Kneser-type oscillation criteria for self-adjoint two-term differential equations

It is proved that the even-order equationy ⁽²ⁿ⁾ +p(t)y=0 is (n,n) oscillatory at if

On local oscillatory integrals with variable Calderón-Zygmund kernels, II

Let , be a real analytic function or a real-C function on ⁿ andk be a variable Calderón-Zygmund kernel. Define the oscillatory singular integral operatorT by

Oscillation of forced neutral differential equations with positive and negative coefficients

In this paper the forced neutral difterential equation with positive and negative coefficients d/dt [x(t)-R(t)x(t-r)] P(t)x(t-x)-Q(t)x(t-σ)=f(t),t≥t0,is considered,where f∈L^1(t0,∞)交集C([t0,∞],R^ )and r,x,σ∈(0,∞),The sufficient conditions to oscillate for all solutions of this equation are studied.     

How big are the increments of lp-valued Gaussian processes?

Let be a sequence of independent Gaussian processes with (h) Put . The large increments forY(·) with bounded σ(p, h) are investigated. As an example the large increments of infinite-dimensional fractional Ornstein-Uhlenbeck process in I^P are given. The method can also be applied to certain processes with stationary increments. Project supported by the National Natural Science Foundation of China and the Natural Science Foundation of Zhejiang Province.     

Differential geometry on almost tangent manifolds

Summary We start from a tensor field Q of type (1, 1) defined in a2n-dimensional manifold M which satisfies Q ²⁼⁰ and has rank n. The tensor field Q defines an almost tangent structure in M. We then introduce another tensor field P of the same type and having properties similar to those of Q. We then define and study the tensors H=PQ, V=QP, J=P−Q, K=P+Q, L=PQ−QP, (J, K, L) defining an almost quaternion structure of the second kind on M. We study the differential geometry on almost tangent manifolds in terms of these tensors. To ProfessorBeniamino Segre on his seventieth birthday Entrata in Redazione il 7 giugno 1973.     

Distribution of Distances and Triangles in a Point Set and Algorithms for Computing the Largest Common Point Sets

This paper considers the following problem, which we call the largest common point set problem (LCP): given two point sets P and Q in the Euclidean plane, find a subset of P with the maximum cardinality that is congruent to some subset of Q . We introduce a combinatorial-geometric quantity λ(P, Q) , which we call the inner product of the distance-multiplicity vectors of P and Q , show its relevance to the complexity of various algorithms for LCP, and give a nontrivial upper bound on λ(P, Q) . We generalize this notion to higher dimensions, give some upper bounds on the quantity, and apply them to algorithms for LCP in higher dimensions. Along the way, we prove a new upper bound on the number of congruent triangles in a point set in four-dimensional space. Received July 17, 1997, and in revised form March 6, 1998.     

Heat content and Brownian motion for some regions with a fractal boundary

LetD be an open, bounded set in euclidean space ^m (m=2, 3, ...) with boundary D. SupposeD has temperature 0 at timet=0, while D is kept at temperature 1 for allt>0. We use brownian motion to obtain estimates for the solution of corresponding heat equation and to obtain results for the asymptotic behaviour ofE _D (t), the amount of heat inD at timet, ast0⁺. For the triadic von Koch snowflakeK our results imply that

Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights

A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedL _p convergence of Hermite series, that is, orthogonal expansions corresponding to the Hermite weight. We generalize these to orthogonal expansions for a class of Freud weightsW ²:=e ^–2Q , by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW ². Our identical necessary and sufficient conditions close a slight gap in Muckenhoupt's conditions for the Hermite weight at least forp>1. Moreover, our necessary conditions apply whenQ(x)=|x|, >1 while our sufficient conditions apply at least for =2,4,6,....Communicated by Vilmos Totik.     

Rang moyen de familles de courbes elliptiques et lois de Sato-Tate

The goal of this note is to give some complements to an article of Fouvery and Pomykala: by anad-hoc method, they bound on average the rank of elliptic curves over in polynomial families:y ²=x ³=a(t)x+b(t) whent varies in under some generic conditions on the polynomials (over a(t),b(t). Here, by a more systematic treatment, we are able to relax most of these conditions, keeping only the natural one (the family is not geometricaly trivial). However, this result, specialized to the case treated by Fouvry and Pomykala, yields a better bound; our method depends on the distribution of the number of points in families of elliptic curves over finite fields (known as the vertical Sato-Tate law), which itself depends on the work of Deligne on the Weil conjectures.

     

Norm behavior and zero distribution for orthogonal polynomials with nonsymmetric weights

The asymptotic behavior of thenth root of the leading coefficient of orthogonal polynomials on (–,) and the distribution of their zeros is studied for nonsymmetric weights that behave like exp(–2B|x|) whenx>0 and exp(–2Ax ) whenx<0,>AB. These results generalize previous investigations of Rakhmanov and Mhaskar and Saff who handle the symmetric caseA=B.Communicated by Paul Nevai.     

