Zur simultanen Diophantischen Approximation. II

An important problem in the field of Diophantine Approximation was solved in 1891 byHurwitz. He could show that any irrational number can be approximated by infinitely many rationals in such a way that the modulus of the difference multiplied with the square of the denominator is less than 5^?1/2, and here 5^?1/2 is best possible. The generalization of this problem in the direction of simultaneous Diophantine Approximation is still unsolved.Furtwängler in 1925 gave lower bounds for the maximum difference, conjecturing that these were best possible. Here this conjecture is disproved by showing the existence of an infinite number of counterexamples.     

Product type bounds on the approximation of values ofE andG functions

This paper obtains effective lower bounds on the absolute values of linear forms, over the integers, in power products of values of certain SiegelE-functions or SiegelG-functions. The bounds obtained are in terms of the product of the absolute values of the coefficients. ForE-functions the bound obtained is a best possible result, up to an arbitrarily small positive epsilon. ForG-functions the result is asymptotically best in the following sense: for each epsilon larger than zero there exists an integerN such that ifz, the point of evaluation, equalsM ^–1 whereM is an integer with absolute value larger thanN, then the bound obtained is within epsilon of a best possible bound. (From the proof it is clear thatz need not be the inverse of an integer. What is necessary that the absolute value of its numerator must be much smaller than the absolute value of its denominator.)Results obtained recently by D. V. andG. V. Chudnovsky bounding the absolute values of similar forms give bounds in terms of the maximum of the absolute values of the coefficients; such lower bounds can be much smaller.Dedicated to Professor E. Hlawka on the occasion of his seventieth birthday     

Linear Forms in two m-adic Logarithms and Applications to Diophantine Problems

We give sharp, explicit estimates for linear forms in two logarithms, simultaneously for several non-Archimedean valuations. We present applications to explicit lower bounds for the fractional part of powers of rational numbers, and to the Diophantine equation (x ⁿ – 1)/(x – 1) = y ^q .     

Rational approximation of maximal commutative subgroups of $${GL(n,\mathbb{R})}$$

How to find “best rational approximations” of maximal commutative subgroups of \({GL(n,\mathbb{R})}\)? In this paper we specify this problem and make first steps in its study. It contains the classical problems of Diophantine and simultaneous approximation as particular subcases but in general is much wider. We prove estimates for n = 2 for both totally real and complex cases and give an algorithm to construct best approximations of a fixed size. In addition we introduce a relation between best approximations and sails of cones and interpret the result for totally real subgroups in geometric terms of sails.     

Linear spaces on the intersection of cubic hypersurfaces

Upper bounds for the number of variables necessary to imply the existence of anm-dimensional linear variety on the intersection ofr cubic hypersurfaces over local and global fields are given. D. J. Lewis was partially supported by a National Science Foundation grant. Rainer Schulze-Pillot was supported by a Deutsche Forschungsgemeinschaft Stipendium.     

The regulator spectrum for totally real cubic fields

Subject to a mild hypothesis, it is proved that there exists a set of totally real cubic fields, with regulatorsR and discriminantsD, such that the corresponding numbersR/log² D are dense in the interval [1/16, 1/12].     

On normal equations for surfaces containing flat totally real submanifolds

This article introduces, for real hyperplanes in C², special coordinates in which a surface is developable along a two-dimensional totally real submanifold. The problem of nonuniqueness for such coordinates is investigated.Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 76–86, July, 1992.     

Locally one-dimensional difference scheme for the convective diffusion equation

We consider the mixed boundary-value problem for the nonstationary convective diffusion equation in a rectangular region. The summation approximation method is applied to construct a locally homogeneous difference scheme with O(t ^1/2 + h^3/2) rate of convergence in the L₂ grid metric.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 64–67, 1992;     

Zum Cauchy-Problem bei der verallgemeinerten Wärmeleitungsgleichung

Cauchy's problem for the equationu _xx +x ^–1 u _x =u _t ( real) was discussed byD. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods ofD. Colton and of this paper are different but the results are similar.     

