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

Orientation of particle attachment and local isotropy in diffusion limited aggregation (DLA)

We simulate 50 off-lattice DLA clusters, one million particles each. The probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically. For increasing cluster size, $N$, the distribution crosses over extremely accurately to a cosine, whose amplitude decreases towards zero as a power-law in $N$. From this viewpoint, asymptotically large DLA clusters are locally $isotropic$. This contradicts previous conclusions drawn from density-density correlation measurements [P. Meakin, and T. Viscek, Phys. Rev. A {\bf 32}, 685 (1985)]. We present an intuitive phenomenological model random process for our numerical findings.Comment: 10 pages, RevTex 3.0, 11-9

Schubert Polynomials for the affine Grassmannian of the symplectic group

We study the Schubert calculus of the affine Grassmannian Gr of the symplectic group. The integral homology and cohomology rings of Gr are identified with dual Hopf algebras of symmetric functions, defined in terms of Schur's P and Q-functions. An explicit combinatorial description is obtained for the Schubert basis of the cohomology of Gr, and this is extended to a definition of the affine type C Stanley symmetric functions. A homology Pieri rule is also given for the product of a special Schubert class with an arbitrary one.Comment: 45 page

Analysis and control of bifurcation and chaos in averaged queue length in TCP/RED model

This paper studies the bifurcation and chaos phenomena in averaged queue length in a developed Transmission Control Protocol (TCP) model with Random Early Detection (RED) mechanism. Bifurcation and chaos phenomena are nonlinear behaviour in network systems that lead to degradation of the network performance. The TCP/RED model used is a model validated previously. In our study, only the average queue size k q − is considered, and the results are based on analytical model rather than actual measurements. The instabilities in the model are studied numerically using the conventional nonlinear bifurcation analysis. Extending from this bifurcation analysis, a modified RED algorithm is derived to prevent the observed bifurcation and chaos regardless of the selected parameters. Our modification is for the simple scenario of a single RED router carrying only TCP traffic. The algorithm neither compromises the throughput nor the average queuing delay of the system

XUV Frequency Combs via Femtosecond Enhancement Cavities

We review the current state of tabletop extreme ultraviolet (XUV) sources based on high harmonic generation (HHG) in femtosecond enhancement cavities (fsEC). Recent developments have enabled generation of high photon flux (1014 photons/sec) in the XUV, at high repetition rates (>50 MHz) and spanning the spectral region from 40 nm - 120 nm. This level of performance has enabled precision spectroscopy with XUV frequency combs and promises further applications in XUV spectroscopic and photoemission studies. We discuss the theory of operation and experimental details of the fsEC and XUV generation based on HHG, including current technical challenges to increasing the photon flux and maximum photon energy produced by this type of system. Current and future applications for these sources are also discussed.Comment: invited review article, 38 page