C 1 surface interpolation with constraints

Givenn pairwise distinct and arbitrarily spaced pointsP _i in a domainD of thex–y plane andn real numbersf _i, consider the problem of computing a bivariate functionf(x, y) of classC ¹ inD whose values inP _i are exactlyf _i,i=1,,n, and whose first or second order partial derivatives satisfy appropriate equality and inequality constraints on a given set ofp pointsQ _l inD.In this paper we present a method for solving the above problem, which is designed for extremely large data sets. A step of this method requires the solution of a large scale quadratic programming (QP) problem.The main purpose of this work is to analyse an iterative method for determining the solution of this QP problem: such a method is very efficient and well suited for parallel implementation on a multiprocessor system.Work supported by MURST Project of Computational Mathematics, Italy.     

Additively decomposed quasiconvex functions

Letf be a real-valued function defined on the product ofm finite-dimensional open convex setsX ₁, ,X _m .Assume thatf is quasiconvex and is the sum of nonconstant functionsf ₁, ,f _m defined on the respective factor sets. Then everyf _i is continuous; with at most one exception every functionf _i is convex; if the exception arises, all the other functions have a strict convexity property and the nonconvex function has several of the differentiability properties of a convex function.We define the convexity index of a functionf _i appearing as a term in an additive decomposition of a quasiconvex function, and we study the properties of that index. In particular, in the case of two one-dimensional factor sets, we characterize the quasiconvexity of an additively decomposed functionf either in terms of the nonnegativity of the sum of the convexity indices off ₁ andf ₂, or, equivalently, in terms of the separation of the graphs off ₁ andf ₂ by means of a logarithmic function. We investigate the extension of these results to the case ofm factor sets of arbitrary finite dimensions. The introduction discusses applications to economic theory.     

Efficiency in integral facility design problems

An example of design might be a warehouse floor (represented by a setS) of areaA, with unspecified shape. Givenm warehouse users, we suppose that useri has a known disutility functionf _isuch thatH _i(S), the integral off _iover the setS (for example, total travel distance), defines the disutility of the designS to useri. For the vectorH(S) with entriesH _i(S), we study the vector minimization problem over the set {H(S) :S a design} and call a design efficient if and only if it solves this problem. Assuming a mild regularity condition, we give necessary and sufficient conditions for a design to be efficient, as well as verifiable conditions for the regularity condition to hold. For the case wheref _iis thel _p-distance from warehouse docki, with 1<p<, a design is efficient if and only if it is essentially the same as a contour set of some Steiner-Weber functionf =₁ f ₁++ _m f _m ,when the _i are nonnegative constants, not all zero.This research was supported in part by the Interuniversity College for PhD Studies in Management Sciences (CIM), Brussels, Belgium; by the Army Research Office, Triangle Park, North Carolina; by a National Academy of Sciences-National Research Council Postdoctorate Associateship; and by the Operations Research Division, National Bureau of Standards, Washington, D.C. The authors would like to thank R. E. Wendell for calling Ref. 16 to their attention.     

Convolution quotients of nonnegative functions

LetG be a locally compact commutative Hausdorff group andf a function belonging toL ₁(G). If the integral off with respect to the Haar measure is positive, then one can find a nonnegative (not identically 0) functiong such that the convolution off andg is also nonnegative.     

On the existence of convex decompositions of partially separable functions

The concept of a partially separable functionf developed in [4] is generalized to include all functionsf that can be expressed as a finite sum of element functionsf _i whose Hessians have nontrivial nullspacesN _i , Such functions can be efficiently minimized by the partitioned variable metric methods described in [5], provided that each element functionf _i is convex. If this condition is not satisfied, we attempt toconvexify the given decomposition by shifting quadratic terms among the originalf _i such that the resulting modified element functions are at least locally convex. To avoid tests on the numerical value of the Hessian, we study the totally convex case where all locally convexf with the separability structureN _i ¹ have a convex decomposition. It is shown that total convexity only depends on the associated linear conditions on the Hessian matrix. In the sparse case, when eachN _i is spanned by Cartesian basis vectors, it is shown that a sparsity pattern corresponds to a totally convex structure if and only if it allows a (permuted) LDL^T factorization without fill-in.     

