The Hausdorff dimension of graphs of Weierstrass functions

The Weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Though there is a simple formula for the Hausdorff dimension of the graph which is widely accepted, it has not been rigorously proved to hold. We prove that if arbitrary phases are included in each term of the summation for the Weierstrass function, the Hausdorff dimension of the graph of the function has the conjectured value for almost every sequence of phases. The argument extends to a much wider class of Weierstrass-like functions.

     

THE ENERGY ASPECT OF THE RECIPROCAL INTERACTIONS OF PAIRS OF TWO DIFFERENT VIBRATION MODES OF A CLAMPED ANNULAR PLATE

The standardized mutual active and reactive sound power of a clamped plate, representing the energy aspect of the reciprocal interactions of two different in vacuo modes, has been computed. It was assumed that the vibrations are axisymmetric, elastic and time harmonic, the plate's transverse deflection is small as compared with the plate's size, and that the vibration velocity is small as compared with the acoustic wavenumber generated. The Kirchhoff-Love theory of a perfectly elastic plate was used. The integral formulae for the mutual sound power were transformed into their Hankel representations which made possible their subsequent computation. A closed path integral was used to express the integral in its Hankel representation to compute the mutual active sound power. The asymptotic stationary phase method was used to compute the two magnitudes, i.e., the mutual active and reactive sound power. The results obtained are the asymptotic formulae valid for the acoustically fast waves. The oscillating as well as the non-oscillating terms have been identified in the formulae to make possible their further separate analysis. The availability of the asymptotic formulae makes possible some fast numerical computations of the mutual sound power. Moreover, the formulae presented herein, together with those for the individual modes known from the literature, make a complete basis for further computations of the total sound power of the plate's damped and forced vibrations in fluid.     

Comparisons of NMR spectral quality and success in crystallization demonstrate that NMR and X-ray crystallography are complementary methods for small protein structure determination

X-ray crystallography and NMR spectroscopy provide the only sources of experimental data from which protein structures can be analyzed at high or even atomic resolution. The degree to which these methods complement each other as sources of structural knowledge is a matter of debate; it is often proposed that small proteins yielding high quality, readily analyzed NMR spectra are a subset of those that readily yield strongly diffracting crystals. We have examined the correlation between NMR spectral quality and success in structure determination by X-ray crystallography for 159 prokaryotic and eukaryotic proteins, prescreened to avoid proteins providing polydisperse and/or aggregated samples. This study demonstrates that, across this protein sample set, the quality of a protein's [15N-1H]-heteronuclear correlation (HSQC) spectrum recorded under conditions generally suitable for 3D structure determination by NMR, a key predictor of the ability to determine a structure by NMR, is not correlated with successful crystallization and structure determination by X-ray crystallography. These results, together with similar results of an independent study presented in the accompanying paper (Yee, et al., J. Am. Chem. Soc., accompanying paper), demonstrate that X-ray crystallography and NMR often provide complementary sources of structural data and that both methods are required in order to optimize success for as many targets as possible in large-scale structural proteomics efforts.     

Iodine(V) reagents in organic synthesis. Part 3. New routes to heterocyclic compounds via o-iodoxybenzoic acid-mediated cyclizations: generality, scope, and mechanism

The discovery and development of the o-iodoxybenzoic acid (IBX) reaction with certain unsaturated N-aryl amides (anilides) to form heterocycles are described. The application of the method to the synthesis of delta-lactams, cyclic urethanes, hydroxy amines, and amino sugars among other important building blocks and intermediates is detailed. In addition to the generality and scope of this cyclization reaction, this article describes a number of mechanistic investigations suggesting a single electron transfer from the anilide functionality to IBX and implicating a radical-based mechanism for the reaction.     

Serum protein immunogenicity: implications for liver xenografting

The objectives of this study were threefold: (i) assess immunogenicity of donor plasma proteins following hepatic xenotransplantation, (ii) identify potential immunogens, and (iii) consider the implications of antibody formation against these plasma proteins in xenograft survival. We studied liver and heart xenografts in a concordant combination, hamster to rat. All grafts were examined at necropsy for evidence of rat immunoglobulin G (IgG) deposition. Cardiac xenografts were placed in recipients who had, or had not, been sensitized with hamster serum. Hepatic xenografts were placed in naive recipients to see if antibodies to hamster serum proteins could be eluted from the rejecting organ. Sera of immunized rats were examined for the presence of anti-hamster antibodies by immunoelectrophoresis and by Western blotting following sodium dodecyl sulfate-polyacrylamide gel electrophoresis (SDS-PAGE) separation of hamster serum. Antibodies in sera of immunized rats were compared with those eluted from rejecting livers. Candidate antigens were identified by tandem mass spectrometry, sequence analysis, and reference to protein databases. Results showed that sera of immunized rats recognized a minimum of four different antigens in hamster serum by immunoelectrophoresis, and a minimum of seven by the more sensitive SDS-PAGE Western blot. IgG eluted from rejecting livers bound three of seven candidate antigens recognized by sera of the immunized animals. Sequence analysis searches revealed proteinase inhibitors in each of the three SDS-PAGE bands common to the above samples. All of these candidate proteinase inhibitor immunogens share a common catabolic fate, uptake via the lipoprotein-related protein (LRP/alpha 2-macroglobulin receptor (CD91). Sensitization to hamster serum proteins hastened cardiac xenograft rejection in 30-50% of recipients (depending on sensitization protocol). Vascular deposition of rat IgG occurred in all rejecting xenografts. Antibody binding to proteinase inhibitors could disturb their functional activity and contribute to the pathogenesis of delayed xenograft rejection.