Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

When Ramification points meet

Summary This paper contains three applications of the technique of limit series (our [1986]) to the theory of ramification of linear series on smooth curves, and curves of compact type, overC.Let {L _t t|<}, be a family of linear series on a smooth family of smooth curves {C _t }, and letp ₁(t),p ₂(t)C _t be sections of the family which coincide (only) att=0. Setp=p ₁())=p ₂(0)C ₀.We first give a condition related to the Schubert calculus which must be satisfied by the ramification series and the . We then take up the converse problem: In what ways can a given ramification point arise as a limit? We show that if the ramification point isdimensionally proper in the sense of our [1986], then families of every kind allowed by the Schubert calculus condition can actually be constructed. Finally, we prove that dimensional propriety is in a strong sense an open condition, so that ramification points constructed as above are again dimensionally proper.In the body of the paper we work not with pairs of points, as above, but with arbitrary finite collections of points approaching (possibly) several points of the limit curve. Further, by their nature, the results are valid for families of curves of compact type.     

Sharpness of the phase transition in percolation models

The equality of two critical points — the percolation thresholdp _H and the pointp _T where the cluster size distribution ceases to decay exponentially — is proven for all translation invariant independent percolation models on homogeneousd-dimensional lattices (d1). The analysis is based on a pair of new nonlinear partial differential inequalities for an order parameterM(,h), which forh=0 reduces to the percolation densityP — at the bond densityp=1–e ^– in the single parameter case. These are: (1)MhM/h+M ²+MM/, and (2) M/|J|MM/h. Inequality (1) is intriguing in that its derivation provides yet another hint of a ³ structure in percolation models. Moreover, through the elimination of one of its derivatives, (1) yields a pair of ordinary differential inequalities which provide information on the critical exponents and . One of these resembles an Ising model inequality of Fröhlich and Sokal and yields the mean field bound 2, and the other implies the result of Chayes and Chayes that . An inequality identical to (2) is known for Ising models, where it provides the basis for Newman's universal relation and for certain extrapolation principles, which are now made applicable also to independent percolation. These results apply to both finite and long range models, with or without orientation, and extend to periodic and weakly inhomogeneous systems.Research supported in part by the NSF Grant PHY-8605164Also in the Physics Department     

Further evidence for magnetic charge from meson spectroscopy

Recently evidence was presented for the existence of magnetic charge from Zeeman splitting in meson states. The model by Akers predicted the existence of a new meson at 1814 MeV withI ^G(J ^PC )=0⁺(0^–+). Experimental evidence for this new meson is cited and discussed.     

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.     

Particle shape in thixotropic aluminium and thorium molybdates gels

Summary Electron microscope examination of thixotropic aluminium and thorium molybdate gels shows that one hour after formation they are constituted by short fibrils of small axial ratio. The fibrils of the aluminium molybdate gels, with ageing at room temperature or with boiling, increase in diameter and length; the fibrous shape of the particles is maintained after two and a half years of ageing; all fibrils are crystalline by electron diffraction. The fibrils of the thorium molybdate gels, except in the gels containing hydrochloric acid, change with ageing at room temperature or with boiling, into plates of hexagonal, elliptical or rectangular profile; the fibrils and plates are crystalline and have the same electron diffraction pattern as the fibrils.This work was aided, in part, by grants from Conselho Nacional de Pesquisas and the Rockefeller Foundation.     

Donor—acceptor interactions of substituted benzenes with molecular chlorine and carbon disulfide

CNDO molecular orbital calculations have been performed to analyze donor—acceptor interactions between molecular chlorine and benzene, toluene, mesitylene and hexamethylbenzene and the, as yet, unreported chlorine—hexafluorobenzene and carbon disulfide—benzene pairs. The stabilization energy and the dipole moment and its derivative (?p/?R_CICI) calculated for the benzene—chlorine complex are in good agreement with the estimated experimental values. The trends in the experimental stabilization energies and the Cl-Cl vibrational frequencies with increasing methyl substitution appear to be well reproduced by the calculations. The charge transferred from the benzene donor is polarized toward the outer chlorine atom or sulfur atom. For hexafluorobenzene-chlorine the direction of electronic charge polarization is reversed from that of the benzene and methylbenzene complexes. The calculated results are discussed within the framework of Muliiken's simplified resonance theory for complexes.     

