A class of quasilinear stochastic partial differential equations of McKean-Vlasov type with mass conservation

Summary A system ofN particles inR ^d with mean field interaction and diffusion is considered. Assuming adiabatic elimination of the momenta the positions satisfy a stochastic ordinary differential equation driven by Brownian sheets (microscopic equation), where all coefficients depend on the position of the particles and on the empirical mass distribution process. This empirical mass distribution process satisfies a quasilinear stochastic partial differential equation (SPDE). This SPDE (mezoscopic equation) is solved for general measure valued initial conditions by extending the empirical mass distribution process from point measure valued initial conditions with total mass conservation. Starting with measures with densities inL ₂(R ^d ,dr), wheredr is the Lebesgue measure, the solution will have densities inL ₂(R ^d ,dr) and strong uniqueness (in the Itô sense) is obtained. Finally, it is indicated how to obtain (macroscopic) partial differential equations as limits of the so constructed SPDE's.This research was supported by NSF grant DMS92-11438 and ONR grant N00014-91J-1386     

Über den Fehler des Runge-Kutta-Verfahrens für die numerische Integration gewöhnlicher Differentialgleichungenn-ter Ordnung

Summary For the numerical solution of an ordinary differential equation of then-th order (1) with the initial conditions (2)Zurmühl has developed general Runge-Kutta formulas which integrate this equation directly without separating it into a system ofn equations of the first order. In the present paper some estimations of the errors of this Runge-Kutta method are given, which are valid for any value ofn. It is assumed that the functionf(x, y, y,...) on the right side of the differential equation and its partial derivatives up to the fourth order are one-valued and continuous in a certain neighbourhood of the initial values and that bounds of these functions are known.     

Über das Reichhaltigkeitsaxiom für partielle affine Räume

S.Meuren and A.Herzer have introduced in [1] an axiom (R). We show that the conjunction of the axioms (R₁) and (R₂) is weaker than (R), i. e., if we replace (R) by (R₁) and (R₂) in their axiom system, we get an axiom system for a larger class of partial affine spaces.     

Problems and methods in partial differential equations

Summary The method of singularities is used to solve theCauchy problem for simple hyperbolic partial differential equations, namely, the wave equation and the damped wave equation. The representation formula for the solution of theCauchy problem is written in terms of finite parts and logarithmic parts of certain divergent integrals. A process of analytic continuation is also used to solve theCauchy problems under consideration. However, to obtain explicitly the representation formulas for the solutions, one must actually perform the analytic continuation. It is shown that this is best achieved by making use of finite and logarithmic parts. Simple examples were purposely chosen so as to show that consideration of finite and logarithmic parts is naturally unavoidable and ? in the very nature of things ?. To Enrico Bompiani on his scientific Jubilee. This work was sponsored in part by the Air Force Office of Scientific Research of the Air Research and Development Command, United States Air Force, through its European Office.     

Minimizing the natural fuel requirement for nuclear reactor power systems: A nonstandard optimal control problem

A class of model problems in nuclear reactor economics is defined and shown to be equivalent to a linear optimal control problem to which present versions of the maximum principle apparently cannot be applied. It is shown that the search for an optimal control can be restricted tocontrols of maximum fuel utilization (Comfu), and that theComfu's are in a one-to-one correspondence with the functions which satisfy certain inequalities and are solutions of a nonlinear Volterra integral equation containing the value of the cost functional as a parameter. In the general case, one can establish an iterative procedure, involving solution of the integral equation at each iteration, for approximating the optimalComfu. For some important special cases, a point on the solution corresponding to the optimalComfu is knowna priori, and thus the optimalComfu can be obtained by solving the integral equation only once. Some possible generalizations of the original economic model are also discussed.This research was sponsored by the US Atomic Energy Commission under contract with the Union Carbide Corporation.     

A restricted computation model on Scott domains and its partial primitive recursive functionals

The paper builds on both a simply typed term system and a computation model on Scott domains via so-called parallel typed while programs (PTWP). The former provides a notion of partial primitive recursive functional on Scott domains supporting a suitable concept of parallelism. Computability on Scott domains seems to entail that Kleene's schema of higher type simultaneous course-of-values recursion (scvr) is not reducible to partial primitive recursion. So extensions and PTWP are studied that are closed under scvr. The twist are certain type 1 G?del recursors for simultaneous partial primitive recursion. Formally, denotes a function , however, is modelled such that is finite, or in other words, a partial sequence. As for PTWP, the concept of type writable variables is introduced, providing the possibility of creating and manipulating partial sequences. It is shown that the PTWP-computable functionals coincide with those definable in plus a constant for sequential minimisation. In particular, the functionals definable in denoted can be characterised by a subclass of PTWP-computable functionals denoted . Moreover, hierarchies of strictly increasing classes in the style of Heinermann and complexity classes are introduced such that . These results extend those for and PTWP [Nig94]. Finally, scvr is employed to define for each type the enumeration functional of all finite elements of . Received January 30, 1996     