The elastic fields and interactions of point defects in isotropic and cubic media

Zusammenfassung Die Formel vonEshelby für die elastische Wechselwirkung zwischen Punktdefekten wird neu hergeleiter. Die elastischen Eigenschaften der Punktdefekte werden behandelt durch Anwendung der mathematischen Methoden der Quantenmechanik des Drehimpulses auf die Gleichungen der linearen Elastizitätslehre des Kontinuums. Für isotrope Medien werden die Multipole beliebiger Ordnung der elastischen Felder exakt berechnet, und für Medien kubischer Symmetrie wird eine Näherung für das elastische Dipolfeld gegeben. Die Wechselwirkungsenergie zweier elastischer Dipole wird für isotrope Medien exakt und für kubische Medien näherungsweise berechnet. Die Formel wird angewendet auf Paare vonO ₂ ^– Zentren und Paare vonOH ^– Zentren inKCl. Numerische Resultate sind gegeben.

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This work is part of a research program suported by the Swiss National Science Foundation under contract No. 4212.     

Zum Cauchy-Problem bei der verallgemeinerten W?rmeleitungsgleichung

Cauchy's problem for the equationu _xx +x ^–1 u _x =u _t ( real) was discussed byD. Colton if –1,–2,–3, ... Now existence and uniqueness theorems and representations of the solutions are given for the cases =–1,–2, –3,... The methods ofD. Colton and of this paper are different but the results are similar.     

Einfluß der Seitenwände auf das Einsetzen der Konvektion in einer horizontalen Flüssigkeitsschicht

Zusammenfassung Der Einfluß der Seitenwände auf das Einsetzen von Naturkonvektion in einem beheizten Fluid wird untersucht, wobei die Wärme sowohl über die Berandung als auch durch innere Wärmequellen dem Medium zugeführt wird. Betrachtet wird eine Fluidschicht, die inx-Richtung unendlich ausgedehnt und durch vier Seiten begrenzt ist.Eine zweidimensionale Approximation dieses dreidimensionalen Problems wird mit Lösungen verglichen, die sich unter Benutzung einer allgemeinen dreidimensionalen Darstellung eines toroidalen und poloidalen Vektorfeldes für den Fall von freien bzw. festen horizontalen und festen vertikalen Rändern ergeben.Für Höhen- zu BreitenverhältnisseA, die zwischen 0.001 und 100 liegen, unterscheiden sich die kritischen Rayleighzahlen bei der zweidimensionalen Approximation nur wenig von den Ergebnissen des allgemeinen dreidimensionalen Falles. Zusätzlich werden für die zweidimensionale Approximation unter Berücksichtigung innerer Wärmequellen die kritischen externen Rayleighzahlen bestimmt, wobei die Rayleighzahlen für interne Wärmequellen in einem Bereich von 0R₁<10⁶ liegen. Soweit ein Vergleich der hier gewonnenen Ergebnisse mit denen aus Experimenten und früheren theoretischen Untersuchungen möglich ist, wurde gute Übereinstimmung festgestellt.

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Summary The influence of sidewalls on the onset of convection in a differentially and internally heated fluid layer is investigated. A two-dimensional approximation to the three-dimensional problem is compared to solutions using a general three-dimensional representation of the toroidal and poloidal vector fields in the case of free and rigid upper and lower boundaries with rigid sidewalls. For all aspect ratios, 0.001A100, it is found that the critical Rayleigh numbers in the two-dimensional approximation do not differ substantially from the general three-dimensional results. Additionally, in the two-dimensional approximation, the problem is investigated with both internal heating and an imposed temperature difference between the upper and lower boundaries. In this case all boundaries are rigid and the critical Rayleigh numbers, R_E, are obtained for aspect ratios between 0.001 and 100 and internal Rayleigh numbers in the range 0R_I<10⁶. Where comparison of the results with experimental and previous theoretical results is possible, good agreement is found.

     

Beispiele dreidimensionaler Hilbertscher Modulmannigfaltigkeiten von allgemeinem Typ

Let k be a totally real algebraic number field of degree n with ring of integers o. By studying certain invariants r_m of the isolated singularities of the usual compact ification X(k) of the quotient ⁿ / Sl (2,0) we construct some example of totally real cubic number fields k such that the field of meromorphic functions of X(k) is not rational.     

