321 research outputs found
Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes
Examples complete our trilogy on the geometric and combinatorial
characterization of global Sturm attractors which consist of a
single closed 3-ball. The underlying scalar PDE is parabolic, on the unit interval with Neumann boundary
conditions. Equilibria are assumed to be hyperbolic. Geometrically, we
study the resulting Thom-Smale dynamic complex with cells defined by the fast
unstable manifolds of the equilibria. The Thom-Smale complex turns out to be a
regular cell complex. In the first two papers we characterized 3-ball Sturm
attractors as 3-cell templates . The
characterization involves bipolar orientations and hemisphere decompositions
which are closely related to the geometry of the fast unstable manifolds. An
equivalent combinatorial description was given in terms of the Sturm
permutation, alias the meander properties of the shooting curve for the
equilibrium ODE boundary value problem. It involves the relative positioning of
extreme 2-dimensionally unstable equilibria at the Neumann boundaries and
, respectively, and the overlapping reach of polar serpents in the
shooting meander. In the present paper we apply these descriptions to
explicitly enumerate all 3-ball Sturm attractors with at most 13
equilibria. We also give complete lists of all possibilities to obtain solid
tetrahedra, cubes, and octahedra as 3-ball Sturm attractors with 15 and 27
equilibria, respectively. For the remaining Platonic 3-balls, icosahedra and
dodecahedra, we indicate a reduction to mere planar considerations as discussed
in our previous trilogy on planar Sturm attractors.Comment: 73+(ii) pages, 40 figures, 14 table; see also parts 1 and 2 under
arxiv:1611.02003 and arxiv:1704.0034
Geodesic bipancyclicity of the Cartesian product of graphs
A cycle containing a shortest path between two vertices and in a graph is called a -geodesic cycle. A connected graph is geodesic 2-bipancyclic, if every pair of vertices of it is contained in a -geodesic cycle of length for each even integer satisfying where is the distance between and In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for every -dimensional torus is a geodesic 2-bipancyclic graph
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
We present determinant formulae for the number of tilings of various domains
in relation with Alternating Sign Matrix and Fully Packed Loop enumeration
Combinatorics and geometry of finite and infinite squaregraphs
Squaregraphs were originally defined as finite plane graphs in which all
inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e.,
the vertices not incident with the outer face) have degrees larger than three.
The planar dual of a finite squaregraph is determined by a triangle-free chord
diagram of the unit disk, which could alternatively be viewed as a
triangle-free line arrangement in the hyperbolic plane. This representation
carries over to infinite plane graphs with finite vertex degrees in which the
balls are finite squaregraphs. Algebraically, finite squaregraphs are median
graphs for which the duals are finite circular split systems. Hence
squaregraphs are at the crosspoint of two dualities, an algebraic and a
geometric one, and thus lend themselves to several combinatorial
interpretations and structural characterizations. With these and the
5-colorability theorem for circle graphs at hand, we prove that every
squaregraph can be isometrically embedded into the Cartesian product of five
trees. This embedding result can also be extended to the infinite case without
reference to an embedding in the plane and without any cardinality restriction
when formulated for median graphs free of cubes and further finite
obstructions. Further, we exhibit a class of squaregraphs that can be embedded
into the product of three trees and we characterize those squaregraphs that are
embeddable into the product of just two trees. Finally, finite squaregraphs
enjoy a number of algorithmic features that do not extend to arbitrary median
graphs. For instance, we show that median-generating sets of finite
squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the
corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure
Variational principles for circle patterns
A Delaunay cell decomposition of a surface with constant curvature gives rise
to a circle pattern, consisting of the circles which are circumscribed to the
facets. We treat the problem whether there exists a Delaunay cell decomposition
for a given (topological) cell decomposition and given intersection angles of
the circles, whether it is unique and how it may be constructed. Somewhat more
generally, we allow cone-like singularities in the centers and intersection
points of the circles. We prove existence and uniqueness theorems for the
solution of the circle pattern problem using a variational principle. The
functionals (one for the euclidean, one for the hyperbolic case) are convex
functions of the radii of the circles. The analogous functional for the
spherical case is not convex, hence this case is treated by stereographic
projection to the plane. From the existence and uniqueness of circle patterns
in the sphere, we derive a strengthened version of Steinitz' theorem on the
geometric realizability of abstract polyhedra.
We derive the variational principles of Colin de Verdi\`ere, Br\"agger, and
Rivin for circle packings and circle patterns from our variational principles.
In the case of Br\"agger's and Rivin's functionals. Leibon's functional for
hyperbolic circle patterns cannot be derived directly from our functionals. But
we construct yet another functional from which both Leibon's and our
functionals can be derived.
We present Java software to compute and visualize circle patterns.Comment: PhD thesis, iv+94 pages, many figures (mostly vector graphics
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