Diffusion times and stability exponents for nearly integrable analytic systems

For a positive integer n and R>0, we set . Given R>1 and n≥4 we construct a sequence of analytic perturbations (H _j ) of the completely integrable Hamiltonian on , with unstable orbits for which we can estimate the time of drift in the action space. These functions H _j are analytic on a fixed complex neighborhood V of , and setting the time of drift of these orbits is smaller than (C(1/ɛ _j )^1/2(n-3)) for a fixed constant c>0. Our unstable orbits stay close to a doubly resonant surface, the result is therefore almost optimal since the stability exponent for such orbits is 1/2(n−2). An analogous result for Hamiltonian diffeomorphisms is also proved. Two main ingredients are used in order to deal with the analytic setting: a version of Sternberg's conjugacy theorem in a neighborhood of a normally hyperbolic manifold in a symplectic system, for which we give a complete (and seemingly new) proof; and Easton windowing method that allow us to approximately localize the wandering orbits and estimate their speed of drift.     

Sharp Jackson-Stechkin inequality inL2 for multidimensional spheres

In this paper we prove the Jackson-Stechkin inequalityE _n−1(f)<ω _n (f, 2τ _n ,λ),n≥1,m≥5,r≥1, f ∈L2( ),f ≢ const, which is sharp for eachn=2, 3, ...; hereE _n−1 (f) is the best approximation of a functionf by spherical polynomials of degree ≤n−1, ω _n (f, τ) is theτth modulus of continuity off based on the translations ,t ∈ ℝ,x ∈ , , is the measure of the unit Euclidean sphere , , andτ _n ,λ is the first positive zero of the Gegenbauer cosine polynomial (cost). Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 333–355, September, 1996. The present paper was discussed at Ural State University in a seminar headed by Professor Arestov. The author is grateful to Professor Arestov and Associate Professor Popov for useful conversations. This research was supported by the State Commission for Higher Education of the Russian Federation under grant No. 2-16-5-31 and by the Russian Foundation for Basic Research under grant No. 93-011-196.     

On generalized local time for the process of brownian motion

We prove that the functionals of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δ_Γ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R ^d , k≥1 and acting on finite continuous functions φ(·) in R ^d according to the rule where ι(·) is a surface measure on Γ.     

Asymptotic behavior of the solutions of an integrodifferential system

Summary We consider the system(L): , t ⩾ p, y(t)=f(t), t⩽0, where y is an n-vector and each A_i, B(t) are n × n matrices. System(L) generates a semigroup by means of T_tf(s)=y (t+s, f), f(s) ∈ BC_l(− ∞, 0]. Under some hypotheses concerning the roots ofdet where is the Laplace transform of B(t), the asymptotic behavior of y(t) is discussed. Two typical results are: Theorem 3.1: suppose ∥B(t)∥ ɛ L₁[0, ∞), thendet forRe λ>0 iff for every ɛ>0 there is an M_ɛ>0 such that ∥T_tf∥_l ⩽ ⩽ M_ɛ exp [ɛt]∥f∥_l for t ⩾ 0. Corollary 3.1.1: suppose exp [at]B(t) ∈ ∈ L₁[0, ∞) for some a>0 anddet forRe λ>−a. Then the solution of(L) is exponentially asymptotically stable. Entrata in Redazione il 21 marzo 1975. The author is grateful to ProfessorC. Corduneanu for suggesting this problem and for many helpful discussions during the preparation of the paper.     

On the Corona problem

We show that if f₁, f₂ are bounded holomorphic functions in the unit ball of ℂⁿ such that , |f₁(z)|² + |f₂(z)2|² ≥ δ² >; 0, then any functionh in the Hardy space ,p < +∞ can be decomposed ash = f₁h₁ + f₂h₂ with . The Corona theorem in would be the same result withp = +∞ and this question is still open forn ≳-2, but the preceding result goes in this direction.     

On one extremal problem for numerical series

Let Γ be the set of all permutations of the natural series and let α = {α _j} _j∈ℕ, ν = {ν_j} _j∈ℕ, and η = {η_j} _j∈ℕ be nonnegative number sequences for which

On matroid intersections

This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr ₁ andr ₂ be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr ₁₂(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r ₁₂(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2 ^E →Z ⁺ which maximizes and , subject toy≧0, ∀j∈E, .     

Some applications of coincidence theory

Summary As a first application we compute the Lefschetz coincidence number of maps between manifolds whose rational singular cohomologyH* has a simple system of generators. For the second let be anH-manifold with multiplicationm. Define for ,m ₂ (x,x)=m(x,x) andm _k (x)=m(x,m _k−1 (x)), for allk>2. All roots of equationm _k (x)=m ₈ (x),k>s such thatm _k (x)=m ₃ (x) butm _i (x)≠m _j (x) for allk>i>j,s≥j≥0 andk−s does not dividei−j, split into a finite number of equivalence classes. We compute precisely the numbers of classes such that their members satisfy the above property. This article was processed by the author using the Springer-Verlag T_EX macro package 1991.     

