Об одной задаче лузина

Резюме В работе предлагается решение одной задачи, принадлежащей н. н. Лузину, о выделении максимальных коэффициентов в ряде Фурье. Ключевое соображение состоит в использовании итерированных сверток, в который максимальные коэффициенты Фурье играют ведущую роль. Результат доведен до формул, содержащих предельный переход по номеру свертки. Предельные значения оказываются априори целыми числами, так что приближенные вычисления с последующим обычным округлением дают абсолютно точный результат.Поступило 12 августа 1997 г, переработанные варианты 12 января 1999, 15 сентября 2000 и 21 мая 2001 гг.     

Some Infinite Families of Locally Partial Geometries Generalizing the Classical Laguerre Planes

Starting with a subgeometry Ω embedded in a β-dimensional projective space PG(β, q), β 1, we construct inductively a series of rank n residually connected geometries Γ^{(n, β, Ω)}, n β, by putting Γ^{(β, β, Ω)} = Ω and extending Γ^{(n - 1, β, Ω)} with a partial geometry.     

Limiting laws for long Brownian bridges perturbed by their one-sided maximum, III

Summary Results of penalization of a one-dimensional Brownian motion ]]>]]>]]>]]>]]>]]>]]>(X_t) $, by its one-sided maximum $\big(S_t=\sup_{0 \leq u \leq t}X_u\big)$, which were recently obtained by the authors are improved with the consideration - in the present paper - of the asymptotic behaviour of the likewise penalized Brownian bridges of length $t$, as $t\rightarrow \infty$, or penalizations by functions of $(S_t,X_t)$, and also the study of the speed of convergence, as $t\rightarrow \infty$, of the penalized distributions at time $t$.     

Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

Non-parametric efficiency, progress and regress measures for panel data: Methodological aspects

This purely methodological paper deals with the rôle of time in non-parametric efficiency analysis. Using both FDH and DEA technologies, it first shows how each observation in a panel can be characterized in efficiency terms vis-à-vis three different kinds of frontiers: (i) ‘contemporaneous’, (ii) ‘sequential’, and (iii) ‘intertemporal’. These are then compared with window analysis. Next, frontier shifts ‘outward’ and ‘inward’, interpreted as progress or regress are considered for the two kinds of technologies, and computational methods are described in detail for evaluating such shifts in either case. These are also contrasted with what is measured by the ‘Malmquist’ productivity index. Finally, an alternative way of identifying progress and regress, independent of the frontier notion and referring instead to some ‘benchmark’ notion, is extended here to panel data.     

Cesàro means of double Walsh-Fourier series

We investigate approximation properties of Cesàro (C; −α, −β)-means of double Walsh-Fourier series with α, β ∈ (0, 1). As an application, we obtain a sufficient condition for the convergence of the means σ _n,m ^/−α,−β (f; x, y) to f(x,y) in the L ^p -metric, p ∈ [1, ∞]. We also show that this sufficient condition cannot be improved.     

The heat flow in nonlinear Hodge theory

We study the nonlinear Hodge system dω=0 and δ(ρ(|ω|²)ω)=0 for an exterior form ω on a compact oriented Riemannian manifold M, where ρ(Q) is a given positive function. The solutions are called ρ-harmonic forms. They are the stationary points on cohomology classes of the functional with e′(Q)=ρ(Q)/2. The ρ-codifferential of a form ω is defined as δ_ρω=ρ⁻¹δ(ρω) with ρ=ρ(|ω|²).We evolve a given closed form ω₀ by the nonlinear heat flow system for a time-dependent exterior form ω(x,t) on M. This system is the differential of the normalized gradient flow for E(ω) with ω=ω₀+du. Under a technical assumption on the function 2ρ′(Q)Q/ρ(Q), we show that the nonlinear heat flow system , with initial condition ω(·,0)=ω₀, has a unique solution for all times, which converges to a ρ-harmonic form in the cohomology class of ω₀. This yields a nonlinear Hodge theorem that every cohomology class of M has a unique ρ-harmonic representative.     

Borel ideals vs. Borel sets of countable relations and trees

For every μ < ω₁, let I_μ be the ideal of all sets S ω^μ whose order type is <ω^μ. If μ = 1, then I₁ is simply the ideal of all finite subsets of ω, which is known to be Σ⁰₂-complete. We show that for every μ < ω₁, I_μ is Σ⁰_2μ-complete. As corollaries to this theorem, we prove that the set WO_ωμ of well orderings Rω × ω of order type <ω^μ is Σ⁰_2μ-complete, the set LP_μ of linear orderings R ω × ω that have a μ-limit point is Σ⁰_2μ+1-complete. Similarly, we determine the exact complexity of the set LT_μ of trees T ^<ωω of Luzin height <μ, the set WR_μ of well-founded partial orderings of height <μ, the set LR_μ of partial orderings of Luzin height <μ, the set WF_μ of well-founded trees T ^<ωω of height <μ(the latter is an old theorem of Luzin). The proofs use the notions of Wadge reducibility and Wadge games. We also present a short proof to a theorem of Luzin and Garland about the relation between the height of ‘the shortest tree’ representing a Borel set and the complexity of the set.     

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤^σ and 𝔥 =Z _𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥^σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)^σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) ^{W(𝔤, 𝔥)} → S(𝔱*) ^{W(𝔨, 𝔱)}, where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) ^{W(𝔤, 𝔱)}. Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤^σ is noncohomologous to zero in 𝔤.