Konvergenz von Mehrschrittverfahren zur Lösung halblinearer Anfangswertaufgaben

Lax andRichtmyer showed, that stability of multi-level difference approximations to linear ordinary or partial initial value problems is sufficient as well as necessary for convergence.Dahlquist showed, that his stability concept for multi-level difference approximations to a system of non-linear first order ordinary differential equations yields a necessary and sufficient condition for convergence too. Here, the Lax-Richtmyer-theory is extended to semi-linear initial value problems, including now both convergence theorems.     

The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation

We study the Cauchy problem for the spatially homogenem Boltzmann equation for true Maxwell molecules. Using the Fourier representation introduced by Bobylev [Bo75],we give a simplified proof of a result proved by Tanaka [Ta78].Moreover, we show by means of simple geometric properties, that Tanaka functional is an entropy decreasing functional for the Boltzmann equation for Maxwell molecules.     

On one approach to the study of the asymptotic behaviour of the Riccati equation with complex-valued coefficients

Summary The asymptotic behaviour of the solutions of the Riccati differential equation with complex-valued coefficients, especially the existence of bounded solutions and solutions satisfying the conditions of a form , is studied. The investigation is based on the reduction of the Riccati equation to the equation (E) z= =zg(t,z)+h(t, z).Here g, h are complex-valued functions, t and z being a real and a complex variable, respectively. The equation (E) is studied by means of various suitable techniques, such as Lyapunov functions, Wazewzki topological principle and by the use of results of M.Cecchi, M.Furi and M.Marini [3] on the solutions of a certain boundary value problem. Even though the obtained results are only of local character, they complete the previous results on the Riccati equation with complex-valued coefficients, which was intensively investigated by M.Ráb, and are effective in some cases not covered by known results.The paper was written during the author's stay at the Mathematical Institute U. Dini in Firenze and the author is very grateful for the kind hospitality.     

Zur Kontinuum-Magneto-Gasdynamik Stationäre rotationssymmetrische Strömung eines vollkommen leitenden Plasmas

Summary In the case of rotational symmetry for the steady motion of a perfectly conducting plasma a differential equation of 2nd order for the stream function is given by completingBernoulli's equation, the equation for the moment of rotation and the equation for the vorticity respectively by the azimutal component of the magnetic field and the azimutal component of electric current density.

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Adolf Busemann zum 60. Geburtstag     

Zur iterativen Auflösung algebraischer Gleichungen

Summary The slow convergence ofNewton's method for the iterative approximation of the roots of an algebraic equation may be improved by using a formula due toLaguerre. It is shown that this formula together with certain generalizations are useful for the numerical calculation of both real and complex zeros. A method is also given for the calculation of a further root of an equation after several roots have already been found, such that errors on the determination of the previous roots will not affect the accuracy with which the present one may be approximated.     

On the Mean Square Formula of the Error Termfor a Class of Arithmetical Functions

 Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1].     

Integral Equation Method for the First and Second Problems of the Stokes System

Using the integral equation method we study solutions of boundary value problems for the Stokes system in Sobolev space H ¹(G) in a bounded Lipschitz domain G with connected boundary. A solution of the second problem with the boundary condition $\partial {\bf u}/\partial {\bf n} -p{\bf n}={\bf g}$ is studied both by the indirect and the direct boundary integral equation method. It is shown that we can obtain a solution of the corresponding integral equation using the successive approximation method. Nevertheless, the integral equation is not uniquely solvable. To overcome this problem we modify this integral equation. We obtain a uniquely solvable integral equation on the boundary of the domain. If the second problem for the Stokes system is solvable then the solution of the modified integral equation is a solution of the original integral equation. Moreover, the modified integral equation has a form f?+?S f?=?g, where S is a contractive operator. So, the modified integral equation can be solved by the successive approximation. Then we study the first problem for the Stokes system by the direct integral equation method. We obtain an integral equation with an unknown ${\bf g}=\partial {\bf u}/\partial {\bf n} -p{\bf n}$ . But this integral equation is not uniquely solvable. We construct another uniquely solvable integral equation such that the solution of the new eqution is a solution of the original integral equation provided the first problem has a solution. Moreover, the new integral equation has a form ${\bf g}+\tilde S{\bf g}={\bf f}$ , where $\tilde S$ is a contractive operator, and we can solve it by the successive approximation.     

