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

Asymptotic bias of some election methods

Consider an election where N seats are distributed among parties with proportions p_1,...,p_m of the votes. We study, for the common divisor and quota methods, the asymptotic distribution, and in particular the mean, of the seat excess of a party, i.e. the difference between the number of seats given to the party and the (real) number Np_i that yields exact proportionality. Our approach is to keep p_1,...,p_m fixed and let N tend to infinity, with N random in a suitable way. In particular, we give formulas showing the bias favouring large or small parties for the different election methods.Comment: 54 page

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

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

Congruence properties of depths in some random trees

Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n tends to infinty. The same is true for the number of vertices of depth divisible by m for m=3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m at least 7, the number is not asymptotically normal, and the variance grows faster than linear in n. The case m=6 is intermediate: the number is asymptotically normal but the variance is of order n log n. This is a simple and striking example of a type of phase transition that has been observed by other authors in several cases. We prove, and perhaps explain, this non-intuitive behavious using a translation to a generalized Polya urn. Similar results hold for a random binary search tree; now the number of vertices of depth divisible by m is asymptotically normal for m at most 8 but not for m at least 9, and the variance grows linearly in the first case both faster in the second. (There is no intermediate case.) In contrast, we show that for conditioned Galton-Watson trees, including random labelled trees and random binary trees, there is no such phase transition: the number is asymptotically normal for every m.Comment: 23 page

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