6,166 research outputs found
The natural algorithmic approach of mixed trigonometric-polynomial problems
The aim of this paper is to present a new algorithm for proving mixed
trigonometric-polynomial inequalities by reducing to polynomial inequalities.
Finally, we show the great applicability of this algorithm and as examples, we
use it to analyze some new rational (Pade) approximations of the function
, and to improve a class of inequalities by Z.-H. Yang. The results
of our analysis could be implemented by means of an automated proof assistant,
so our work is a contribution to the library of automatic support tools for
proving various analytic inequalities
On Kahan's Rules for Determining Branch Cuts
In computer algebra there are different ways of approaching the mathematical
concept of functions, one of which is by defining them as solutions of
differential equations. We compare different such approaches and discuss the
occurring problems. The main focus is on the question of determining possible
branch cuts. We explore the extent to which the treatment of branch cuts can be
rendered (more) algorithmic, by adapting Kahan's rules to the differential
equation setting.Comment: SYNASC 2011. 13th International Symposium on Symbolic and Numeric
Algorithms for Scientific Computing. (2011
Can Computer Algebra be Liberated from its Algebraic Yoke ?
So far, the scope of computer algebra has been needlessly restricted to exact
algebraic methods. Its possible extension to approximate analytical methods is
discussed. The entangled roles of functional analysis and symbolic programming,
especially the functional and transformational paradigms, are put forward. In
the future, algebraic algorithms could constitute the core of extended symbolic
manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure
Trigonometric series and self-similar sets
Let be a self-similar set on associated to contractions
, , for some finite ,
such that is not a singleton. We prove that if is
irrational for some , then is a set of multiplicity, that is,
trigonometric series are not in general unique in the complement of . No
separation conditions are assumed on . We establish our result by showing
that every self-similar measure on is a Rajchman measure: the Fourier
transform as . The rate of
is also shown to be logarithmic if is diophantine for some . The proof is based on quantitative
renewal theorems for random walks on .Comment: 18 pages, v2: improved the main theore
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