Discussione del problema di Cauchy per le equazioni di tipo ellittico

Sunto Il problema diCauchy per le equazioni a derivate parziali di tipo ellittico è riesaminato in questa Memoria principalmente dal punto di vista delle sue possibili applicazioni fisiche. Vi è data una analisi critica di alcuni classici concetti e impostazioni della Fisica matematica e si considerano tali problemi diCauchy anche con l'introduzione di una ulteriore condizione sulla soluzione, una sua limitazione a priori nella regione considerata. In particolare si prova che le soluzioni positive dei problemi diCauchy relativi alla equazione diLaplace dipendono con continuità dai dati diCauchy.

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Summary TheCauchy problem for ellittic partial differential equations is studied in this paper mainly from the point of view of its physical applications. We discuss some classical conceptions of Applied Mathematics and we consider theseCauchy problems also with the introduction of another condition: an ? a priori ? bound of the solutions. In particular we prove that the positive solutions ofCauchy problems for theLaplace equation depend continuously on the data

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Lavoro eseguito nell'Institute for Fluid Dynamics and Applied Mathematics della Università del Maryland e nell'Istituto Nazionale per le Applicazioni del Calcolo.     

On the Mean Square Formula of the Error Termfor a Class of Arithmetical Functions

 Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1]. (Received 29 June 1998; in final form 11 November 1998)     

Convergence of the newton process to multiple solutions

In 1870, E.Schröder showed that the convergence of the Newton process of successive approximations to a multiple solution of a scalar equation was geometric in character, and that quadratic convergence could be restored by multiplying the ordinary corrections by a constant. Here, this result is extended to finite systems, and it is shown that there exist various subspaces of the given space in which the convergence is geometric with a rate characteristic of the given subspace. Quadratic convergence may be restored by applying a given fixed linear operator to the ordinary corrections. The conditions under which these results apply to equations in infinite-dimensional Banach spaces are given. Numerical examples involving scalar equations and a simple 2 × 2 system are presented.Sponsored by the Mathematics Research Center, United States Army, Madison, Wisconsin, under Contract No. DA-11-022-ORD-2059.     

On a regular cauchy problem for the Euler-Poisson-Darboux equation

Summary The explicit solution of a particularCauchy problem for the n-dimensionalEuler-Poisson-Darboux equation is found. To obtain the solution the method ofM. Riesz is extended to include non self-adjoint equations. Existence and uniqueness are shown. This research was supported in part by the United States Air Force under Contract No. AF18(600)-573 — monitored by the Office of Scientific Research, Air Research and Development Command.     

An eigenvalue comparison theorem for second order self-adjoint differential equations

Summary This paper is concerned with a comparison of the eigenvalues for pairs of self-adjoint differential systems which arise from a single ordinary second-order differential equation. The sistems differ in the boundary conditions which are imposed, and it is these boundary conditions which will draw most of the attention. The principal results obtained deal with predicting alternation of the eigenvalues for two such systems from the boundary conditions alone, without special consideration of the differential equation. This paper is part of a thesis submitted to Carnegie Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy. The author wishes to express his thanks to the thesis director, ProfessorAllan D. Martin Jr. Presented to the American Mathematical Society November 17, 1962.     

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;     

Das Taylor-Deansche Stabilitätsproblem für beliebige Spaltbreiten

Summary The stability of a viscous flow between two co-axial circular cylinders, produced by the simultaneous effect of rotation of the interior cylinder and by a constant pressure gradient in peripherical direction, has been examined. (The two single problems had been studied first byTaylor [1] resp.Dean [2].) The perturbations investigated are vortices of the Taylor-Görtler-type and are assumed to be small. It therefore becomes possible to operate on the basis of the linear differential equations for perturbations.The present stability problem had been treated under the same assumptions byDiPrima [5] for small gap-distances by using the series method ofTaylor. In this paper, however, arbitrary gap-distances are admitted, and the integral equation method ofGörtler [3] andHämmerlin [4] has been applied. The calculated dependences of the critical Reynolds number upon the pressure gradient are in good qualitative agreement with the results ofDiPrima. However, our computations performed by more exact methods yielded somewhat different results with regard to the dependence of the critical vortex number upon the pressure gradient.