Singular solutions of some quasilinear elliptic equations

We study isolated singularities of the quasilinear equation in an open set of ^N , where 1 < p N, p -1 q < N(p — 1)/ (N -p). We prove that, for any positive solution, if a singularity at the origin is not removable then either or u(x)/(x) any positive constant as x 0 where is the fundamental solution of the p-harmonic equation: . Global positive solutions are also classified.     

Framework for three-dimensional coherent diffraction imaging by focused beam x-ray Bragg ptychography

We present the framework for convergent beam Bragg ptychography, and, using simulations, we demonstrate that nanocrystals can be ptychographically reconstructed from highly convergent x-ray Bragg diffraction. The ptychographic iterative engine is extended to three dimensions and shown to successfully reconstruct a simulated nanocrystal using overlapping raster scans with a defocused curved beam, the diameter of which matches the crystal size. This object reconstruction strategy can serve as the basis for coherent diffraction imaging experiments at coherent scanning nanoprobe x-ray sources.     

Analyticity for the Navier-Stokes equations governed by surface tension on the free boundary

A free boundary problem for an incompressible viscous fluid is considerd; the boundary is to be determined by equilibrium conditions involving the fluid's stress tensor and its surface tension. It is proved that if the data of the problem are analytic then the free boundary, the velocity vector, and the pressure are all analytic.     

Identification of small inhomogeneities of extreme conductivity by boundary measurements: a theorem on continuous dependence

We consider an electrostatic problem for a conductor consisting of finitely many small inhomogeneities of extreme conductivity, embedded in a spatially varying reference medium. Firstly we establish an asymptotic formula for the voltage potential in terms of the reference voltage potential, the location of the inhomogeneities and their geometry. Secondly we use this representation formula to prove a Lipschitz-continuous dependence estimate for the corresponding inverse problem. This estimate bounds the difference in the location and in certain geometric properties of two sets of inhomogeneities by the difference in the boundary voltage potentials corresponding to a fixed current distribution.     

Computations of cuspidal cohomology of congruence subgroups of SL(3, Z)

Algorithms are presented which find a basis of the vector space of cuspidal cohomology of certain congruence subgroups of SL(3, $\text{Z}$) and which determine the action of the Hecke operators on this space. These algorithms were implemented on a computer. Four pairs of cuspidal classes were found with prime level less than 100. Tables are given of the eigenvalues of the first few Hecke operators on these classes.     

A Conjecture Concerning the Spectrum of a Sentence

We give a plausible-sounding conjecture involving the number of n-equivalence classes of structures of size m which would imply that the complement of a spectrum is also a spectrum. Mathematics Subject Classification : 03B10, 03D15, 68Q15.     

The shape of axisymmetric rotating fluid

The equations for axisymmetric self-gravitating rotating fluid have been studied extensively since Poincaré. The model derives its primary interest from celestial mechanics, where it can be used to study the geometry of stars and planets. Existence of a solution for both the compressible and the incompressible cases is known. The smoothness of the boundary of the fluid is studied, and, in particular, it is proved that the rotating fluid has at most a finite number of rings.     

Separating chemical shift and quadrupolar anisotropies via multiple-quantum NMR spectroscopy

Chemical shift anisotropy (CSA) has been an invaluable probe of structure and dynamics for a variety of systems in NMR spectroscopy. Unfortunately, the presence of strong quadrupolar couplings has severely limited the ability to measure CSA in nuclei with spins I > 1/2. Here we show that these two interactions can be refocused at different times in a 2D multiple-quantum NMR experiment on polycrystalline samples. Combining this experiment with appropriate affine transformations allows these interactions to be cleanly separated into orthogonal dimensions. The 1D projection onto each axis can be fit to extract the respective principal tensor components. These components can then be used to fit the 2D spectrum for the relative orientation between the CSA and quadrupolar-coupling tensors. The necessary affine transformation parameters are given for all possible I values. Illustrative examples of spectra and analyses are given for 63Cu in K3[Cu(CN)4], 59Co in K3[Co(CN)6], and 87Rb in RbCrO4.