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 $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$u_t = u_{xx} + f(x,u,u_x)\,,$$ on the unit interval $0 < x<1$ with Neumann boundary conditions. Equilibria $v_t=0$ 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 $\mathcal{A}$ as 3-cell templates $\mathcal{C}$. 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 $x=0$ and $x=1$, 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 $\mathcal{A}$ 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

On vertices enforcing a Hamiltonian cycle

A nonempty vertex set X ⊆ V(G) of a hamiltonian graph G is called an of G if every X-cycle of G (i.e. a cycle of G containing all vertices of X) is hamiltonian. The h(G) of a graph G is defined to be the smallest cardinality of an H-force set of G. In the paper the study of this parameter is introduced and its value or a lower bound for outerplanar graphs, planar graphs, k-connected graphs and prisms over graphs is determined

A Gap-{ETH}-Tight Approximation Scheme for Euclidean {TSP}

We revisit the classic task of finding the shortest tour of $n$ points in $d$-dimensional Euclidean space, for any fixed constant $d \geq 2$. We determine the optimal dependence on $\varepsilon$ in the running time of an algorithm that computes a $(1+\varepsilon)$-approximate tour, under a plausible assumption. Specifically, we give an algorithm that runs in $2^{\mathcal{O}(1/\varepsilon^{d-1})} n\log n$ time. This improves the previously smallest dependence on $\varepsilon$ in the running time $(1/\varepsilon)^{\mathcal{O}(1/\varepsilon^{d-1})}n \log n$ of the algorithm by Rao and Smith (STOC 1998). We also show that a $2^{o(1/\varepsilon^{d-1})}\text{poly}(n)$ algorithm would violate the Gap-Exponential Time Hypothesis (Gap-ETH). Our new algorithm builds upon the celebrated quadtree-based methods initially proposed by Arora (J. ACM 1998), but it adds a simple new idea that we call \emph{sparsity-sensitive patching}. On a high level this lets the granularity with which we simplify the tour depend on how sparse it is locally. Our approach is (arguably) simpler than the one by Rao and Smith since it can work without geometric spanners. We demonstrate the technique extends easily to other problems, by showing as an example that it also yields a Gap-ETH-tight approximation scheme for Rectilinear Steiner Tree

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

On the Spectral Gap of a Quantum Graph

We consider the problem of finding universal bounds of "isoperimetric" or "isodiametric" type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature

Maximal diameter of integral circulant graphs

Integral circulant graphs are proposed as models for quantum spin networks that permit a quantum phenomenon called perfect state transfer. Specifically, it is important to know how far information can potentially be transferred between nodes of the quantum networks modelled by integral circulant graphs and this task is related to calculating the maximal diameter of a graph. The integral circulant graph $ICG_n (D)$ has the vertex set $Z_n = \{0, 1, 2, \ldots, n - 1\}$ and vertices $a$ and $b$ are adjacent if $\gcd(a-b,n)\in D$, where $D \subseteq \{d : d \mid n,\ 1\leq d<n\}$. Motivated by the result on the upper bound of the diameter of $ICG_n(D)$ given in [N. Saxena, S. Severini, I. Shparlinski, \textit{Parameters of integral circulant graphs and periodic quantum dynamics}, International Journal of Quantum Information 5 (2007), 417--430], according to which $2|D|+1$ represents one such bound, in this paper we prove that the maximal value of the diameter of the integral circulant graph $ICG_n(D)$ of a given order $n$ with its prime factorization $p_1^{\alpha_1}\cdots p_k^{\alpha_k}$, is equal to $r(n)$ or $r(n)+1$, where $r(n)=k + |\{ i \ | \alpha_i> 1,\ 1\leq i\leq k \}|$, depending on whether $n\not\in 4N+2$ or not, respectively. Furthermore, we show that, for a given order $n$, a divisor set $D$ with $|D|\leq k$ can always be found such that this bound is attained. Finally, we calculate the maximal diameter in the class of integral circulant graphs of a given order $n$ and cardinality of the divisor set $t\leq k$ and characterize all extremal graphs. We actually show that the maximal diameter can have the values $2t$, $2t+1$, $r(n)$ and $r(n)+1$ depending on the values of $t$ and $n$. This way we further improve the upper bound of Saxena, Severini and Shparlinski and we also characterize all graphs whose diameters are equal to $2|D|+1$, thus generalizing a result in that paper.Comment: 29 pages, 1 figur

Strongly Correlated Random Interacting Processes

The focus of the workshop was to discuss the recent developments and future research directions in the area of large scale random interacting processes, with main emphasis in models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics. This report contains extended abstracts of the presentations, which featured research in several directions including selfinteracting random walks, spatially growing processes, strongly dependent percolation, spin systems with long-range order, and random permutations

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum