BMO-scale of distribution on ℝ n

Let S′ be the class of tempered distributions. For ƒ ∈ S′ we denote by J ^−α ƒ the Bessel potential of ƒ of order α. We prove that if J ^−α ƒ ∈ BMO, then for any λ ∈ (0, 1), J ^−α (f)_λ ∈ BMO, where (f)_λ = λ⁻ⁿ f(φ(λ⁻¹)), φ ∈ S. Also, we give necessary and sufficient conditions in order that the Bessel potential of a tempered distribution of order α > 0 belongs to the VMO space.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

The Riesz Basis Property of an Indefinite Sturm-Liouville Problem with a Non-Odd Weight Function

For the Sturm-Liouville eigenvalue problem − f′′ = λrf on [−1, 1] with Dirichlet boundary conditions and with an indefinite weight function r changing its sign at 0 we discuss the question whether the eigenfunctions form a Riesz basis of the Hilbert space L ² _|r|[− 1, 1]. So far a number of sufficient conditions on r for the Riesz basis property are known. However, a sufficient and necessary condition is only known in the special case of an odd weight function r. We shall here give a generalization of this sufficient and necessary condition for certain generally non-odd weight functions satisfying an additional assumption.      

Absolute upper semicontinuity

It is proved that the following conditions are equivalent: the function ϕ [a, b]→R is absolutely upper semicontinuous (see [1]); ϕ is a function of bounded variation with decreasing singular part; there exists a summable function g: [a, b] → R such that for anyt′∈[a, b] andt″∈[t′, b], we have ϕ(t″)−ϕ(t′)⩽∫ _t′ ^t″ g (s) ds. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 395–399, September, 1977.     

Nonuniqueness ing-functions

We give an example of two distinct stationary processes {X _n} and {X′ _n} on {0, 1} for whichP[X₀=1|X₋₁=a₋₁,X₋₂=a₋₂, …]=P[X′₀=1|X′₋₁=a₋₁,X′₋₂=a₋₂, …] for all {a _i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a _i}. Supported in part by NSF Grant DMS 89-01545. Supported in part by the US Army Research Office.     

On Fractional Part Sums: A Mean-Square Asymptotics over Short Intervals

 This article is concerned with sums 𝒮(t) = ∑ _n  ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ?, f is a C ⁴-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫ _T−Λ ^T+Λ (𝒮(t))²dt for any Λ = Λ(T) growing faster than log T. Received April 30, 2001; in revised form February 15, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Harmonic Functions,Entropy, and a Characterization of the Hyperbolic Space

Let (M ⁿ ,g) be a compact Riemannian manifold with Ric ≥−(n−1). It is well known that the bottom of spectrum λ ₀ of its universal covering satisfies λ ₀≤(n−1)²/4. We prove that equality holds iff M is hyperbolic. This follows from a sharp estimate for the Kaimanovich entropy. The author was partially supported by NSF Grant 0505645.     

Polynomials of best approximation inC[−1,1]

Equivalences between the condition |P _n ^(k) (x)|≦K(n ⁻¹√1−x ²+1/n ²) ^k n ^-a, whereP _n(x) is the bestn-th degree polynomial approximation tof(x), and the Peetre interpolation space betweenC[−1,1] and the space (1−x ²) ^k f ^(2k)(x)∈C[−1,1] is established. A similar result is shown forE _n(f)= ‖f−P _n‖ _C[−1,1]. Rates other thann ^-a are also discussed. Supported by NSERC grant A4816 of Canada.     

Remarks on g-functions on H^n

In this paper, the authors study some properties of Littlewood-Paley g-functions gψ（f）,Lusin area functions Sψ,α（f） and Littlewood-Paley gψ^＊,λ（f） functions defined on H^n, where α,λ 〉 0 and ψ, f are suitable functions. They are the generalization of the corresponding operators on R^n.     

