13,039 research outputs found
Rotation Numbers, Boundary Forces and Gap labelling
We review the Johnson-Moser rotation number and the -theoretical gap
labelling of Bellissard for one-dimensional Schr\"odinger operators. We compare
them with two further gap-labels, one being related to the motion of Dirichlet
eigenvalues, the other being a -theoretical gap label. We argue that the
latter provides a natural generalisation of the Johnson-Moser rotation number
to higher dimensions.Comment: 10 pages, version accepted for publicatio
On stronger conjectures that imply the Erd\"os-Moser conjecture
The Erd\"os-Moser conjecture states that the Diophantine equation , where , has no solution for positive integers
and with . We show that stronger conjectures about
consecutive values of the function , that seem to be more naturally, imply
the Erd\"os-Moser conjecture.Comment: 7 page
Formal Solutions of a Class of Pfaffian Systems in Two Variables
In this paper, we present an algorithm which computes a fundamental matrix of
formal solutions of completely integrable Pfaffian systems with normal
crossings in two variables, based on (Barkatou, 1997). A first step was set in
(Barkatou-LeRoux, 2006) where the problem of rank reduction was tackled via the
approach of (Levelt, 1991). We give instead a Moser-based approach. And, as a
complementary step, we associate to our problem a system of ordinary linear
singular differential equations from which the formal invariants can be
efficiently derived via the package ISOLDE, implemented in the computer algebra
system Maple.Comment: Keywords: Linear systems of partial differential equations, Pfaffian
systems, Formal solutions, Moser-based reduction, Hukuhara- Turritin normal
for
What Moser Could Have Asked: Counting Hamilton Cycles in Tournaments
Moser asked for a construction of explicit tournaments on vertices having
at least Hamilton cycles. We show that he could have asked
for rather more
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