The convergence of padé approximants and the size of the power series coefficients

Let f be a power series ∑a_izⁱ with complex coefficients. The (n. n) Pade approximant to f is a rational function P/Q where P and Q are polynomials, Q(z) ? 0, of degree ≦ n such that f(z)Q(z)-P(z) = Az²ⁿ⁺¹ + higher degree terms. It is proved that if the coefficients a_i satisfy a certain growth condition, then a corresponding subsequence of the sequence of (n, n) Pade approximants converges to f in the region where the power series f converges, except on an exceptional set E having a certain Hausdorff measure 0. It is also proved that the result is best possible in the sense that we may have divergence on E. In particular,there exists an entire function f such that the sequence of (ny n) Pade approximants diverges everywhere (except at 0)     

Area of Julia sets of holomorphic self-maps onC *

It is a general problem to study the measure of Julia sets. There are a lot of results for rational and entire functions. In this note, we describe the measure of Julia set for some holomorphic self-maps onC ^*. We'll prove thatJ(f) has positive area, wheref:C ^*→C ^*,f(z)=z ^m c ^P(z)+Q(1/z) ,P(z) andQ(z) are monic polynomials of degreed, andm is an integer.     

A proof of Freud's conjecture for exponential weights

LetW (x) be a function nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ²(x) are finite. Let {p _n (W ²;x)} ₀ denote the sequence of orthonormal polynomials with respect to the weightW ²(x), and let {A _n } ₁ and {B _n } ₁ denote the coefficients in the recurrence relation

Inverse scattering in one-dimensional nonconservative media

The inverse scattering problem arising in wave propagation in one-dimensional non-conservative media is analyzed. This is done in the frequency domain by considering the Schrödinger equation with the potentialikP(x)+Q(x), wherek ² is the energy andP(x) andQ(x) are real integrable functions. Using a pair of uncoupled Marchenko integral equations,P(x) andQ(x) are recovered from an appropriate set of scattering data including bound-state information. Some illustrative examples are provided.Dedicated to M.G. Kreîn, one of the founding fathers of inverse scattering theory.     

Nearest Neighbor Queries in Metric Spaces

Given a set S of n sites (points), and a distance measure d , the nearest neighbor searching problem is to build a data structure so that given a query point q , the site nearest to q can be found quickly. This paper gives data structures for this problem when the sites and queries are in a metric space. One data structure, D(S) , uses a divide-and-conquer recursion. The other data structure, M(S,Q) , is somewhat like a skiplist. Both are simple and implementable. The data structures are analyzed when the metric space obeys a certain sphere-packing bound, and when the sites and query points are random and have distributions with an exchangeability property. This property implies, for example, that query point q is a random element of . Under these conditions, the preprocessing and space bounds for the algorithms are close to linear in n . They depend also on the sphere-packing bound, and on the logarithm of the distance ratio of S , the ratio of the distance between the farthest pair of points in S to the distance between the closest pair. The data structure M(S,Q) requires as input data an additional set Q , taken to be representative of the query points. The resource bounds of M(S,Q) have a dependence on the distance ratio of S Q . While M(S,Q) can return wrong answers, its failure probability can be bounded, and is decreasing in a parameter K . Here K≤ |Q|/n is chosen when building M(S,Q) . The expected query time for M(S,Q) is O(Klog n)log , and the resource bounds increase linearly in K . The data structure D(S) has expected O( log n) ^O(1) query time, for fixed distance ratio. The preprocessing algorithm for M(S,Q) can be used to solve the all nearest neighbor problem for S in O(n(log n) ² (log ϒ(S)) ² ) expected time. Received September 17, 1996, and in revised form November 1, 1998.     

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space on the n-dimensional Heisenberg group ℍ ⁿ = ℂ ⁿ × ℝ, where 1 = p < Q and Q = 2n + 2 is the homogeneous dimension of ℍn. Suppose that the subelliptic gradient is gloablly L ^p integrable, i.e., is finite. We prove a Poincaré inequality for f on the entire space ℍ ⁿ . Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of under the norm of

On the singularities of residual intertwining operators

Let G be a reductive algebraic group defined over \Bbb Q {\Bbb Q} . Let P, P' be parabolic subgroups of G, defined over \Bbb Q {\Bbb Q} , and let _boxclose_boxclose, a_P') t \in W({\frak a}_{P}, {\frak a}_{P'}) . In this paper we study the intertwining operator M_P￠|P(t,l), l ? \frak a^*_{P,\Bbb C} M_{P' \vert P}(t,\lambda),\,\lambda \in {\frak a}^*_{P,{\Bbb C}} , acting in corresponding spaces of automorphic forms. One of the main results states that each matrix coefficient of M_P￠|P(t,l) M_{P' \vert P}(t,\lambda) is a meromorphic function of order ￡ n + 1 \le n + 1 , where n = dim G. Using this result, we further investigate the rank one intertwining operators, in particular, we study the distribution of their poles.     

On Intersection of Maximal Orthogonally k-Starshaped Polygons

Let F be a simply connected orthogonal polygon in R ² and let P denote the intersection of all maximal orthogonally k-starshaped polygons in F for any fixed integer k,k2. If P and for every x,y P which are joined in F by a staircase path having two segments there is a similar staircase path from x to y in P, then there exists a maximal orthogonally k-starshaped polygon Q in F such that the staircase k-kernel of Q is a subset of the staircase k-kernel of P. In particular, F is either an orthogonally k-starshaped simply connected polygon in F or empty.     

Renormalization and blow up for charge one equivariant critical wave maps

We prove the existence of equivariant finite time blow-up solutions for the wave map problem from ℝ²⁺¹→S ² of the form where u is the polar angle on the sphere, is the ground state harmonic map, λ(t)=t ^-1-ν, and is a radiative error with local energy going to zero as t→0. The number can be prescribed arbitrarily. This is accomplished by first “renormalizing” the blow-up profile, followed by a perturbative analysis. Mathematics Subject Classification (1991) 35L05, 35Q75, 35P25