On vanishing sums of m \,m\,th roots of unity in finite fields

In an earlier work, the authors have determined all possible weights $n$ for which there exists a vanishing sum $\zeta_1+\cdots +\zeta_n=0$ of $m$th roots of unity $\zeta_i$ in characteristic 0. In this paper, the same problem is studied in finite fields of characteristic $p$. For given $m$ and $p$, results are obtained on integers $n_0$ such that all integers $n\geq n_0$ are in the ``weight set'' $W_p(m)$. The main result $(1.3)$ in this paper guarantees, under suitable conditions, the existence of solutions of $x_1^d+\cdots+x_n^d=0$ with all coordinates not equal to zero over a finite field

Elementary Proofs Of Two Theorems Involving Arguments Of Eigenvalues Of A Product Of Two Unitary Matrices

We give elementary proofs of two theorems concerning bounds on the maximum argument of the eigenvalues of a product of two unitary matrices --- one by Childs \emph{et al.} [J. Mod. Phys., \textbf{47}, 155 (2000)] and the other one by Chau [arXiv:1006.3614]. Our proofs have the advantages that the necessary and sufficient conditions for equalities are apparent and that they can be readily generalized to the case of infinite-dimensional unitary operators.Comment: 8 pages in Revtex 4.1 preprint format, to appear in Journal of Inequalities and Application

The Low Column Density Lyman-alpha Forest

We develop an analytical method based on the lognormal approximation to compute the column density distribution of the Lyman-alpha forest in the low column density limit. We compute the column density distributions for six different cosmological models and found that the standard, COBE-normalized CDM model cannot fit the observations of the Lyman-alpha forest at z=3. The amplitude of the fluctuations in that model has to be lowered by a factor of almost 3 to match observations. However, the currently viable cosmological models like the lightly tilted COBE-normalized CDM+Lambda model, the CHDM model with 20% neutrinos, and the low-amplitude Standard CDM model are all in agreement with observations, to within the accuracy of our approximation, for the value of the cosmological baryon density at or higher than the old Standard Bing Bang Nucleosynthesis value of 0.0125 for the currently favored value of the ionizing radiation intensity. With the low value for the baryon density inferred by Hogan & Rugers (1996), the models can only marginally match observations.Comment: three postscript figures included, submitted to ApJ

Instabilities in nematic liquid crystal films and droplets

The dynamics of thin films of nematic liquid crystal (NLC) are studied. Nematic liquid crystals are a type of non-Newtonian fluid with anisotropic viscous effects (due to the shape of the molecules) and elasticity effects (due to interacting electrical dipole moments). Exploiting the small aspect ratio in the geometry of interest, a fourth-order non-linear partial differential equation is used to model the free surface of the thin films. Particular attention is paid to the interplay between the bulk elasticity and the preferred orientation (boundary condition) of NLC molecules at the two interfaces: the substrate and the free surface. This work is a collection of three previously published papers and some recent unpublished work. Two main topics are covered: 1) the flow of thin films of NLC down an inclined substrate under gravity, and 2) the stability of thin NLC films on a horizontal substrate under the influence of surface tension, internal elastic effects, and fluid/solid interactions. Using a combination of analytical and computational techniques allows for a novel understanding of relevant instability mechanisms, and of their influence on transient and fully developed fluid film morphologies. While the analytical results in this thesis focus on NLC films, these results may be extended to a variety of other thin film models. Finally, a numerical code that utilizes a graphics processor unit (GPU) is presented, and the significant performance gains are discussed

Fault detection for fuzzy systems with intermittent measurements

This paper investigates the problem of fault detection for Takagi-Sugeno (T-S) fuzzy systems with intermittent measurements. The communication links between the plant and the fault detection filter are assumed to be imperfect (i.e., data packet dropouts occur intermittently, which appear typically in a network environment), and a stochastic variable satisfying the Bernoulli random binary distribution is utilized to model the unreliable communication links. The aim is to design a fuzzy fault detection filter such that, for all data missing conditions, the residual system is stochastically stable and preserves a guaranteed performance. The problem is solved through a basis-dependent Lyapunov function method, which is less conservative than the quadratic approach. The results are also extended to T-S fuzzy systems with time-varying parameter uncertainties. All the results are formulated in the form of linear matrix inequalities, which can be readily solved via standard numerical software. Two examples are provided to illustrate the usefulness and applicability of the developed theoretical results. © 2009 IEEE.published_or_final_versio