593 research outputs found
Non-Rectangular Convolutions and (Sub-)Cadences with Three Elements
The discrete acyclic convolution computes the 2n-1 sums sum_{i+j=k; (i,j) in
[0,1,2,...,n-1]^2} (a_i b_j) in O(n log n) time. By using suitable offsets and
setting some of the variables to zero, this method provides a tool to calculate
all non-zero sums sum_{i+j=k; (i,j) in (P cap Z^2)} (a_i b_j) in a rectangle P
with perimeter p in O(p log p) time.
This paper extends this geometric interpretation in order to allow arbitrary
convex polygons P with k vertices and perimeter p. Also, this extended
algorithm only needs O(k + p(log p)^2 log k) time.
Additionally, this paper presents fast algorithms for counting sub-cadences
and cadences with 3 elements using this extended method
Euler systems for Rankin--Selberg convolutions of modular forms
We construct an Euler system in the cohomology of the tensor product of the
Galois representations attached to two modular forms, using elements in the
higher Chow groups of products of modular curves. We use this Euler system to
prove a finiteness theorem for the strict Selmer group of the Galois
representation when the associated p-adic Rankin--Selberg L-function is
non-vanishing at s = 1.Comment: Revised version with many updates and correction
Orthogonal transforms and their application to image coding
Imperial Users onl
Euler systems for modular forms over imaginary quadratic fields
We construct an Euler system attached to a weight 2 modular form twisted by a
Groessencharacter of an imaginary quadratic field, and apply this to bounding
Selmer groups.Comment: Revised version, to appear in Compositio Mat
An annotated bibliography for comparative prime number theory
The goal of this annotated bibliography is to record every publication on the
topic of comparative prime number theory together with a summary of its
results. We use a unified system of notation for the quantities being studied
and for the hypotheses under which results are obtained. We encourage feedback
on this manuscript (see the end of Section~1 for details).Comment: 98 pages; supersedes "Comparative prime number theory: A survey"
(arXiv:1202.3408
Higher rank lamplighter groups are graph automatic
We show that the higher rank lamplighter groups, or Diestel-Leader groups
for , are graph automatic. This introduces a new family
of graph automatic groups which are not automatic
- …