310 research outputs found

    Sturm 3-ball global attractors 3: Examples of Thom-Smale complexes

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    Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors A\mathcal{A} which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, ut=uxx+f(x,u,ux) , u_t = u_{xx} + f(x,u,u_x)\,, on the unit interval 0<x<10 < x<1 with Neumann boundary conditions. Equilibria vt=0v_t=0 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 A\mathcal{A} as 3-cell templates C\mathcal{C}. 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 x=0x=0 and x=1x=1, 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 A\mathcal{A} 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

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    A cycle containing a shortest path between two vertices uu and vv in a graph GG is called a (u,v)(u,v)-geodesic cycle. A connected graph GG is geodesic 2-bipancyclic, if every pair of vertices u,vu,v of it is contained in a (u,v)(u,v)-geodesic cycle of length ll for each even integer ll satisfying 2d+2≤l≤∣V(G)∣,2d + 2\leq l \leq |V(G)|, where dd is the distance between uu and v.v. 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 n≥2n \geq 2 every nn-dimensional torus is a geodesic 2-bipancyclic graph

    Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops

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    We present determinant formulae for the number of tilings of various domains in relation with Alternating Sign Matrix and Fully Packed Loop enumeration

    Combinatorial Structures in Hypercubes

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    Combinatorics and geometry of finite and infinite squaregraphs

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    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

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    Variational principles for circle patterns

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    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

    The Complexity of Change

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    Many combinatorial problems can be formulated as "Can I transform configuration 1 into configuration 2, if certain transformations only are allowed?". An example of such a question is: given two k-colourings of a graph, can I transform the first k-colouring into the second one, by recolouring one vertex at a time, and always maintaining a proper k-colouring? Another example is: given two solutions of a SAT-instance, can I transform the first solution into the second one, by changing the truth value one variable at a time, and always maintaining a solution of the SAT-instance? Other examples can be found in many classical puzzles, such as the 15-Puzzle and Rubik's Cube. In this survey we shall give an overview of some older and more recent work on this type of problem. The emphasis will be on the computational complexity of the problems: how hard is it to decide if a certain transformation is possible or not?Comment: 28 pages, 6 figure
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