A new method for the direct calculation of resonance parameters with application to the quasibound states of the H2X1Σ g + system

A new method for the direct calculation of resonance parameters is presented. It is based upon searching for poles of the scattering matrix at complex energies. This search is expedited by the use of analytic derivatives of the scattering matrix with respect to the total energy. This procedure is applied initially to a single channel problem, but is generalizable to more complicated systems. Using the most accurate available potential energy data, we calculate resonance parameters for all of the physically important quasibound states of the ground electronic state of the hydrogen molecule. Corrections to the Born-Oppenheimer potential are included and assessed. The new method has no difficulty locating resonances with widths greater than about 1×10^–7 cm^–1. It is easier to find narrow resonances by monitoring the dependence of the imaginary part of the reactance matrix on the real part of a complex energy than to monitor the dependence of the eigenphase sum on energy at real energies.     

Interaction of dextran sulfates with positively and negatively charged silver bromide hydrosols

Summary The effects of dextran sulfates of different molecular weights and charge densities on the stability of positively and negatively charged silver bromide solsin statu nascendi have been studied by means of light scattering.The plots of scattering intensities of a positively charged sol against the concentration of the added dextran sulfates show one pronounced maximum, the position which is independent on the molecular weight of the polyelectrolyte used. The maximum shifts to lower polyion concentrations with dilution of the sol.The effect of polyanions on the stability of silver bromide sols of the same sign of charge were investigated as a function of molecular weight of the polymers, the sol concentration, the charge and the concentration of counterions, and the addition of methanol. Destabilization only occurred if sufficient amount of indifferent electrolyte was present in the system. However, the concentration of counterions necessary to produce a flocculation maximum was lower in the presence of the polyelectrolytes than in their absence. The results obtained were compared with the data on sols containing nonionic dextrans reported earlier.

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Zusammenfassung Der Einfluß der Dextransulfate von verschiedenen Molekulargewichten und Ladungsdichten auf die Stabilität der positiv und negativ geladenen Silberbromid-Solein statu nascendi wurden durch Lichtstreuung untersucht. Wenn die Intensitäten der Lichtstreuung eines positiv geladenen Silberbromid-Sols gegen die Konzentration der zugesetzten DextranSulfate aufgetragen werden, tritt ein ausgeprägtes Maximum auf, dessen Lage von dem Molekulargewicht des angewandten Polyelektrolyten unabhängig ist. Bei Verdünnung des Sols wird das Maximum zur niedrigeren Konzentration des Polyanions verschoben.Der Einfluß der Polyanionen auf die Stabilität der Silberbromid-Sole mit gleichem Ladungsvorzeichen wurde in Abhängigkeit von dem Molekulargewicht des zugesetzten Dextransulfats, der Solkonzentration, der Ladung und Konzentration der Gegenionen und dem Zusatz von Methylalkohol untersucht. Destabilisierung wird nur in Anwesenheit einer genügenden Menge von Neutralelektrolyten in den untersuchten Systemen beobachtet. Die Konzentration der Gegenionen, die ein Flockungsmaximum verursachen, ist immer niedriger als die kritische Koagulationsmenge derselben Ionen für das gleiche Sol in Abwesenheit von Makroionen. Die Resultate wurden mit den Wechselwirkungen der Silberbromid-Sole mit nichtionogenen Dextranen verglichen, und der Mechanismus der Flockung durch Polyelektrolyte wurde diskutiert.

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5 figures and 1 table

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Supported by the NSF Grant GP 42331 X.

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A part of the Ph.D. Thesis by David Lindsay.     

Rapid and sensitive liquid chromatographic determination of carbamazepine suitable for use in monitoring multiple-drug anticonvulsant therapy.

A rapid, sensitive and selective method for the determination of carbamazepine and its major metabolite in plasma has been developed. Other commonly used anticonvulsants can be determined in the same procedure without interference. After extraction with dichloromethane, the components are separated by high-pressure liquid chromatography without further clean-up or concentration on a column packed with small-particle silica gel. The mean recovery from plasma is 98.6% with a relative standard deviation of 1.6%. The detection limit for carbamazepine is approximately 2 ng/ml, requiring 1 ml of plasma.