Upper tails for triangles

With $\xi$ the number of triangles in the usual (Erd\H{o}s-R\'enyi) random graph $G(m,p)$, $p>1/m$ and $\eta>0$, we show (for some $C_{\eta}>0$) \Pr(\xi> (1+\eta)\E \xi) < \exp[-C_{\eta}\min{m^2p^2\log(1/p),m^3p^3}]. This is tight up to the value of $C_{\eta}$.Comment: 10 page

Digital learning objects: A need for educational leadership

Despite increasing interest in technology-assisted education, technology-based instructional design still lacks support from a reliable body of empirical research. This dearth of reliable information hampers its integration into mainstream school systems. In fact, many teachers remain resistant to using technology in the classroom. In order to overcome teacher resistance to technology in the classroom, we have sought to follow a process described by Friesen to evaluate the advantages and disadvantages of the educational use of digital learning objects (DLOs) from the teachers' point of view. This article explores the opportunities and challenges inherent in using digital learning objects and reports on the impact of DLO use at both the classroom and school levels. By providing research that links students' use of DLOs with the development of key competencies, we hope to sharpen teachers' visions of how DLOs can help them achieve their educational goals, and to encourage DLO uptake for educational purposes. Finally, we envision a DLO that can assist school principals in the facilitation of educational leadership and help transform teachers' attitudes toward technology-based teaching

Roots of polynomials of degrees 3 and 4

We present the solutions of equations of degrees 3 and 4 using Galois theory and some simple Fourier analysis for finite groups, together with historical comments on these and other solution methods.Comment: 29 page

Conditioned Galton-Watson trees do not grow

An example is given which shows that, in general, conditioned Galton-Watson trees cannot be obtained by adding vertices one by one, as has been shown in a special case by Luczak and Winkler.Comment: 5 pages, 2 figure

The largest component in a subcritical random graph with a power law degree distribution

It is shown that in a subcritical random graph with given vertex degrees satisfying a power law degree distribution with exponent $\gamma>3$, the largest component is of order $n^{1/(\gamma-1)}$. More precisely, the order of the largest component is approximatively given by a simple constant times the largest vertex degree. These results are extended to several other random graph models with power law degree distributions. This proves a conjecture by Durrett.Comment: Published in at http://dx.doi.org/10.1214/07-AAP490 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas

This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz.\spacefactor =1000 Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright's constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which appears in slightly different ways in formulas for all seven random variables. We further want to give explicit relations between the many different similar notations and definitions that have been used by various authors. There are also some new results, mainly to fill in gaps left in the literature. Some short proofs are given, but most proofs are omitted and the reader is instead referred to the original sources.Comment: Published at http://dx.doi.org/10.1214/07-PS104 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org