Identifying combinations of tetrahedra into hexahedra: a vertex based strategy

Indirect hex-dominant meshing methods rely on the detection of adjacent tetrahedra an algorithm that performs this identification and builds the set of all possible combinations of tetrahedral elements of an input mesh T into hexahedra, prisms, or pyramids. All identified cells are valid for engineering analysis. First, all combinations of eight/six/five vertices whose connectivity in T matches the connectivity of a hexahedron/prism/pyramid are computed. The subset of tetrahedra of T triangulating each potential cell is then determined. Quality checks allow to early discard poor quality cells and to dramatically improve the efficiency of the method. Each potential hexahedron/prism/pyramid is computed only once. Around 3 millions potential hexahedra are computed in 10 seconds on a laptop. We finally demonstrate that the set of potential hexes built by our algorithm is significantly larger than those built using predefined patterns of subdivision of a hexahedron in tetrahedral elements.Comment: Preprint submitted to CAD (26th IMR special issue

The Cube Recurrence

We construct a combinatorial model that is described by the cube recurrence, a nonlinear recurrence relation introduced by Propp, which generates families of Laurent polynomials indexed by points in $\mathbb{Z}^3$. In the process, we prove several conjectures of Propp and of Fomin and Zelevinsky, and we obtain a combinatorial interpretation for the terms of Gale-Robinson sequences. We also indicate how the model might be used to obtain some interesting results about perfect matchings of certain bipartite planar graphs

Space Saving by Dynamic Algebraization

Dynamic programming is widely used for exact computations based on tree decompositions of graphs. However, the space complexity is usually exponential in the treewidth. We study the problem of designing efficient dynamic programming algorithm based on tree decompositions in polynomial space. We show how to construct a tree decomposition and extend the algebraic techniques of Lokshtanov and Nederlof such that the dynamic programming algorithm runs in time $O^*(2^h)$, where $h$ is the maximum number of vertices in the union of bags on the root to leaf paths on a given tree decomposition, which is a parameter closely related to the tree-depth of a graph. We apply our algorithm to the problem of counting perfect matchings on grids and show that it outperforms other polynomial-space solutions. We also apply the algorithm to other set covering and partitioning problems.Comment: 14 pages, 1 figur

Generic Power Sum Decompositions and Bounds for the Waring Rank

A notion of open rank, related with generic power sum decompositions of forms, has recently been introduced in the literature. The main result here is that the maximum open rank for plane quartics is eight. In particular, this gives the first example of $n,d$, such that the maximum open rank for degree $d$ forms that essentially depend on $n$ variables is strictly greater than the maximum rank. On one hand, the result allows to improve the previously known bounds on open rank, but on the other hand indicates that such bounds are likely quite relaxed. Nevertheless, some of the preparatory results are of independent interest, and still may provide useful information in connection with the problem of finding the maximum rank for the set of all forms of given degree and number of variables. For instance, we get that every ternary forms of degree $d\ge 3$ can be annihilated by the product of $d-1$ pairwise independent linear forms.Comment: Accepted version. The final publication is available at link.springer.com: http://link.springer.com/article/10.1007/s00454-017-9886-

Perfect State Transfer in Laplacian Quantum Walk

For a graph $G$ and a related symmetric matrix $M$, the continuous-time quantum walk on $G$ relative to $M$ is defined as the unitary matrix $U(t) = \exp(-itM)$, where $t$ varies over the reals. Perfect state transfer occurs between vertices $u$ and $v$ at time $\tau$ if the $(u,v)$-entry of $U(\tau)$ has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer: (1) If a $n$-vertex graph has perfect state transfer at time $\tau$ relative to the Laplacian, then so does its complement if $n\tau$ is an integer multiple of $2\pi$. As a corollary, the double cone over any $m$-vertex graph has perfect state transfer relative to the Laplacian if and only if $m \equiv 2 \pmod{4}$. This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009). (2) If a graph $G$ has perfect state transfer at time $\tau$ relative to the normalized Laplacian, then so does the weak product $G \times H$ if for any normalized Laplacian eigenvalues $\lambda$ of $G$ and $\mu$ of $H$, we have $\mu(\lambda-1)\tau$ is an integer multiple of $2\pi$. As a corollary, a weak product of $P_{3}$ with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of $P_{3}$ has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).Comment: 26 pages, 5 figures, 1 tabl