5 research outputs found
Potts models with magnetic field: arithmetic, geometry, and computation
We give a sheaf theoretic interpretation of Potts models with external
magnetic field, in terms of constructible sheaves and their Euler
characteristics. We show that the polynomial countability question for the
hypersurfaces defined by the vanishing of the partition function is affected by
changes in the magnetic field: elementary examples suffice to see
non-polynomially countable cases that become polynomially countable after a
perturbation of the magnetic field. The same recursive formula for the
Grothendieck classes, under edge-doubling operations, holds as in the case
without magnetic field, but the closed formulae for specific examples like
banana graphs differ in the presence of magnetic field. We give examples of
computation of the Euler characteristic with compact support, for the set of
real zeros, and find a similar exponential growth with the size of the graph.
This can be viewed as a measure of topological and algorithmic complexity. We
also consider the computational complexity question for evaluations of the
polynomial, and show both tractable and NP-hard examples, using dynamic
programming.Comment: 16 pages, LaTeX; v2: final version with small correction
Persistent Topology of Syntax
We study the persistent homology of the data set of syntactic parameters of
the world languages. We show that, while homology generators behave erratically
over the whole data set, non-trivial persistent homology appears when one
restricts to specific language families. Different families exhibit different
persistent homology. We focus on the cases of the Indo-European and the
Niger-Congo families, for which we compare persistent homology over different
cluster filtering values. We investigate the possible significance, in
historical linguistic terms, of the presence of persistent generators of the
first homology. In particular, we show that the persistent first homology
generator we find in the Indo-European family is not due (as one might guess)
to the Anglo-Norman bridge in the Indo-European phylogenetic network, but is
related to the position of Ancient Greek and the Hellenic branch within the
network.Comment: 15 pages, 25 jpg figure
Persistent Topology of Syntax
We study the persistent homology of a data set of syntactic parameters of world languages. We show that, while homology generators behave erratically over the whole data set, non-trivial persistent homology appears when one restricts to specific language families. Different families exhibit different persistent homology. We focus on the cases of the Indo-European and the Niger–Congo families, for which we compare persistent homology over different cluster filtering values. The persistent components appear to correspond to linguistic subfamilies, while the meaning, in historical linguistic terms, of the presence of persistent generators of the first homology is more mysterious. We investigate the possible significance of the persistent first homology generator that we find in the Indo-European family. We show that it is not due to the Anglo-Norman bridge (which is a lexical, not syntactic phenomenon), but is related instead to the position of Ancient Greek and the Hellenic branch within the Indo-European phylogenetic network