On the beta expansion for Salem numbers of degree 6

For a given , the beta transformation is defined for by (mod ). The number is said to be a beta number if the orbit is finite, hence eventually periodic. It is known that all Pisot numbers are beta numbers, and it is conjectured that this is true for Salem numbers, but this is known only for Salem numbers of degree . Here we consider some computational and heuristic evidence for the conjecture in the case of Salem numbers of degree , by considering the set of such numbers of trace at most . Although the orbit is small for the majority of these numbers, there are some examples for which the orbit size is shown to exceed and for which the possibility remains that the orbit is infinite. There are also some very large orbits which have been shown to be finite: an example is given for which the preperiod length is and the period length is . This is in contrast to Salem numbers of degree where the orbit size is bounded by . An heuristic probabilistic model is proposed which explains the difference between the degree- and degree- cases. The model predicts that all Salem numbers of degree and should be beta numbers but that degree- Salem numbers can have orbits which are arbitrarily large relative to the size of . Furthermore, the model predicts that a positive proportion of Salem numbers of any fixed degree will not be beta numbers. This latter prediction is not tested here.

     

On vanishing of characteristic numbers in Poincaré complexes

Let be the evaluation subgroup as defined by Gottlieb. Assume the Hurewicz map is non-trivial and is a field. We will prove: if is a Poincaré complex oriented in -coefficient, all the characteristic numbers of in -coefficient vanish. Similarly, if and is a -Poincaré complex, then all the mod Wu numbers vanish. We will also show that the existence of a non-trivial derivation on with some suitable conditions implies vanishing of mod Wu numbers.

     

Limit theorems for the maximum likelihood estimate under general multiply Type II censoring

Assume n items are put on a life-time test, however for various reasons we have only observed the r ₁-th,..., r _k-th failure times % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaamiEamaaBaaaleaamiaadkhadaWgaaqaaSGaaGymaiaacYcaaWqa% baGaamOBaiaacYcacaGGUaGaaiOlaiaac6caaSqabaGccaGGSaGaam% iEamaaBaaaleaamiaadkhadaWgaaqaaSGaam4AaiaacYcaaWqabaGa% amOBaaWcbeaaaaa!48BB!\[x_{r_{1,} n,...} ,x_{r_{k,} n} \]with % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaGimaiabgsMiJkaadIhadaWgaaWcbaadcaWGYbWaaSbaaeaaliaa% igdacaGGSaaameqaaiaad6gaaSqabaGccqGHKjYOcqWIVlctcqGHKj% YOcaWG4bWaaSbaaSqaaWGaamOCamaaBaaabaWccaWGRbGaaiilaaad% beaacaWGUbaaleqaaeXatLxBI9gBaGqbaOGae8hpaWJaeyOhIukaaa!521B!\[0 \le x_{r_{1,} n} \le \cdots \le x_{r_{k,} n} > \infty \]. This is a multiply Type II censored sample. A special case where each x _ri ,n goes to a particular percentile of the population has been studied by various authors. But for the general situation where the number of gaps as well as the number of unobserved values in some gaps goes to , the asymptotic properties of MLE are still not clear. In this paper, we derive the conditions under which the maximum likelihood estimate of is consistent, asymptotically normal and efficient. As examples, we show that Weibull distribution, Gamma and Logistic distributions all satisfy these conditions.This research was supported in part by the Designated Research Initiative Fund, University of Maryland Baltimore County.     

Formula that accounts a number of atoms within a period of the periodic table

The formula that determines a number of atoms within a period of the Periodic Table is derived and compared with the formula offered by Tomkeieff in 4 . © 2006 Wiley Periodicals, Inc. Int J Quantum Chem 2007     

Spectrophotometric studies on the solvatochromism of the Fe(CN)2(phen)2 complex

Summary The solvatochromic behaviour ofbis-(cyano)-bis-(9,10-phenanthroline)-iron(II) has been reexamined spectrophotometrically in a series of 19 solvents including water, alcohols and dipolar. aprotic media. Satisfactory correlations between the wave numbers of the long-wavelength band and solvent acceptor numbers are obtained in all media; only values, for carboxylic acids deviate from the regression line.

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Spektrophotometrische Untersuchungen über die Solvatochromie der Komplexverbindung Fe(CN)₂(phen)₂

Zusammenfassung Das solvatochrome Verhalten vonbis-(Cyano)-bis-(9,10-phenanthrolin)-eisen(II) wurde in 19 Lösungsmitteln, darunter Wasser, Alkohole und dipolare aprotische Medien, spektrophotometrisch überprüft. In allen Medien wurden zufriedenstellende Korrelationen zwischen den Wellenzahlen der langwelligen Bande und den Akzeptorzahlen der Lösungsmittel erhalten; lediglich die Werte für Carbonsäuren weichen von der Ausgleichgeraden ab.

     

Nuclear magnetic resonance study of speciation of water at low concentrations in hydrogen bond acceptor solvents

Equilibrium constants for the formation of 1:1 and 1:2 complexes of monomeric water with twelve hydrogen bond acceptor solvents were evaluated from the chemical shifts of the water protons in ternary mixtures containing a low concentration of water and varying proportions of the acceptor solvent and carbon tetrachloride. Formation constants of the 1:1 complexes (with the exception of those of dioxane and dimethoxyethane) correlate well with Kamlet-Taft -parameters. A good correlation is also obtained with Gutmann donor numbers if nitromethane and pyridine are excluded. Stepwise formation constants of the 1:2 complexes vary relatively little; this is attributed to a leveling effect of first hydrogen bond formation.     

A radio-frequency moving-boundary detector

A new type of moving-boundary detector is described in which boundaries are detected at platinum ring electrodes situated on the outside of a moving-boundary tube. These rings are part of a bridge circuit operating at radio frequencies. Impedance changes in the region of the rings are converted by a phase-sensitive detector to a voltage output which, when operating in the differential mode, takes the form of a sharp peak as a boundary passes through the rings. Boundary displacements of less then 20 m can be detected, and a precision of better than 0.02% is obtained in the measurement of boundary velocities in typical transference measurements. The fact that only three electrical leads are required in the detecting circuit gives this detector a decided advantage for high-pressure work. Equally important is the elimination of electrode polarization problems associated with other types of electrical detectors.Taken, in part, from a Ph.D. thesis, Carnegie-Mellon University, 1971.     

Ion-solvent interactions of silver(I)-18-crown-6 perchlorate complex in methanol-acetonitrile mixtures

The standard Gibbs transfer energies of the silver(I)-18-crown-6 perchlorate complex salt from methanol to various compositions of methanol-acetonitrile mixtures were determined from solubility measurements at 30°C and these data were separated into the corresponding ionic contributions by employing the negligible liquid junction potential method of Parkeret al. The solvent transport numbers _AN, for the salt were also determined at various solvent compositions using a concentration cell with transference.The Gibbs transfer energy of the silver(I)-18-crown-6 complex cation is negative and decreases with the addition of acetonitrile but the transfer energy of the anion is positive and increases under the same conditions. The solvent transport number, _AN, increases and passes through a maximum value of 5.48 at _AN=0.55. These results indicate that the complex salt is heteroselectively solvated in these mixtures with the cation being preferentially solvated by acetonitrile and the anion by methanol molecules.     

Spectroscopic investigations on the binding of ammonium salt of 8-anilino-1-naphthalene sulfonic acid with non-ionic surfactant micelles in aqueous media

The anionic dye 8-anilino-1-napthalensulfonic acid ammonium salt, or ANS, was used as a fluorescent probe to investigate the behaviour of dye-surfactant interactions in aqueous solutions of Triton X-100 and the Brij and polyoxyethylene tridecyl ether (POE TDE) series of polyoxyethylene non-ionic surfactants. The fluorescence behaviour of the dye with the non-ionic surfactants was examined in micellar media. The concentration of surfactant was kept well above the cmc to investigate the interaction of the dye with surfactant micelles. In this investigation, the relative fluorescence enhancements, binding constants of the dye to the surfactant micelles and aggregation numbers of the micelles were determined, from the analysis of spectroscopic data.     

Single-ion and molar conductances. Application to transference number data analysis

Several of the present molar conductance equations have been examined in detail in order to obtain equations for the concentration dependence of the single-ion conductances which, when combined with existing conductance and transference data, permit three independent evaluations of the conductance parameters, in particular the association constant and the distance of closest approach of two ions of opposite charge. In order to make use of existing transference data it was necessary to develop a suitable extrapolation equation. It is shown that the procedures outlined constitute a useful test of the concentration dependence of transference numbers found experimentally.Equipe de recherche associée au C.N.R.S., ERA 310.