Complément de Schur et sous-différentiel du second ordre d'une fonction convexe

Summary The Schur complement relative to the linear mappingA of a functionf is denotedAf and defined as the image off underA. In this paper we give some estimates for the second-order differential ofAf whenf is either a partially quadratic convex function or aC ² convex function with a nonsingular second-order differential. We then consider an arbitrary convex functionf and study the second-order differentiability ofAf in a more general sense.

     

Solvability for differential equations on fractals

We show that nonlinear differential equations based on the Laplacian have local solutions on pcf self-similar fractals. However, even linear equations may fail to have global solutions. The equation Δu =f may be solved on an arbitrary proper open set for any functionf continuous there. Research supported in part by the National Science Foundation, Grant DMS-0140194.     

On the value distribution of a certain type of differential polynomials

Some quantitative estimations on the value distribution of the functionafg ⁿ(n2) are obtained, whereg is a linear differential polynomial inf anda is a small function off.The research was partially supported by a U.G.C grant of Hong Kong     

On the Wl q-Regularity of Solutions to Systems of Differential Equations in the Case When the Equations Are Constructed from Discontinuous Functions

Some solution, final in a sense from the standpoint of the theory of Sobolev spaces, is obtained to the problem of regularity of solutions to a system of (generally) nonlinear partial differential equations in the case when the system is locally close to elliptic systems of linear equations with constant coefficients. The main consequences of this result are Theorems 5 and 8. According to the first of them, the higher derivatives of an elliptic C ^l -smooth solution to a system of lth-order nonlinear partial differential equations constructed from C ^l -smooth functions meet the local Hoelder condition with every exponent , 0<<1. Theorem 8 claims that if a system of linear partial differential equations of order l with measurable coefficients and right-hand sides is uniformly elliptic then, under the hypothesis of a (sufficiently) slow variation of its leading coefficients, the degree of local integrability of lth-order partial derivatives of every W ^l _q,loc-solution, q>1, to the system coincides with the degree of local integrability of lower coefficients and right-hand sides.     

Convexity and Bernstein polynomials onk-simploids

This paper is concerned with Bernstein polynomials onk-simploids by which we mean a cross product ofk lower dimensional simplices. Specifically, we show that if the Bernstein polynomials of a given functionf on ak-simploid form a decreasing sequence thenf +l, wherel is any corresponding tensor product of affine functions, achieves its maximum on the boundary of thek-simploid. This extends recent results in [1] for bivariate Bernstein polynomials on triangles. Unlike the approach used in [1] our approach is based on semigroup techniques and the maximum principle for second order elliptic operators. Furthermore, we derive analogous results for cube spline surfaces.This work was partially supported by NATO Grant No. DJ RG 639/84.     

Norm bounds for rational matrix functions

Summary If the field of values of a matrixA is contained in the left complex halfplaneH and a functionf mapsH into the unit disc then f(A)₂1 by a theorem of J.v. Neumann. We prove a theorem of this type, only the field of values ofA is used for functions which are absolutely bounded by one in only part ofH. An extension can be used to show norm-stability of single step methods for stiff differential equations. The results are applicable among others to several subdiagonal Padé approximations which are notA-stable.     

A sixth order tridiagonal finite difference method for non-linear two-point boundary value problems

We present a new sixth order finite difference method for the second order differential equationy=f(x,y) subject to the boundary conditionsy(a)=A,y(b)=B. An interesting feature of our method is that each discretization of the differential equation at an interior grid point is based onfive evaluations off; the classical second order method is based on one and the well-known fourth order method of Noumerov is based on three evaluations off. In case of linear differential equations, our finite difference scheme leads to tridiagonal linear systems. We establish, under appropriate conditions, the sixth order convergence of the finite difference method. Numerical examples are considered to demonstrate computationally the sixth order of the method.     

Approximation by non-negative algebraic polynomials

Theorem 1 gives an estimate for the approximation of a continuous functionf by polynomials resulting from the convolution off with non-negative algebraic polynomialsp _n . Jackson's theorem can be deduced from it by choosing a particularp _n whose second Chebyshev-Fourier coefficient is sufficiently close to –1.Work supported in part by the Atomic Energy Commission under contract U.S. AEC AT (11-1) 1469, and in part by the National Science Foundation under grant NSF-GJ-812.     

