Spectral properties of certain operators on Hardy spaces of planar regions

If G is any region in the complex plane and H^P (G), 1p, denotes the Hardy space of analytic functions on G, then the principal aim of this paper is to present the spectral properties of the multiplication operators M defined on H^P(G) by Mf=f, where is any bounded analytic function on G. In order to do this, the speccial case in which G is bounded and (z)=z is studied in detail. In addition an operator called the truncation operator that is not a multilication operator and acts on the Hardy spaces of regions of the form /K for a compact set K is studied.Dedicated to the memory of my brotherThis research was partially supported by National Science Foundation grant MCS 83-204-26.     

Gray codes for reflection groups

LetG be a finite group generated by reflections. It is shown that the elements ofG can be arranged in a cycle (a Gray code) such that each element is obtained from the previous one by applying one of the generators. The case G =A ₁ ⁿ yields a conventional binary Gray code. These generalized Gray codes provide an efficient way to run through the elements of any finite reflection group.     

Indefinite integrals of Lommel functions from an inhomogeneous Euler–Lagrange method

A method given recently for deriving indefinite integrals of special functions which satisfy homogeneous second-order linear differential equations has been extended to include functions which obey inhomogeneous equations. The extended method has been applied to derive indefinite integrals for the Lommel functions, which obey an inhomogeneous Bessel equation. The method allows integrals to be derived for the inhomogeneous equation in a manner which closely parallels the homogeneous case, and a number of new Lommel integrals are derived which have well-known Bessel analogues. Results will be presented separately for other special functions which obey inhomogeneous second-order linear equations.     

Experimental observations of the hydrodynamic behavior of solvent systems in high-speed counter-current chromatography. III. Effects of physical properties of the solvent systems and operating temperature on the distribution of two-phase solvent systems

Statistical studies were made to correlate the hydrodynamic behavior of two-phase solvent system in counter-current chromatography (CCC) to their physical properties including interfacial tension, viscosity, and the difference in density of the two phases. Settling time measured under unit gravity provided a reliable numerical index for the hydrodynamic behavior of the solvent systems in a centrifugal force field. Viscosity and settling time were strongly correlated (correlation coefficient, r = +0.88) while interfacial tension (r = -0.65) and phase density difference (r = -0.45) showed moderate and weak correlation, respectively. Studies of the effect of temperature on settling time as well as a preliminary apparatus operated at higher temperature show that raising the temperature will improve the performance of high-speed CCC.     

Ion hydration co-sphere interactions in the double-layer and ionic solutions

Two-dimensional concentrations of adsorbed ions in double-layers at charged interfaces, especially when appreciable specific adsorption obtains, are equivalent to quite substantial (1–4 M) three-dimensional concentrations in regular electrolyte solutions. Under such conditions, ion-specific Gurney co-sphere overlap interactions give an important contribution to the non-ideal free energy of electrolytes in solution. It is proposed that similar interaction effects arise two-dimensionally in double-layers, giving rise to a contribution to the lateral interaction energy in monolayer arrays of ions. Three types of calculations are described by which these interaction effects can be evaluated. One is applied to some recent data on tetrapropylammonium ion adsorption at Hg, where hydrophobic interactions arise.Related problems concerned with solvent dipole orientation in the inner layer, when appreciable surface concentrations of hydrated ions are present, are referred to. The probable role of field-gradient/quadrupole interactions is noted.     

Investigation of removal of Cr(VI), Mo(VI), W(VI), V(IV), and V(V) oxy-ions from industrial waste-waters by adsorption and electrosorption at high-area carbon cloth

Adsorption and electrosorption of Cr(VI), Mo(VI), W(VI), V(IV), and V(V) ions from water samples at low concentration were studied at high-area C-cloth electrodes. The concentrations of ions in the solution were monitored using in situ UV spectroscopy. All the investigated ions, except V(IV), showed better adsorption in acidic media. Positive polarization of the C-cloth caused increased adsorption of Cr(VI), Mo(VI), and V(V) ions. When previously adsorbed, Mo(VI) and V(V) ions were shown to be largely desorbable by negative polarization of the C-cloth. Since V(IV) does not become adsorbed significantly at the C-cloth in acidic media, the method provides an interesting means for separation of V(V) and V(IV) species in solution.     

Chemical and X-Ray crystallographic characterization of the reaction products in the 1,3-dipolar cycloaddition of diazomethane to p-toluquinone

The product 2 in the 1,3-dipolar cycloaddition of one equivalent of diazomethane to p-toluquinone (1) was determined by 250 MHz nmr spectra to be approximately 85% 6-methyl-1-H-indazole-4,7-dione (2b). X-ray crystallographic analysis was employed in the characterization of 1,6-dimethyl-1-H-indazole-4,7-dione (4a), which was the major 1-N-methyl regioisomer in the methylation of the cycloaddition mixture 2 with diazomethane. Methylation of the cycloaddition product 2 with diazomethane also provided a regioisomeric mixture of the 2-N-methyl derivatives 5. This mixture was synthesized for characterization by an independent method which utilized the cycloaddition of 3-methylsydnone (10) to toluquinone (1). 1,5,6-Trimethyl-1-H-ind-azole-4,7-dione (9) was found to be a minor product in the reaction of diazomethane with the cycloaddition product 2.     

Ion hydration near air/water interfaces and the structure of liquid surfaces

Negative adsorption of ions, commonly observed at air-water interfaces, is examined in terms of models of restricted polarization of the solvent by ions at the interface and the structure of the liquid interface. The Born and other models of ionic hydration are applied to evaluate the self-energy of the ion arising in the region of solvent near its interface and in the vacuum or vapour beyond. The adsorption energy of an ion varies substantially with distance from the liquid interface so that a distribution of ions arises as a function of distance from the interface. Integration of this distribution gives an expression, and results, for the ionic surface excess. The diffuse-layer potential, which an unequal distribution of cations and anions give rise to, gives a contribution to the surface potential of the electrolyte solution at finite concentrations.Structural aspects of the liquid interface at which ions are negatively adsorbed are discussed in terms of Stefan's ratio and the superficial excess entropies of various liquid surfaces. These entropies are related to the cohesive energy densities of the bulk liquids. Ion solvent-structure co-sphere interactions with structured interfaces will lead to specificity of negative adsorption of ions.