Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers

This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number , with minimal polynomial , such that , and where has only one real root, then there exists a , explicitly given here, such that:

(1)

For all 0$">, all but finitely many integer quadratics satisfy

where is the height function.

(2)

For all 0$"> there exists a sequence of integer quadratics such that

Furthermore, for all in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.

     

On minimum delta set values in block monoids over cyclic groups

The difference in length between two distinct factorizations of an element in a Dedekind domain or in the corresponding block monoid is an object of study in the theory of non-unique factorizations. It provides an alternate way, distinct from what the elasticity provides, of measuring the degree of non-uniqueness of factorizations. In this paper, we discuss the difference in consecutive lengths of irreducible factorizations in block monoids of the form where . We will show that the greatest integer r, denoted by , which divides every difference in lengths of factorizations in can be immediately determined by considering the continued fraction of . We then consider the set including necessary and sufficient conditions (which depend on p) for a value to be an element of . 2000 Mathematics Subject Classification Primary—20M14, 11A55, 20D60, 11A51 Parts of this work are contained in the first author’s Doctoral Dissertation written at the University of North Carolina at Chapel Hill under the direction of the third author.     

Approximation to Real Numbers by Cubic Algebraic Integers I

In 1969, H. Davenport and W. M. Schmidt studied the problemof approximation to a real number by algebraic integers ofdegree at most 3. They did so, using geometry of numbers, byresorting to the dual problem of finding simultaneous approximationsto and ² by rational numbers with the same denominator. Inthis paper, we show that their measure of approximation forthe dual problem is optimal and that it is realized for a countableset of real numbers . We give several properties of these numbersincluding measures of approximation by rational numbers, byquadratic real numbers and by algebraic integers of degree atmost 3. 2000 Mathematics Subject Classification 11J04 (primary),11J13, 11J82 (secondary).     

An application of Diophantine approximation to the construction of rank-1 lattice quadrature rules

Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in and are generalizations of `number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)---themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit , which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order (that is, having nodes) then becomes that of finding rules with large values of . This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.

     

Continued-fraction expansions for the Riemann zeta function and polylogarithms

It appears that the only known representations for the Riemann zeta function in terms of continued fractions are those for and 3. Here we give a rapidly converging continued-fraction expansion of for any integer . This is a special case of a more general expansion which we have derived for the polylogarithms of order , , by using the classical Stieltjes technique. Our result is a generalisation of the Lambert-Lagrange continued fraction, since for we arrive at their well-known expansion for . Computation demonstrates rapid convergence. For example, the 11th approximants for all , , give values with an error of less than 10.

     

Wall and Siegmund Duality Relations for Birth and Death Chains with Reflecting Barrier

For a birth and death chain on the nonnegative integers with birth and death probabilities p _i and q _i 1 –p _i and reflecting barrier at 0, it is shown that the right tails of the probability of the first return from state 0 to state 0 are simple transition probabilities of a dual birth and death chain obtained by switching p _iand q _i. Combinatorial and analytical proofs are presented. Extensions and relations to other concepts of duality in the literature are discussed.     

Yield stress and maximum packing fraction of concentrated suspensions

The yield stress and features of the structure of concentrated suspensions based on silica flour, with particles of average diameter around 4 m, were investigated in terms of a phenomenological model. The yield stress of a concentrated suspension of known solid volume concentration is estimated by employing a shear-dependent maximum packing fraction _m which is obtained by model fitting equilibrium viscosity data, and by incorporating a first-order kinetic equation. The model proposed was examined by using several mineral suspensions in which silica flour was mixed with metal oxide particles so that microstructural features of the suspensions could be adjusted. A cocoa fat suspension was also used as a test sample having radically different chemistry. The agreement between the model prediction and independently obtained experimental evidence is acceptable. Furthermore, a qualitative explanation is obtained by a scaling analysis in an effort to relate the model parameters with the suspension structure that stems from interactions among the suspension constituents.     

