arXiv subjects

Mathieu Dutour

Publications and source records attributed to Mathieu Dutour.

At least 19 recordsLinked to original sources

arXiv2025

A Minkowski-type theorem on distances to cusps: the general case

In a previous paper, we studied the connection between points in $\mathbb{H}^n$ and $2$-dimensional rigid adelic spaces on a totally real number field $K$ with class number $h_K = 1$. This last assumption was needed to link heights and distances to cusps. In this paper, we remove this hypothesis to obtain, without restriction on $K$ totally real, an analogue of Minkowski's second theorem on the Roy--Thunder minima of a $2$-dimensional rigid adelic space in the framework of distances between a point $\tau \in \mathbb{H}^n$ and its two closest cusps.

math.NT↗

arXiv2025

A Minkowski-type theorem on distances to cusps: the class number one case

In the study of Euclidean lattices, the product of the successive minima is bounded from above and below by explicit quantities. This result is known as Minkowski's second theorem, and can be refined to include Hermite's constant in the upper bound, which measures how short a non-zero vector can be in a given lattice. A version of this result exists in the context of number fields, where lattices are replaced with rigid adelic spaces, and successive minima with the Roy--Thunder minima. In this paper, drawing on the analogy between rank $2$ Euclidean lattices and points in $\mathbb{H}$, we will see an analogy between $2$-dimensional rigid adelic spaces and points in $\mathbb{H}^n$, and use that to translate the Minkowski-type theorem on Roy--Thunder minima into a theorem on the distances to cusps in $\mathbb{H}^n$.

math.NT↗

arXiv2022

Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Alvarez--Wentworth boundary conditions

A cuspidal end is a type of metric singularity, described as a product $S^1 \times \left] a, +\infty \right[$ with the Poincar\'e metric. The underlying set can also be seen as $\mathbb{R} \times \left] a, +\infty \right[$ subject to the action of the translation $T : \left( x,y \right) \longrightarrow \left( x+1, y \right)$. On it, one may consider a holomorphic line bundle $L$, coming from a unitary character of the group generated by $T$. The complex modulus induces a flat metric on $L$, and a pseudo-Laplacian $\Delta_{L,0}$ acting on functions can be associated to the Chern connection. One needs to specify boundary conditions, and they are here chosen to be the Alvarez--Wentworth boundary conditions, which are a combination of Dirichlet and Neumann boundary conditions. The aim of this paper is to find the asymptotic behavior of the zeta-regularized determinant $\det \left( \Delta_{L,0} + \mu \right)$, as $\mu > 0$ goes to infinity for any $a$, and also as $a$ goes to infinity for $\mu = 0$.

math.DG↗

arXiv2022

Infinite-rank Euclidean Lattices and Loop Groups

In this paper, we associate a family of infinite-rank pro-Euclidean lattices to elements of a formal loop group and a highest weight representation of the underlying affine Kac--Moody algebra. In the case that the element has a polynomial representative, we can prove our lattices are theta-finite in the sense of Bost, allowing us to attach to each of our lattices a well-defined theta-like function.

math.RT↗

arXiv2021

Pseudo-Laplacian on a cuspidal end with a flat unitary line bundle: Dirichlet boundary conditions

A cuspidal end is a type of metric singularity, described as a product $S^1 \times \left] a, +\infty \right[$ with the Poincar\'e metric. The underlying set can also be seen as $\mathbb{R} \times \left] a, +\infty \right[$ subject to the action of the translation $T : \left( x,y \right) \longrightarrow \left( x+1, y \right)$. On it, one may consider a holomorphic line bundle $L$, coming from a unitary character of the group generated by $T$. The complex modulus induces a flat metric on $L$, and a pseudo-Laplacian $\Delta_{L,0}$ can be associated to the Chern connection, with Dirichlet boundary conditions. The aim of this paper is to find the asymptotic behavior of the zeta-regularized determinant $\det \left( \Delta_{L,0} + \mu \right)$, as $\mu > 0$ goes to infinity for any $a$, and also as $a$ goes to infinity for $\mu = 0$.

math.DG↗

arXiv2008

Hypercube embedding of Wythoffians

The Wythoff construction takes a $d$-dimensional polytope $P$, a subset $S$ of $\{0,..., d\}$ and returns another $d$-dimensional polytope $P(S)$. If $P$ is a regular polytope, then $P(S)$ is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want to determine, which of those Wythoffians $P(S)$ with regular $P$ have their skeleton or dual skeleton isometrically embeddable into the hypercubes $H_m$ and half-cubes ${1/2}H_m$. We find six infinite series, which, we conjecture, cover all cases for dimension $d>5$ and some sporadic cases in dimension 3 and 4 (see Tables \ref{WythoffEmbeddable3} and \ref{WythoffEmbeddable4}). Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable $P(S)$ are addressed throughout the text.

math.CO↗

arXiv2007

A New Algorithm in Geometry of Numbers

A lattice Delaunay polytope P is called perfect if its Delaunay sphere is the only ellipsoid circumscribed about P. We present a new algorithm for finding perfect Delaunay polytopes. Our method overcomes the major shortcomings of the previously used method. We have implemented and used our algorithm for finding perfect Delaunay polytopes in dimensions 6, 7, 8. Our findings lead to a new conjecture that sheds light on the structure of lattice Delaunay tilings.

