6,075 research outputs found
From the Ising and Potts models to the general graph homomorphism polynomial
In this note we study some of the properties of the generating polynomial for
homomorphisms from a graph to at complete weighted graph on vertices. We
discuss how this polynomial relates to a long list of other well known graph
polynomials and the partition functions for different spin models, many of
which are specialisations of the homomorphism polynomial.
We also identify the smallest graphs which are not determined by their
homomorphism polynomials for and and compare this with the
corresponding minimal examples for the -polynomial, which generalizes the
well known Tutte-polynomal.Comment: V2. Extended versio
Diffeomorphisms of Stein structures
We prove that a pseudoholomorphic diffeomorphism between two almost complex
manifolds with boundaries satisfying some pseudoconvexity type condition cannot
map a pseudoholomorphic disc in the boundary to a single point. This can be
viewed as an almost complex analogue of a well-known theorem of J. E. Fornaess
What makes nonholonomic integrators work?
A nonholonomic system is a mechanical system with velocity constraints not
originating from position constraints; rolling without slipping is the typical
example. A nonholonomic integrator is a numerical method specifically designed
for nonholonomic systems. It has been observed numerically that many
nonholonomic integrators exhibit excellent long-time behaviour when applied to
various test problems. The excellent performance is often attributed to some
underlying discrete version of the Lagrange--d'Alembert principle. Instead, in
this paper, we give evidence that reversibility is behind the observed
behaviour. Indeed, we show that many standard nonholonomic test problems have
the structure of being foliated over reversible integrable systems. As most
nonholonomic integrators preserve the foliation and the reversible structure,
near conservation of the first integrals is a consequence of reversible KAM
theory. Therefore, to fully evaluate nonholonomic integrators one has to
consider also non-reversible nonholonomic systems. To this end we construct
perturbed test problems that are integrable but no longer reversible (with
respect to the standard reversibility map). Applying various nonholonomic
integrators from the literature to these problems we observe that no method
performs well on all problems. This further indicates that reversibility is the
main mechanism behind near conservation of first integrals for nonholonomic
integrators. A list of relevant open problems is given.Comment: 27 pages, 9 figure
- …