15,455 research outputs found
Jordan's Inequality: Refinements, Generalizations, Applications and Related Problems
This is an expository article. Some developments on refinements,
generalizations, applications of Jordanâs inequality and related problems, including
some estimates for three classes of complete elliptic integrals and several
proofs of Wilkerâs inequality, are summarized
Combinatorial modulus and type of graphs
Let a be the 1-skeleton of a triangulated topological annulus. We
establish bounds on the combinatorial modulus of a refinement , formed by
attaching new vertices and edges to , that depend only on the refinement and
not on the structure of itself. This immediately applies to showing that a
disk triangulation graph may be refined without changing its combinatorial
type, provided the refinement is not too wild. We also explore the type problem
in terms of disk growth, proving a parabolicity condition based on a
superlinear growth rate, which we also prove optimal. We prove our results with
no degree restrictions in both the EEL and VEL settings and examine type
problems for more general complexes and dual graphs.Comment: 24 pages, 12 figure
Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)
The generating series of a radix-rational sequence is a rational formal power
series from formal language theory viewed through a fixed radix numeration
system. For each radix-rational sequence with complex values we provide an
asymptotic expansion for the sequence of its Ces\`aro means. The precision of
the asymptotic expansion depends on the joint spectral radius of the linear
representation of the sequence; the coefficients are obtained through some
dilation equations. The proofs are based on elementary linear algebra
Artinian algebras and Jordan type
The Jordan type of an element of the maximal ideal of an Artinian
k-algebra A acting on an A-module M of k-dimension n, is the partition of n
given by the Jordan block decomposition of the multiplication map on
M. In general the Jordan type has more information than whether the pair
is strong or weak Lefschetz. We develop basic properties of the
Jordan type and their loci for modules over graded or local Artinian algebras.
We as well study the relation of generic Jordan type of to the Hilbert
function of . We introduce and study a finer invariant, the Jordan degree
type.
In our last sections we give an overview of topics such as the Jordan types
for Nagata idealizations, for modular tensor products, and for free extensions,
including examples and some new results. We as well propose open problems.Comment: 53 pages. Added results, examples for Jordan degree type (Section
2.4) and Jordan type and initial ideal (Section 2.5
Expanding Thurston Maps
We study the dynamics of Thurston maps under iteration. These are branched
covering maps of 2-spheres with a finite set of
postcritical points. We also assume that the maps are expanding in a suitable
sense. Every expanding Thurston map gives rise to a type of
fractal geometry on the underlying sphere . This geometry is represented
by a class of \emph{visual metrics} that are associated with the map.
Many dynamical properties of the map are encoded in the geometry of the
corresponding {\em visual sphere}, meaning equipped with a visual metric
. For example, we will see that an expanding Thurston map is
topologically conjugate to a rational map if and only if is
quasisymmetrically equivalent to the Riemann sphere . We
also obtain existence and uniqueness results for -invariant Jordan curves
containing the set . Furthermore, we
obtain several characterizations of Latt\`{e}s maps.Comment: 492 pages, 51 figure
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