2,928 research outputs found
Parametric Euler Sum Identities
We consider some parametrized classes of multiple sums first studied by
Euler. Identities between meromorphic functions of one or more variables
generate reduction formulae for these sums.Comment: 12 page
Minimization of entropy functionals
Entropy functionals (i.e. convex integral functionals) and extensions of
these functionals are minimized on convex sets. This paper is aimed at reducing
as much as possible the assumptions on the constraint set. Dual equalities and
characterizations of the minimizers are obtained with weak constraint
qualifications
Densities of short uniform random walks
We study the densities of uniform random walks in the plane. A special focus
is on the case of short walks with three or four steps and less completely
those with five steps. As one of the main results, we obtain a hypergeometric
representation of the density for four steps, which complements the classical
elliptic representation in the case of three steps. It appears unrealistic to
expect similar results for more than five steps. New results are also presented
concerning the moments of uniform random walks and, in particular, their
derivatives. Relations with Mahler measures are discussed.Comment: 32 pages, 9 figure
Log-sine evaluations of Mahler measures, II
We continue the analysis of higher and multiple Mahler measures using
log-sine integrals as started in "Log-sine evaluations of Mahler measures" and
"Special values of generalized log-sine integrals" by two of the authors. This
motivates a detailed study of various multiple polylogarithms and worked
examples are given. Our techniques enable the reduction of several multiple
Mahler measures, and supply an easy proof of two conjectures by Boyd.Comment: 35 page
Efficient implementation of the Hardy-Ramanujan-Rademacher formula
We describe how the Hardy-Ramanujan-Rademacher formula can be implemented to
allow the partition function to be computed with softly optimal
complexity and very little overhead. A new implementation
based on these techniques achieves speedups in excess of a factor 500 over
previously published software and has been used by the author to calculate
, an exponent twice as large as in previously reported
computations.
We also investigate performance for multi-evaluation of , where our
implementation of the Hardy-Ramanujan-Rademacher formula becomes superior to
power series methods on far denser sets of indices than previous
implementations. As an application, we determine over 22 billion new
congruences for the partition function, extending Weaver's tabulation of 76,065
congruences.Comment: updated version containing an unconditional complexity proof;
accepted for publication in LMS Journal of Computation and Mathematic
M\"untz spaces and Remez inequalities
Two relatively long-standing conjectures concerning M\"untz polynomials are
resolved. The central tool is a bounded Remez type inequality for non-dense
M\"untz spaces.Comment: 5 page
Maximality of the sum of a maximally monotone linear relation and a maximally monotone operator
The most famous open problem in Monotone Operator Theory concerns the maximal
monotonicity of the sum of two maximally monotone operators provided that
Rockafellar's constraint qualification holds.
In this paper, we prove the maximal monotonicity of provided that are maximally monotone and is a linear relation, as soon as
Rockafellar's constraint qualification holds: \dom A\cap\inte\dom
B\neq\varnothing. Moreover, is of type (FPV).Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:1010.4346, arXiv:1005.224
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