Some applications of the condensation of the singularities of families of nonnegative functions

RecentlyPrus-Wi?niowski [14] has proved that the continuous functions of Λ-bounded variation on [0, 1] form a set of the first category in the Banach space C[0, 1] and also in each Banach space CΓBV[0, 1] of continuous functions of Γ-bounded variation on [0, 1] provided that the sequence Γ is adequate. In the present paper these results are extended and completed by using a principle of the condensation of the singularities of a family of nonnegative functions that follows from the theorems given byBreckner [3]. It is shown that Baire category properties similar to those stated in [14] are valid for two very large classes of real-valued functions called functions of bounded λ-variation of orderp and functions of boundedmth variation of orderp, respectively. The benefit of considering these classes is that they comprise several classes of functions of bounded variation type which have occurred so far in Fourier analysis or real analysis; in particular, the functions investigated in [14]. Thus, by specializing the results derived in the present paper, they give at once Baire category information concerning a number of well-known sets of real-valued functions.     

Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: 2. Improved error bounds for eigenfunctions

The application of the Rayleigh-Ritz method for approximating the eigenvalues and eigenfunctions of linear eigenvalue problems in several dimensions is investigated. The object is to improve upon known error estimates for the approximate eigenfunctions. Results for the Galerkin approximation of the eigenfunctions are developed under varying assumptions on the boundary conditions and domain of definition of the eigenvalue problem. These results, coupled with a previous result relating Galerkin and Rayleigh-Ritz approximation of the eigenfunctions, are then used to obtain improved error estimates for the approximate eigenfunctions in theL ² and uniform norms.This research was supported in part by AEC Grant (11-1)-2075.     

Nematic phase transitions in mixtures of thin and thick colloidal rods

We report experimental measurements of the phase behavior of mixtures of thin (charged semiflexible fd virus) and thick (fd-PEG, fd virus covalently coated with polyethylene glycol) rods with diameter ratio varying from 3.7 to 1.1. The phase diagrams of the rod mixtures reveal isotropic-nematic, isotropic-nematic-nematic, and nematic-nematic coexisting phases with increasing concentration. In stark contrast to predictions from earlier theoretical work, we observe a nematic-nematic coexistence region bound by a lower critical point. Moreover, we show that a rescaled Onsager-type theory for binary hard-rod mixtures qualitatively describes the observed phase behavior.     

Adsorption of 1-octanol at the free water surface as studied by Monte Carlo simulation

The adsorption of 1-octanol at the free water surface has been investigated by Monte Carlo computer simulation. Six different systems, built up by an aqueous and a vapor phase, the latter also containing various number of octanol molecules, have been simulated. The number of the octanol molecules has been chosen in such a way that the octanol surface density varies in a broad range, between 0.27 and 7.83 micromol/m(2) in the six systems simulated. For reference, the interfacial system containing bulk liquid octanol in the apolar phase has also been simulated. The results have shown that the formation of hydrogen bonds between the interfacial water and adsorbed octanol molecules is of key importance in determining the properties of the adsorbed layer. At low octanol surface concentration values all the octanol molecules are strongly (i.e., by hydrogen bonds) bound to the aqueous phase, whereas their interaction with each other is negligibly small. Hence, they are preferentially oriented in such a way that their own binding energy (and thus their own free energy) is minimized. In this preferred orientation the O-H bond of the octanol molecule points flatly toward the aqueous phase, declining by about 30 degrees from the interfacial plane, irrespectively from whether the octanol molecule is the H-donor or the H-acceptor partner in the hydrogen bond. Hence, in its preferred orientation the octanol molecule can form at least two low energy hydrogen bonds with water: one as a H-donor and another one as a H-acceptor. Moreover, the preferred orientation of the hydrogen bonded water partners is close to one of the two preferred interfacial water alignments, in which the plane of the water molecule is parallel with the interface. When increasing the octanol surface density, the water surface gets saturated with hydrogen bonded octanols, and hence any further octanol molecule can just simply condense to the layer of the adsorbed octanols. The surface density value at which this saturation occurs is estimated to be about 1.7 micromol/m(2). Above this surface density value the hydrogen bonded octanols and their water partners are oriented in such a way that the number of the water-octanol hydrogen bonds is maximized. Hence, the preferred alignment of the O...O axes of these hydrogen bonds is perpendicular to the interface. This orientation is far from the optimal alignment of the individual octanol molecules, which is also reflected in the observed fact that, unlike in the case of many other adsorbents, the average molecular binding energy of the adsorbed octanol molecules increases (i.e., becomes less negative) with increasing octanol surface density.     

First Measurement of the Gerasimov-Drell-Hearn Sum Rule for 1H from 0.7 to 1.8 GeV at ELSA

To verify the fundamental Gerasimov-Drell-Hearn (GDH) sum rule for the first time experimentally, we measured the helicity dependent total photoabsorption cross section with circularly polarized real photons and longitudinally polarized nucleons in the photon energy range 0.68-1.82 GeV with the tagged photon facility at ELSA. The experiment was carried out with a 4pi detection system, a circularly polarized tagged photon beam, and a frozen spin polarized proton target. The contribution to the GDH sum rule in this photon energy range is [49.9+/-2.4(stat)+/-2.2(syst)] microb.     

On zeros of polynomials orthogonal over a convex domain

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain.