Diameters of some classes of differentiable periodic functions in the space L

In this paper, diameters in the sense of A. N. Kolmogorov are found for the class of 2π-periodic functions W^(r)H_ω in the space L, that is, d_2n-1(W^(r)H_ω,L), where ω(t) is an upper-convex regular modulus of continuity (r, n=1, 2, ...). An estimate from below is found for diameters in the sense of I. M. Gel'fand, that is, d^2n-!(W^(r)H_ω,L).     

Sous-espaces invariants par translations bilatérales de certains espaces de Hilbert de suites quasi-analytiquement pondérées

We use the theory of asymptotically holomorphic functions in the disc to study the lattice of left-invariant subspaces of ℓ_ω², where ω is nonincreasing, in the quasianalytic case log ω(n)/n² = +∞. When (ω(−n))_n≥0 satisfies suitable growth and regularity conditions, we show in particular that all bilaterally invariant subspaces of ℓ_ω² are generated by their intersection with ℓ_ω². When ω(n) = 1 for n ≥ 0 and ω(n) = ¦n¦/e^{1+log ¦n¦} for n < 0 this shows that all nontrivial bi-invariant subspaces of ℓ_ω² are generated by the Fourier transform of a nonconstant singular inner function.     

Moments of Two-Variable Functions and the Uniqueness of Graph Limits

For a symmetric bounded measurable function W on [0, 1]² and a simple graph F, the homomorphism density $t(F,W) = \int _{[0,1]^{V (F)}} \prod_ {i j\in E(F)} W(x_i, x_j)dx .$ can be thought of as a “moment” of W. We prove that every such function is determined by its moments up to a measure preserving transformation of the variables. The main motivation for this result comes from the theory of convergent graph sequences. A sequence (G _n ) of dense graphs is said to be convergent if the probability, t(F, G _n ), that a random map from V(F) into V(G _n ) is a homomorphism converges for every simple graph F. The limiting density can be expressed as t(F, W) for a symmetric bounded measurable function W on [0, 1]². Our results imply in particular that the limit of a convergent graph sequence is unique up to measure preserving transformation.     

A dual form of Ramsey's Theorem

Let k?ω, where ? is the set of all natural numbers. Ramsey's Theorem deals with colorings of the k-element subsets of ω. Our dual form deals with colorings of the k-element partitions of ω. Let (ω)^k (respectively (ω)^ω) be the set of all partitions of ω having exactly k (respectively infinitely many) blocks. Given X? (ω)^ω let (X)^k be the set of all Y? (ω)^k such that Y is coarser than X. Dual Ramsey Theorem. If (ω)^k = C₀ ∪ … ∪ C_t?1 where each C_i is Borel then there exists X? (ω)^ω such that (X)^k ? C_i for some i <l. Dual Galvin-Prikry Theorem. Same as before with k replaced by ω. We also obtain dual forms of theorems of Ellentuck and Mathias. Our results also provide an infinitary generalization of the Graham-Rothschild “parameter set” theorem [Trans. Amer. Math. Soc.159 (1971), 257–292] and a new proof of the Halpern-Läuchli Theorem [Trans. Amer. Math. Soc.124 (1966), 360–367].     

Solving stochastic structural optimization problems by RSM-based stochastic approximation methods — gradient estimation in case of intermediate variables

Reliability-based structural optimization methods use mostly the following basic design criteria: I) Minimum weight (volume or costs) and II) high strength of the structure. Since several parameters of the structure, e.g. material parameters, loads, manufacturing errors, are not given, fixed quantities, but random variables having a certain probability distribution P,stochastic optimization problems result from criteria (I), (II), which can be represented by (1) $$\mathop {\min }\limits_{x \in D} F(x)withF(x): = Ef(\omega ,x).$$ Here,f=f(ω,x) is a function on ? ^r depending on a random element ω, “E” denotes the expectation operator andD is a given closed, convex subset of ? ^r . Stochastic approximation methods are considered for solving (1), where gradient estimators are obtained by means of the response surface methodology (RSM). Moreover, improvements of the RSM-gradient estimator by using “intermediate” or “intervening” variables are examined.     

