Kinetics of the photoaquation of Hexacyanoferrate(II) ion

Kinetics of the photoaquation of hexacyanoferrate(II) ion in aqueous solution were studied potentiometrically and spectrophotometrically. Supposing the simplest mechanism (see Fig. 3. in text), the photoaquation in alkaline medium can be well described. The value of the constants at pH = ll.0 are: ø = 0.8-1.0, k₆ = (3.0 ± 0.5) × 10^?8 s^?1 and k_?6 = 1.5 ± 0.2 mol^?1 dm₃ s^?1. To describe the photoaquation in neutral medium t was extended (k′ = 3.33 x 10² mol^?1 dm³s^?1). The quantum yield in acidic medium can be calculated by combination of ø values of different protonated complexes. The reversibility of photoaquation in alkaline medium is also explained by the scheme.     

Notes on Fourier series in homogeneous Banach spaces

В первой части работы устанавливается при нцип двойственности для с ходимости операторо в свертки в некоторых ф ункциональных прост ранствах. Вторая часть содержи т несколько характеристик после довательностей коэф фициентов Фурье функций из этих пространств.     

On the strong approximation by the (C, α)-means of Fourier series. I

Статья продолжает ис следования Г. Суноути, Л. Лейндлера и В. Ленског о; она посвящена изучению средних вида . Здесь α> — 1/2 и рα> — 1. Для а>0 полностью иссл едован случай неотри цательной, регулярной матрицы {t _nk } и классаH ^ω (ω — модуль не прерывности). При 0>а> ?1/2 получены результаты для весьма широкого клас са матриц. Доказано, чт о в каждом случае порядки указа нных величин для функций изH ^ω совпа дают с . Кроме того, устанавли вается, что не существ ует естественных обобще ний этих утверждений на классыW ^r H ^ω (r≧1). Из результатов стать и вытекают решения не которых ранее поставленных з адач.     

Unique expansions of real numbers

It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci∈[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.In this paper we consider for each fixed q>1 the set Uq of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which Uq is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci∈[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.     

Discrete Ingham type inequalities and simultaneous observability of strings or beams

Harmonic analysis has been an efficient tool in control theory for a long time, see, e.g., Russell [D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. Recent progress and open questions, SIAM Rev. 20 (1978) 639-739] and its numerous references. Here we establish discrete Ingham type and Haraux type inequalities for exponential sums satisfying a weakened gap condition. They enable us to obtain discrete simultaneous observability theorems for systems of vibrating strings or beams.     

Some problems of A. Kroó on multiple Chebyshev polynomials

Three problems of A. Kroó on multiple Chebyshev polynomials are solved using the Borsuk–Ulam antipodal theorem.     

Uniform Spacing of Zeros of Orthogonal Polynomials

It is shown that for doubling weights, the zeros of the associated orthogonal polynomials are uniformly spaced in the sense that if cos θ _m,k , θ _m,k ∈[0,π] are the zeros of the m-th orthogonal polynomial associated with w, then θ _m,k −θ _m,k+1∼1/m. It is also shown that for doubling weights, neighboring Cotes numbers are of the same order. Finally, it is proved that these two properties are actually equivalent to the doubling property of the weight function.     

Synthesis and x-ray crystal structure of a dinuclear copper(I) thiobenzoate triphenylphosphine complex

Summary In 9,10-phenanthrenesemiquinonatobis(triphenylphosphine)copper(I) (1) the semiquione ligand is oxidized by dibenzoyl disulphide to 9,10-phenanthrenequinone and the complex [(PPh₃)Cu(-PhCOS)₂ Cu(PPh₃)₂] (3) is formed. Crystals of the complex are triclinic, space group , with cell parametersa=10.712(1),b=12.887(1).c=22.064 Å, =78.29(1), =81.88(1), =80.83(1), and Z=2. The copper atoms are bridged by the sulphur atoms of the two thiobenzoate groups to form binuclear molecules. The Cu₂S₂ rhombus is strictly co-planar (Cu–S distances from 2.309 to 2.424 Å). The trigonal planar co-ordination around the one and the tetrahedral coordination around the other copper atom are completed by the phosphorus atoms of the triphenylphosphine molecules (Cu–P distances from 2.238 to 2.287 Å). The copper-copper distance (Cu–Cu=2.620 Å) is 0.6 Å longer than that in metallic copper.     

Asymptotics for the ratio of leading coefficients of orthonormal polynomials on the unit circle

Consider a system {φ _n } of polynomials orthonormal on the unit circle with respect to a measuredμ, withμ′>0 almost everywhere. Denoting byk _n the leading coefficient ofφ _n , a simple new proof is given for E. A. Rakhmanov's important result that lim _n→∞,k _n /k _n+1=1; this result plays a crucial role in extending Szegö's theory about polynomials orthogonal with respect to measuresdμ with logμ′∈L ¹ to a wider class of orthogonal polynomials.