5,179 research outputs found
Acceleration of generalized hypergeometric functions through precise remainder asymptotics
We express the asymptotics of the remainders of the partial sums {s_n} of the
generalized hypergeometric function q+1_F_q through an inverse power series z^n
n^l \sum_k c_k/n^k, where the exponent l and the asymptotic coefficients {c_k}
may be recursively computed to any desired order from the hypergeometric
parameters and argument. From this we derive a new series acceleration
technique that can be applied to any such function, even with complex
parameters and at the branch point z=1. For moderate parameters (up to
approximately ten) a C implementation at fixed precision is very effective at
computing these functions; for larger parameters an implementation in higher
than machine precision would be needed. Even for larger parameters, however,
our C implementation is able to correctly determine whether or not it has
converged; and when it converges, its estimate of its error is accurate.Comment: 36 pages, 6 figures, LaTeX2e. Fixed sign error in Eq. (2.28), added
several references, added comparison to other methods, and added discussion
of recursion stabilit
Summation of rational series twisted by strongly B-multiplicative coefficients
We evaluate in closed form series of the type , where
is a strongly -multiplicative sequence and a (well-chosen)
rational function. A typical example is: where is the sum of the
binary digits of the integer . Furthermore closed formulas for series
involving automatic sequences that are not strongly -multiplicative, such as
the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for
example, for integer : where is the regular paperfolding sequence and is an Euler number.Comment: Typo in a crossreference corrected in Example 9, page 6. Remark added
top of Page 9 about the relation between paperfolding and the
Jacobi-Kronecker symbo
Convergence of a greedy algorithm for high-dimensional convex nonlinear problems
In this article, we present a greedy algorithm based on a tensor product
decomposition, whose aim is to compute the global minimum of a strongly convex
energy functional. We prove the convergence of our method provided that the
gradient of the energy is Lipschitz on bounded sets. The main interest of this
method is that it can be used for high-dimensional nonlinear convex problems.
We illustrate this method on a prototypical example for uncertainty propagation
on the obstacle problem.Comment: 36 pages, 9 figures, accepted in Mathematical Models and Methods for
Applied Science
Extending pseudo-Anosov maps to compression bodies
We show that a pseudo-Anosov map on a boundary component of an irreducible
3-manifold has a power that partially extends to the interior if and only if
its (un)stable lamination is a projective limit of meridians. The proof is
through 3-dimensional hyperbolic geometry, and involves an investigation of
algebraic limits of convex cocompact compression bodies.Comment: 29 page
Special values of multiple polylogarithms
Historically, the polylogarithm has attracted specialists and non-specialists alike with its lovely evaluations. Much the same can be said for Euler sums (or multiple harmonic sums), which, within the past decade, have arisen in combinatorics, knot theory and high-energy physics. More recently, we have been forced to consider multidimensional extensions encompassing the classical polylogarithm, Euler sums, and the Riemann zeta function. Here, we provide a general framework within which previously isolated results can now be properly understood. Applying the theory developed herein, we prove several previously conjectured evaluations, including an intriguing conjecture of Don Zagier
Convergence of Entropic Schemes for Optimal Transport and Gradient Flows
Replacing positivity constraints by an entropy barrier is popular to
approximate solutions of linear programs. In the special case of the optimal
transport problem, this technique dates back to the early work of
Schr\"odinger. This approach has recently been used successfully to solve
optimal transport related problems in several applied fields such as imaging
sciences, machine learning and social sciences. The main reason for this
success is that, in contrast to linear programming solvers, the resulting
algorithms are highly parallelizable and take advantage of the geometry of the
computational grid (e.g. an image or a triangulated mesh). The first
contribution of this article is the proof of the -convergence of the
entropic regularized optimal transport problem towards the Monge-Kantorovich
problem for the squared Euclidean norm cost function. This implies in
particular the convergence of the optimal entropic regularized transport plan
towards an optimal transport plan as the entropy vanishes. Optimal transport
distances are also useful to define gradient flows as a limit of implicit Euler
steps according to the transportation distance. Our second contribution is a
proof that implicit steps according to the entropic regularized distance
converge towards the original gradient flow when both the step size and the
entropic penalty vanish (in some controlled way)
Derived induction and restriction theory
Let be a finite group. To any family of subgroups of ,
we associate a thick -ideal of the
category of -spectra with the property that every -spectrum in
(which we call -nilpotent) can be
reconstructed from its underlying -spectra as varies over .
A similar result holds for calculating -equivariant homotopy classes of maps
into such spectra via an appropriate homotopy limit spectral sequence. In
general, the condition implies strong
collapse results for this spectral sequence as well as its dual homotopy
colimit spectral sequence. As applications, we obtain Artin and Brauer type
induction theorems for -equivariant -homology and cohomology, and
generalizations of Quillen's -isomorphism theorem when is a
homotopy commutative -ring spectrum.
We show that the subcategory contains many
-spectra of interest for relatively small families . These
include -equivariant real and complex -theory as well as the
Borel-equivariant cohomology theories associated to complex oriented ring
spectra, any -local spectrum, the classical bordism theories, connective
real -theory, and any of the standard variants of topological modular forms.
In each of these cases we identify the minimal family such that these results
hold.Comment: 63 pages. Many edits and some simplifications. Final version, to
appear in Geometry and Topolog
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