Single step methods for general second order singular initial value problems with spherical symmetry

A one parameter family of self-starting explicit Runge-Kutta-Nyström methods has been obtained for the solution of the general second order singular initial value problem with spherical symmetry of the formu+2r ^–1 u=f(r, u, u),u(0)=A, u(0)=0. The methods are exact foru(r)=r ^–1, l,r, r ²,r ³ andr ⁴.     

Existenzaussagen für gewisse Zweipunkt-Randwertaufgaben

Zusammenfassung Für Randwertaufgaben der Form–u–l ₀ ...u–l ₀ ...u=f(x, u) mitl ₀R,lR,f definiert und stetig auf {a<-x<-b, |u|<} wird eine Existenzaussage gewonnen, fallsf inu linear durch die aufeinanderfolgenden Eigenwerte der zugehörigen linearen Aufgabe beschränkt ist. Zum Beweis betrachtet man die äquivalente Hammersteinsche Integralgleichung mit nichtsymmetrischem Kern. Mit Hilfe des Schauderschen Fixpunktsatzes erhält man für diese Integralgleichung Existenzaussagen, welche Ergebnisse von Dolph verallgemeinern.

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Summary This note contains an existence theorem for a two-point boundary value problem of the form–u–l ₀ u–l ₀ u=f(x, u) wherel ₀R,l ₀R,f defined and continuous on {a<-xb, |u|<} iff is linear bounded inu by the successive eigenvalues of the corresponding linear problem. To prove this result we consider the equivalent Hammerstein integral equation with non-symmetric kernel. Schauders fixpoint principle supplies existence theorems for integral equations of this type which generalize results of Dolph in some sense.

     

Zur Aufllösung linearer homogener Differentialgleichungen 2. Ordnung

Summary By the transformationy(x)=v(u),u = exp (–G(x) dx) dx the differential equationDy+G(x)y+H(x)y=0 turns toT(u) ² v ^**+H v=0, wherev ^** signifiesd ² v/du ², andu=du/dx andH=H(x) should be expressed as functions ofu.From the solutionv(u) ofT follows immediately the solutiony(x) ofD, and vice versa.In this paper there are treated some of the types of differential equations, that may be solved by this method.     

On the dynamics of turbulent vortical flow

Zusammenfassung Die Wirbelströmung in einer turbulenten kompressiblen Flüssigkeit wird behandelt. Die Bewegungsgleichungen werden durch die Vernachlässigung der kleineren Glieder vereinfacht. Die experimentellen Ergebnisse erlauben eine Fehlerrechnung für einen Fall. Ein wichtiger Schritt in der Vereinfachung ist die Voraussetzung, dass die Radialgradienten viel grösser sind als die Axialgradienten. Für die kräftigen Wirbel erwies sich diese Annahme sowohl durch die Erfahrung als auch aus den wesentlichen theoretischen Betrachtungen heraus als begründet. Die vereinfachten Gleichungen zeigen einige wichtige Aspekte der schnelldrehenden Strömungen und geben einen Anhaltspunkt für die ausführlicheren Studien.

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Notation r radial co-ordinate - z axial co-ordinate - u, U, u radial velocity component, its time-mean value, and the turbulent perturbation to it - v, V, v tangential or swirl velocity component, etc. - w, W, w axial velocity component, etc. - p, P, p fluid pressure, etc. - ,R, fluid density, etc. - , , , numbers small with respect to unity - C, F, G, G ₀ constants of integration     

A symmetry problem in the calculus of variations

Let be a ball in ^N, centered at zero, and letu be a minimizer of the nonconvex functional over one of the classesC _M := {w W _loc ^1, () 0 w(x) M in,w concave} orE _M := {w W _loc ^1,2 () 0 w(x) M in,w 0 inL()}of admissible functions. Thenu is not radial and not unique. Therefore one can further reduce the resistance of Newton's rotational body of minimal resistance through symmetry breaking.     

Stabilization of ill-posed Cauchy problems for parabolic equations

Summary In this paper we study the noncharacteristic Cauchy problem, u_t–(a(x)u_x)_x=0, x(0, l), t.(0, T], u(0, t)=(t), u_x(0,t)=0, 0tT, assuming only L for a. In the case of weak a priori bounds on u, we derive stability estimates on u of Hölder type in the interior and of logarithmic type at the boundary. Also the continuous dependence on a is considered.

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Sunto Nel presente lavoro consideriamo il problema di Cauchy non ben posto u_t= (a(x)u_x)_x, x(0, l), t(0, T), u(0, t)=(t), u_x(0, t)=0, 0tT. Supponiamo che a sia misurabile e limitato inferiormente e superiormente da constanti positive. Introduciamo delle limitazioni a priori su u e dimostriamo la dipendenza continua di u rispetto al dato sia in (0, l)×(0, T) (di tipo hölderiano) sia per x=l (di tipo logaritmico). Consideriamo, inoltre, la dipendenza continua di u da a.

