Sobolev Spaces with only Trivial Isometries

We will give some conditions for Sobolev spaces on bounded Lipschitz domains to admit only trivial isometries.     

The umbral calculus and orthogonal polynomials

We determine all orthogonal polynomials having Boas-Buck generating functions g(t)(xf(t)), where% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqqHOo% qwcaGGOaGaamiDaiaacMcacqGH9aqpruqqYLwySbacfaGaa8hiamaa% BeaaleaacaaIWaaabeaakiaadAeacaqGGaWaaSbaaSqaaiaabgdaae% qaaOGaaeikaiaadggacaGGSaGaa8hiaiaadshacaqGPaGaaeilaiaa% bccacaqGGaGaaeiiaiaadggacqGHGjsUcaaIWaGaaiilaiaa-bcacq% GHsislcaaIXaGaaiilaiaa-bcacqGHsislcaaIYaGaaiilaiablAci% ljaacUdaaeaacqqHOoqwcaGGOaGaamiDaiaacMcacqGH9aqpcaWFGa% WaaSraaSqaaiaaicdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOm% aaqabaGccaGGOaWaaSqaaSqaaiaaigdaaeaacaaIZaaaaOGaaiilai% aa-bcadaWcbaWcbaGaaGOmaaqaaiaaiodaaaGccaGGSaGaa8hiaiaa% dshacaGGPaGaa8hiamaaBeaaleaacaaIWaaabeaakiaadAeacaqGGa% WaaSbaaSqaaiaabkdaaeqaaOGaaeikamaaleaaleaacaaIYaaabaGa% aG4maaaakiaacYcacaWFGaWaaSqaaSqaaiaaisdaaeaacaaIZaaaaO% Gaaiilaiaa-bcacaWG0bGaaiykaiaacYcacaWFGaWaaSraaSqaaiaa% icdaaeqaaOGaamOraiaabccadaWgaaWcbaGaaeOmaaqabaGccaGGOa% WaaSqaaSqaaiaaisdaaeaacaaIZaaaaOGaaiilaiaa-bcadaWcbaWc% baGaaGynaaqaaiaaiodaaaGccaGGSaGaa8hiaiaadshacaGGPaGaai% 4oaaqaaiabfI6azjaacIcacaWG0bGaaiykaiabg2da9iaa-bcadaWg% baWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaacaqGZaaabe% aakiaacIcadaWcbaWcbaGaaGymaaqaaiaaisdaaaGccaGGSaGaa8hi% amaaleaaleaacaaIYaaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaS% qaaiaaiodaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGaaiykaiaa% -bcadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiamaaBaaaleaaca% qGZaaabeaakiaabIcadaWcbaWcbaGaaGOmaaqaaiaaisdaaaGccaGG% SaGaa8hiamaaleaaleaacaaIZaaabaGaaGinaaaakiaacYcacaWFGa% WaaSqaaSqaaiaaiwdaaeaacaaI0aaaaOGaaiilaiaa-bcacaWG0bGa% aiykaiaacYcaaeaadaWgbaWcbaGaaGimaaqabaGccaWGgbGaaeiiam% aaBaaaleaacaqGZaaabeaakiaacIcadaWcbaWcbaGaaG4maaqaaiaa% isdaaaGccaGGSaGaa8hiamaaleaaleaacaaI1aaabaGaaGinaaaaki% aacYcacaWFGaWaaSqaaSqaaiaaiAdaaeaacaaI0aaaaOGaaiilaiaa% -bcacaWG0bGaaiykaiaacYcacaGGUaGaa8hiamaaBeaaleaacaaIWa% aabeaakiaadAeacaqGGaWaaSbaaSqaaiaabodaaeqaaOGaaeikamaa% leaaleaacaaI1aaabaGaaGinaaaakiaacYcacaWFGaWaaSqaaSqaai% aaiAdaaeaacaaI0aaaaOGaaiilaiaa-bcadaWcbaWcbaGaaG4naaqa% aiaaisdaaaGccaGGSaGaa8hiaiaadshacaGGPaGaaiOlaaaaaa!C1F3!\[\begin{gathered}\Psi (t) = {}_0F{\text{ }}_{\text{1}} {\text{(}}a, t{\text{), }}a \ne 0, - 1, - 2, \ldots ; \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{2}} (\tfrac{1}{3}, \tfrac{2}{3}, t) {}_0F{\text{ }}_{\text{2}} {\text{(}}\tfrac{2}{3}, \tfrac{4}{3}, t), {}_0F{\text{ }}_{\text{2}} (\tfrac{4}{3}, \tfrac{5}{3}, t); \hfill \\\Psi (t) = {}_0F{\text{ }}_{\text{3}} (\tfrac{1}{4}, \tfrac{2}{4}, \tfrac{3}{4}, t) {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{2}{4}, \tfrac{3}{4}, \tfrac{5}{4}, t), \hfill \\{}_0F{\text{ }}_{\text{3}} (\tfrac{3}{4}, \tfrac{5}{4}, \tfrac{6}{4}, t),. {}_0F{\text{ }}_{\text{3}} {\text{(}}\tfrac{5}{4}, \tfrac{6}{4}, \tfrac{7}{4}, t). \hfill \\\end{gathered}\]We also determine all Sheffer polynomials which are orthogonal on the unit circle. The formula for the product of polynomials of the Boas-Buck type is obtained.     