Diophantine representations of linear recurrent sequences. II

Direct constructions of Diophantine representations for linear recurrent sequences are considered. Diophantine representations of the sets of values for third-order sequences with negative discriminants are found. As an auxiliary problem, we study the structure of the multiplicative group of the ringZ[λ], where λ is an invertible algebraic integer (unit) in a real quadratic field or in a cubic field of negative discriminant. The index of the subgroup {±λ ⁿ ∣n ∈Z} in the group (Z[λ])^* and the generator of (Z[λ])^* are evaluated explicitly. Bibliography: 14 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 241, 1997, pp. 5–29. This research was supported in part by the ISSEP, grants a96-1965 and a97-2261. Translated by M. A. Vsemirnov     

Quadratic forms which have only large zeros

We give lower bounds for zeros of quadratic forms. For example, givenn2d>0 there are infinitely many nonsingular quadratic formsF with integral coefficients inn variables which vanish on ad-dimensional rational subspace such that anyn linearly independent integral zerosx ₁,...,x _n satisfy |x ₁|...|x _n |F(ⁿ²/^2d)–d, whereF is the maximum modulus of the coefficients ofF. This complements a recent result of the authors in the opposite direction and shows that it is best possible.Written whileSchmidt had a Senior Scientist Award from the Alexander von Humboldt-Stiftung in the Federal Republic of Germany.     

Ein Oszillationssatz für algebraische Eigenwertprobleme

Summary Conditions on a linear algebraic eigenvalue problem are given, under which there are exactlyk–1 changes of sign in the sequence of components of thek-th eigenvector. This is analogous to the oscillation theorems of differential equations. A class of difference equations which satisfies these conditions is defined. Finally a modification of a method ofCollatz is given, by means of which upper and lower bounds for thek-th eigenvalue may be derived from a trial vector havingk–1 sign changes in the sequence of its components. This paper is merely a summary of results; no proofs are given.     

Eine bemerkung zur topologischen entropie

We study fractional quadratic transformationsT of the sphere and try to determine their topological entropy. In the case whereT is a constant mapping or a homeomorphism, the topological entropy is of course zero. In the other cases, we have the following results. IfT has only one fixed point, its entropy is log 2. IfT has exactly two fixed points, it can be written asT _z =z–z ^–1 +v, and ifv is real, then the entropy ofT is again log 2. A general result ofMisiurewicz andPrzytycki shows that the entropy ofT is at least log2, and we conjecture that this entropy is always equal to log2 in the remaining cases, i. e. two fixed points andv not real, and three fixed points.     

Differential Equations and Diophantine Approximationin Positive Characteristic

 We prove that the Osgood-Thue Theorem, about Diophantine Approximation in function fields, holds under a more general condition when the ground field is finite. Received 28 January 1998     

Flächen mit gleichen Summen der Hauptkrümmungsradien

Ohne ZusammenfassungDen Zusammenhang mit den infinitesimalen Kugelverbiegungen hat schon W.Süss bemerkt: Eindeutige Bestimmung von Eihyperflächen durch die Summe ihrer Hauptkrümmungsradien. Arch. Math.8, 352–354 (1957). Übrigens folgen die angegebenen Sätze auch aus der Formel vonBlaschke.     

A duality theorem for continuous-time linear programming problems

Summary In dealing with dynamic economic policy models one encounters optimization problems whose objective function is an integral of a linear function of a finite number of continuous variables and whose constraints are linear integral inequalities. A set of intertemporal efficiency conditions (equilibrium conditions) yielding the optimal policy are given. By approximating the continuous problem by a set of discrete problems and appealing to a well known convergence theorem in functional analysis a continuous analog of the duality theorem is proved.

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Zusammenfassung Bei der Beschäftigung mit dynamischen Modellen der ökonomischen Politik stößt man auf Optimierungsprobleme, deren Zielfunktion ein Integral einer linearen Funktion von einer endlichen Anzahl stetiger Variablen ist und deren Beschränkungen lineare Integral-Ungleichungen sind. Eine Menge intertemporaler Effizienz-Bedingungen (Gleichgewichtsbedingungen), die zur optimalen Politik führen, sind gegeben. Durch Approximation des kontinuierlichen Problems mittels einer Menge von diskreten Problemen und Berufung auf einen wohlbekannten Konvergenzsatz aus der Funktionalanalysis wird ein stetiges Analogon des Dualitätstheorems bewiesen.

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The author is indebted to Mr.Arnold Faden for helpful suggestions and to ProfessorKarl A. Fox andGerhard Tintner for encouragement during the preparation of the paper. This research has been partially supported by a grant from the Ford Foundation to the School of Business Administration administered by the Center for Research in Management Science, University of California, Berkeley.

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Vorgel. v.:G. Tintner.