Notes on Interpolation. X (Two Boundary Value Problems)

In this paper we use (0, 2) interpolational polynomials to give an approximate solution of the differential equation y(x) + A(x)y(x) = F(x), x I := [-1, 1] j in case when the boundary values are y(-1) = and y(1) = , , R.     

Efficiency of Proximal Bundle Methods

We give efficiency estimates for proximal bundle methods for finding f_*min_Xf, where f and X are convex. We show that, for any accuracy <0, these methods find a point x^kX such that f(x^k)–f^* after at most k=O(1/³) objective and subgradient evaluations.     

Posterior Sensitivity to the Sampling Distribution and the Prior: More than One Observation

Sensitivity of a posterior quantity (f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of (f, P) when f and P vary in nonparametric classes ^f and ^P respectively. Direct and iterative methods are described which obtain the range of (f, P) over f ^f when prior P is fixed, and also the overall range over f ^f and P ^P . When multiple i.i.d. observations X ₁,...,X _k are observed from f, the posterior quantity (f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of (f, P). Several examples illustrate applications of these methods.     

Parabolic Differential Equations in Ordered Banach Spaces of Bounded Functions

Let A be a set, and let E be the Banach space of bounded functions : A R, equipped with its natural order. With a rectangle R = (a,b) × (0,T] let F(x,t,) : R × E E be a bounded, continuous function satisfying a local Hölder condition and being quasimonotone increasing with respect to . Then there exists a solution u: [a,b] × [0,T] E of the problem ut(x,t) – uxx(x,t) = F(x,t,u(x,t)) ((x,t) R), u(x,t) = 0 ((x,t) R R).     

Generalized sequencing problem “Towers of Hanoi”

ForpN certain integer-valued functionsA _p (x), defined forx N {0}, are studied. These functions occur in a functional equation system corresponding to a generalized version of the transportation game Towers of Hanoi and their values may be interpreted as minimum numbers of moves. An explicit representation ofA _p (x) is given and so-called minimum partitions ofx with respect top are determined for allx N. The minimum partitions ofx are of interest concerning the realisation of the minimum number of moves by optimal policies.

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Zusammenfassung Es werden fürp N gewisse ganzzahlige für allex N {0} erklärte FunktionenA _p (x) untersucht, die bei einer Verallgemeinerung des unter dem Namen Türme zu Hanoi bekannten Transportspiels in einem entsprechenden Funktionalgleichungssystem auftreten und deren Funktionswerte sich als Mindestzugzahlen interpretieren. Es werden fürA _p (x) eine explizite Darstellung und sogenannte Minimalzerlegungen vonx bezüglichp für allex N bestimmt. Die Minimalzerlegungen vonx spielen eine besondere Rolle bei der Angabe von optimalen Strategien zur Realisierung der Mindestzugzahl.

     

Partial applicability of Moser's method to nonlinear elliptic systems

We consider nonlinear elliptic systems of divergent-type second-order partial differential equations with solutionsu W _p ¹ . It is proved thatDu L _q with someq (p; +) and it is explicitly shown howq depends on the ellipticity modulus of the system. Some conditions on the ellipticity modulus are obtained under which the solutions satisfy the Hölder conditions and the Liouville theorem holds.Translated fromMatematicheskie Zametki, Vol. 58, No. 4, pp. 547–557, October, 1995.     

On the complete solution of ∈y″=y 3

In Ref. 1, the author claimed that the problem y=y ³ is soluble only for a certain range of the parameter . An analytic approach, as adopted in the following contribution, reveals that a unique solution exists for any positive value of . The solution is given in closed form by means of Jacobian elliptic functions, which can be numerically computed very efficiently. In the limit 0+, the solutions exhibit boundary-layer behavior at both endpoints. An easily interpretable approximate solution for small is obtained using a three-variable approach.     

Approximate solution of a class of singular control problems

A class of singular control problems involving amplitude constraints on the controls is examined. IfL is the space of control functionsU, the control constraint setS can be identified with the unit ball inL . Now, for anyn (1, ), an analogous problem may be set up withL ⁿ forU and the unit ball inL ⁿ forS. This modified problem is necessarily nonsingular for controllable systems. It is shown that, by takingn sufficiently large, the solution to the modified problem also solves the original problem arbitrarily closely (in a sense made precise). Behavior asn is investigated.This research was supported by the Science Research Council of Great Britain and the Commonwealth Fund (Harkness Fellowship).     

Some Area-Diameter Inequalities for Two-Dimensional Lattices

We obtain some inequalities concerning area and diameter of planar convex sets verifying the assumptions of Minkowski's Theorem, establishing the best possible upper bounds for the ratio A/D, (0,2], and for A/D when some lattice points are included.     