Determination of the poles of the topological zeta function for curves

Tof ∈ℂ[x ₁…,x _n ] one associates thetopological zeta function which is an invariant of (the germ of)f at 0, defined in terms of an embedded resolution of (the germ of)f ⁻¹{0} inf ⁻¹{0}. By definition the topological zeta function is a rational function in one variable, and it is related to Igusa’s local zeta function. A major problem is the study of its poles. In this paper we exactly determine all poles of the topological zeta function forn=2 and anyf ∈ℂ[x ₁,x ₂]. In particular there exists at most one pole of order two, and in this case it is the pole closest to the origin. Our proofs rely on a new geometrical result which makes the embedded resolution graph of the germ off into an ‘ordered tree’ with respect to the so-callednumerical data of the resolution. The author is a Postdoctoral Fellow of the Belgian National Fund for Scientific Research N.F.W.O.     

On boundary behavior of generalized quasi-isometries

We establish a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in [`(ⁿ)] = ⁿ è{ ￥} ,n 3 2\overline {{^n}} = {^{^n}} \cup \{ \infty \} ,n \ge 2, under integral constraints of the type ∫ Φ(Q ⁿ⁻¹(x))dm(x) < ∞ with a convex non-decreasing function Φ: [0,∞]→[0,∞]. Integral conditions on Φ are found that are necessary and sufficient for a continuous extension of f to the boundary. Our results are applied to finitely bi-Lipschitz mappings, which are a far-reaching generalization of isometries as well as quasi-isometries in ℝ ⁿ . In particular, a generalization and strengthening of the well-known theorem of Gehring-Martio on homeomorphic extension to boundaries of quasi-conformal mappings between QED (quasi-extremal distance) domains is obtained.     

On the order of growth of solutions of algebraic differential equations

Assume thatf is an integer transcendental solution of the differential equationP _n (z, f, f′)=P _n−1(z, f, f′, ... f ^(p)), whereP _n andP _n−1 are polynomials in all variables, the degree ofP _n with respect tof andf′ is equal ton, and the degree ofP _n−1 with respect tof, f′, ... f ^(p) is at mostn−1. We prove that the order ρ of growth off satisfies the relation 1/2≤ρ<∞. We also prove that if ρ=1/2, then, for a certain real ν, in the domain {z: ν<argz<ν+2π}/E _*, whereE _* is a certain set of disks with finite sum of radii, the estimate lnf(z)=z ^1/2 (β+o(1)), β∈C, holds forz=re ^iϕ,r≥r(ϕ)≥0. Furthermore, on the ray {z: argz=ν}, the following relation is true: ln‖f(re ^iν)‖=o(r ^1/2),r→+∞,r>0, , where Δ is a certain set on the semiaxisr>0 with mes Δ<∞. “L'vivs'ka Politekhnika” University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 69–77, January, 1999.     

On the error terms for representation numbers of quadratic forms

Let f be a primitive positive integral binary quadratic form of discriminant −D, and r _f (n) the number of representations of n by f up to automorphisms of f. We first improve the error term E(x) of $ \sum\limits_{n \leqq x} {r_f (n)^\beta } $ \sum\limits_{n \leqq x} {r_f (n)^\beta } for any positive integer β. Next, we give an estimate of ∫₁ ^T |E(x)|² x ^−3/2 dx when β = 1.     

Existence principles for higher-order nonlocal boundary-value problems and their applications to singular Sturm-Liouville problems

We present existence principles for the nonlocal boundary-value problem (φ(u^(p−1)))′=g(t,u,...,u^(p−1), α_k(u)=0, 1≤k≤p−1, where p ≥ 2, π: ℝ → ℝ is an increasing and odd homeomorphism, g is a Carathéodory function that is either regular or has singularities in its space variables, and α _k: C ^p−1[0, T] → ℝ is a continuous functional. An application of the existence principles to singular Sturm-Liouville problems (−1)ⁿ(φ(u⁽²ⁿ⁻⁾))′=f(t,u,...,u⁽²ⁿ⁻¹⁾), u^(2k)(0)=0, α_ku^(2k)(T)+b_ku^(2k=1)(T)=0, 0≤k≤n−1, is given. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 240–259, February, 2008.     

Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous

Summary Asymptotic properties of the mean integrated squared error (MISE) of kernel estimators of a density function, based on a sampleX ₁, …,X _n, were obtained by Rosenblatt [4] and Epanechnikov [1] for the case when the densityf and its derivativef′ are continuous. They found, under certain additional regularity conditions, that the optimal choiceh _n0 for the scale factorh _n=Kn^−α is given byh _n0=K₀n^−1/5 withK ₀ depending onf and the kernel; they also showed that MISE(h _n0)=O(n^−4/5) and Epanechnikov [1] found the optimal kernel. In this paper we investigate the robustness of these results to departures from the assumptions concerning the smoothness of the density function. In particular it is shown, under certain regularity conditions, that whenf is continuous but its derivativef′ is not, the optimal value of α in the scale factor becomes 1/4 and MISE(h _n0)=O(n^−3/4); for the case whenf is not continuous the optimal value of α becomes 1/2 and MISE(h _n0)=O(n^−1/2). For this last case the optimal kernel is shown to be the double exponential density. Supported by the Natural Sciences and Engineering Research Council of Canada under Grant Nr. A 3114 and by the Gouvernement du Québec, Programme de formation de chercheurs et d'action concertée.     

The perfectB-splines and a time-optimal control problem

Letx _v =cos (πν/n) (v=0, 1, …,n). It is shown that theB-splineM(x)=M(x; x ₀ ,x ₁ ,…, x _n ) is such thatM _n ⁽ⁿ⁾ (x) has a constant absolute value (=2 ⁿ⁻² (n−1)!) in [−1, 1]. Its integralf ₀(x)=∫ ₋₁ ^x M(t)dt is shown to have an optimal property that allows to solveexplicitly a certain time-optimal control problem.     

环域上一类拟线性椭圆型方程的正径向解的存在性

The existence of positive radial solutions of the equation -din( |Du|p-2Du)=f(u) is studied in annular domains in Rn,n≥2. It is proved that if f(0)≥0, f is somewherenegative in (0,∞), limu→0^ f‘ (u)=0 and limu→∞ (f(u)/u^p-1)=∞, then there is alarge positive radial solution on all annuli. If f(0)≤0 and satisfies certain conditions, then the equation has no radial solution if the annuli are too wide.     

The total reducibility order of a polynomial in two variables

LetK be an algebraically closed field of characteristic zero. ForA ∈K[x, y] let σ(A) = {λ ∈K:A − λ is reducible}. For λ ∈ σ(A) letA − λ = ∏ _i=1 ^n(λ) A _iλ ^k _μ whereA _iλ are distinct primes. Let ϱ_λ(A) =n(λ) − 1 and let ρ(A) = Σ_λɛσ(A)ϱ_λ(A). The main result is the following: Theorem.If A ∈ K[x, y] is not a composite polynomial, then ρ(A) < degA.     

A counterexample in copositive approximation

The present paper gives a converse result by showing that there exists a functionf ∈C _[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE _n ⁽⁰⁾ (f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E _n(f) is the ordinary best polynomial approximation off of degreen.     

Vague convergence of sums of independent random variables

A sequence (μ _n) of probability measures on the real line is said to converge vaguely to a measureμ if∫ fdμ _n →∫ fdμ for every continuous functionf withcompact support. In this paper one investigates problems analogous to the classical central limit problem under vague convergence. Let ‖μ‖ denote the total mass ofμ andδ ₀ denote the probability measure concentrated in the origin. For the theory of infinitesimal triangular arrays it is true in the present context, as it is in the classical one, that all obtainable limit laws are limits of sequences of infinitely divisible probability laws. However, unlike the classical situation, the class of infinitely divisible laws is not closed under vague convergence. It is shown that for every probability measureμ there is a closed interval [0,λ], [0,e ⁻¹] ⊂ [0,λ] ⊂ [0, 1], such thatβμ is attainable as a limit of infinitely divisible probability laws iffβ ε [0,λ]. In the independent identically distributed case, it is shown that if (x ₁ + ... +x _n)/a _n, a_n → ∞, converges vaguely toμ with 0<‖μ‖<1, thenμ=‖μ‖δ ₀. If furthermore the ratiosa _n+1/a _n are bounded above and below by positive numbers, thenL(x)=P[|X ₁|>x] is a slowly varying function ofx. Conversely, ifL(x) is slowly varying, then for everyβ ε (0, 1) one can choosea _n → ∞ so that the limit measure=βδ ₀. To the memory of Shlomo Horowitz This research was partially supported by the National Science Foundation.