Photochemical reduction of organic electron acceptors in aqueous solutions

Conclusions The formation of OH radicals in the photoreduction of aqueous solutions of anthraquinonesulfonic acid as the result of electron transfer from water (OH^–) to a triplet molecule has been demonstrated.Translated from Izvestiya Akademii Nauk, Seriya Khimicheskaya, No. 11, pp. 2615–2617, November, 1969.     

The mechanism of oxidation of indoxyl and its analogs to indigo and indigoides

Conclusions Intermediate formation of 3-oxynaphthenyl-2 radical during oxidation of 3-oxythionaphthene to thioindigo was demonstrated by the inhibitor method. The 1:1 recombination product of this radical and diphenylpicrylhydrazyl was isolated. It is assumed that the postulated oxidation mechanism is general for compounds possessing a methylene group bonded to an electron-donating X group and an electron-accepting C=O group.Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 5, pp. 1168–1170, May, 1970.     

Element neighbourhoods in twofold triple systems

The method of differences is used to establish that every 2-regular multigraph onv– 10,2 (mod 3) points occurs as the neighbourhood graph of an element in a twofold triple system of orderv, with two exceptions: C₂C₃and C₃C₃.Dedicated to Professor Hanfried Lenz on the occasion of his seventieth birthday     

Unramified forcing preserving the law of double negation

Shoenfield's unramified version of Cohen's forcing is defined in two stages: one which does not preserve double negation and the other which modifies the former so as to preserve double negation. Here we express the unramified forcing, which preserves double negation, in a single stage. Surprisingly enough, the corresponding definition of forcing for equality acquires a rather simple form. In [2] forcing ∥- is expressed in terms of strong forcing \( \Vdash * \) viap∥-Q iffp \( \Vdash * \) ¬ ¬Q for every formulaQ ofZF set theory and every elementp of a partially ordered set (P, ≦). In its turn,p \( \Vdash * \) Q is defined by the following five clauses: (1) $$p \Vdash * a \in biff(\exists c)(\exists q \geqq p)((c,q) \in b \wedge p \Vdash * a = c)$$ (2) $$\begin{gathered} p \Vdash * a \ne biff(\exists c)(\exists q \geqq p)(((c,q) \in a \wedge p \Vdash * c \notin b) \hfill \\ ((c,q) \in b \wedge p \Vdash * c \notin a)) \hfill \\ \end{gathered} $$ (3) $$p \Vdash * \neg Qiff(\forall q)(q \leqq p \to \neg (q \Vdash * Q))$$ (4) $$p \Vdash * (Q \vee S)iff(p \Vdash * Q) \vee (p \Vdash * S)$$ (5) $$p \Vdash * (\exists x)Q(x)iff(\exists b)(p \Vdash * Q(b))$$ .     

Photosensitization of electronic transitions by dye molecules in systems consisting of a semiconductor and adsorbed molecules of p-benzoquinone

Electrophysical methods and ESR measurements have been used to investigate the changes, photosensitized by molecules of rhodamine B (RB), in the charge states of electron traps created by molecules of p-benzoquinone (pBQ) in the Ge/GeO₂ system. The results obtained in studying the quenching of fluorescence of RB molecules indicate that the photodestruction of electron traps is due to transfer of electronic excitation energy traps is due to transfer of electronic oxidation energy from the RB molecules to charged complexes formed upon adsorption of the pBQ. In order to select optimal conditions for recharging of such complexes, a study has been made of the relationship between the relative change in ESR signal and the concentration of adsorbed RB molecules.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 25, No. 5, pp. 545–550, September–October, 1989.     

Use of long-chain alkylamines for preconcentration and determination of traces of molybdenum, tungsten and rhenium by atomic-absorption spectroscopy-II: molybdenum in soils, sediments and natural waters

Molybdenum is extracted as the thiocyanate complex with the quaternary long-chain aliphatic amine Aliquat 336 in chloroform, followed by evaporation of the solvent, dissolution in MIBK, and atomic-absorption spectroscopy. The method is simple, rapid and sensitive, with few interference problems for the determination of the Mo content of soils and sediments in the range 0.1-1.0 ppm with a relative standard deviation better than 5% when 1-g samples are used. Quantitative extraction from large volumes of aqueous solution has also been confirmed, allowing the determination of Mo in natural waters in the ppM range.     

Potentiometric response of graphite electrodes coated with modified polymer films

Graphite electrodes coated with chemically-modified polymer films are described. Several different polymers were used, including poly(acrylic acid), poly[triethyl(vinylbenzyl)ammonium chloride], poly[trihexyl(vinylbenzyl)ammonium chloride], and poly[trihexyl(vinylbenzyl)ammonium thiocyanate]. A cation-responsive electrode can be prepared from poly(acrylic acid)-coated graphite. Anion-responsive electrodes can be prepared from graphite coated with polymeric quaternary amines. In these electrodes, the ion-sensing species is irreversibly attached to the polymer (rather than physically entrapped within a polymer matrix); this factor eliminates leaching of the active component, and the addition of a plasticizer is unnecessary. A selective sensor for thiocyanate is described; it yields a Nernstian response over the concentration range 1 × 10^?1–1 × 10^?5 M sodium thiocyanate.