Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

Summary Conventional numerical methods, when applied to the ordinary differential equations of motion of classical mechanics, conserve the total energy and angular momentum only to the order of the truncation error. Since these constants of the motion play a central role in mechanics, it is a great advantage to be able to conserve them exactly. A new numerical method is developed, which is a generalization to arbitrary order of the discrete mechanics described in earlier work, and which conserves the energy and angular momentum to all orders. This new method can be applied much like a corrector as a modification to conventional numerical approximations, such as those obtained via Taylor series, Runge-Kutta, or predictor-corrector formulae. The theory is extended to a system of particles in Part II of this work.     

Parameter error estimate of near alternation approximation

In practical computation of the discrete best uniform approximation, we usually only get near best (i.e., with the -near alteration property) approximation. We need to estimate the error between the (unknown) best approximation and the achieved approximation. In this paper we estimate the parameter error by means of the generalized strong unicity constants.Visiting scholar from Department of Mathematics, Shanghai University of Science and Technology, Shanghai, P.R. China 201800.     

Subgroups of a finite group whose algebra of invariants is a complete intersection

Let G be a finite subgroup of GL(V), where V is a finite-dimensional vector space over the field K and char KG. We show that if the algebra of invariants K(V)^G of the symmetric algebra of V is a complete intersection then K(V)^H is also a complete intersection for all subgroups H of G such that H={ Gv (v)=v for all v V^H}.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 116, pp. 63–67, 1982.     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

Über eine Verallgemeinerung der Nicoletti-Aufgabe für Funktional-Differentialgleichungen mit voreilendem Argument

We consider a functional differential equation (1) (t)=F(t,) fort[0,+) together with a generalized Nicoletti condition (2)H()=. The functionF: [0,+)×C ⁰[0,+)B is given (whereB denotes the Banach space) and the value ofF (t, ) may depend on the values of (t) fort[0,+);H: C ⁰[0,+)B is a given linear operator and B. Under suitable assumptions we show that when the solution :[0,+)B satisfies a certain growth condition, then there exists exactly one solution of the problem (1), (2).     

On the exactness of a theorem of G.A. Fomin

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Some special results on convergent sequences of radon measures

Two problems will be considered. In Part I we consider a class of subsets of a topological space X and a Radon measure on X; if it is known that, for sufficiently many , the restrictions of the sets in constitutes a uniformity class in T w.r.t. the restriction of the given measure, then we ask if it follows that is a uniformity class in X.Part II, which can be read independently of Part I, is concerned with the question whether, to a given convergent sequence of Radon measures, say _n, there always exist sufficiently many compact sets K such that _n(K)(K).     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

Асимптоический Анализ Распределения Статистики Диксона

. u 60-u u     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

. - . .. . . . , 340 048 . 1      

On the admissibility of function spaces of typeB p,q s andF p,q s,and boundary value problems for non-linear partial differential equations

u=f(x)+S(u), S — , u-G(u), G — . B _p,q ^s () -F _p,q ^s (). R _n — . — . _p,q ^s F _p,q ^s .     

On equivalence of rearrangements of the Haar system in dyadic Hardy and BMO spaces

H={h ₁,I } — , . : , I ¦(I)¦=¦I¦, ¦I¦ — I. H H ={h _(I),I} . , , . L ^p .

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Dedicated to Professor B. Szökefalvi-Nagy on his 75th birthday

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This research was supported in part by MTA-NSF Grants INT-8400708 and 8620153.     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

О лагранжевом интерполировании с узлами в корнях многочленов сонина-маркова

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Generation of alternating groups by pairs of conjugates

Considering the conjugacy classes of the alternating group of degreen, those classes that contain a pair of generators are in the majority. In fact, the proportion of such classes is 1 –(n), and(n) 0 asn .     

Role of information in the stochastic zero-sum differential game

In this paper, a linear-quadratic Gaussian zero-sum differential game is studied. Maneuverability is defined to measure players' strength. It is shown that a more maneuverable player would prefer a more observable information system. An example is given to show that a more controllable player might not prefer more observable measurements in the stochastic environment.The research reported in this paper was made possible through support extended to the Division of Engineering and Applied Physics, Harvard University, by the US Office of Naval Research under the Joint Services Electronics Program by Contract No. N00014-75-c-0648 and by the National Science Foundation under Grant No. GK31511.     

OnU ϕ-sets of trigonometric integrals

, (t) >0 E(–, +),E<, , ¦f(t)¦ (t) xE, f(t)=0 (–, +).     

Interpolation and best polynomial approximation in the domain ofp-adic integers

, , . , . Lip

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The authors are indebted to Professor R. Bojanic for his valuable remarks and suggestions, especially for the simplification of the proof of Theorem 4.