29 research outputs found
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 and prime level, and discuss
under which conditions the argument will apply to general reasonable family of
automorphic -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 of complex
random matrices are obtained in terms of the characteristic polynomial of
the Hermitian matrix for the class of matrices where is a
given matrix and 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 âdistinct significant eventsâ 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