Discrete Fourier analysis, Cubature and Interpolation on a Hexagon and a Triangle

Several problems of trigonometric approximation on a hexagon and a triangle are studied using the discrete Fourier transform and orthogonal polynomials of two variables. A discrete Fourier analysis on the regular hexagon is developed in detail, from which the analysis on the triangle is deduced. The results include cubature formulas and interpolation on these domains. In particular, a trigonometric Lagrange interpolation on a triangle is shown to satisfy an explicit compact formula, which is equivalent to the polynomial interpolation on a planer region bounded by Steiner's hypocycloid. The Lebesgue constant of the interpolation is shown to be in the order of $(\log n)^2$. Furthermore, a Gauss cubature is established on the hypocycloid.Comment: 29 page

A higher moment formula for the Siegel--Veech transform over quotients by Hecke triangle groups

We compute higher moments of the Siegel--Veech transform over quotients of $SL(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on $SL(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W.~Schmidt on the variance of the Siegel transform over $SL(2,\mathbb{R})/SL(2,\mathbb{Z})$.Comment: 23 pages, 5 figure

Computing a discrete Morse gradient from a watershed decomposition

We consider the problem of segmenting triangle meshes endowed with a discrete scalar function f based on the critical points of f . The watershed transform induces a decomposition of the domain of function f into regions of influence of its minima, called catchment basins. The discrete Morse gradient induced by f allows recovering not only catchment basins but also a complete topological characterization of the function and of the shape on which it is defined through a Morse decomposition. Unfortunately, discrete Morse theory and related algorithms assume that the input scalar function has no flat areas, whereas such areas are common in real data and are easily handled by watershed algorithms. We propose here a new approach for building a discrete Morse gradient on a triangulated 3D shape endowed by a scalar function starting from the decomposition of the shape induced by the watershed transform. This allows for treating flat areas without adding noise to the data. Experimental results show that our approach has significant advantages over existing ones, which eliminate noise through perturbation: it is faster and always precise in extracting the correct number of critical elements

Quantum simplicial geometry in the group field theory formalism: reconsidering the Barrett-Crane model

A dual formulation of group field theories, obtained by a Fourier transform mapping functions on a group to functions on its Lie algebra, has been proposed recently. In the case of the Ooguri model for SO(4) BF theory, the variables of the dual field variables are thus so(4) bivectors, which have a direct interpretation as the discrete B variables. Here we study a modification of the model by means of a constraint operator implementing the simplicity of the bivectors, in such a way that projected fields describe metric tetrahedra. This involves a extension of the usual GFT framework, where boundary operators are labelled by projected spin network states. By construction, the Feynman amplitudes are simplicial path integrals for constrained BF theory. We show that the spin foam formulation of these amplitudes corresponds to a variant of the Barrett-Crane model for quantum gravity. We then re-examin the arguments against the Barrett-Crane model(s), in light of our construction.Comment: revtex, 24 page

Non-commutative quantum geometric data in group field theories

We review briefly the motivations for introducing additional group-theoretic data in tensor models, leading to the richer framework of group field theories, themselves a field theory formulation of loop quantum gravity. We discuss how these data give to the GFT amplitudes the structure of lattice gauge theories and simplicial gravity path integrals, and make their quantum geometry manifest. We focus in particular on the non-commutative flux/algebra representation of these models.Comment: 10 pages; to appear in the proceedings of the workshop "Non-commutative field theory and gravity", Corfu', Greece, EU, September 201

Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices

An integrable self-adjoint 7-point scheme on the triangular lattice and an integrable self-adjoint scheme on the honeycomb lattice are studied using the sublattice approach. The star-triangle relation between these systems is introduced, and the Darboux transformations for both linear problems from the Moutard transformation of the B-(Moutard) quadrilateral lattice are obtained. A geometric interpretation of the Laplace transformations of the self-adjoint 7-point scheme is given and the corresponding novel integrable discrete 3D system is constructed.Comment: 15 pages, 6 figures; references added, some typos correcte