243 research outputs found
Stratified graphs for imbedding systems
AbstractTwo imbeddings of a graph G are considered to be adjacent if the second can be obtained from the first by moving one or both ends of a single edge within its or their respective rotations. Thus, a collection of imbeddings S of G, called a ‘system’, may be represented as a ‘stratified graph’, and denoted SG; the focus here is the case in which S is the collection of all orientable imbeddings. The induced subgraph of SG on the set of imbeddings into the surface of genus k is called the ‘kth stratum’, and the cardinality of that set of imbeddings is called the ‘stratum size’; one may observe that the sequence of stratum sizes is precisely the genus distribution for the graph G. It is known that the genus distribution is not a complete invariant, even when the category of graphs is restricted to be simplicial and 3-connected. However, it is proved herein that the link of each point — that is, the subgraph induced by its neighbors — of SG is a complete isomorphism invariant for the category of graphs whose minimum valence is at least three. This supports the plausibility of a probabilistic approach to graph isomorphism testing by sampling higher-order imbedding distribution data. A detailed structural analysis of stratified graphs is presented
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Graph Imbeddings and Overlap Matrices (Preliminary Report)
Mohar has shown an interesting relationship between graph imbeddings and certain boolean matrices. In this paper, we show some interesting properties of this kind of matrices. Using these properties, we give the distributions of nonorietable imbeddings of several interesting infinite families of graphs, including cobblestone paths, closed-end ladders for which the distributions of orientable imbeddings are known
Wave splitting and imbedding equations for a spherically symmetric dispersive medium
The direct problem of time dependent electromagnetic scattering in the dispersive sphere is solved by a wave splitting technique. The electric field is expanded in a series involving vector spherical harmonics, leading to a system of wave equations for each term. These systems are reduced to scalar wave equations for each term, which are solved via reflection operators. Some preliminary numerical results are presented
Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents
This paper concerns the dynamics of polynomial automorphisms of .
One can associate to such an automorphism two currents and the
equilibrium measure . In this paper we study some
geometric and dynamical properties of these objects. First, we characterize
as the unique measure of maximal entropy. Then we show that the measure
has a local product structure and that the currents have a
laminar structure. This allows us to deduce information about periodic points
and heteroclinic intersections. For example, we prove that the support of
coincides with the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of non-uniformly hyperbolic
dynamical systems
Discrete Morse theory for computing cellular sheaf cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology,
greatly generalizing persistent homology. We develop an algorithm for
simplifying the computation of cellular sheaf cohomology via (discrete)
Morse-theoretic techniques. As a consequence, we derive efficient techniques
for distributed computation of (ordinary) cohomology of a cell complex.Comment: 19 pages, 1 Figure. Added Section 5.
Chow rings of stacks of prestable curves I
We study the Chow ring of the moduli stack of prestable
curves and define the notion of tautological classes on this stack. We extend
formulas for intersection products and functoriality of tautological classes
under natural morphisms from the case of the tautological ring of the moduli
space of stable curves. This paper provides
foundations for the second part of the paper.
In the appendix (joint with J. Skowera), we develop the theory of a proper,
but not necessary projective, pushforward of algebraic cycles. The proper
pushforward is necessary for the construction of the tautological rings of
and is important in its own right. We also develop
operational Chow groups for algebraic stacks.Comment: This paper is the first part of the previous version which has been
split off due to length. An appendix (joint with Jonathan Skowera) about
proper pushforwards for Chow groups of Artin stacks has been added. 65 pages.
Comments are very welcome
Euler flag enumeration of Whitney stratified spaces
The flag vector contains all the face incidence data of a polytope, and in
the poset setting, the chain enumerative data. It is a classical result due to
Bayer and Klapper that for face lattices of polytopes, and more generally,
Eulerian graded posets, the flag vector can be written as a cd-index, a
non-commutative polynomial which removes all the linear redundancies among the
flag vector entries. This result holds for regular CW complexes.
We relax the regularity condition to show the cd-index exists for Whitney
stratified manifolds by extending the notion of a graded poset to that of a
quasi-graded poset. This is a poset endowed with an order-preserving rank
function and a weighted zeta function. This allows us to generalize the
classical notion of Eulerianness, and obtain a cd-index in the quasi-graded
poset arena. We also extend the semi-suspension operation to that of embedding
a complex in the boundary of a higher dimensional ball and study the simplicial
shelling components.Comment: 41 pages, 3 figures. Final versio
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