664 research outputs found
Forbidden ordinal patterns in higher dimensional dynamics
Forbidden ordinal patterns are ordinal patterns (or `rank blocks') that
cannot appear in the orbits generated by a map taking values on a linearly
ordered space, in which case we say that the map has forbidden patterns. Once a
map has a forbidden pattern of a given length , it has forbidden
patterns of any length and their number grows superexponentially
with . Using recent results on topological permutation entropy, we study in
this paper the existence and some basic properties of forbidden ordinal
patterns for self maps on n-dimensional intervals. Our most applicable
conclusion is that expansive interval maps with finite topological entropy have
necessarily forbidden patterns, although we conjecture that this is also the
case under more general conditions. The theoretical results are nicely
illustrated for n=2 both using the naive counting estimator for forbidden
patterns and Chao's estimator for the number of classes in a population. The
robustness of forbidden ordinal patterns against observational white noise is
also illustrated.Comment: 19 pages, 6 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
A lifting and recombination algorithm for rational factorization of sparse polynomials
We propose a new lifting and recombination scheme for rational bivariate
polynomial factorization that takes advantage of the Newton polytope geometry.
We obtain a deterministic algorithm that can be seen as a sparse version of an
algorithm of Lecerf, with now a polynomial complexity in the volume of the
Newton polytope. We adopt a geometrical point of view, the main tool being
derived from some algebraic osculation criterions in toric varieties.Comment: 22 page
Computing Chebyshev knot diagrams
A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t);
y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the
Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no
double points, it defines a polynomial knot. We determine all possible knots
when a, b and c are given.Comment: 8
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