n of ordered sets such that for every ordered setP the setJ(P) of ideals ofP, ordered by inclusion, contains a chain of type if and only ifP contains a subset isomorphic to one of theA ₁ ^#x03B1; , ...,A _n . The finiteness of the list relies on the notion of better quasi-ordering introduced by Nash-Williams and the properties of scattered chains obtained by Laver.The results presented. here constitute the second chapter of the third cycle thesis presented by the second author before the Claude Bernard University, Lyon (July 1983).     

Growth of length of sequential derivation transformed into natural one

We consider the (&, )-fragment of the intuitionistic propositional calculus. It is proved that under the standard transformation of a Gentzen derivation into a natural derivation(), the length of (())2^2·length( ). There is constructed a sequence of Gentzen derivations of length _i, for which the length of (( _i))2^{1/3·length(i}), which shows that the upper bound obtained is not too weak.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 192–196, 1979.     

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.     

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.     

The evolution of a population for a certain viability system

Summary For the Fisher-Wright-Haldane selection model with fitness parametersf _ij =1+ _{i ij} ( _i –1) a complete global analysis is performed.

---

Zusammenfassung Für das Fisher-Wright-Haldane-Selektionsmodell mit den Fitneßparameternf _ij =1+ _{i ij} ( _i –1) wird eine vollständige globale Analyse, durchgeführt.

     

Black holes on the plane drawn by a Wiener process

Summary We say that the discD()R ², of radius , located around the origin isp-covered in timeT by a Wiener processW(·) if for anyzD() there exists a 0tT such thatW(t) is a point of the disc of radiusp, located aroundz. The supremum of those 's (0) is studied for which,D() isp-covered inT.     

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.     

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].     

On monotone and convex approximation by algebraic polynomials

The following results are obtained: If >0, 2, [3, 4], andf is a nondecreasing (convex) function on [–1, 1] such thatE _n (f) n ^– for any n>, then E _n ⁽¹⁾ (f)Cn ^– (E _n ⁽²⁾ (f)Cn ^–) for n>, where C=C(), E_n(f) is the best uniform approximation of a continuous function by polynomials of degree (n–1), and E _n ⁽¹⁾ (f) (E _n ⁽²⁾ (f)) are the best monotone and convex approximations, respectively. For =2 ( [3, 4]), this result is not true.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1266–1270, September, 1994.     

Untere und obere Schranken für den tiefsten Eigenwert der elastisch gestützten Membran

Zusammenfassung Es werden untere und obere Schranken für den tiefsten Eigenwert ₁() der elastisch gestützten schwingenden Membran hergeleitet. Die elastische Bindung der Membran am Rande wird durch charakterisiert, und wird als Parameter betrachtet.Die Verwendung des klassischen Rayleigh-Prinzipes liefert obere Schranken, mit Hilfe eines konvexen FunktionalsJ() erhält man obere und untere Schranken. Eine Zerlegungsmethode endlich gibt eine untere Schranke für ₁().

---

Summary This article is concerned with the determination of upper and lower bounds for the lowest eigenvalue ₁() of the elastically supported vibrating membrane. The elastic support on the boundary is characterized by which is regarded as a parameter.The classical Rayleigh-Principle gives upper bounds. The use of a convex functionalJ() yields upper and lower bounds for ₁(). A method of decomposition leads to a lower bound for ₁().

---

---

Neu-Technikum, Buchs SG     

Solution of integral equations in fractional Sobolev spaces

We consider linear integral equations and Urysohn equations with constant integration limits. Sufficient conditions are given for the solutions of these equations to be in Sobolev spacesW ₂ (0,1), 0 2. Finite-difference schemes are constructed for approximate solution of the original equation by special averaging of the right-hand side kernel. The rate of convergence of the approximate solution to the averaged exact solution is shown to beO(h|ln h|(1/2,)⁺(3/2,)).Translated from Vychislitel'naya i Prikladnaya Matematika, No. 63, pp. 3–19, 1987.     

A generalized sandwich theorem

It is proved for an arbitrary commutative ring A with identity and any integer n3 that if H is a subgroup of GL_n(A) normalized by E _n(A,q), then there is an ideal of A such that E _n(A,) H GL _n (A, (:q⁴⁰).Furthermore, is uniquely determined up to a certain equivalence relation on the set of ideals of A. The result extends a theorem of Bak, by removing a stability condition he uses on A.     

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.     

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.     

Similarity solutions of radially symmetric two-phase flow

When both the diffusivityD and fractional flow functionf have a power law dependence on the water content , i.e.D=D _o andf=⁺¹, the nonlinear transport equation for radially symmetric two phase flow can, in certain circumstances, be reduced to a weakly coupled system of two first order nonlinear ordinary differential equations. Numerical solutions of these equations for a constant flux boundary conditionV _wo and comparison with experimental data are given. In particular, when the fluxV _wo and a are related byV _wo( + 1)/D _o=2, a new fully explicit analytical solution is found as (r, t)=(1 – r ²/4D _ot)^1/ forr ² < 4D _ot/ and (r, t)=0 forr ² 4D _ot/ We show that the existence of this exact soution is due to the presence of a Lagrangian symmetry.     

Application of Hermite-Mahler polynomials to the approximation of the values ofp-adic exponential function

We shall give a further application of Hermite-Mahler polynomials to the consideration ofp-adic exponential function. An effective lower bound is obtained for max {| – | _p ,P(e )| _p }, where is an algebraic number satisfying || _p <p ^–/(p–1), and 0 is ap-adic number with | | _p depending on the degree of the polynomialPZ[y]. The bound obtained implies the transcendence ofe if ap-adic number satisfying 0 < || _p <p ^–/(p–1) is algebraic or can be well approximated by algebraic numbers.This work was carried out while the author was a research fellow of the Alexander von Humboldt Foundation.     

Continuity of the Multifractal Spectrum of a Random Statistically Self-Similar Measure

Until now [see Kahane;⁽¹⁹⁾ Holley and Waymire;⁽¹⁶⁾ Falconer;⁽¹⁴⁾ Olsen;⁽²⁹⁾ Molchan;⁽²⁸⁾ Arbeiter and Patzschke;⁽¹⁾ and Barral⁽³⁾] one determines the multifractal spectrum of a statistically self-similar positive measure of the type introduced, in particular by Mandelbrot,^{(26, 27)} only in the following way: let be such a measure, for example on the boundary of a c-ary tree equipped with the standard ultrametric distance; for 0, denote by E the set of the points where possesses a local Hölder exponent equal to , and dim E the Hausdorff dimension of E ; then, there exists a deterministic open interval I *₊ and a function f: I *₊ such that for all in I, with probability one, dim E =f(). This statement is not completely satisfactory. Indeed, the main result in this paper is: with probability one, for all I, dim E =f(). This holds also for a new type of statistically self-similar measures deduced from a result recently obtained by Liu.⁽²²⁾ We also study another problem left open in the previous works on the subject: if =inf(I) or =sup(I), one does not know whether E is empty or not. Under suitable assumptions, we show that E ø and calculate dim E .