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.     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

Computation of Feynman integrals of functionals that are functions of quadratic functionals

If the arguments of a function G: ^{d 1} are taken as quadratic functional defined on a space C of continuous functions, we obtain a functional G: C ¹.We give a formula for computing analytic Feynman integrals of such functionals. We also propose a method of approximate computation of sequential Feynman integrals based on replacing the kernel of an integral operator by a degenerate kernel.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 58–61.     

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

An Elementary Approach to Quasi-Isometries of Tree x ℝ n

We prove by elementary means a regularity theorem for quasi-isometries of T x ⁿ (where T denotes an infinite tree), and of many other metric spaces with similar combinatorial properties, e.g. Cayley graphs of Baumslag–Solitar groups. For quasi-isometries of T x ⁿ, it states that the image of {x} x ⁿ (xT) is uniformly close to {y} x ⁿ for some yT, and there is a well-defined surjection . Even stronger, the image of a quasi-isometric embedding of ⁿ⁺¹ in T x ⁿ is close to (a geodesic in T)T)x ⁿ.     

Computation of exponential integrals

Let P ^d be a convex polyhedron and f: ^d a linear function. One studies the computational complexity of the integral _pexp f(xdx. It is shown that these integrals satisfy nontrivial algebraic relations, which makes possible the construction of polynomial algorithms for certain polyhedra. Examples are given of the application of exponential integrals to the calculation of volume and nonlinear programming.Translated from Zapiski Nauchnykh Seminarov Leningradksogo Otdeleniya Matematicheskogo Instituta im. V. A. Skeklova AN SSSR, Vol. 192, pp. 149–162, 1991.     

Variations on the theme of repeated distances

We give an asymptotically sharp estimate for the error term of the maximum number of unit distances determined byn points in ^{d, d}4. We also give asymptotically tight upper bounds on the total number of occurrences of the favourite distances fromn points in ^{d, d}4. Related results are proved for distances determined byn disjoint compact convex sets in ².At the time this paper was written, both authors were visiting the Technion — Israel Institute of Technology.     

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).     

Finite-dimensional discrete linear stochastic accelerated-time systems and their application to quadratic stochastic dynamical systems

The limiting behavior of the trajectories {x ⁽ⁿ⁾ } of linear discrete stochastic systems of the form (K, P ^an+b ) _nN , whereK is the standard simplex in ^N ,P: ^{N N} is a linear operator,PK K,a ft,b ,a+b>0, is described. An application to a class of quadratic stochastic dynamical systems is considered.Translated fromMatematicheskie Zametki, Vol. 59, No. 5, pp. 709–718, May, 1996.     

Une caracterisation du type de la loi de Cauchy-conforme sur ℝ n

Résumé La loi de Cauchy-conforme est la mesure de probabilité sur ⁿ de densitéC/(1+X²)ⁿ. Le type d'une mesure sur ⁿ étant l'ensemble des mesures images de par les similitudes-translations de ⁿ et étant une mesure de probabilité sans atome, on démontre que le type de est invariant par les inversions de ⁿ si et seulement si est du type de la loi de Cauchy-conforme.

---

The conformal Cauchy law is the probability on ⁿ with densityC/(1+X²)ⁿ. It is shown that for a non-atomic measure on ⁿ the following is true: its type is invariant under inversions of ⁿ if and only if it is the type of a conformal Cauchy law. (The type of a measure is defined as the set of its images under similarities and translations.)

     

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 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.     

Brownian charges around loops

Summary Let (X _t ⁿ ) be a Poisson sequence of independent Brownian motions in ^d ,d3; Let be a compact oriented submanifold of d, of dimensiond–2 and volume ; let t be the sum of the windings of (X _s ⁿ , 0st) around ; then t/t converges in law towards a Cauchy variable of parameter /2. A similar result is valid when the winding is replaced by the integral of a harmonic 1-form in ^d .     

On homogenization of periodic parabolic systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in ^d and prove that the solutions converge in L ₂(^d) to the solution of the homogenized problem for each t > 0. For the L₂(^d)-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L ₂(^d)-norm of the initial value.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 86–90, 2004Original Russian Text Copyright © by T. A. SuslinaSupported by RFBR grant No. 02-01-00798.     

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.     

Symmetriegerechte Basisvektoren für Permutationsdarstellungen aller endlichen Punktgruppen der Dimensionen 3 und 2

Zusammenfassung Es seiG eine endliche Untergruppe der orthogonalen Gruppe (det=±1) des ^k mitk=2 oder 3 undN eine endliche Menge von Punkten des ^k , welche unterG invariant ist. Dies gibt Anlass zu einer Permutationsdarstellung vonG im Vektorraum der komplexen Funktionen aufN.In Abschn. 3 wird für eine symmetriegerechte Basis angegeben. Dabei sind die Funktionswerte jeweils exakt tabelliert.

---

Let G be a finite subgroup of the orthogonal group (det=±1) of ^k wherek=2 or 3 and letN be a finite set of points of ^k , which is invariant underG. In this way one gets a permutation representation ofG in the vector space of the complex functions onN.In Section 3, a symmetry adapted basis is given for , where the function values are tabulated exactly.

---

---

Im Buch [1] wurden lediglich die Diedergruppen behandelt (in Abschn. 3.1).     

Embedded minimal annuli solving an exterior problem

Given a compact, strictly convex body in ³ and a closed Jordan curve ³ satisfying several additional assumptions, the existence of a parametric, annulus type minimal surface is proved, which parametrizes along one boundary component, has a free boundary onX along the other boundary component, and which stays in ³. As a consequence of this and a reasoning developed by W. H. Meeks and S. -T. Yau we find an embedded minimal surface with these properties. Another application is the existence of an embedded minimal surface with a flat end, free boundary onX and controlled topology.This article was processed by the author using the LATEX style filepljourlm from Springer-Verlag.     

Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on a triangular mesh

Summary We study here the discretisation of the nonlinear hyperbolic equationu _t +div(vf(u))=0 in ³ × ₊, with given initial conditionu(.,0)=u ₀(.) in ², wherev is a function from ² × ₊ to ² such that divv=0 andf is a given nondecreasing function from to . An explicit Euler scheme is used for the time discretisation of the equation, and a triangular mesh for the spatial discretisation. Under a usual stability condition, we prove the convergence of the solution given by an upstream finite volume scheme towards the unique entropy weak solution to the equation.     

Asymptotic behaviour of Gaussian random fields

Summary Let X={X(t), t ^N} be a centred Gaussian random field with covariance X(t)X(s)=r(t–s) continuous on ^N×^N and r(0)=1. Let (t,s)=((X(t)–X(s)) ²)^1/2; (t,s) is a pseudometric on ^N. Assume X is -separable. Let D ₁ be the unit cube in ^N and for 0<k, D _k= {x^N: k ^–1 xD₁}, Z(k)=sup{X(t),tD _k}. If X is sample continuous and ¦r(t)¦ =o(1/log¦t¦) as ¦t¦8 then Z(k)-(2Nlogk) ^1/20 as k a.s.