7,947 research outputs found
Commensurable continued fractions
We compare two families of continued fractions algorithms, the symmetrized
Rosen algorithm and the Veech algorithm. Each of these algorithms expands real
numbers in terms of certain algebraic integers. We give explicit models of the
natural extension of the maps associated with these algorithms; prove that
these natural extensions are in fact conjugate to the first return map of the
geodesic flow on a related surface; and, deduce that, up to a conjugacy, almost
every real number has an infinite number of common approximants for both
algorithms.Comment: 41 pages, 10 figure
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
Melvin Models and Diophantine Approximation
Melvin models with irrational twist parameter provide an interesting example
of conformal field theories with non-compact target space, and localized states
which are arbitrarily close to being delocalized. We study the torus partition
sum of these models, focusing on the properties of the regularized dimension of
the space of localized states. We show that its behavior is related to
interesting arithmetic properties of the twist parameter , such as the
Lyapunov exponent. Moreover, for in a set of measure one the
regularized dimension is in fact not a well-defined number but must be
considered as a random variable in a probability distribution.Comment: 26pp. harvmac(b); v2: Some clarifications and references added. One
small error corrected; v3: reference adde
Status of the differential transformation method
Further to a recent controversy on whether the differential transformation
method (DTM) for solving a differential equation is purely and solely the
traditional Taylor series method, it is emphasized that the DTM is currently
used, often only, as a technique for (analytically) calculating the power
series of the solution (in terms of the initial value parameters). Sometimes, a
piecewise analytic continuation process is implemented either in a numerical
routine (e.g., within a shooting method) or in a semi-analytical procedure
(e.g., to solve a boundary value problem). Emphasized also is the fact that, at
the time of its invention, the currently-used basic ingredients of the DTM
(that transform a differential equation into a difference equation of same
order that is iteratively solvable) were already known for a long time by the
"traditional"-Taylor-method users (notably in the elaboration of software
packages --numerical routines-- for automatically solving ordinary differential
equations). At now, the defenders of the DTM still ignore the, though much
better developed, studies of the "traditional"-Taylor-method users who, in
turn, seem to ignore similarly the existence of the DTM. The DTM has been given
an apparent strong formalization (set on the same footing as the Fourier,
Laplace or Mellin transformations). Though often used trivially, it is easily
attainable and easily adaptable to different kinds of differentiation
procedures. That has made it very attractive. Hence applications to various
problems of the Taylor method, and more generally of the power series method
(including noninteger powers) has been sketched. It seems that its potential
has not been exploited as it could be. After a discussion on the reasons of the
"misunderstandings" which have caused the controversy, the preceding topics are
concretely illustrated.Comment: To appear in Applied Mathematics and Computation, 29 pages,
references and further considerations adde
Continued fractions, modular symbols, and non-commutative geometry
Using techniques introduced by D. Mayer, we prove an extension of the
classical Gauss-Kuzmin theorem about the distribution of continued fractions,
which in particular allows one to take into account some congruence properties
of successive convergents. This result has an application to the Mixmaster
Universe model in general relativity. We then study some averages involving
modular symbols and show that Dirichlet series related to modular forms of
weight 2 can be obtained by integrating certain functions on real axis defined
in terms of continued fractions. We argue that the quotient
should be considered as
non-commutative modular curve, and show that the modular complex can be seen as
a sequence of -groups of the related crossed-product -algebras.
This paper is an expanded version of the previous "On the distribution of
continued fractions and modular symbols". The main new features are Section 4
on non-commutative geometry and the modular complex and Section 1.2.2 on the
Mixmaster Universe.Comment: AMS-TeX, 50 pages, 2 figures (eps
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