A note on functions whose local minima are global

In this note, we introduce a new class of generalized convex functions and show that a real functionf which is continuous on a compact convex subsetM of ⁿ and whose set of global minimizers onM is arcwise-connected has the property that every local minimum is global if, and only if,f belongs to that class of functions.     

Über ein simultanes Differenzenverfahren zur Abschätzung der Torsionssteifigkeit und der Kapazität nach beiden Seiten

Summary We consider the torsion of a prism. Let its cross-sectionQ be covered by a square network (lattice) of mesh sizeh.Pólya defined a functionu _i on all lattice points by means of a partial difference equation, and then a functionp onQ by bilinear continuation ofu _i in every square. Top he appliedDirichlet's principle and obtained thus lower bounds for the torsional rigidityP ofQ.—Here we define (§ 3) acell function U _i , attributing a real value to each square; usingU _i , we construct a vector field , to which we can applyThomson's principle and get upper bounds forP.—A simultaneous method for upper and lower bounds is then indicated and discussed (§ 4) and an example is given (§ 5).—Evaluations for capacity may be calculated in a very similar way (§ 6).     

Sur les espaces analytiques quasi-compacts de dimension 1 sur un corps valué complet ultramétrique

Summary We study the structure of the one dimensional analytic quasi-compact spaces over a complete non archimedean valued field. An affinoid open subset U of a one dimensional analytic quasi-compact space X is defined by a meromorphic function f on X;i.e. U is the set of all x in X such that f is holomorphic at x and ¦f(x)¦1.The set of the meromorphic functions on X which are holomorphic on U is dense in the ring of all holomorphic functions on U. An irreducible, one dimensional quasi-compact space is either affinoid, or projective. An analytic reduction of X is defined by a meromorphic invertible function f on X;i.e. the reduction is isomorphic to the reduction associated to the covering ¦f(x)¦1and ¦f(x)¦1.     

A note on certain functional determinants

Summary We consider the problem when a scalar function ofn variables can be represented in the form of a determinant det(f _i (x _j )), the so-called Casorati determinant off ₁,f ₂,,f _n . The result is applied to the solution of some functional equations with unknown functionsH of two variables that involve determinants det(H(x _i ,x _j )).     

A variable dimension algorithm with the Dantzig-Wolfe decomposition for structured stationary point problems

Given a set ofR ⁿ and a functionf from intoR ⁿ we consider a problem of finding a pointx ^* in such that(x–x ^*) ^t f(x ^*)0 holds for every pointx in. This problem is called the stationary point problem and the pointx ^* is called a stationary point. We present a variable dimension algorithm for solving the stationary point problem with an affine functionf on a polytope defined by constraints of linear equations and inequalities. We propose a system of equations whose solution set contains a piecewise linear path connecting a trivial starting point in with a stationary point. The path can be followed by solving a series of linear programs which inherit the structure of constraints of. The linear programs are solved efficiently with the Dantzig-Wolfe decomposition method by exploiting fully the structure.Part of this research was carried out when the first author was supported by the Center for Economic Research, Tilburg University, The Netherlands and the third author was supported by the Alexander von Humboldt-Foundation, Federal Republic of Germany.     

МЕРы ВИНЕРА И НЕкОтОР ыЕ ВОпРОсы АппРОксИМАцИИ В БАНА хОВых пРОстРАНстВАх

Let (i, H, B) be an abstract Wiener space, whereH is a real separable Hilbert space,B a Banach space (H is dense inB),i is the inclusion mapping ofH inB. A functionf: R( is an open subset ofB) is calledH-analytic if it is analytic alongH. It is proved thatf can be approximated byH-analytic functions.     

A-Statistical extension of the Korovkin type approximation theorem

In this paper, using the concept ofA-statistical convergence which is a regular (non-matrix) summability method, we obtain a general Korovkin type approximation theorem which concerns the problem of approximating a functionf by means of a sequenceL _n f of positive linear operators.