Distal criterion of transitive transformation groups

Let G be a topological group, H a closed subgroup of the group G, and G/H a homogeneous space of cosets Hg(g G). The group G acts naturally on G/H, defining a transitive transformation group (G/H, G,gp), (Ha, g)=Hag(a G, g G). Necessary and sufficient conditions for the distalness of the transformation group (G/H, G,) are indicated.Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 77–80, January, 1969.     

Left linear theories — A generalization of module theory

In this paper we introduce left linear theories of exponentN (a set) on the setL as mapsL ×L ^N (l, ) l · L such that for alll L and , L ^N the relation (l · ) =l( · ) holds, where · L ^N is given by ( · )(i) = (i),i N. We assume thatL has a unit, that is an element L ^N withl · =l, for alll L, and · = , for all L ^N . Next, left (resp. right)L-modules andL-M-bimodules and their homomorphisms are defined and lead to categoriesL-Mod, Mod-L, andL-M-Mod. These categories are algebraic categories and their free objects are described explicitly. Finally, Hom(X, Y) andX Y are introduced and their properties are investigated.Herrn Professor Dr. D. Pumplün zum 60. Geburtstag gewidmet     

Kite-freeP- andQ-polynomial schemes

LetY = (X, {R _i } _oid) denote aP-polynomial association scheme. By a kite of lengthi (2 i d) inY, we mean a 4-tuplexyzu (x, y, z, u X) such that(x, y) R ₁,(x, z) R ₁,(y, z) R ₁,(u, y) R _i–1,(u, z) R _i–1,(u, x) R _i. Our main result in this paper is the following.     

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.     

On a one-dimensional version of the dynamical Marguerre-Vlasov system

A one-dimensional version of the so-called Marguerre-Vlasov system of equations describing the vibrations of shallow shells is considered. The system depends on a parameter 0 in a singular way and undergoes the effect of damping mechanisms. We show that the system converges to a nonlinear beam equation while the energy decays exponentially uniformly (on 0) as time goes to infinity.     

Some fractal sets determined by stable processes

Summary LetY _i be independent stable subordinators in (, ,P) with indices 0< _i <1 andR _i are the ranges ofY _i ,i=1, 2. We are able to find the exact Hausdorff measure and packing measure results for the product setsR ₁×R ₂, and whenever ₁ + ₂ 1/2, we deduce results for the vector sumR ₁R ₂={x+y:xR ₁,yR ₂}.     

On Gauss quadrature formulas with equal weights

Summary It is well known that the Chebyshev weight function (1–x ²)^–1/2 is the only weight function (up to a linear transformation) for which then point Gauss quadrature formula has equal weights for alln. In this paper we describe all weight functions for which thenm point Gauss quadrature formula has equal weights for alln, wherem is fixed.     

Galerkin methods for parabolic equations with nonlinear boundary conditions

A variety of Galerkin methods are studied for the parabolic equationu _t =(a(x) u),x ⁿ ,t (O,T], subject to the nonlinear boundary conditionu _v =g(x,t,u),x,t (O,T] and the usual initial condition. Optimal order error estimates are derived both inL ² () andH ¹ () norms for all methods treated, including several that produce linear computational procedures.The authors were partially supported by The National Science Foundation during the preparation of this paper.     

Chaotic solutions of systems with almost periodic forcing

Summary In a recent paper [J. Diff. Equat. 55 (2) (1984), 225–256], J. Palmer proved Smale's theorem on the embedding of the Bernoulli shift in the context of a periodic differential system (^*) =f (t, x),x ⁿ , using a nonautonomous shadow lemma. By means of this lemma, we show that one does get a similar kind of chaotic motion whenf is almost periodic int. Actually, we do not consider the equation (^*). In order to show that the hypotheses can be satisfied, we rather consider a parameter-dependent equation of the formx=g(x)+h(t,x,), whereIR is the parameter.

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Zusammenfassung In einer kürzlich erschienenen Arbeit [J. Diff. Equat. 55 (2) (1984), 225–256] bewies J. Palmer den Satz von Smale über die Einbettung des Bernoullischifts für periodische Differentialgleichungssysteme der Form (^*) =f (t, x),x ⁿ , unter Verwendung eines Schattenlemmas für nicht-autonome Systeme. Mit Hilfe dieses Lemmas zeigen wir, daß man eine ähnliche chaotische Bewegung erhält, wennf fast-periodisch int ist. Genau genommen betrachten wir nicht die Gleichung (^*). Um zu zeigen, daß die Voraussetzungen erfüllt werden können, betrachten wir vielmehr eine parameterabhängige Gleichung der Form =g(x)+h(t,x,), wobeiIR der Parameter ist.

     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323