Derivative moments for characteristic polynomials from the CUE

We calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer. The calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity. The latter asymptotic calculation allows us to prove a long-standing conjecture from random matrix theory.Comment: 31 pages, 3 figure

Bagchi's Theorem for families of automorphic forms

We prove a version of Bagchi's Theorem and of Voronin's Universality Theorem for family of primitive cusp forms of weight $2$ and prime level, and discuss under which conditions the argument will apply to general reasonable family of automorphic $L$-functions.Comment: 15 page

Exponential sums with coefficients of certain Dirichlet series

Under the generalized Lindel\"of Hypothesis in the t- and q-aspects, we bound exponential sums with coefficients of Dirichlet series belonging to a certain class. We use these estimates to establish a conditional result on squares of Hecke eigenvalues at Piatetski-Shapiro primes.Comment: 13 page

Elliptic curves of large rank and small conductor

For r=6,7,...,11 we find an elliptic curve E/Q of rank at least r and the smallest conductor known, improving on the previous records by factors ranging from 1.0136 (for r=6) to over 100 (for r=10 and r=11). We describe our search methods, and tabulate, for each r=5,6,...,11, the five curves of lowest conductor, and (except for r=11) also the five of lowest absolute discriminant, that we found.Comment: 16 pages, including tables and one .eps figure; to appear in the Proceedings of ANTS-6 (June 2004, Burlington, VT). Revised somewhat after comments by J.Silverman on the previous draft, and again to get the correct page break

On absolute moments of characteristic polynomials of a certain class of complex random matrices

Integer moments of the spectral determinant $|\det(zI-W)|^2$ of complex random matrices $W$ are obtained in terms of the characteristic polynomial of the Hermitian matrix $WW^*$ for the class of matrices $W=AU$ where $A$ is a given matrix and $U$ is random unitary. This work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results in this context are discussed.Comment: 41 page, typos correcte

Random matrix ensembles of time correlation matrices to analyze visual lifelogs

Visual lifelogging is the process of automatically recording images and other sensor data for the purpose of aiding memory recall. Such lifelogs are usually created using wearable cameras. Given the vast amount of images that are maintained in a visual lifelog, it is a significant challenge for users to deconstruct a sizeable collection of images into meaningful events. In this paper, random matrix theory (RMT) is applied to a cross-correlation matrix C, constructed using SenseCam lifelog data streams to identify such events. The analysis reveals a number of eigenvalues that deviate from the spectrum suggested by RMT. The components of the deviating eigenvectors are found to correspond to “distinct significant events” in the visual lifelogs. Finally, the cross-correlation matrix C is cleaned by separating the noisy part from the non-noisy part. Overall, the RMT technique is shown to be useful to detect major events in SenseCam images