18 research outputs found
Decomposable polynomials in second order linear recurrence sequences
We study elements of second order linear recurrence sequences of polynomials in which are decomposable, i.e.
representable as for some satisfying
. Under certain assumptions, and
provided that is not of particular type, we show that
may be bounded by a constant independent of , depending only on the
sequence.Comment: 26 page
Positivity Problems for Low-Order Linear Recurrence Sequences
We consider two decision problems for linear recurrence sequences (LRS) over
the integers, namely the Positivity Problem (are all terms of a given LRS
positive?) and the Ultimate Positivity Problem} (are all but finitely many
terms of a given LRS positive?). We show decidability of both problems for LRS
of order 5 or less, with complexity in the Counting Hierarchy for Positivity,
and in polynomial time for Ultimate Positivity. Moreover, we show by way of
hardness that extending the decidability of either problem to LRS of order 6
would entail major breakthroughs in analytic number theory, more precisely in
the field of Diophantine approximation of transcendental numbers
Elliptic hypergeometric terms
General structure of the multivariate plain and q-hypergeometric terms and
univariate elliptic hypergeometric terms is described. Some explicit examples
of the totally elliptic hypergeometric terms leading to multidimensional
integrals on root systems, either computable or obeying non-trivial symmetry
transformations, are presented.Comment: 20 pp., version to appear in a workshop proceeding
On the Positivity Problem for Simple Linear Recurrence Sequences
Given a linear recurrence sequence (LRS) over the integers, the Positivity
Problem} asks whether all terms of the sequence are positive. We show that, for
simple LRS (those whose characteristic polynomial has no repeated roots) of
order 9 or less, Positivity is decidable, with complexity in the Counting
Hierarchy.Comment: arXiv admin note: substantial text overlap with arXiv:1307.277
Universal Calabi-Yau Algebra: Towards an Unification of Complex Geometry
We present a universal normal algebra suitable for constructing and
classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach
includes natural extensions of reflexive weight vectors to higher dimensions,
related to Batyrev's reflexive polyhedra, and their n-ary combinations. It also
includes a `dual' construction based on the Diophantine decomposition of
invariant monomials, which provides explicit recurrence formulae for the
numbers of Calabi-Yau spaces in arbitrary dimensions with Weierstrass, K3,
etc., fibrations. Our approach also yields simple algebraic relations between
chains of Calabi-Yau spaces in different dimensions, and concrete
visualizations of their singularities related to Cartan-Lie algebras. This
Universal Calabi-Yau Algebra is a powerful tool for decyphering the Calabi-Yau
genome in all dimensions.Comment: 81 pages LaTeX, 8 eps figure
Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties
This is a continuation of our study concerning q-tori, i.e. tori of low
dimensionality in the phase space of nonlinear lattice models like the
Fermi-Pasta-Ulam (FPU) model. In our previous work we focused on the beta FPU
system, and we showed that the dynamical features of the q-tori serve as an
interpretational tool to understand phenomena of energy localization in the FPU
space of linear normal modes. In the present paper i) we employ the method of
Poincare - Lindstedt series, for a fixed set of frequencies, in order to
compute an explicit quasi-periodic representation of the trajectories lying on
q-tori in the alpha model, and ii) we consider more general types of initial
excitations in both the alpha and beta models. Furthermore we turn into
questions of physical interest related to the dynamical features of the q-tori.
We focus on particular q-tori solutions describing low-frequency `packets' of
modes, and excitations of a small set of modes with an arbitrary distribution
in q-space. In the former case, we find formulae yielding an exponential
profile of energy localization, following an analysis of the size of the
leading order terms in the Poincare - Lindstedt series. In the latter case, we
explain the observed localization patterns on the basis of a rigorous result
concerning the propagation of non-zero terms in the Poincare - Lindstedt series
from zeroth to subsequent orders. Finally, we discuss the extensive (i.e.
independent of the number of degrees of freedom) properties of some q-tori
solutions.Comment: To appear in Physica D, 34 pages, 9 figure
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life
Number Theory, Analysis and Geometry: In Memory of Serge Lang
Serge Lang was an iconic figure in mathematics, both for his own important work and for the indelible impact he left on the field of mathematics, on his students, and on his colleagues. Over the course of his career, Lang traversed a tremendous amount of mathematical ground. As he moved from subject to subject, he found analogies that led to important questions in such areas as number theory, arithmetic geometry and the theory of negatively curved spaces. Lang's conjectures will keep many mathematicians occupied far into the future.
In the spirit of Lang’s vast contribution to mathematics, this memorial volume contains articles by prominent mathematicians in a variety of areas, namely number theory, analysis and geometry, representing Lang’s own breadth of interests. A special introduction by John Tate includes a brief and engaging account of Serge Lang’s life