11,446 research outputs found
Random Dynamics of Transcendental Functions
This work concerns random dynamics of hyperbolic entire and meromorphic
functions of finite order and whose derivative satisfies some growth condition
at infinity. This class contains most of the classical families of
transcendental functions and goes much beyond. Based on uniform versions of
Nevanlinna's value distribution theory we first build a thermodynamical
formalism which, in particular, produces unique geometric and fiberwise
invariant Gibbs states. Moreover, spectral gap property for the associated
transfer operator along with exponential decay of correlations and a central
limit theorem are shown. This part relies on our construction of new positive
invariant cones that are adapted to the setting of unbounded phase spaces. This
setting rules out the use of Hilbert's metric along with the usual contraction
principle. However these cones allow us to apply a contraction argument
stemming from Bowen's initial approach.Comment: Final Version, to appear in J. d'Analyse Math, 35 page
Satisfiability Modulo Transcendental Functions via Incremental Linearization
In this paper we present an abstraction-refinement approach to Satisfiability
Modulo the theory of transcendental functions, such as exponentiation and
trigonometric functions. The transcendental functions are represented as
uninterpreted in the abstract space, which is described in terms of the
combined theory of linear arithmetic on the rationals with uninterpreted
functions, and are incrementally axiomatized by means of upper- and
lower-bounding piecewise-linear functions. Suitable numerical techniques are
used to ensure that the abstractions of the transcendental functions are sound
even in presence of irrationals. Our experimental evaluation on benchmarks from
verification and mathematics demonstrates the potential of our approach,
showing that it compares favorably with delta-satisfiability /interval
propagation and methods based on theorem proving
- XSummer - Transcendental Functions and Symbolic Summation in Form
Harmonic sums and their generalizations are extremely useful in the
evaluation of higher-order perturbative corrections in quantum field theory. Of
particular interest have been the so-called nested sums,where the harmonic sums
and their generalizations appear as building blocks, originating for example
from the expansion of generalized hypergeometric functions around integer
values of the parameters. In this Letter we discuss the implementation of
several algorithms to solve these sums by algebraic means, using the computer
algebra system Form.Comment: 21 pages, 1 figure, Late
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