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.     

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) $$p \Vdash * (\exists x)Q(x)iff(\exists b)(p \Vdash * Q(b))$$ .     

On [Na(POCl3)4]+ Complex Ion Formation: Conductivity,Raman, and NMR studies in the system phosphoryl chloride–sodium tetrachloro aluminate and related compounds

The system POCl₃–NaAlCl₄ was investigated by measuring the conductivity and the Raman and NMR spectra (²⁷Al, ²³Na and ³¹P) as a function of the mol fraction x of NaAlCl₄ in POCl₃. Additionally, Raman spectra of POCl₃ solutions of NaFeCl₄, LiAlCl₄, LiFeCl₄, and KAlCl₄ were recorded. In solutions containing Li⁺ or Na⁺ ions a liquid to solid (or jelly) phase transition was observed under certain conditions, dependent on salt concentration and temperature. Observed changes in the Raman spectra of the electrolyte solutions in comparison to the pure solvent POCl₃ demonstrate the existence of interactions. Clearly, the POCl₃ eigenfrequencies and hence the molecules are pertubed. The formation of [M(POCl₃)₄]⁺ complexes (M = Li, Na) can be deduced from the Raman measurements. NMR investigations support this conclusion. For assigning of Raman spectra, (Li⁺, K⁺) cation and ([FeCl₄]^?, [SbCl₆]^?) anion substitutions were employed.     

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.     

Protection of immobilized sulfhydryl groups against autooxidation by alterations in their microenvironment

Immobilized sulfhydryl groups were prepared by partial thiolation of NH₂-glass beads. The microenvironment of the immobilized SH groups was varied by different chemical modifications of neighboring NH₂ groups. Introduction of a strong charge in the surroundings of immobilized sulfhydryls results in their dramatic stabilization against autooxidation. This effect is due to the salting of O₂ from the surface microlayer of the thiolated beads.