О полноте систем функ ций и систем ортогона льных многочленов

The paper is devoted to the study of completeness problem of systems { ⁿ (x)} _n=0 inL ^p (a, b), where –a<b+,(x) is a weight function subject to mild assumptions, and(x) is a continuous function on (a,b), either bounded or unbounded in the neighbourhood of the end-points of (a,b). It turns out that this problem is connected with that of quasianalyticity of certain additive set of functions at a given point. As the most important application of the general results, the completeness problem is treated for systems of orthogonal polynomials.     

О разложениях в ряд ы о бобщенно-абсолютно-м онотонных функций

In an earlier paper [1] the notion of the so-called 〈?, GL_J>-absolutely monotonie functions was introduced, where ?≧1, {λ_{k k=0} ^∞ is an arbitrary non-increasing sequence of positive numbers. It was found that the condition \(\sum\limits_{\lambda _{k > 0} } {\lambda _k^{ - 1} } = + \infty \) is necessary in order to have the series expansion for any function f(x)∈〈?, λ_j). HereL ^k/? f/(x) are special integro-differential operators of fractional order, is a system of functions associated with the Mittag-Leffler type functions \(E_\varrho (z;\mu ) = \sum\limits_{k = 0}^\infty {z^n /\Gamma (\mu + \kappa /\varrho )} \) and with the sequence {λ_k}. In the present paper it is proved (in particular, see Theorem 3.2) that the expansion (*) is valid almost everywhere on (0,l) if ∑ λ _k ^?1 =+∞. This result contains, as a special case (when ?=1 and λ_k=0,k≧0) the known theorem of S. N. Bernstein on absolutely monotonic functions.     

Thermal decomposition of 1-alkyl-3-methylpyridinium based ionic liquids

Products of thermodegradation of 1-alkyl-3-methylpyridinium halides are investigated by the method of gas chromatomass spectrometry. The main products of the reaction are 3-methylpyridine and the corresponding alkyl halide. It was supposed that thermodecomposition of pyridinium ionic liquids proceeds similar to that of imidazolium ionic liquids by dealkylation with elimination of the substituent at the nitrogen atom of the pyridinium ring via S_N2 mechanism. The presence of 5 wt % of cellulose decreases the onset temperature of thermodecomposition by 15–20°С.     

Об аппроксимации свермками и баэисах иэ сдвигов функции

For 2π-periodic functions f ∈ L ^p ( $ \mathbb{T} $ ), 1 ≤ p < ∞, σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), we consider the convolutions $$ (f*d\sigma )_T (x) = \int_0^{2\pi } {f(x - t)d\sigma (t), } (f*g)_T (x) = \int_0^{2\pi } {f(x - t)g(t)dt.} $$ For fixed functions σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), necessary and sufficient conditions are obtained for the density of the ranges of these operators in L ^p . Similar result is proved for the dyadic convolution $$ (f*g)_2 (x) = \int_0^1 {f(x \oplus t)g(t)dt,} $$ where ⊕ is the operation of dyadic addition on [0, 1). Moreover, it is proved that in the spaces L ^p ( $ \mathbb{T} $ ), 1 ≤ p ∞, and C( $ \mathbb{T} $ ) there exist no bases of shifts of a function. Similar results are obtained for the spaces L ^p [0, 1]*, 1 ≤ p < ∞, and C[0, 1]* relative to dyadic shifts, where [0, 1]* is the modified segment [0, 1]. It is also proved that in the space L(?₊) there exists no basis of dyadic shifts of a function.     

Reactions of Dichloroethenes with Sulfur in the System Hydrazine Hydrate–KОН

Vinylidene chloride and 1,2-dichloroethene react with sulfur in the system hydrazine hydrate–KОН with the formation of polyvinylenesulfide oligomer of molecular mass 1750–6120 Da containing also vinylenehydrazine and vinylenechloride links in the Z-configuration, and 1,4-dithiine in the yield up to 46%. In aqueous-hydrazine layer polysulfide anions S _n ^2– (n = 1–4), mainly S₂^2– were found. The mechanism of formation of oligomers and 1,4-dithiine is suggested, which includes in the first stage dehydrochlorination of dichloroethenes and generation of chloroacetylene.     

Synthesis and Structure of Trichlorogermane Aminate

Trichlorogermane triethylaminate [HGeCl₃·NEt₃] has been synthesized and its structure investigated using the methods of NMR and IR spectroscopy and quantum chemistry. Possible structures of complexes of triethylamine with trichlorogermane are considered. The presence of two minima on the potential energy surface is shown; the global minimum corresponds to the structure with proton transfer to the nitrogen atom, [Et₃NH]⁺[GeCl₃]^–, while the local minimum lying by 25.8 kcal/mol above is characterized by the presence of very weak van-der-Waals interaction N···Ge. The comparison of the experimental and calculated chemical shifts of the proton is indicative of the formation of complex with proton transfer to the nitrogen atom in the liquid phase.     

Кусочно-монотонная и рациональная аппрок симации и равномерная сходим ость рядов Фурье

Let $$R_n [f] = \inf \left\{ {\mathop {\max }\limits_{ - \pi \leqq x \leqq \pi } \left| {f(x) - \frac{{P(x)}}{{Q(x)}}} \right|} \right\}$$ , whereP andQ denote polynomials (algebraic or trigonometric) of degree ≦n. Theorem 2a. If for a continuous 2π-periodic function f the condition $$\sum\limits_{n = 1}^\infty {\frac{1}{n}R_n [f]< \infty } $$ holds, then the Fourier series of f converges to f(x) uniformly. Theorem 2b. Let {β _n } be a non-increasing sequence of positive numbers such that $$\sum\limits_{n = 1}^\infty {\frac{1}{n}\beta _n = \infty } $$ Then there exists a continuous 2π-periodic function f₀ for which R_n[f₀]≦β_n for all n=1 and yet the Fourier series of f₀ diverges at x=0.R _n [f]may be replaced in these theorems byM _n [f], whereM _n [f] is the minimal uniform deviation off(x) from piecewise monotonie functionsМ _n (х) of order ≦n.     

Ambidentate and redox-properties of 4,7-phenanthroline-5,6-dione in cobalt complexes: a quantum chemical study

Mono- and dinuclear adducts of cobalt diketonates with tetradentate 4,7-phenanthroline- 5,6-dione were modeled within the framework of DFT UB3LYP*/6-311++G(d,p) approximation. A competitive coordination of the metal ion to different donor centers of the redox-active ligand was studied. Variation of substituents in the diketone moieties allowed one to reveal compounds than can undergo thermally initiated one- and two-step valence tautomeric rearrangements. The calculated energy and magnetic characteristics of the dinuclear complexes give reasons to consider them as potential basis of molecular electronics and spintronics devices.