Equivariant D-modules attached to nilpotent orbits in a semisimple Lie algebra

Let _u be a compact Lie algebra and let _u be its complexification. Let ζ^−1/2 be the inverse on the set of regular elements of _u of a square root of the discriminant of . Generalizing a result of W. Lichtenstein in the case _u = (n, ℂ) or (nℝ), we prove that ∂(q).ζ^1/2 is non zero for all harmonic polynomialsq ∈S( ) \ {0}. This fact is deduced from results about equivariantD-modules supported on the nilpotent cone of .     

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

Comparison of exact constants in Kolmogorov-type inequalities for periodic and nonperiodic functions of many variables

We investigate the correlation between the constants K(ℝⁿ) and , where

Exact constants in inequalities of the jackson type for quadrature formulas

We prove that if is the error of a simple quadrature formula and ω(ε, δ)₁ is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true: whereD _r is the Bernoulli kernel.     

Eine Verallgemeinerung derCW-Komplexe

In the definition ofCW-complexes, the one-point spaceP, respectively the spaceP∪* with basepoint *, play the roll of the only “building-stone”. Let be a family of compact spaces. Then the definition of a generalizedCW-complex over is obtained from the definition of aCW-complex by replacingP by the spaces of and formation of the mapping cone by a slightly modified construction. LetCW * denote the category of all pointed spaces which have the homotopy type of a generalizedCW-complex over . If , thenCW * is the category of all pointedCW-spaces.CW * is closed under the formation of direct sums and of mapping cones, cylinders and tori, and is formally characterized as the smallest such subcategory of Top * containing the spaces W∪*, . Following the methods of E. H. Brown, it is proved, that any half exact homotopy functor onCW * is representable, and any cohomology theory onCW is naturally equivalent to the cohomology theory of an Ω-spectrum; for example, the singular cohomo logy is representable onCW for any family of compact spaces.      

Order comparison of norms of polynomials in regions of the complex plane

For an arbitrary Jordan region G, inequalities are obtained which relate the norm with the norms and of a polynomial P_n(z) of degree n.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1277–1280, September, 1991.     

Inequalities for polynomial splines

We establish exact estimates for the variation on a period of the derivative s ^(r)(t) of a periodic polynomial spline s(t) of degree r and defect 1 with respect to a fixed partition of [0, 2π) under the condition that , where X=C or L ₁     

Asymptotic expansion of the log-likelihood function based on stopping times defined on a Markov process

Consider the parameter space Θ which is an open subset of ℝ ^k ,k≧1, and for each θ∈Θ, let the r.v.′sY _n ,n=0, 1, ... be defined on the probability space (X,A,P _θ) and take values in a Borel setS of a Euclidean space. It is assumed that the process {Y _n },n≧0, is Markovian satisfying certain suitable regularity conditions. For eachn≧1, let υ _n be a stopping time defined on this process and have some desirable properties. For 0 < τ _n → ∞ asn→∞, set h _n →h ∈R ^k , and consider the log-likelihood function of the probability measure with respect to the probability measure . Here is the restriction ofP _θ to the σ-field induced by the r.v.′sY ₀,Y ₁, ..., . The main purpose of this paper is to obtain an asymptotic expansion of in the probability sense. The asymptotic distribution of , as well as that of another r.v. closely related to it, is obtained under both and . This research was supported by the National Science Foundation, Grant MCS77-09574. Research supported by the National Science Foundation, Grant MCS76-11620.     

Singly generatedC *-algebras

We consider aC ^*-algebraA generated byk self-adjoint elements. We prove that, for , the algebraM _n (A) is singly generated, i.e., generated by one non-self-adjoint element. We present an example of algebraA for which the property thatM _n (A) is singly generated implies the relation . Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1136–1141, August, 1999.     

Minimal number of idempotent generators for certain algebras

It is known [KRS] that for each finitely generated Banach algebra there exists a numberN such that for eachn>N the matrix algebras can be generated by three idempotents. In this paper we show that the same statement is true for direct sums and , where is a finitely generated free algebra, i.e. polynomials in several non-commuting variables. These results are new even for algebras because the numberN we obtain here improves known estimates (see for example [R]). We show that the algebra can be generated by two idempotents if and only ifn _j =2 for eachj and is singly generated. Also we give an example of a free singly generated algebra for which can not be generated by two idempotents. But% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOfdaryqr1ngBPrginfgDObYtUvgaiuaacuWFSeIqgaacaaaa!409A!\[{\tilde {\cal B}}\] can be generated by three idempotents for each singly generated free algebra .     

Autostability of hyperarithmetical models

Let be a Δ ₁ ¹ -constructivizable model. If its Scott rank is strictly less than ω ₁ ^CK , then it is proved autostable. But if , then there exists an ordinal α<ω ₁ ^CK for which is not autostable in any degree O^(γ+1) for all γ>α. We also consider some problems concerning Δ ₁ ¹ -autostability of Δ ₁ ¹ -constructivizable Boolean algebras. Supported through the FP “Integration” and the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 198–205, March–April, 2000.