The Hyperdeterminant and Triangulations of the 4-Cube

The hyperdeterminant of format 2 x 2 x 2 x 2 is a polynomial of degree 24 in 16 unknowns which has 2894276 terms. We compute the Newton polytope of this polynomial and the secondary polytope of the 4-cube. The 87959448 regular triangulations of the 4-cube are classified into 25448 D-equivalence classes, one for each vertex of the Newton polytope. The 4-cube has 80876 coarsest regular subdivisions, one for each facet of the secondary polytope, but only 268 of them come from the hyperdeterminant.Comment: 30 pages, 6 figures; An author's name changed, typos fixe

On the maximal number of real embeddings of minimally rigid graphs in R2\mathbb{R}^2, R3\mathbb{R}^3 and S2S^2

Rigidity theory studies the properties of graphs that can have rigid embeddings in a euclidean space $\mathbb{R}^d$ or on a sphere and which in addition satisfy certain edge length constraints. One of the major open problems in this field is to determine lower and upper bounds on the number of realizations with respect to a given number of vertices. This problem is closely related to the classification of rigid graphs according to their maximal number of real embeddings. In this paper, we are interested in finding edge lengths that can maximize the number of real embeddings of minimally rigid graphs in the plane, space, and on the sphere. We use algebraic formulations to provide upper bounds. To find values of the parameters that lead to graphs with a large number of real realizations, possibly attaining the (algebraic) upper bounds, we use some standard heuristics and we also develop a new method inspired by coupler curves. We apply this new method to obtain embeddings in $\mathbb{R}^3$. One of its main novelties is that it allows us to sample efficiently from a larger number of parameters by selecting only a subset of them at each iteration. Our results include a full classification of the 7-vertex graphs according to their maximal numbers of real embeddings in the cases of the embeddings in $\mathbb{R}^2$ and $\mathbb{R}^3$, while in the case of $S^2$ we achieve this classification for all 6-vertex graphs. Additionally, by increasing the number of embeddings of selected graphs, we improve the previously known asymptotic lower bound on the maximum number of realizations. The methods and the results concerning the spatial embeddings are part of the proceedings of ISSAC 2018 (Bartzos et al, 2018)

Quartic Parameters for Acoustic Applications of Lattice Boltzmann Scheme

Using the Taylor expansion method, we show that it is possible to improve the lattice Boltzmann method for acoustic applications. We derive a formal expansion of the eigenvalues of the discrete approximation and fit the parameters of the scheme to enforce fourth order accuracy. The corresponding discrete equations are solved with the help of symbolic manipulation. The solutions are explicited in the case of D3Q27 lattice Boltzmann scheme. Various numerical tests support the coherence of this approach.Comment: 23 page

Note on Ward-Horadam H(x) - binomials' recurrences and related interpretations, II

We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials'$ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly, we supply a review of selected related combinatorial interpretations of generalized binomial coefficients. We then propose also a kind of transfer of interpretation of $p,q-binomial$ coefficients onto $q-binomial$ coefficients interpretations thus bringing us back to $Gy{\"{o}}rgy P\'olya$ and Donald Ervin Knuth relevant investigation decades ago.Comment: 57 pages, 8 figure