OnL 1-vorticity for 2-D incompressible flow

We prove the existence of a classical weak solution for the 2-D incompressible Euler equations with initial vorticity ω₀=ω ₀ ^′ + ω ₀ ^″ , where ω ₀ ^′ is inL ¹(R ²)⌢H ⁻¹(R ²), compactly supported, and ω ₀ ^″ is a compactly supported positive Radon measure inH ⁻¹(R ²).     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

Inverse problems for Sturm—Liouville operators with potentials in Sobolev spaces: Uniform stability

Two inverse problems for the Sturm-Liouville operator Ly = s-y″ + q(x)y on the interval [0, fy] are studied. For θ ⩾ 0, there is a mapping F:W ₂^θ → l _B ^θ, F(σ) = {s _k }₁^∞, related to the first of these problems, where W ₂^∞ = W ₂^∞[0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q, and l _B ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂^θ, where we place the regularized spectral data s = {s _k }₁^∞ in the problem of reconstruction from two spectra. The main result is uniform lower and upper bounds for ∥σ - σ₁∥_θ via the l _B ^θ-norm ∥s − s₁∥_θ of the difference of regularized spectral data. A similar result is obtained for the second inverse problem, that is, the problem of reconstructing the potential from the spectral function of the operator L generated by the Dirichlet boundary conditions. The result is new even for the classical case q ∈ L ₂, which corresponds to θ = 1.     

Symmetric block bases of sequences with large average growth

We show that if 0<ε≦1, 1≦p<2 andx ₁, …,x _n is a sequence of unit vectors in a normed spaceX such thatE ‖∑ _l ⁿ ε_i x _l‖≧n ^1/p, then one can find a block basisy ₁, …,y _m ofx ₁, …,x _n which is (1+ε)-symmetric and has cardinality at leastγn ^2/p-1(logn)⁻¹, where γ depends on ε only. Two examples are given which show that this bound is close to being best possible. The first is a sequencex ₁, …,x _n satisfying the above conditions with no 2-symmetric block basis of cardinality exceeding 2n ^2/p-1. This sequence is not linearly independent. The second example is a sequence which satisfies a lowerp-estimate but which has no 2-symmetric block basis of cardinality exceedingCn ^2/p-1(logn)^4/3, whereC is an absolute constant. This applies when 1≦p≦3/2. Finally, we obtain improvements of the lower bound when the spaceX containing the sequence satisfies certain type-condition. These results extend results of Amir and Milman in [1] and [2]. We include an appendix giving a simple counterexample to a question about norm-attaining operators.     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China     

On the Cantor-Bendixson derivative, resolvable ranks, and perfect set theorems of A. H. Stone

A Borel derivative on the hyperspace 2 ^X of a compactumX is a Borel monotone mapD: 2 ^X →2 ^X . The derivative determines a Cantor-Bendixson type rank δ:2^X → ω₁ ∪ {∞} . We show that ifA⊂2 ^X is analytic andZ⊂A intersects stationary many layers δ₋₁({ξ}), then for almost all σ,F∩δ₋₁({ξ}) cannot be separated fromZ ∩∪ _a<ξ ^δ−1({a}) (and also fromZ ∩∪ _a>ξ ^δ−1({a}) by anyF _σ-set. Another main result involves a natural partial order on 2 ^X related to the derivative. The results are obtained in a general framework of “resolvable ranks” introduced in the paper. During our work on this paper the second author was a Visiting Professor at the Miami University, Ohio. This author would like to express his gratitude to the Department of Mathematics and Statistics for the hospitality.     

On the degree of iterates of automorphisms¶of the affine plane

For a polynomial automorphism f of ?² _ℂ, we set τ = deg f ²)/(deg f). We prove that τ≤ 1 if and only if f is triangularizable. In this situation, we show (by using a deep result from number theory known as the theorem of Skolem–Mahler–Lech) that the sequence (deg f ⁿ ) _{n ∈ℕ} is periodic for large n. In the opposite case, we prove that τ is an integer (τ≥ 2) and that the sequence (deg f ⁿ ) _{n ∈ℕ} is a geometric progression of ratio τ. In particular, if f is any automorphism, we obtain the rationality of the formal series . Received: 1 December 1997     

Conservative finite difference scheme for a singularly perturbed elliptic reaction-diffusion equation: Approximation of solutions and derivatives

