75,935 research outputs found
Partition function of the Potts model on self-similar lattices as a dynamical system and multiple transitions
We present an analytic study of the Potts model partition function on two
different types of self-similar lattices of triangular shape with non integer
Hausdorff dimension. Both types of lattices analyzed here are interesting
examples of non-trivial thermodynamics in less than two dimensions. First, the
Sierpinski gasket is considered. It is shown that, by introducing suitable
geometric coefficients, it is possible to reduce the computation of the
partition function to a dynamical system, whose variables are directly
connected to (the arising of) frustration on macroscopic scales, and to
determine the possible phases of the system. The same method is then used to
analyse the Hanoi graph. Again, dynamical system theory provides a very elegant
way to determine the phase diagram of the system. Then, exploiting the analysis
of the basins of attractions of the corresponding dynamical systems, we
construct various examples of self-similar lattices with more than one critical
temperature. These multiple critical temperatures correspond to crossing phases
with different degrees of frustration.Comment: 16 pages, 12 figures, 1 table; title changed, references and
discussion on multiple transitions adde
Genus Computing for 3D digital objects: algorithm and implementation
This paper deals with computing topological invariants such as connected
components, boundary surface genus, and homology groups. For each input data
set, we have designed or implemented algorithms to calculate connected
components, boundary surfaces and their genus, and homology groups. Due to the
fact that genus calculation dominates the entire task for 3D object in 3D
space, in this paper, we mainly discuss the calculation of the genus. The new
algorithms designed in this paper will perform:
(1) pathological cases detection and deletion, (2) raster space to point
space (dual space) transformation, (3) the linear time algorithm for boundary
point classification, and (4) genus calculation.Comment: 12 pages 7 figures. In Proceedings of the Workshop on Computational
Topology in image context 2009, Aug. 26-28, Austria, Edited by W. Kropatsch,
H. M. Abril and A. Ion, 200
- …