On local oscillatory integrals with variable Calderón-Zygmund kernels

Letp(1, ). In this paper, the authors investigate the uniformL ^p ( ⁿ ) in of the oscillatory singular integral operatorT defined by

On Non-Negative Elliptic-Type Differential Inequalities in ℝ n , Vanishing Almost Everywhere on Some Open Subset in ℝ n

It is proved in this article that any generalized solution of a sufficiently general class of elliptic-type differential inequalities in  ⁿ that is non-negative almost everywhere in  ⁿ and vanishes almost everywhere on an open set ⁿ is trivial in  ⁿ .     

On a Class of Convex Sets and Functions

Given a convex subset C of ⁿ, the set-valued mapping C (where 0C is, by convention, the recession cone of C) is increasing on ₊ if and only if C contains the origin, and decreasing on ₊ if and only if C is contained in its recession cone. This simple fact enables us to define a binary operation which combines a concave or convex function on ^m with a convex subset of ⁿ to produce a convex subset of ^n+m. This binary operation is the set theoretic counterpart of a functional operation introduced by the author. In this paper, we present a detailed study of the class of convex subsets which are contained in their recession cones, and we establish some remarkable properties of our binary operation.Mathematics Subject Classifications (2000) 26A51, 26B25, 26E25.     

Über den algorithmischen Nachweis von Ungleichungen I

We shall develop a method to prove inequalities in a unified manner. The idea is as follows: It is quite often possible to find a continuous functional : ⁿ , such that the left- and the right-hand side of a given inequality can be written in the form (u)(v) for suitable points,v=v(u). If one now constructs a map ^{n n} , which is functional increasing (i.e. for each x ⁿ (which is not a fixed point of ) the inequality (x)<((x)) should hold) one specially gets the chain (u)( u))( ²(u))... ⁿ (u)). Under quite general conditions one finds that the sequence { ⁿ (u)} _n converges tov=v(u). As a consequence one obtains the inequality (u)(v).     

Reducible Dupin submanifolds

Pinkall's standard constructions for obtaining a Dupin hypersurface W in ^N from a Dupin hypersurface M in ⁿ , N>n, are studied in the context of Lie sphere geometry. It is shown that a compact Dupin hypersurface W in ^N with g distinct principal curvatures at each point is reducible to a compact Dupin hypersurface M in ⁿ if and only if g=2.This research was supported by NSF Grant No. DMS 87-06015.     

The Newton Bracketing Method for Convex Minimization

An iterative method for the minimization of convex functions f :ⁿ , called a Newton Bracketing (NB) method, is presented. The NB method proceeds by using Newton iterations to improve upper and lower bounds on the minimum value. The NB method is valid for n = 1, and in some cases for n > 1 (sufficient conditions given here). The NB method is applied to large scale Fermat–Weber location problems.     

On the domain dependence of solutions to the two-phase Stefan problem

We prove that solutions to the two-phase Stefan problem defined on a sequence of spatial domains _n ^N converge to a solution of the same problem on a domain where is the limit of _n in the sense of Mosco. The corresponding free boundaries converge in the sense of Lebesgue measure on ^N.     

Regularized spectral distributions

For C a bounded, injective operator with dense image, we define a C-regularized spectral distribution. This produces a functional calculus, f f(B), from C() into the space of closed densely defined operators, such that f(B)C is bounded when f has compact support. As an analogue of Stone's theorem, we characterize certain regularized spectral distributions as corresponding to generators of polynomially bounded C-regularized groups. We represent the regularized spectral distribution in terms of the regularized group and in terms of the C-resolvent. Applications include the Schrödinger equation with potential, and symmetric hyperbolic systems, all on L^p(ⁿ) (1p<), C _o(ⁿ), BUC(ⁿ), or any space of functions where translation is a bounded strongly continuous group.     

Fibred links and a construction of real singularities via complex geometry

This note gives a method for constructing real analytic maps from ²ⁿ into ², with an isolated critical point at 0 ²ⁿ , for alln>1. This provides infinite families of real singularities which fiber a la Milnor.Research partially supported by CONACYT, Mexico, grant 1206-E92103.     