Additional results on the strong approximation of Fourier series

Первая цель работы — п олучить достаточное условие, выраженное в термина х некоторых свойств сильной аппр оксимации функций, пр и котором она принадлежит клас суW ^r H ^ω без каких-либо ограни чений на модуль непре рывностиω. Вторая теорема также дает достаточное усл овие для того, чтобы функция пр инадлежалаW ^r H ^ω α, гдеH ^ω α обозначает ра сширенный класс Липш ица.     

A definable failure of the singular cardinal hypothesis

We show first that it is consistent that κ is a measurable cardinal where the GCH fails, while there is a lightface definable wellorder of H(κ ⁺). Then with further forcing we show that it is consistent that GCH fails at ? _ω , ? _ω strong limit, while there is a lightface definable wellorder of H(? _ω+1) (“definable failure” of the singular cardinal hypothesis at ? _ω ). The large cardinal hypothesis used is the existence of a κ ⁺⁺-strong cardinal, where κ is κ ⁺⁺-strong if there is an embedding j: V → M with critical point κ such that H(κ ⁺⁺) ? M. By work of M. Gitik and W. J. Mitchell [12], [20], our large cardinal assumption is almost optimal. The techniques of proof include the “tuning-fork” method of [10] and [3], a generalisation to large cardinals of the stationary-coding of [4] and a new “definable-collapse” coding based on mutual stationarity. The fine structure of the canonical inner model L[E] for a κ ⁺⁺-strong cardinal is used throughout.     

On weaker forms of the chain (F) condition and metacompactness-like covering properties in the product spaces

We introduce the concept of a family of sets generating another family. Then we prove that if X is a topological space and X has W = {W(x): x ∈ X} which is finitely generated by a countable family satisfying (F) which consists of families each Noetherian of ω-rank, then X is metaLindelöf as well as a countable product of them. We also prove that if W satisfies ω-rank (F) and, for every x ∈ X, W(x) is of the form W ₀(x) ∪ W ₁(x), where W ₀(x) is Noetherian and W ₁(x) consists of neighbourhoods of x, then X is metacompact.     

Classification theory for non-elementary classes I: The number of uncountable models ofψ ∈L ω_1, ω. Part A

Assuming that 2^Nn < 2^Nn+1 forn < ω, we prove that everyψ ∈L _{ω_1, ω} has many non-isomorphic models of powerN _n for somen>0or has models in all cardinalities. We can conclude that every such Ψ has at least 2 _N 1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countableT (we can reduce general cases to this) we find a cutting line named “excellent”. Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in someN _n (provided {ie212-7}).     

Products without remote points

It is shown that ω × Y^ω does not have remote points if Y is a compact space with cellularity larger than ω₁. It is also shown that it is consistent that ω × Y^ω does not have remote points if Y is compact with uncountable cellularity. As an application we construct a compact space with weight ω₂ · c which can be covered by nowhere dense P-sets and a compact space with weight c for which it is independent that it can be covered by nowhere dense P-sets.     

Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems

In this paper we study linear differential systems (1) $\text{x′ =}\text{A}\text{?}\text{(θ + ωt)x,}\text{where}\text{A}\text{?}\text{(θ)}\text{is an}\text{(n × n)}$ matrix-valued function defined on the k-torus T^k and (θ, t) → θ + ωt is a given irrational twist flow on T^k. First, we show that if A ? C^N(T^k), where N ? {0, 1, 2,…; ∞; ω}, then the spectral subbundles are of class C^N on T^k. Next we assume that à is sufficiently smooth on T^k and ω satisfies a suitable “small divisors” inequality. We show that if (1) satisfies the “full spectrum” assumption, then there is a quasi-periodic linear change of variables x = P(t)y that transforms (1) to a constant coefficient system y′ =By. Finally, we study the case where the matrix $\text{A}\text{?}\text{(θ + ωt)}$ in (1) is the Jacobian matrix of a nonlinear vector field $\text{?(x)}$ evaluated along a quasi-periodic solution x = φ(t) of (2) $\text{x′ = ?(x)}$. We give sufficient conditions in terms of smoothness and small divisors inequalities in order that there is a coordinate system (z, ?) defined in the vicinity of $\text{Ω = H(φ)}$, the hull of φ, so that the linearized system (1) can be represented in the form z′ = Dz, ?′ = ω, where D is a constant matrix. Our results represent substantial improvements over known methods because we do not require that à be “close to” a constant coefficient system.     

