A mathematical approach to the study of the United States Code

The United States Code (Code) is a document containing over 22 million words that represents a large and important source of Federal statutory law. Scholars and policy advocates often discuss the direction and magnitude of changes in various aspects of the Code. However, few have mathematically formalized the notions behind these discussions or directly measured the resulting representations. This paper addresses the current state of the literature in two ways. First, we formalize a representation of the United States Code as the union of a hierarchical network and a citation network over vertices containing the language of the Code. This representation reflects the fact that the Code is a hierarchically organized document containing language and explicit citations between provisions. Second, we use this formalization to measure aspects of the Code as codified in October 2008, November 2009, and March 2010. These measurements allow for a characterization of the actual changes in the Code over time. Our findings indicate that in the recent past, the Code has grown in its amount of structure, interdependence, and language.     

Minimal Submanifolds

This paper provides a survey of both old and new results aboutminimal surfaces and submanifolds. 2000 Mathematics SubjectClassification 54C40, 14E20 (primary), 46E25, 20C20 (secondary).     

Chirped-pulse Fourier transform microwave spectroscopy of perfluoroiodoethane

The pure rotational spectrum of perfluoroiodoethane between 8.0 and 11.9 GHz has been measured on a search accelerated, correct intensity Fourier transform microwave (SACI-FTMW) spectrometer. The spectra is dense with 247 measured transitions in the given region. Only the anti conformer was observed for which rotational constants are reported. Nuclear electric quadrupole coupling constants due to the iodine-127 were determined and are reported. Also, two dipole forbidden/quadrupole allowed ΔJ=2 transitions were observed in the spectra. The observation of these transitions has been rationalized on the basis of near degeneracies between energy levels connected by χ_ab.     

Solid support synthesis of 14- and 17-membered macrocycles via the SNAr methodology

Efficient assembly of 14-membered macrocycles utilizing the S_NAr of fluorine in 3-fluoro-4-nitrobenzoic acid with the OH of 3-hydroxytyrosine on solid support is reported. The flexibility of this synthesis, as well as the excellent purity (>90%) of the final products are the distinctive characteristic of the resultant library.     

Sublattices of regular elements

Let L be an r-lattice, i.e., a modular multiplicative lattice that is compactly generated, principally generated, and has greatest element 1 compact. We consider certain subsets of L consisting of regular elements: there is a compact element with there is a principal element is a principal element with (0:X_{\alpha })=0\} . The first three subsets L_{f}, L_{sr}, and L_{r} are augmented filters on L, i.e., is a multiplicatively closed subset of $L$ with $A\in and $B\geq A$ with $B\in L$ implies $B\in and hence are sublattices of $L$ closed under multiplication. We first consider the more general situation of augmented filters on $L.$ These results are then applied to study the four previously defined subsets for $L$ an $r$-lattice or Noether lattice (i.e., an $r$-lattice with ACC). Finally, we give a brief discussion of how the results for augmented lattices can be applied to subsets of $L$ which are regular with respect to an $L$-module.     

Studies of substituted dihydroxy-p-benzoquinones as precipitate-exchange reagents

Studies of the barium salts of bromanilic acid, nitranilic acid and polyporic acid show that they are not superior to barium chloranilate as reagents for determining sulfate. Improved syntheses of polyporic acid and atromentin are presented; two new syntheses of barium bromanilate, and one new synthesis of barium nitranilate and of barium polyporate are also presented. Ultraviolet absorption spectra show that polyporic acid forms a dimer in dimethylsulfoxide solution.     

An investigation of the critical liquid-vapor properties of dilute KCl solutions

The three parameters that define the critical point, temperature, pressure, and volume have been experimentally determined by means of filling studies in a platinum-lined system for five KCl solutions ranging from 0.006 to 0.568m. The platinum-lined vessels were used to overcome the problems with corrosion experienced by earlier workers. The critical temperature (t _c), pressure (P _c), and volume (V _c) were found to fit the equations $\begin{gathered} {\text{t}}_c = 374.14{\text{ }} + {\text{ }}16.602\sqrt {\text{m}} {\text{ }} + {\text{ }}41.740{\text{m }} \pm 0.5^ \circ C \hfill \\ {\text{P}}_c = 220.9 {\text{ }} + {\text{ }}135.164{\text{m }} + {\text{ }}41.173{\text{m}}^{\text{2}} {\text{ }} \pm {\text{ }}0.5 bars \hfill \\ {\text{V}}_c = 3.155{\text{ }} - {\text{ }}1.373\sqrt m {\text{ }} + {\text{ }}0.507{\text{m }} \pm {\text{ }}0.008cm^3 - g^{ - 1} \hfill \\ \end{gathered} $ from infinite dilution to 1.0m.     

Variational derivation and extensions of distributed approximating functionals

Summary Extensions of the "Distributed Approximating Functional (DAF) approach to approximating functions and their derivatives are given. The method, although inherently approximate, can be made arbitrarily accurate, numerically stable, and computationally efficient by appropriate choice of parameters. It also provides approximate representations of quantum operators which are analytic and which can be accurate. Differences between the DAFs and more standard basis set approaches are discussed in order to clarify the properties of the DAFs. Some illustrative examples are given.Supported in part under National Science Foundation Grant CHE92-01967.The Ames Laboratory is operated for the Department of Energy by Iowa State University under Contract No 2-7405-ENG82.Supported under National Science Foundation Grants CHE-8907429 and CHE-9403416.Supported under R.A. Welch Foundation Grant E-0608.Supported under National Science Foundation Grants CHE-8907429 and CHE-9403416.     

Novel one pot synthesis of polysubstituted pyrazolo[1,5-a]- and imidazo[1,2-a]pyrimidines

We have described a convenient regioselective one-pot approach to pyrazolo[1,5-a]- and imidazo[1,2-a]pyrimidine derivatives from α,β-unsaturated imines generated in situ and amino heterocycles. Reaction is general with respect to all three components, namely (i) nitrile, (ii) aldehyde, and (iii) amino heterocycle reagents. Good yields (52-77%), convenient isolation of the targeted molecules are the distinct characteristics of the developed protocol.