A boundary value problem for a singularly perturbed elliptic reaction-diffusion equation in a vertical strip is considered. The derivatives are written in divergent form. The derivatives in the differential equation are multiplied by a perturbation parameter ɛ², where ɛ takes arbitrary values in the interval (0, 1]. As ɛ → 0, a boundary layer appears in the solution of this problem. Using the integrointerpolational method and the condensing grid technique, conservative finite difference schemes on flux grids are constructed that converge ɛ-uniformly at a rate of O(N ₁⁻²ln² N ₁ + N ₂⁻²), where N ₁ + 1 and N ₂ + 1 are the number of mesh points on the x ₁-axis and the minimal number of mesh points on a unit interval of the x ₂-axis respectively. The normalized difference derivatives ɛ ^k (∂ ^k /∂x ₁ ^k )u(x) (k = 1, 2), which are ɛ-uniformly bounded and approximate the normalized derivatives in the direction across the boundary layer, and the derivatives along the boundary layer (∂ ^k /∂ x ₂ ^k )u(x) (k = 1, 2) converge ɛ-uniformly at the same rate.     

The speed of convergence of a martingale

LetX _n, n≧0, be a martingale with respect to the σ-fieldsF _n and letB _n ² =Σ_1≧n E{(X ₁−X ₁₋₁)²|F ₁₋₁} It is known that ifB ₁ ² <∞ on some set Ω₀ thenX _∞=limX _n exists and is finite a.e. on Ω₀ We show that under suitable conditions there exists a constant ν<∞ for which lim supB _n ⁻¹ {log logB _n ² }^−1/2|X _∞−X _n−1 | ≦ √2(η+1). If “the fluctuations ofB _n are small” (in the sense of the Corollary) then ν=0 and the usual upper bound of a law of the iterated logrithm results. This upper bound is not necessarily achieved, though. Research supported in part by the NSF under Grant No. MCS 72-04534A04.     

S-forcing,I. A “black-box” theorem for morasses,with applications to super-Souslin trees

We formulate, for regular μ>ω, a “forcing principle” S_μ which we show is equivalent to the existence of morasses, thus providing a new and systematic method for obtaining applications of morasses. Various examples are given, notably that for infinitek, if 2 ^k =k ⁺ and there exists a (k ⁺, 1)-morass, then there exists ak ⁺⁺-super-Souslin tree: a normalk ⁺⁺ tree characterized by a highly absolute “positive” property, and which has ak ⁺⁺-Souslin subtree. As a consequence we show that CH+SH_{ℵ 2}⟹ℵ₂ is (inaccessible)^L. This author thanks the US-Israel Binational Science Foundation for partial support of this research.     

On the comparison of PBIB designs with two associate classes

A method to compare two-associate-class PBIB designs is discussed. As an application, it is shown that ifd ^* is a group-divisible design withλ ₂=λ₁+1, a group divisible design with group size two andλ ₂=λ₁+1>1, a design based on a triangular scheme andv=10 andλ ₁=λ₂+1, a design with anL ₂ scheme andλ ₂=λ₁+1, a design with anL _s scheme,v=(s+1) ², andλ ₂=λ₁+1, wheres is a positive integer, or a design with a cyclic schemev=5, andλ ₁=λ₂±1, thend ^* is optimum with respect to a very general class of criteria over all the two-associate-class PBIB designs with the same values ofv, b andk asd ^*. The best two-associate-class PBIB design, however, is not necessarily optimal over all designs. This paper was prepared with the support of Office of Naval Research Contract No. N00014-75-C-0444/NR 042-036 and National Science Foundation Grant No. MCS-79-09502.     

On the bestL 1-approximations of functional classes by splines under restrictions imposed on their derivatives

We find the exact asymptotics (asn→∞) of the bestL ₁-approximations of classesW ₁ ^r of periodic functions by spliness∈S _{2n, r∼-1} (S _{2n, r∼-1} is a set of 2π-periodic polynomial splines of orderr−1, defect one, and with nodes at the pointskπ/n,k∈ℤ) such that V ₀ ^2π s( ^r-1)≤1+ɛ _n , where {ɛ _n } _n=1 ^∞ is a decreasing sequence of positive numbers such that ɛ _n n ²→∞ and ɛ _n →0 asn→∞. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 435–444, April, 1999.     

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Summary Given a random sample of sizen from a densityf ₀ on the real line satisfying certain regularity conditions, we propose a nonparametric estimator forψ ₀=−f ₀ ^′ /f⁰. The estimate is the minimizer of a quadratic functional of the formλJ(ψ)+∫[ψ ²−2ψ′]dF_n where λ>0 is a smoothing parameter,J(·) is a roughness penalty, andF _n is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which is related to spline functions. A more complete study of the caseJ(ψ)=∫[d ²ψ/dx²]² is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness limit (λ→∞). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem is simple in comparison with the analogous penalized likelihood estimators. This research was supported by the Office of Naval Research under Grant Number N00014-82-C-0062.     

