48,419 research outputs found
Computing functions on Jacobians and their quotients
We show how to efficiently compute functions on jacobian varieties and their
quotients. We deduce a quasi-optimal algorithm to compute isogenies
between jacobians of genus two curves
Numerical Evaluation of Six-Photon Amplitudes
We apply the recently proposed amplitude reduction at the integrand level
method, to the computation of the scattering process 2 photons -> 4 photons,
including the case of a massive fermion loop. We also present several
improvements of the method, including a general strategy to reconstruct the
rational part of any one-loop amplitude and the treatment of vanishing
Gram-determinants.Comment: 21 pages, 3 figures. Version accepted for publication in JHE
Continued fractions of certain Mahler functions
We investigate the continued fraction expansion of the infinite products
where polynomials satisfy
and . We construct relations between partial quotients of
which can be used to get recurrent formulae for them. We provide that
formulae for the cases and . As an application, we prove that for
where is an arbitrary rational number except 0 and 1, and for
any integer with such that the irrationality exponent
of equals two. In the case we provide a partial analogue of the
last result with several collections of polynomials giving the
irrationality exponent of strictly bigger than two.Comment: 25 page
Accurate and Efficient Expression Evaluation and Linear Algebra
We survey and unify recent results on the existence of accurate algorithms
for evaluating multivariate polynomials, and more generally for accurate
numerical linear algebra with structured matrices. By "accurate" we mean that
the computed answer has relative error less than 1, i.e., has some correct
leading digits. We also address efficiency, by which we mean algorithms that
run in polynomial time in the size of the input. Our results will depend
strongly on the model of arithmetic: Most of our results will use the so-called
Traditional Model (TM). We give a set of necessary and sufficient conditions to
decide whether a high accuracy algorithm exists in the TM, and describe
progress toward a decision procedure that will take any problem and provide
either a high accuracy algorithm or a proof that none exists. When no accurate
algorithm exists in the TM, it is natural to extend the set of available
accurate operations by a library of additional operations, such as , dot
products, or indeed any enumerable set which could then be used to build
further accurate algorithms. We show how our accurate algorithms and decision
procedure for finding them extend to this case. Finally, we address other
models of arithmetic, and the relationship between (im)possibility in the TM
and (in)efficient algorithms operating on numbers represented as bit strings.Comment: 49 pages, 6 figures, 1 tabl
Block Toeplitz determinants, constrained KP and Gelfand-Dickey hierarchies
We propose a method for computing any Gelfand-Dickey tau function living in
Segal-Wilson Grassmannian as the asymptotics of block Toeplitz determinant
associated to a certain class of symbols. Also truncated block Toeplitz
determinants associated to the same symbols are shown to be tau function for
rational reductions of KP. Connection with Riemann-Hilbert problems is
investigated both from the point of view of integrable systems and block
Toeplitz operator theory. Examples of applications to algebro-geometric
solutions are given.Comment: 35 pages. Typos corrected, some changes in the introductio
- …