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.     

Gewichtete Volumsmittelwerte von Simplices,welche zufällig in einem konvexen Körper des ℝ gewählt werden

For any set ofn+1 pointsx ₁, ...,x _n+1F we denote byv(C(x ₁,...,x _n+1)) then-dimensional oriented volume of the convex hullC(x ₁,...,x _n+1) of these points. With a fixed symmetric functionf: >> strictly monotone increasing on the nonnegative real line, we consider the real functional RODEL on the set of all convex bodiesK of ⁿ with absolute volume |v(K)|=1 and assert, that it takes its minimal value on the ellipsoids with absolute volume 1.     

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.     

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 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 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     

On the boundedness of Weyl sums

Leta be irrational and letf:[0,1] be Riemann-integrable with integral zero. Letf _n (x) denote the Weyl sumf _n (x):= _k=0 ^n–1 f({x k>}),x/[0,1[,n. We prove criteria for the boundedness of the sequence (f _n ) _n1 and discuss the relation of this question to irregularities of the distribution of sequences.     

Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

Let _i(L), _i(L^*) denote the successive minima of a latticeL and its reciprocal latticeL ^*, and let [b₁,..., b _n ] be a basis ofL that is reduced in the sense of Korkin and Zolotarev. We prove that and, where and _j denotes Hermite's constant. As a consequence the inequalities are obtained forn7. Given a basisB of a latticeL in ^m of rankn andx ^m , we define polynomial time computable quantities(B) and(x,B) that are lower bounds for ₁(L) and(x,L), where(x,L) is the Euclidean distance fromx to the closest vector inL. If in additionB is reciprocal to a Korkin-Zolotarev basis ofL ^*, then ₁(L) _n ^* (B) and.The research of the second author was supported by NSF contract DMS 87-06176. The research of the third author was performed at the University of California, Berkeley, with support from NSF grant 21823, and at AT&T Bell Laboratories.     

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.     

On a class of Hankel operators: Fredholm properties and invertibility

A study is presented on the Fredholm properties and invertibility of a Hankel integral operator inL ₊ ² () with a kernel function inS whose Fourier transformK is a measurable essentially bounded function in . This study is based on the properties of a Wiener-Hopf operator with a matrix valued symbol naturally associated with the operator mentioned above. Further results are obtained for the case whereK PC(), and an application to a diffraction problem is presented.     

Random continued fractions and inverse Gaussian distribution on a symmetric cone

In this paper we introduce the inverse Gaussian and Wishart distributions on the cone of real (n, n) symmetric positive definite matricesH _n ⁺ () and more generally on an irreducible symmetric coneC. Then we study the convergence of random continued fractions onH _n ⁺ () andC by means of real Lagrangians forH _n ⁺ () and by new algebraic identities on symmetric cones forC. Finally we get a characterization of the inverse Gaussian distribution onH _n ⁺ () andC.     

A solution of Hadwiger's covering problem for zonoids

For a convex body M ⁿ byb(M) the least integerp is denoted, such that there are bodiesM ₁, ...,M _p each of which is homothetic toM with a positive ratiok<1 andM ₁...M _p M. H. Martini has proved [7] thatb(M)<-3·2 ^n–2 for every zonotope M ⁿ , which is not a parallelotope.In the paper this Martini's result is extended to zonoids. In the proof some notions and facts of real functions theory are used (points of density, approximative continuity).     

On the lowest eigenvalue of the Laplacian for the intersection of two domains

IfA andB are two bounded domains in _n and (A), (B) are the lowest eigenvalues of – with Dirichlet boundary conditions then there is some translate,B _x, ofB such that (AB _x)<(A)+(B). A similar inequality holds for .There are two corollaries of this theorem: (i) A lower bound for sup _x {volume (AB _x)} in terms of (A), whenB is a ball; (ii) A compactness lemma for certain sequences inW ^1,p ( _n ).Work partially supported by U.S. National Science Foundation grant PHY-8116101 A01. AMS(MOS) Classification: 35P15     