Separation and identification of the light harvesting proteins contained in grana and stroma thylakoid membrane fractions

This paper presents the results of a study performed to develop a rapid and straightforward method to resolve and simultaneously identify the light-harvesting proteins of photosystem I (LHCI) and photosystem II (LHCII) present in the grana and stroma of the thylakoid membranes of higher plants. These hydrophobic proteins are embedded in the phospholipid membrane, and their extraction usually requires detergent and time consuming manipulations that may introduce artifacts. The method presented here makes use of digitonin, a detergent which causes rapid (within less than 3 min) cleavage of the thylakoid membrane into two subfractions: appressed (grana) and non-appressed (stroma) membranes, the former enriched in photosystem II and the latter containing mainly photosystem I. From these two fractions identification of the protein components was performed by separating them by reversed-phase high-performance liquid chromatography (RP-HPLC) and determining the intact molecular mass by electrospray ionization mass spectrometry (ESI-MS). By this strategy the ion suppression during ESI-MS that normally occurs in the presence of membrane phospholipids was avoided, since RP-HPLC removed most phospholipids from the analytes. Consequently, high quality mass spectra were extracted from the reconstructed ion chromatograms. The specific cleavage of thylakoid membranes by digitonin, as well as the rapid identification and quantification of the antenna composition of the two complexes facilitate future studies of the lateral migration of the chlorophyll-protein complexes along thylakoid membranes, which is well known to be induced by high intensity light or other environmental stresses. Such investigations could not be performed by sodium dodecylsulfate-polyacrylamide gel electrophoresis because of insufficient resolution of the proteins having molecular masses between 22,000 and 25,000.     

Selectable one-dimensional or two-dimensional gas chromatography-olfactometry/mass spectrometry with preparative fraction collection for analysis of ultra-trace amounts of odor compounds

A novel selectable one-dimensional ((1)D) or two-dimensional ((2)D) gas chromatography-olfactometry/mass spectrometry with preparative fraction collection (selectable (1)D/(2)D GC-O/MS with PFC) system was developed. The main advantages of this system are the simple and fast selection of (1)D GC-O/MS or (2)D GC-O/MS or (1)D GC-PFC or (2)D GC-PFC operation with a mouse click (without any instrumental set-up change), and total transfer of enriched compounds with thermal desorption (TD) on the same system for identification with (2)D GC-O/MS analysis. Recovery of PFC enrichment with 20 injection cycles of 15 model compounds at 500pg each (e.g. alcohol, aldehyde, ester, lactone, and phenol) was very good with recoveries in the range of 98-116%. The feasibility and benefit of the proposed system was demonstrated with an identification of off-flavor compounds (e.g. 2,4,6-trichloroanisole (TCA), 2-isobutyl-3-methoxypyrazine (IBMP), and geosmin) in spiked wine at odor perception threshold level (5-50ngL(-1)). After parallel stir bar sorptive extraction (SBSE) for 20 aliquots of a sample and subsequent PFC enrichment for the odor-active fractions from the 20 stir bars, three off-flavor compounds were clearly resolved and detected with TD-(2)D GC-O/MS in scan mode. The good efficiency of SBSE-PFC enrichment in the range of 71-78% shows that all analytical steps, e.g. SBSE, TD, (1)D/(2)D GC-O/MS, and PFC, are quantitative and identification of off-flavor compounds at ngL(-1) level in wine is possible.     

Microcalorimetric studies of the adsorption of N-ethylpyrrolidone,oligomeric and polymeric vinylpyrrolidone from CHCl3 on silica

Adsorption enthalpies of N-ethylpyrrolidone, oligomeric and polymeric vinylpyrrolidone from CHCl₃ solution on silica interfaces (Aerosil 200) have been measured at 25 °C by microcalorimetry. The dependence on surface coverage has been examined using measured adsorption isotherms. Binding enthalpies are calculated with fractions of adhered segments obtained by IR spectrometry and wetting enthalpies received from calorimetry. Fractions of adhered segments are derived from the adsorption enthalpies assuming equal binding enthalpies of monomers and polymer segments and considering desorption enthalpies of the solvent depending on the surface area of the polymer segments. The resulting enthalpies are compared with literature values from other solvents and the incongruity of the fraction of adhered segments obtained by IR, ESR, NMR and microcalorimetry is explained.Herrn Professor Dr. Robert Kerber zum 60. Geburtstag gewidmet.