The set of idempotents in the weakly almost periodic compactification of the integers is not closed

This paper answers negatively the question of whether the sets of idempotents in the weakly almost periodic compactifications of and are closed.

     

Weakly Almost Periodic Functions on N with a Negative Base

Weakly almost periodic compactifications have been seriouslystudied for over 30 years. In the pioneering papers of de Leeuwand Glicksberg [4] and [5], the approach adopted was operator-theoretic.The current definition is more likely to be created from theperspective of universal algebra (see [1, Chapter 3]). For adiscrete group or semigroup S, the weakly almost periodic compactificationwS is the largest compact semigroup which (i) contains S asa dense subsemigroup, and (ii) has multiplication continuousin each variable separately (where largest means that any othercompact semigroup with the properties (i) and (ii) is a quotientof wS). A third viewpoint is to envisage wS as the Gelfand spaceof the C*-algebra of bounded weakly almost periodic functionson S (for the definition of such functions, see below). In this paper, we are concerned only with the simplest semigroup(N, +). The three approaches described above give three methodsof obtaining information about wN. An early striking resultabout wN, that it contains more than one idempotent, was obtainedby T. T. West using operator theory [13]. He considered theweak operator closure of the semigroup {T, T², T³, ...} of iteratesof a single operator T on the Hilbert space L²(µ) fora particular measure µ on [0, 1]. Brown and Moran, ina series of papers culminating in [2], used sophisticated techniquesfrom harmonic analysis to produce measures µ that permittedthe detection of further structure in wN; in particular, theyfound 2^cdistinct idempotents. However, for many years, no otherway of showing the existence of more than one idempotent inwN was found. The breakthrough came in 1991, and it was made by Ruppert [11].In his paper, he created a direct construction of a family ofweakly almost periodic functions which could detect 2^c differentidempotents in wN. His method was very ingenious (he used aunique variant of the p-adic expansion of integers) and rathercomplicated. Our main aim in this paper is to construct weaklyalmost periodic functions which are easy to describe and soappear more ‘natural’ than Ruppert's. We also showthat there are enough functions of our type to distinguish 2^cidempotentsin wN.     

Equation of State of Newborn Neutron Star Matter with Untrapped Neutrinos

The equation of state of the newborn neutron star matter with untrapped neutrinos is calculated with the AV₁₈ potential along isentropic paths. The same calculations are done with the AV₁₄ potential for the sake of comparison. Temperature–density correlation, proton fraction, adiabatic index, and the velocity of sound are also obtained at different entropies. It is shown that the proton fraction (adiabatic index) increases (decreases) by increasing entropy. We have shown that our calculated equations of state obey the causality condition. The results are compared with those of others in the literature.     

Interaction of a homologous series of <Emphasis Type="Italic">n</Emphasis>-alkyl trimethyl ammonium bromides with eggwhite lysozyme

The interaction of a series of n-alkyl trimethyl ammonium bromides (C₁₂, C₁₄ and C₁₆) with egg white lysozyme have been studied using fluorescence and UV-Vis spectroscopies and isothermal titration calorimetry (ITC). The trend of variation of molar absorptivity at 281 nm, quantum yields (λ_ex=281 nm) and heat of interaction with respect to surfactant concentration, were measured. The spectrophotometric results show that the hydrophobic interactions have a major role in denaturation mechanism and it would be increased with increasing in hydrocarbon tail length of surfactant. The ITC results indicated the two-step mechanism for unfolding of lysozyme due to its interaction with surfactants.     

Isothermal titration calorimetric study on the interaction of apo-human transferrin with sodium n-dodecyl sulfate

Isothermal titration calorimetry (ITC) was utilized at conditions close to physiologic (50 mM HEPES buffer, pH 7.4 and 160 mM NaCl) and at various temperatures (20, 25, 30, 35, and 40 °C) to evaluate the enthalpy and heat capacity changes for the interactions of sodium n-dodecyl sulfate (SDS) as an anionic surfactant with apo-human transferrin (apo-hTf). The obtained results are very informative due to importance of heat capacity change as a major thermodynamic quantity that is one of the richest potential sources of information in physical terms. The obtained precise curves and heat capacity curves were interpreted in terms of molecular events such as specific and non-specific binding and the unfolding process. A three step mechanism for the interaction of SDS with apo-hTf has been figured out on basis of ITC studies: $ N \leftrightarrow I_{1} \leftrightarrow I_{2} \leftrightarrow D $ , where N, I, and D correspond to native, partially unfolded, and denatured states, respectively.     

Energitics of micellizaion of sodium n-dodecyl sulfate at physiological conditions using isothermal titration calorimetry

The curing kinetics of nanocomposites based on phenolic resol cured with triethylamine (TEA) containing different amounts of organic montmorillonite was analyzed by differential scanning calorimetry. Kissinger-Akahira-Sunose (KAS) model-free kinetics has been applied to correlate the dynamic cure behaviour in the presence of modified montmorillonite. The effect in the curing of the use of different clay modifiers has also been studied. A commercial clay with hydroxyl groups (Cloisite 30B) and a customized montmorillonite (PheMMT) whose reactive groups induce condensation reactions with the resol matrix have been used. Strong dependency of activation energy on apparent conversion has been observed for all compounds.