24,246 research outputs found
Expansion around half-integer values, binomial sums and inverse binomial sums
I consider the expansion of transcendental functions in a small parameter
around rational numbers. This includes in particular the expansion around
half-integer values. I present algorithms which are suitable for an
implementation within a symbolic computer algebra system. The method is an
extension of the technique of nested sums. The algorithms allow in addition the
evaluation of binomial sums, inverse binomial sums and generalizations thereof.Comment: 21 page
- 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
Brushing the hairs of transcendental entire functions
Let f be a hyperbolic transcendental entire function of finite order in the
Eremenko-Lyubich class (or a finite composition of such maps), and suppose that
f has a unique Fatou component. We show that the Julia set of is a Cantor
bouquet; i.e. is ambiently homeomorphic to a straight brush in the sense of
Aarts and Oversteegen. In particular, we show that any two such Julia sets are
ambiently homeomorphic.
We also show that if f\in\B has finite order (or is a finite composition of
such maps), but is not necessarily hyperbolic, then the Julia set of f contains
a Cantor bouquet.
As part of our proof, we describe, for an arbitrary function f\in\B, a
natural compactification of the dynamical plane by adding a "circle of
addresses" at infinity.Comment: 19 pages. V2: Small number of minor corrections made from V
On the complexity of algebraic number I. Expansions in integer bases
Let be an integer. We prove that the -adic expansion of every
irrational algebraic number cannot have low complexity. Furthermore, we
establish that irrational morphic numbers are transcendental, for a wide class
of morphisms. In particular, irrational automatic numbers are transcendental.
Our main tool is a new, combinatorial transcendence criterion
Rational approximation for solving an implicitly given Colebrook flow friction equation
The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Pade approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.Web of Science81art. no. 2
Accurate and efficient explicit approximations of the Colebrook flow friction equation based on the Wright omega-function
The Colebrook equation is a popular model for estimating friction loss coefficients in water and gas pipes. The model is implicit in the unknown flow friction factor, f. To date, the captured flow friction factor, f, can be extracted from the logarithmic form analytically only in the term of the Lambert W-function. The purpose of this study is to find an accurate and computationally efficient solution based on the shifted Lambert W-function also known as the Wright omega-function. The Wright omega-function is more suitable because it overcomes the problem with the overflow error by switching the fast growing term, y = W (e(x)), of the Lambert W-function to series expansions that further can be easily evaluated in computers without causing overflow run-time errors. Although the Colebrook equation transformed through the Lambert W-function is identical to the original expression in terms of accuracy, a further evaluation of the Lambert W-function can be only approximate. Very accurate explicit approximations of the Colebrook equation that contain only one or two logarithms are shown. The final result is an accurate explicit approximation of the Colebrook equation with a relative error of no more than 0.0096%. The presented approximations are in a form suitable for everyday engineering use, and are both accurate and computationally efficient.Web of Science71art. no. 3
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