Sharp nonasymptotic bounds on the norm of random matrices with independent entries

We obtain nonasymptotic bounds on the spectral norm of random matrices with independent entries that improve significantly on earlier results. If $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that $$\mathbf{E}\Vert X\Vert \lesssim\max_i\sqrt{\sum_jb_{ij}^2}+\max _{ij}\vert b_{ij}\vert \sqrt{\log n}.$$ This bound is optimal in the sense that a matching lower bound holds under mild assumptions, and the constants are sufficiently sharp that we can often capture the precise edge of the spectrum. Analogous results are obtained for rectangular matrices and for more general sub-Gaussian or heavy-tailed distributions of the entries, and we derive tail bounds in addition to bounds on the expected norm. The proofs are based on a combination of the moment method and geometric functional analysis techniques. As an application, we show that our bounds immediately yield the correct phase transition behavior of the spectral edge of random band matrices and of sparse Wigner matrices. We also recover a result of Seginer on the norm of Rademacher matrices.Comment: Published at http://dx.doi.org/10.1214/15-AOP1025 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Classical Kloosterman sums: representation theory, magic squares, and Ramanujan multigraphs

We study the representation theory of a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of "magical" combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.Comment: 20 page

Formal Scheduling Constraints for Time-Sensitive Networks

In recent years, the IEEE 802.1 Time Sensitive Networking (TSN) task group has been active standardizing time-sensitive capabilities for Ethernet networks ranging from distributed clock synchronization and time-based ingress policing to frame preemption, redundancy management, and scheduled traffic enhancements. In particular the scheduled traffic enhancements defined in IEEE 802.1Qbv together with the clock synchronization protocol open up the possibility to schedule communication in distributed networks providing real-time guarantees. In this paper we formalize the necessary constraints for creating window-based IEEE~802.1Qbv Gate Control List schedules for Time-sensitive Networks (TSN). The resulting schedules allow a greater flexibility in terms of timing properties while still guaranteeing deterministic communication with bounded jitter and end-to-end latency

Musical Actions of Dihedral Groups

The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.Comment: 27 pages, 11 figures. To appear in the American Mathematical Monthly

Ensuring a More Evidence-based Policy for Basic Education

Can the Philippines achieve its goal on two key international commitments--the Millennium Development Goals (MDGs) on Education and the Education for All (EFA) Initiative--for the achievement of education for everyone? A closer look at the figures would show that the country may be at risk of not achieving its goal of attaining universal primary education by 2015. Given this scenario, the authors suggest that the government should intensify its efforts in improving basic education by having evidence-based policies and actions. In particular, they note that the Department of Education (DepEd) should be required to have a sound monitoring and evaluation system to regularly assess the conditions of basic education.basic education, millennium development goal (MDG), education indicators, monitoring and evaluation

On the nature of the AGILE galactic transient sources

The Italian gamma-ray satellite AGILE has recently reported the detection of some variable high-energy sources likely of galactic origin. These sources do not have any obvious counterpart at lower energies. We propose that these sources are produced in proton-dominated jets of galactic microquasars. We develop a model for microquasar jets that takes into account both primary leptons and protons and all relevant radiative processes, including secondary particle emission and gamma-ray attenuation due to pair creation in the jet. We obtain spectral energy distributions that correspond to what is observed by AGILE, with most of the power concentrated between 100 MeV and 10 GeV and reaching luminosities of $10^{34-35}$ erg s$^{-1}$. We make detailed spectral predictions that can be tested by the Fermi gamma-ray telescope in the immediate future. We conclude that hadronic jets in galactic accreting sources can be responsible for the variable unidentified gamma-ray sources detected by AGILE.Comment: 4 pages, 2 figures. Accepted for publication in Astronomy & Astrophysics (Letters