48,945 research outputs found
Families of Graphs with W_r({G},q) Functions That Are Nonanalytic at 1/q=0
Denoting as the chromatic polynomial for coloring an -vertex
graph with colors, and considering the limiting function , a fundamental question in graph theory is the
following: is analytic or not at the origin
of the plane? (where the complex generalization of is assumed). This
question is also relevant in statistical mechanics because
, where is the ground state entropy of the
-state Potts antiferromagnet on the lattice graph , and the
analyticity of at is necessary for the large- series
expansions of . Although is analytic at for many
, there are some for which it is not; for these, has no
large- series expansion. It is important to understand the reason for this
nonanalyticity. Here we give a general condition that determines whether or not
a particular is analytic at and explains the
nonanalyticity where it occurs. We also construct infinite families of graphs
with functions that are non-analytic at and investigate the
properties of these functions. Our results are consistent with the conjecture
that a sufficient condition for to be analytic at is
that is a regular lattice graph . (This is known not to be a
necessary condition).Comment: 22 pages, Revtex, 4 encapsulated postscript figures, to appear in
Phys. Rev.
Special curves and postcritically-finite polynomials
We study the postcritically-finite (PCF) maps in the moduli space of complex
polynomials . For a certain class of rational curves in
, we characterize the condition that contains infinitely
many PCF maps. In particular, we show that if is parameterized by
polynomials, then there are infinitely many PCF maps in if and only if
there is exactly one active critical point along , up to symmetries; we
provide the critical orbit relation satisfied by any pair of active critical
points. For the curves in the space of cubic
polynomials, introduced by Milnor (1992), we show that
contains infinitely many PCF maps if and only if
. The proofs involve a combination of number-theoretic methods
(specifically, arithmetic equidistribution) and complex-analytic techniques
(specifically, univalent function theory). We provide a conjecture about
Zariski density of PCF maps in subvarieties of the space of rational maps, in
analogy with the Andr\'e-Oort Conjecture from arithmetic geometry.Comment: Final version, appeared in Forum of Math. P
Singularities and Topology of Meromorphic Functions
We present several aspects of the "topology of meromorphic functions", which
we conceive as a general theory which includes the topology of holomorphic
functions, the topology of pencils on quasi-projective spaces and the topology
of polynomial functions.Comment: 21 pages, 1 figur
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