1,338 research outputs found
A note on Diophantine systems involving three symmetric polynomials
Let and be -th elementary symmetric polynomial. In this
note we prove that there are infinitely many triples of integers such
that for each the system of Diophantine equations
\begin{equation*}
\sigma_{i}(\bar{X}_{2n})=a, \quad \sigma_{2n-i}(\bar{X}_{2n})=b, \quad
\sigma_{2n}(\bar{X}_{2n})=c \end{equation*} has infinitely many rational
solutions. This result extend the recent results of Zhang and Cai, and the
author. Moreover, we also consider some Diophantine systems involving sums of
powers. In particular, we prove that for each there are at least
-tuples of integers with the same sum of -th powers for .
Similar result is proved for and .Comment: to appear in J. Number Theor
Algorithmic Algebraic Geometry and Flux Vacua
We develop a new and efficient method to systematically analyse four
dimensional effective supergravities which descend from flux compactifications.
The issue of finding vacua of such systems, both supersymmetric and
non-supersymmetric, is mapped into a problem in computational algebraic
geometry. Using recent developments in computer algebra, the problem can then
be rapidly dealt with in a completely algorithmic fashion. Two main results are
(1) a procedure for calculating constraints which the flux parameters must
satisfy in these models if any given type of vacuum is to exist; (2) a stepwise
process for finding all of the isolated vacua of such systems and their
physical properties. We illustrate our discussion with several concrete
examples, some of which have eluded conventional methods so far.Comment: 41 pages, 4 figure
Pendulum: separatrix splitting
An exact expression for the determinant of the splitting matrix is derived:
it allows us to analyze the asympotic behaviour needed to amend the large
angles theorem proposed in Ann. Inst. H. Poincar\'e, B-60, 1, 1994. The
asymptotic validity of Melnokov's formulae is proved for the class of models
considered, which include polynomial perturbations.Comment: 30 pages, one figur
Linear independence of time frequency translates for special configurations
We prove that for any 4 points in the plane that belong to 2 parallel lines,
there is no linear dependence between the associated time-frequency translates
of any nontrivial Schwartz function. If mild Diophantine properties are
satisfied, we also prove linear independence in the category of
functions.Comment: Inaccuracies in Section 3 have been correcte
Condition number bounds for problems with integer coefficients
An apriori bound for the condition number associated to each of the following
problems is given: general linear equation solving, minimum squares,
non-symmetric eigenvalue problems, solving univariate polynomials, solving
systems of multivariate polynomials. It is assumed that the input has integer
coefficients and is not on the degenerate locus of the respective problem (i.e.
the condition number is finite). Then condition numbers are bounded in terms of
the dimension and of the bit-size of the input.
In the same setting, bounds are given for the speed of convergence of the
following iterative algorithms: QR without shift for the symmetric eigenvalue
problem, and Graeffe iteration for univariate polynomials
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