Perturbation theory in large order

For many quantum mechanical models, the behavior of perturbation theory in large order is strikingly simple. For example, in the quantum anharmonic oscillator, which is defined by−″ + (x²/4+εx⁴/4−E)y=0, y(±∞)=0,the perturbation coefficients A_n in the expansion for the ground-state energysimplify dramatically as n → ∞:.We use the methods of applied mathematics to investigate the nature of perturbation theory in quantum mechanics and we show that its large-order behavior is determined by the semiclassical content of the theory. In quantum field theory the perturbation coefficients are computed by summing Feynman graphs. We present a statistical procedure in a simple λ⁴ model for summing the set of all graphs as the number of vertices → ∞. Finally, we discuss the connection between the large-order behavior of perturbation theory in quantum electrodynamics and the value of α, the charge on the electron.     

Almost all rooted maps have large representativity

Let M be a map on a surface S. The edge-width of M is the length of a shortest noncontractible cycle of M. The face-width (or, representativity) of M is the smallest number of intersections a noncontractible curve in S has with M. (The edge-width and face-width of a planar map may be defined to be infinity.) A map is a large-edge-width embedding (LEW-embedding) if its maximum face valency is less than its edge-width. For several families of rooted maps on a given surface, we prove that there are positive constants C₁ and C₂, depending on the family and the surface, such that

• 1 almost all maps with n edges have face-width and edge-width greater than c₁ log n, and
• 2 the fraction of such maps that are LEW-embeddings and the fraction that are not LEW-embeddings both exceed n^{? >C2}.

     

Enthalpies of solution of polyvinylacetate in chloroform,acetone, and methanol using titration calorimetry

Heats of mixing of the solvents chloroform, methanol, and acetone with their solutions in polyvinylacetate have been measured at 25°C using a titration calorimeter. Titration of pure solvent to solutions as well as titration of solutions to the pure solvent or to diluted solutions have been performed. The second method, called reverse titration, proved to be more sensitive than normal dilution titration at sufficiently low polymer concentration. Equations describing the relative sensitivity of both titration methods have been developed. The concentration range covered by the experiments is restricted to volume fractions of polyvinylacetate not higher than 0.1–0.15 due to the high viscosities of concentrated solutions. The interaction parameterX _H of the Flory-Huggins theory has been determined from the calorimetric data.X _H is positive for methanol solutions and negative for chloroform and acetone solutions reflecting endothermic and exothermic heat effects respectively.     

General entries to C-aryl glycosides. Formal synthesis of galtamycinone

Utilizing a general entry we had developed for the synthesis of C-aryl glycosides, we have prepared the juglone derivatives 18-20 as well as the juglone precursor 13. Because 19 had been previously converted in two steps by Suzuki into galtamycinone (1), its preparation constitutes a total synthesis of 1.     

Investigation of the Nature of Iron Oxide Species Present in Ruby Glasses Prepared by the Sol-Gel Method

Mössbauer spectroscopy in an external magnetic field, electron diffraction microscopy and electron spin resonance (ESR) measurements were used in an attempt to characterize the iron oxide particles present in ruby glasses prepared by the sol-gel method. The obtained results lead to the conclusion that particles are present as a ferromagnetic phase that shows superparamagnetic behaviour.     

2H-pyrrole-1-oxides by selenium dioxide dehydrogenation of pyrroline-1-oxides1)

Five different 2$\text{H}$-pyrrole-1-oxides have been prepared by dehydrogenation of their corresponding pyrroline-1-oxide presursors and characterized by 1,3-dipolar cycloadditions with dimethyl ethyne dicarboxylate and/or N-phenyl maleic imide.     

The rotational barrier in dimethyl acetylene and related molecules

The rotational barriers in dimethyl acetylene (CH₃CCCH₃), diamino acetylene (H₂NCCNH₂), dihydroxy acetylene (HOCCOH), methyl trifluoro methyl acetylene (CF₃CCCH₃), silyl methyl acetylene (SiH₃CCH₃), propene, cis and trans 2-butene and ethane were studied by ab initio molecular orbital methods using various basis sets. The eclipsed structure for dimethyl acetylene and its CF₃ and SiH₃ analogs was found to be the most stable, as has been inferred from experimental work, and the barrier height for these compounds has been predicted. The barriers in the OH and NH₂ substituted acetylenes, propene, butene and ethane were studied in order to more clearly understand the important influences in determining the barrier mechanism; specifically, the delocalized molecular orbital and Pauling VB model have been compared.