Kähler structures on complex torus

Consider the complex torus T _C under the natural action of the compact real torus T. In this paper, we study T-invariant Kähler structures ω on T_C. For each ω, we consider the corresponding line bundleL on T _C. Namely, the Chern class ofL is [ω], and L is equipped with a connection ? whose curvature is ω. We construct a canonical T-invariant L ²-structure on the sections ofL,and let H _ω be the square-integrable holomorphic sections ofL.Then the Hilbert space H _ω is a unitary T-representation, and we study the multiplicity of the (l-dimensional) irreducible unitary T-representations in H_ω. We shall see that the multiplicity is controlled by the image of the moment map corresponding to the T-action preserving ω.     

On approximation of a class by another class and extremal subspaces inL 1

Получена оценка (в опр еделенном смысле неу лучшаемая) наилучшего приближе ния в метрикеL ₁=L₁(0,2π) 2π-перио дических функций кла сса W^rH^ω = {f:f∈C^r,ω(f ^(r),δ) ≦ ω(δ)}, r = 0, 1, ..., (ω(δ) — выпуклый вверх мо дуль непрерывности) ф ункциями класса W ₁ ^r+v N = {?: ?^r+v?1)(t) —локально абсолютн о непрерывна, ∥?(^r+v∥L₁≦N}, v≧2. Доказано, что каждое п одпространство нече тной размерности, реализу ющее поперечник (по Колмогорову) класс а W ₁ ^r+v в L₁, обладает аналог ичным свойством относител ьно класса W^rH^ω при любом выпуклом вверх ω(δ).     

On the exact values of the best approximations of classes of differentiable periodic functions by splines

We obtain the exact values of the best L ₁-approximations of classes W ^r F, r ∈ ?, of periodic functions whose rth derivative belongs to a given rearrangement-invariant set F, as well as of classes W ^r H ^ω of periodic functions whose rth derivative has a given convex (upward) majorant ω(t) of the modulus of continuity, by subspaces of polynomial splines of order m ≥ r + 1 and of deficiency 1 with nodes at the points 2kπ/n and 2kπ/n + h, n ∈ ?, k ∈ ?, h ∈ (0, 2π/n). It is shown that these subspaces are extremal for the Kolmogorov widths of the corresponding functional classes.     

An extremum problem on a class of differentiable functions of several variables

On the multidimensional classW ₀ ^r H _ω ⁽ⁿ⁾ of continuous periodic functionsF with therth derivativeD ^r F from $$H_\omega ^{(n)} = \left\{ {f \in C| |f(x) - f(y)| \leqslant \sum\limits_{i = 1}^n {\omega _i } (|x_i - y_i |)\forall x, y \in \mathbb{R}^n } \right\}$$ (where the ω _i (x _i ) are the convex moduli of continuity) and zero mean with respect to each variable, we obtain the exact value of $$M_r (\omega ) = \mathop {\sup }\limits_{F \in W_0^r H_\omega ^{(n)} } \left\| F \right\|c$$ .     

A partition theorem for the complete graph on ω

This paper is a proof of the following theorem:

ω^ω→(ω^ω, 3)²

Variational measures and the Kurzweil-Henstock integral

For a given continuous function F on a compact interval E in the set ? of reals the problem is how to describe the “total change” of F on a set M ? E. Full variational measures W _F (M) and V _F (M) (see Section 2) in the sense presented by B. S. Thomson are introduced in this work to this aim. They are generated by two slightly different interval functions, namely the oscillation of F over an interval and the value of the additive interval function generated by F, respectively. They coincide with the concept of classical total variation if M is an interval and they are zero if on the set M the function F is of negligible variation. The Kurzweil-Henstock integration is shortly described and some of its properties are studied using the variational measure W _F (M) for the indefinite integral F of an integrable function f.     

F2Δr−A2Πi band system of CN

The spectrum of CN has been studied in condensed electrical discharge through flowing CH₃CN vapour. Ten new bands of F²?,?A²Π_i system in the region 2100–2700 Å are obtained. The vibrational constantsω ₀′ = 1229.7 cm.^?1 and ω₀′χ₀′= 14.0 cm.^?1 are obtained for the first time for the²Δ state of CN.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.