Congruences for overpartitions with restricted odd differences

The Ramanujan Journal - In a recent work, Bringmann, Dousse, Lovejoy, and Mahlburg defined the function $$\overline{t}(n)$$ to be the number of overpartitions of weight n where (i) the difference...     

Automation in the chemical analysis of suspect arson samples by GC/MS. A systematic approach

A method ist described that permits direct comparison of ion chromatograms between an isolate from a suspect fire debris sample and corresponding accelerant profiles. The system is highly automated and produces a diagnostic one page summary report for each sample. Side by side comparison of ion profiles is carried out between 4 common fuel types, e. g. gasoline, naphthenic type charcoal lighter fluid, mineral spirits, and kerosene, and the sample. Both qualitative and semiquantitative information is available. An on-line search can also be carried out in which mass spectra are compared between components in a suspect sample and standards that are contained in an accelerant library. The system is flexible and requires only minimal interaction with the analyst.     

Stanislaw M. Ulam's contributions to theoretical theory

S. M. Ulam's contributions to biology are surveyed. The survey covers cellular automata theory, population biology, Fermi-Pasta-Ulam results, pattern recognition, and sequence similarity.     

Limits in arson debris analysis by capillary column gas chromatography/mass spectrometry

Sample enrichment technology and instrumental sensitivity are no longer limiting factors in arson debris analysis. When pyrolyzed, petroleum based composite materials may produce artifacts. Urban air is another major source of interference since some of the components typically found in air, in particular alkylated aromatic hydrocarbons, are customarily used as indicators for the presence of gasoline. It is suggested to replace qualitative analysis with a quantitative approach, taking the sample matrix into consideration.     

Reactions of tetrafluoroethylene oligomers. Part 1. Some pyrolytic reactions of the pentamer and hexaher and of the fluorine adducts of the tetramer and pentamer

Pyrolyses of these highly branched fluorocarbons over glass beads caused the preferential thermolyses of CC bonds where there is maximum carbon substitution. Fluorinations of perfluoro-3,4-dimethylhex-3-ene (tetramer) (I) and perfluoro-4-ethyl-3,4-dimethylhex- 2-ehe (pentamer) (II) over cobalt (III) fluoride at 230° and 145° respectively afforded the corresponding saturated fluorocarbons (III) and (IV), though II gave principally the saturated tetramer (III) at 250°. Pyrolysis of III alone at 500—520° gave perfluoro-2-methylbutane (V), whilst pyrolysis of III in the presence of bromine or toluene afforded 2-bromononafluorobutane (VI) and 2H-nonafluorobutane (VII) respectively. Pyrolysis of perfluoro-3-ethyl-3, 4-dimethylhexane (IV) alone gave a mixture of perfluoro-2-methylbutane (V), perfluoro-2-methylbut-1-ene (VIII), perfluoro-3-methylpentane (IX), perfluoro-3,3-dimethylpentane (X), and perfluoro-3,4- dimethylhexane (III). Pyrolysis of IV in the presence of bromine gave (VI) and 3-bromo-3-trifluoromethyl-decafluoropentane (XI): with toluene, pyrolysis gare VlI and 3H-3-trifluoromethyldecafluoropentane (XII). Pyrolysis of II at 500° over glass gave perfluoro-1,2,3-trimethylcyclobutene (XIII) and perfluoro-2,3-dimethylpenta-1,3(E)- and (Z)-diene (XIV) and (XV) respectively. The diene mixture (XIV and XV) was fluorinated with CoF₃ to give perfluoro-2,3-dimethylpentane (XVI) and was cyclised thermally to give the cyclobutene (XIII). Pyrolysis of perfluoro-2- (1′-ethyl-1′-methylpropyl)-3-methylpent-1-ene (XVII) (TFE hexamer major isomer) at 500° gave perfluoro-1-methyl-2-(1′-methylpropyl)cyclobut-1-ene (XVIII) and perfluoro-2-methyl-2-(1′-methylpropyl)buta-1,3-diene (XIX). Fluorination of XVIII over CoF₃ gave perfluoro-1-methyl-2- (1′-methylpropyl)cyclobutane (XX), which on co-pyrolysis with bromine gave VI. XIX on heating gave XVIII. Reaction of XVIII with ammonia in ether gave a mixture of E and Z 1′-trifluoromethyl-2-(1′-trifluoromethyl- pentafluoropropyliden-1′-yl)tetrafluorocyclobutylamine (XXI) which on diazotisation and hydrolysis afforded 2-(2′trifluoromethyl- tetrafluorocyclobut-1-en-1′-yl)-octafluorobutan-2-ol (XXII).     

Free-energy density functions for nematic elastomers

The recently proposed neo-classical theory for nematic elastomers generalizes standard molecular-statistical Gaussian network theory to allow for anisotropic distributions of polymer chains. The resulting free-energy density models several of the novel properties of nematic elastomers. In particular, it predicts the ability of nematic elastomers to undergo large deformations with exactly zero force and energy cost—so called soft elasticity. Although some nematic elastomers have been shown to undergo deformations with unusually small applied forces, not all do so, and none deform with zero force. Further, as a zero force corresponds to infinitely many possible deformations in the neo-classical theory, this non-uniqueness leads to serious indeterminacies in numerical schemes. Here we suggest that the neo-classical free-energy density is incomplete and propose an alternative derivation that resolves these difficulties. In our approach, we use the molecular-statistical theory to identify appropriate variables. This yields the choice for the microstructural degrees of freedom as well as two independent strain tensors (the overall macroscopic strain plus a relative strain that indicates how the deformation of the elastomeric microstructure deviates from the macroscopic deformation). We then propose expressions for the free-energy density as a function of the three quantities and show how the material parameters can be measured by two simple tests. The neo-classical free-energy density can be viewed as a special case of our expressions in which the free-energy density is independent of the overall macroscopic strain, thus supporting our view that the neo-classical theory is incomplete.     

Incompatible strains associated with defects in nematic elastomers

In a nematic elastomer the deformation of the polymer network chains is coupled to the orientational order of the mesogenic groups. Statistical arguments have derived the so-called neoclassical free energy that models this coupling. Here we show that the neoclassical model supplemented by the usual Frank energy predicts incompatible network strains associated with the formation of standard nematic textures. The incompatibility is measured by the Riemann curvature tensor, which we find to be nonzero for both radial hedgehog defects and escaped disclinations of strength +1 in circular cylinders. Analogous problems for conventional nonlinearly elastic solids do not possess solutions with such incompatibilities. Compatibility in nematic elastomers would require either more complicated nematic textures in elastomers than in conventional (polymeric and low molecular weight) liquid crystals or a free-energy density more complicated than the neoclassical expression.