The gonality of smooth curves with plane models

LetC be the normalization of an integral plane curve of degreed with δ ordinary nodes or cusps as its singularities. If δ=0, then Namba proved that there is no linear seriesg _d ^−2/1 and that everyg _d ^−1/1 is cut out by a pencil of lines passing through a point onC. The main purpose of this paper is to generalize his result to the case δ>0. A typical one is as follows: Ifd≥2(k+1), and δ<kd−(k+1)²+3 for somek>0, thenC has no linear seriesg _d ^−3/1 . We also show that ifd≥2k+3 and δ<kd−(k+1)²+2, then each linear seriesg _d ^−2/1 onC is cut out by a pencil of lines. We have similar results forg _d ^−1/1 andg _2d ^−9/1 . Furthermore, we also show that all of our theorems are sharp.     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

Renormalization group, causality, and nonpower perturbation expansion in QFT

The structure of the QFT expansion is studied in the framework of a new “invariant analytic” version of the perturbative QCD. Here, an invariant coupling constant α(Q ² /Λ ² ) = β ₁ α_s(Q ² )/(4π) becomes a Q ² -analytic invariant function α _an (Q²/Λ ² ) ≡A(x), which, by construction, is free of ghost singularities because it incorporates some nonperturbative structures. In the framework of the “analyticized” perturbation theory, an expansion for an observable F, instead of powers of the analytic invariant charge A(x), may contain specific functions A_n(x)=[aⁿ(x)] _an , the “nth power of a(x) analyticized as a whole.” Functions A _n>2(x) for small Q² ≤Λ ² oscillate, which results in weak loop and scheme dependences. Because of the analyticity requirement, the perturbation series for F(x) becomes an asymptotic expansion à la Erdélyi using a nonpower set {A _n (x)}. The probable ambiguities of the invariant analyticization procedure and the possible inconsistency of some of its versions with the renormalization group structure are also discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 55–66, April, 1999.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

The asymptotic behavior of Chern-Simons Higgs model on a compact Riemann surface with boundary

We study the self-dual Chern-Simons Higgs equation on a compact Riemann surface with the Neumann boundary condition.In the previous paper,we show that the Chern-Simons Higgs equation with parameter λ0 has at least two solutions(uλ1,uλ2) for λ sufficiently large,which satisfy that uλ1→u0 almost everywhere as λ→∞,and that uλ2→∞ almost everywhere as λ→∞,where u 0 is a(negative) Green function on M.In this paper,we study the asymptotic behavior of the solutions as λ→∞,and prove that uλ2-uλ2 converges to a solution of the Kazdan-Warner equation if the geodesic curvature of the boundary M is negative,or the geodesic curvature is nonpositive and the Gauss curvature is negative where the geodesic curvature is zero.     

Computable numberings of families of low sets and Turing jumps in the Ershov hierarchy

If ν and μ are some Δ₂⁰-computable numberings of families of sets of the naturals then P(x,y) ⇔ ν(x)′ ≠ μ(y) is a Σ₂⁰-predicate. Deriving corollaries from this result, we obtain a sufficient condition for existence of a Δ₂⁰-computable numbering of the subfamily of all sets in a given family with the Turing jumps belonging to a fixed level of the Ershov hierarchy, and we deduce existence of a Σ_ω⁻¹-computable numbering of the family of all superlow sets.     

Joint ergodicity and mixing

The study of jointly ergodic measure preserving transformations of probability spaces, begun in [1], is continued, and notions of joint weak and strong mixing are introduced. Various properties of ergodic and mixing transformations are shown to admit analogues for several transformations. The case of endomorphisms of compact abelian groups is particularly emphasized. The main result is that, given such commuting endomorphisms σ₁σ₂,...,σ, ofG, the sequence ((1/N)Σ _n=0 ^N−1 σ ₁ ⁿ f ¹·σ ₂ ⁿ f ²· ··· · σ _s ⁿ f ^sconverges inL ²(G) for everyf ₁,f ₂,…,f _s∈L ^∞(G). If, moreover, the endomorphisms are jointly ergodic, i.e., if the limit of any sequence as above is Π _i=1 ^s ∫ _G f ₁ d μ, where μ is the Haar measure, then the convergence holds also μ-a.e.