On solvability of linear difference equations in smooth and real analytic vector functions of several variables

We investigate the multidimensional equations _j=1 ^q A_j(x)y(x+e _j )=f(x),e _j ⁿ wherex ⁿ andA _j : ⁿ Hom( ^p , ^m ),f : ^{n m} are given maps. Sufficient conditions for smooth and analytic solvability for anyf C ^k ,k are found.Research partially supported by the Israel Ministry of ScienceAMS classification 39B Functional equations     

Existence of strong solutions for Itô's stochastic equations via approximations

Summary Given strong uniqueness for an Itô's stochastic equation with discontinuous coefficients, we prove that its solution can be constructed on any probability space by using, for example, Euler's polygonal approximations. Stochastic equations in ^d and in domains in ^d are considered.Research supported by the Hungarian National Foundation of Scientific Research No. 2990.Supported in part by NSF Grant DMS-9302516     

Constraint decomposition algorithms in global optimization

Many global optimization problems can be formulated in the form min{c(x, y): x X, y Y, (x, y) Z, y G} where X, Y are polytopes in ^p , ⁿ , respectively, Z is a closed convex set in ^p+n, while G is the complement of an open convex set in ⁿ . The function c: ^p+n is assumed to be linear. Using the fact that the nonconvex constraints depend only upon they-variables, we modify and combine basic global optimization techniques such that some new decomposition methods result which involve global optimization procedures only in ⁿ . Computational experiments show that the resulting algorithms work well for problems with smalln.     

On the relationship between differentiability conditions and existence of a strong gradient

It is proved that for any n 2 there exists continuous function f : ⁿ which is differentiable almost everywhere, but has no strong gradient almost everywhere.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 93–98.Original Russian Text Copyright © 2005 by G. G. Oniani.This revised version was published online in April 2005 with a corrected issue number.     

On the semiboundedness ofδ-perturbations of the laplacian supported by curves with angle points

Elementary self-adjoint perturbations of the Laplacian supported by curves with singular angle points in ³ and ⁴ are studied. The perturbations are shown to be semibounded in ³ and not semibounded in ⁴. In the latter case semiboundedness may take place in subspaces with a given symmetry, as simple examples illustrate.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 1, pp. 3–17, October, 1995.     

Bochner-Riesz means of functions in weak-L p

The Bochner-Riesz means of order 0 for suitable test functions on ^N are defined via the Fourier transform by . We show that the means of the critical index , do not mapL ^p,( ^N ) intoL ^p,( ^N ), but they map radial functions ofL ^p,( ^N ) intoL ^p,( ^N ). Moreover, iff is radial and in theL ^p,( ^N ) closure of test functions,S _R f(x) converges, asR+, tof(x) in norm and for almost everyx in ^N . We also observe that the means of the function|x| ^–N/p, which belongs toL ^p,( ^N ) but not to the closure of test functions, converge for nox.     

Perturbation of the Laplacian by the Coulomb potential and a point interaction in L p (ℝ)3

Perturbations of -+/|x| (with >0) by a point interaction centered at zero are defined in L ^p(³). This is done for 3/20 (³{0}), such that the extension is the negative generator of an analytic semigroup on L ^p(³).     

The caratheodory number for thek-core

Thek-core of the setS ⁿ is the intersection of the convex hull of all setsA S with ¦SA¦<-k. The Caratheodory number of thek-core is the smallest integerf (d,k) with the property thatx core _{kS, S} ⁿ implies the existence of a subsetT S such thatx core_kT and ¦T¦f (d, k). In this paper various properties off(d, k) are established.Research of this author was partially supported by Hungarian National Science Foundation grant no. 1812.     

Approximation of the efficient point set by perturbation of the ordering cone

A vector optimization problem is given by a feasible setZ ⁿ , a vector-valued objective functionf: ^{n l} , and an ordering coneC ^l . We perturb the ordering cone in such a way that the weakly efficient points of the perturbed vector optimization problem given byZ, f, and the perturbed cone are efficient points of the original problem. Especially this means that scalarization methods, which compute in general only weakly efficient points, determine efficient points of the original problem, when they were applied to the perturbed problem.It turns out that the efficient points are the limits of weakly efficient points of the perturbed problems, letting the perturbation tend to zero. On the basis of this, a reference point algorithm is formulated. Finally, we apply this algorithm to a structural optimization problem.     

On Two-Point Boundary Conditions in Optimal Control Problems

Let x=g(t,x(t),u(t)) be the governing equation of an optimal control problem with two-point boundary conditions h ₀(x(a))+h ₁(x(b)) = 0, where x: [a,b] ⁿ is continuous, u: [a,b] ^k-n is piecewise continuous and left continuous, h₀,h₁: ^{n q} are continuously differentiable, and g:[a,b]× ^{k n} is continuous. The paper finds functions _i C¹([a,b]× ⁿ ) such that (x(t),u(t)) is a solution of the governing equation if and only if

Complete controllability

A linear autonomous control system in ⁿ is said to be completely controllable iff there existsT>0 such that eachx ⁿ can be steered to anyy ⁿ in timeT. This paper presents a geometric characterization of this property in the case in which there are constraints on the values which the control maps can assume. A necessary and sufficient condition to get instant controllability (i.e., complete controllability for anyT>0) is also derived. This condition generalizes the well-known Kalman condition to the constrained case.