2,348 research outputs found
Periodic representations and rational approximations of square roots
In this paper the properties of R\'edei rational functions are used to derive
rational approximations for square roots and both Newton and Pad\'e
approximations are given as particular cases. As a consequence, such
approximations can be derived directly by power matrices. Moreover, R\'edei
rational functions are introduced as convergents of particular periodic
continued fractions and are applied for approximating square roots in the field
of p-adic numbers and to study periodic representations. Using the results over
the real numbers, we show how to construct periodic continued fractions and
approximations of square roots which are simultaneously valid in the real and
in the p-adic field
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Branch points, m. Fractions, and rational functions matching both derivatives and values
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
Algebr
Transition probabilities for general birth-death processes with applications in ecology, genetics, and evolution
A birth-death process is a continuous-time Markov chain that counts the
number of particles in a system over time. In the general process with
current particles, a new particle is born with instantaneous rate
and a particle dies with instantaneous rate . Currently no robust and
efficient method exists to evaluate the finite-time transition probabilities in
a general birth-death process with arbitrary birth and death rates. In this
paper, we first revisit the theory of continued fractions to obtain expressions
for the Laplace transforms of these transition probabilities and make explicit
an important derivation connecting transition probabilities and continued
fractions. We then develop an efficient algorithm for computing these
probabilities that analyzes the error associated with approximations in the
method. We demonstrate that this error-controlled method agrees with known
solutions and outperforms previous approaches to computing these probabilities.
Finally, we apply our novel method to several important problems in ecology,
evolution, and genetics
Rational approximations to algebraic Laurent series with coefficients in a finite field
In this paper we give a general upper bound for the irrationality exponent of
algebraic Laurent series with coefficients in a finite field. Our proof is
based on a method introduced in a different framework by Adamczewski and
Cassaigne. It makes use of automata theory and, in our context, of a classical
theorem due to Christol. We then introduce a new approach which allows us to
strongly improve this general bound in many cases. As an illustration, we give
few examples of algebraic Laurent series for which we are able to compute the
exact value of the irrationality exponent
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