140,719 research outputs found
Computing Jacobi's in quasi-linear time
Jacobi's function has numerous applications in mathematics and
computer science; a naive algorithm allows the computation of ,
for verifying certain conditions, with precision in
bit operations, where denotes the
number of operations needed to multiply two complex -bit numbers. We
generalize an algorithm which computes specific values of the function
(the \textit{theta-constants}) in asymptotically faster time; this gives us an
algorithm to compute with precision in bit operations, for any and reduced using
the quasi-periodicity of
Singular value decay of operator-valued differential Lyapunov and Riccati equations
We consider operator-valued differential Lyapunov and Riccati equations,
where the operators and may be relatively unbounded with respect to
(in the standard notation). In this setting, we prove that the singular values
of the solutions decay fast under certain conditions. In fact, the decay is
exponential in the negative square root if generates an analytic semigroup
and the range of has finite dimension. This extends previous similar
results for algebraic equations to the differential case. When the initial
condition is zero, we also show that the singular values converge to zero as
time goes to zero, with a certain rate that depends on the degree of
unboundedness of . A fast decay of the singular values corresponds to a low
numerical rank, which is a critical feature in large-scale applications. The
results reported here provide a theoretical foundation for the observation
that, in practice, a low-rank factorization usually exists.Comment: Corrected some misconceptions, which lead to more general results
(e.g. exponential stability is no longer required). Also fixed some
off-by-one errors, improved the presentation, and added/extended several
remarks on possible generalizations. Now 22 pages, 8 figure
The Quaternion-Based Spatial Coordinate and Orientation Frame Alignment Problems
We review the general problem of finding a global rotation that transforms a
given set of points and/or coordinate frames (the "test" data) into the best
possible alignment with a corresponding set (the "reference" data). For 3D
point data, this "orthogonal Procrustes problem" is often phrased in terms of
minimizing a root-mean-square deviation or RMSD corresponding to a Euclidean
distance measure relating the two sets of matched coordinates. We focus on
quaternion eigensystem methods that have been exploited to solve this problem
for at least five decades in several different bodies of scientific literature
where they were discovered independently. While numerical methods for the
eigenvalue solutions dominate much of this literature, it has long been
realized that the quaternion-based RMSD optimization problem can also be solved
using exact algebraic expressions based on the form of the quartic equation
solution published by Cardano in 1545; we focus on these exact solutions to
expose the structure of the entire eigensystem for the traditional 3D spatial
alignment problem. We then explore the structure of the less-studied
orientation data context, investigating how quaternion methods can be extended
to solve the corresponding 3D quaternion orientation frame alignment (QFA)
problem, noting the interesting equivalence of this problem to the
rotation-averaging problem, which also has been the subject of independent
literature threads. We conclude with a brief discussion of the combined 3D
translation-orientation data alignment problem. Appendices are devoted to a
tutorial on quaternion frames, a related quaternion technique for extracting
quaternions from rotation matrices, and a review of quaternion
rotation-averaging methods relevant to the orientation-frame alignment problem.
Supplementary Material covers extensions of quaternion methods to the 4D
problem.Comment: This replaces an early draft that lacked a number of important
references to previous work. There are also additional graphics elements. The
extensions to 4D data and additional details are worked out in the
Supplementary Material appended to the main tex
An Adaptive Mechanism for Accurate Query Answering under Differential Privacy
We propose a novel mechanism for answering sets of count- ing queries under
differential privacy. Given a workload of counting queries, the mechanism
automatically selects a different set of "strategy" queries to answer
privately, using those answers to derive answers to the workload. The main
algorithm proposed in this paper approximates the optimal strategy for any
workload of linear counting queries. With no cost to the privacy guarantee, the
mechanism improves significantly on prior approaches and achieves near-optimal
error for many workloads, when applied under (\epsilon, \delta)-differential
privacy. The result is an adaptive mechanism which can help users achieve good
utility without requiring that they reason carefully about the best formulation
of their task.Comment: VLDB2012. arXiv admin note: substantial text overlap with
arXiv:1103.136
Heegner points on Cartan non-split curves
Let be an elliptic curve of conductor , and let be an imaginary
quadratic field such that the root number of is . Let be an order
in and assume that there exists an odd prime , such that , and is inert in . Although there are no Heegner points on
attached to , in this article we construct such points on Cartan non-split
curves. In order to do that we give a method to compute Fourier expansions for
forms in Cartan non-split curves, and prove that the constructed points form a
Heegner system as in the classical case.Comment: 25 pages, revised versio
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