For the discrete solution of the dual mixed formulation for the p-Laplace equation, we define two residues R and r. Then we bound the norm of the errors on the two unknowns in terms of the norms of these two residues. Afterwards, we bound the norms of these two residues by functions of two error estimators whose expressions involve at the very most the datum and the computed quantities. We next...

Cortical bone, as the principal structural component of the human body, functions in both a load-bearing and protective capacity. As a material, cortical bone has been the subject of mechanical investigation and characterization for well over a century, and the time dependent nature of this material’s mechanical response to loading is well documented. To model this time-dependent behavior, many...

Recall that all the “Real-World” graphs we examined in Lecture 2 had one component containing a large constant fraction of the vertices. The second-largest component was smaller by many orders of magnitude. This property is shared by Erdös-Rényi random graphs and by many other graph models. In this lecture, we will see (mostly) why Erdös-Rényi random graphs have this property. The large compone...

This paper presents results of an exhaustive computer search for the maximum spectral value in full period two-term kth-order multiple recursive generators (MRGs) with efficient multipliers. For orders k = 2, 3, . . . , 7, we provide a table of kth-order MRGs with largest spectral values. These MRGs not only have the best spectral values found so far but also can be implemented efficiently by a...

Term-by-term Fourier-expansion series, each made up of components having element-specific phases and amplitudes acquired with x-ray standing wave measurements on successive orders of Bragg reflections, are used to reconstruct impurity atom distributions in muscovite mica with respect to the (001) lattice without a priori assumptions on their structures.

We consider the maximal size of families of k-element subsets of an n element set [n] = {1, 2, . . . , n} that satisfy the properties that every r subsets of the family have non-empty intersection, and no ` subsets contain [n] in their union. We show that for large enough n, the largest such family is the trivial one of all ( n−2 k−1 ) subsets that contain a given element and do not contain ano...

Abstract. We investigate, for linear and higher-order elements, various ways of calculating the advective limit of the stabilization parameter (“τ”) used in the streamlineupwind/Petrov-Galerkin (SUPG) formulation of flow problems. In the context of a pure advection test problem, we compare the “UGN-based” and element-matrix-based calculations and the calculations from the quadrature-point compo...

We study the dispersion and dissipation of the numerical scheme obtained by taking a weighted averaging of the consistent (finite element) mass matrix and lumped (spectral element) mass matrix for the small wavenumber limit. We find and prove that for the optimum blending the resulting scheme (a) provides 2p+4 order accuracy for p-th order method (two orders more accurate compared with finite a...

For a β-Jacobi ensemble determined by parameters a1, a2 and n, under the restriction that the three parameters go to infinity with n and a1 being of small orders of a2, we obtain some limit theorems about the eigenvalues. In particular, we derive the asymptotic distributions for the largest and the smallest eigenvalues, the central limit theorems of the eigenvalues, and the limiting distributio...

Uniform period sub-micron gratings have been fabricated using focused ion beam milling on hafnium oxide waveguides. Atomic force microscopy indicates that the gratings have smooth and uniform profiles. At the period of 330 nm, the largest peak-to-peak height that was achieved was 85 nm. Scattering at the grating imperfections was found to be at least two orders of magnitude weaker than the inte...