     

Theorem of Bartle,Dunford, and Schwartz concerning vector measures

We show the existence, for an arbitrary vector measure: x (where X is a Banach space and gs is a-algebra of subsets of a set S) of a functional x X (X is the conjugate space of X) such that is absolutely continuous with respect to _x, _x (E)=(E)>, E gs.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 247–254, February, 1970.     

Problems with one quarter

In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation (r(t)u(t)) + p(t)u(t) = 0 are derived. One of them deals with the case dt/r(t) = , and the other with the case dt/r(t) < .This work was supported by the grant VGA of Slovak Republic No. 1/7466/20.     

Quadratic spline interpolation on uniform meshes

The interpolation problem at uniform mesh points of a quadratic splines(x _i)=f _i,i=0, 1,...,N ands(x ₀)=f₀ is considered. It is known that s–f=O(h ³) and s–f=O(h ²), whereh is the step size, and that these orders cannot be improved. Contrary to recently published results we prove that superconvergence cannot occur for any particular point independent off other than mesh points wheres=f by assumption. Best error bounds for some compound formulae approximatingf _i andf _i ⁽³⁾ are also derived.     

Finite Extensions of Topogenities

Let X = Y Z, Y Z = Ø, < be a topogenity on Y, a topology on X. A (<, )-extension is a topogenity < on X such that < ¦Y = <, (<) = . We establish some properties of (<, )-extensions and construct all of them in the case of a finite Z.     

On quadratic spline interpolation

Using the quadratic spline interpolates(x) fitting the data (x _i,y _i), 0in and satisfying the end conditions_o=y_o, we give formulae approximatingy andy at selected knots by orders up toO(h ⁴).     

Embeddings of Lorentz—Marcinkiewicz spaces with mixed norms

X(Y) f -:X(Y)={fM(×): f_{X(Y)=f(x,.)YX<} . =(0, ), M (×) — , ×, X, Y, Z— . X(Y) Z(×).     

A finite element method for a singularly perturbed boundary value problem

Summary We examine the problem:u+a(x)u–b(x)u=f(x) for 0<x<1,a(x)>0,b(x)>, ² = 4>0,a, b andf inC ² [0, 1], in (0, 1],u(0) andu(1) given. Using finite elements and a discretized Green's function, we show that the El-Mistikawy and Werle difference scheme on an equidistant mesh of widthh is uniformly second order accurate for this problem (i.e., the nodal errors are bounded byCh ², whereC is independent ofh and ). With a natural choice of trial functions, uniform first order accuracy is obtained in theL (0, 1) norm. On choosing piecewise linear trial functions (hat functions), uniform first order accuracy is obtained in theL ¹ (0, 1) norm.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

Conditions for non-oscillation of singular linear differential equations of second order

Conditions are found in the fulfillment of which each non-trivial solution of the equation u+ (t)u+(t)u=0, where(t) L(a, b) and (t–a)(t–b)(t) L(a, b) has not more than one zero on the interval atb.Translated from Matematicheskie Zametki, Vol. 6, No. 5, pp. 633–639, November, 1969.     

An analog of the generalized Hardy-Littlewood problem with almost prime numbers

We obtain an asymptotic formula for the number of solutions of the equationm–(u, v)=a, wherem is an almost prime number,(u, v) is a given binary quadratic form, anda is an arbitrary fixed integer.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 81–94, January, 1996.     

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.     

Über einen Typ von injektiven Klassen

If u is a morphism, A an object of some category, we say, A is injective relative to u, iff Hom (u, A) is epi. In this paper we consider classesU of monomorphisms in categories of modules with the following property: if u, u are monos, such that uu exists and is inU, then u is inU. We show, that the closure a(U) ofU is an injective class in the sence of S. Eilenberg and J. C. Moore, that is, for each module A there is a mono a:AB, such that B is injective relative to u for all uU and that each module C, that is injective relative to u for all uU is injective relative to a.     

Error bounds for the solution of nonlinear two-point boundary value problems by Galerkin's method

Letu be the solution of the differential equationLu(x)=f(x, u(x)) forx(0,1) (with appropriate boundary conditions), whereL is an elliptic differential operator. Letû be the Galerkin approximation tou with polynomial spline trial functions. We obtain error bounds of the form , where 0jm andmk2m+q,p=2 orp=,h is the mesh size andq is a non negative integer depending on the splines being used.This research was supported in part by the Office of Naval Research under Contract N00014-69-A0200-1017.     

High order approximation of implicitly defined maps

Summary Approximations for the function implicitly defined by (u)=(u, (u)) are obtained via the iterative scheme _n(u)=(u, _n–1(u)). In this paper the uniform convergence of high order derivatives of _n to the corresponding derivatives of is proved. This result yields a high order approximation theorem for the input-output map generated by a nonlinear control system, using linear combinations of iterated integrals of the control.Lavoro eseguito nell'ambito del G.N.A.F.A. del C.N.R.