Lower functions for asymmetric Lévy processes

Summary Let {X _t } be a ¹ process with stationary independent increments and its Lévy measurev be given byv{yy>x}=x ^–L ₁ (x), v{yy<–x}=x ^–L ₂ (x) whereL ₁,L ₂ are slowly varying at 0 and and 0<1. We construct two types of a nondecreasing functionh(t) depending on 0<<1 or =1 such that lim inf a.s. ast 0 andt for some positive finite constantC.This research is partialy supported by a grant from Korea University     

A note on quasi-periodic solutions of some elliptic systems

We extend a recent method of proof of a theorem by Kolmogorov on the conservation of quasi-periodic motion in Hamiltonian systems so as to prove existence of (uncountably many) real-analytic quasi-periodic solutions for elliptic systems u=f _x (u, y), whereu y ^M u(y) ^N ,f=f(x, y) is a real-analytic periodic function and is a small parameter. Kolmogorov's theorem is obtained (in a special case) whenM=1 while the caseN=1 is (a special case of) a theorem by J. Moser on minimal foliations of codimension 1 on a torusT ^M +1. In the autonomous case,f=f(x), the above result holds for any .     

Punkt- und Geradenabbildungen,die das volumen 1 vonn- dimensionalen simplices im ℝn erhalten

A mappingf of ⁿ ,n3, into itself such thatf(x ₁),f(x ₂, ...,f(x _n+1 ) are the vertices of a simplex of volume 1 ifx ₁,...,x _n+1 are the vertices of a simplex of volume 1, must be equi-affine. (This theorem is also true in casen= 2 as it was proved by Gil Martin, see W. Benz [4].)LetM ⁿ be the set of lines of ⁿ . A mapping: M ⁿ M ⁿ ,n3 such that(a ₁ ),...,(a _n(n+1)/2 ) are the edges of a simplex of volume 1 ifa ₁,...,a _n(n+1)/2 are the edges of a simplex of volume 1, must be induced by an equi-affine mapping of ⁿ .     

On strong maximal operators corresponding to different frames

The problem is posed and solved whether the conditionsf L(1+1n⁺ L)²(²) and are equivalent for functionsf L(²) (whereM _2, denotes the strong maximal operator corresponding to the frame {OX ,OY }).The results obtained represent a general solution of M. de Guzmán's problem that was previously studied by various authors.     

Sets determined by finitely many X-rays

A measurable set in _n which is uniquely determined among all measurable sets (modulo null sets) by its X-rays in a finite set L of directions, or more generally by its X-rays parallel to a finite set L of subspaces, is called L-unique, or simply unique. Some subclasses of the L-unique sets are known. The L-additive sets are those measurable sets E which can be written E {x _n : _i f _i (x) * 0}. Here, denotes equality modulo null sets, * is either or >, and the terms in the sum are the values of ridge functions f _i orthogonal to subspaces S _i in L. If n=2, the L-inscribable convex sets are those whose interiors are the union of interiors of inscribed convex polygons, all of whose sides are parallel to the lines in L. Relations between these classes are investigated. It is shown that in _n each L-inscribable convex set is L-additive, but L-additive convex sets need not be L-inscribable. It is also shown that every ellipsoid in _n is unique for any set of three directions. Finally, some results are proved concerning the structure of convex sets in _n , unique with respect to certain families of coordinate subspaces.     

Multiplication operator on a matrix polynomial

It is proved that the study of a perturbed multiplication operator on a matrix polynomial in the space L₂(, ⁿ) may be reduced to the study of a perturbed multiplication operator with independent variable in the space L₂(, , ^N) with weight satisfying the Mackenhaupt condition.Translated from Ukrayins'kyy Matemarychnyy Zhurnal, Vol. 44, No. 9, pp. 1287–1289, September, 1992.     

Weak solutions of the Stokes problem in weighted Sobolev spaces

For n2 we consider the Stokes problem in ⁿ, -u + p=f, -divu=g, in weighted Soboiev spaces H ₆ ^m,r , where the weights are proportional to (1+|x|). We prove the existence of weak solutions for any K, whereK is a discrete set of critical values. Furthermore, we characterize the solutions of the homogeneous problem.This research was supported by the DFG research group Equations of Hydrodynamics, Universities of Bayreuth and Paderborn.