math.NT↗

arXiv2007

Perfect Delaunay Polytopes in Low Dimensions

A lattice Delaunay polytope is known as perfect if the only ellipsoid, that can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay polytopes are in one-to-one correspondence with arithmetic equivalence classes of positive quadratic functions on the n-dimensional integral lattice that can be recovered, up to a scale factor, from the representations of its minimum. We develop a structural theory of such polytopes and describe all known perfect Delaunay polytopes in dimensions one through eight. We suspect that this list is complete.

math.MG↗

arXiv2005

Cube packings, second moment and holes

We consider tilings and packings of $\RR^d$ by integral translates of cubes $[0,2[^d$, which are $4\ZZ^d$-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimension $d\leq 4$. For higher dimension, we use random methods for generating some examples. Such a cube packing is called {\em non-extendible} if we cannot insert a cube in the complement of the packing. In dimension 3, there is a unique non-extendible cube packing with 4 cubes. We prove that $d$-dimensional cube packings with more than $2^d-3$ cubes can be extended to cube tilings. We also give a lower bound on the number $N$ of cubes of non-extendible cube packings. Given such a cube packing and $z\in \ZZ^d$, we denote by $N_z$ the number of cubes inside the $\4t$-cube $z+[0,4[^d$ and call {\em second moment} the average of $N_z^2$. We prove that the regular tiling by cubes has maximal second moment and give a lower bound on the second moment of a cube packing in terms of its density and dimension.

math.CO↗

arXiv2005

Elementary elliptic $(R,q)$-polycycles

We consider the following generalization of the decomposition theorem for polycycles. A {\em $(R,q)$-polycycle} is, roughly, a plane graph, whose faces, besides some disjoint {\em holes}, are $i$-gons, $i \in R$, and whose vertices, outside of holes, are $q$-valent. Such polycycle is called {\em elliptic}, {\em parabolic} or {\em hyperbolic} if $\frac{1}{q} + \frac{1}{r} - {1/2}$ (where $r={max_{i \in R}i}$) is positive, zero or negative, respectively. An edge on the boundary of a hole in such polycycle is called {\em open} if both its end-vertices have degree less than $q$. We enumerate all elliptic {\em elementary} polycycles, i.e. those that any elliptic $(R,q)$-polycycle can be obtained from them by agglomeration along some open edges.

math.CO↗

arXiv2005

Some six-dimensional rigid forms

One can always decompose Dirichlet-Voronoi polytopes of lattices non-trivially into a Minkowski sum of Dirichlet-Voronoi polytopes of rigid lattices. In this report we show how one can enumerate all rigid positive semidefinite quadratic forms (and thereby rigid lattices) of a given dimension d. By this method we found all rigid positive semidefinite quadratic forms for d = 5 confirming the list of 7 rigid lattices by Baranovskii and Grishukhin. Furthermore, we found out that for d <= 5 the adjacency graph of primitive L-type domains is an infinite tree on which GL_d(Z) acts. On the other hand, we demonstrate that in d = 6 we face a combinatorial explosion.

math.MG↗

arXiv2004

Zigzag structure of complexes

Inspired by Coxeter's notion of Petrie polygon for $d$-polytopes (see \cite{Cox73}), we consider a generalization of the notion of zigzag circuits on complexes and compute the zigzag structure for several interesting families of $d$-polytopes, including semiregular, regular-faced, Wythoff Archimedean ones, Conway's 4-polytopes, half-cubes, folded cubes. Also considered are regular maps and Lins triality relations on maps.

math.CO↗

arXiv2004

Adjacency method for extreme Delaunay polytopes

The {\em hypermetric cone} is defined as the cone of semimetrics satisfying the {\em hypermetric inequalities}. Every {\em Delaunay polytope} corresponds to a ray of this polyhedral cone. The Delaunay polytopes, which correspond to extreme rays are called {\em extreme}. We use this polyhedral cone and the {\em closest vector problem} to present a new technique that allow to find, from a given extreme Delaunay polytope, some new ones. Then, we show some examples of applications of this technique in low-dimensions.

math.MG↗

arXiv2003

On simplicial and cubical complexes with short links

We consider closed simplicial and cubical $n$-complexes in terms of link of their $(n-2)$-faces. Especially, we consider the case, when this link has size 3 or 4, i.e., every $(n-2)$-face is contained in 3 or 4 $n$-faces. Such simplicial complexes with {\em short} (i.e. of length 3 or 4) links are completely classified by their {\em characteristic partition}. We consider also embedding into hypercubes of the skeletons of simplicial and cubical complexes.

math.GT↗

arXiv2003

A result on the phase diagram of a Ginzburg-Landau problem

Working with a particular modelization of Ginzburg-Landau phenomenological theory (see \cite{dutourII}, \cite{dutour} and Section \ref{change-var}), we first recall the form of the phase diagram of this modelization as it usually drawn in the physical literature (\cite{tink}, \cite{Kittel}, \cite{sarma} and \cite{PG-de-Gennes}). We then study in detail the special case, when the critical Ginzburg Landau parameter $k$ is equal to $\frac{1}{\sqrt{2}}$. This allows us to prove that the critical magnetic field $H_{c1}(k)$ is strictly decreasing at $k=\frac{1}{\sqrt{2}}$. PACS: 01.30.Cc, 02.30.Jr, 74.25.Dw

math-ph↗

arXiv2002

The hypermetric cone on seven vertices

The hypermetric cone $HYP_n$ is the set of vectors $(d_{ij})_{1\leq i< j\leq n}$ satisfying the inequalities $\sum_{1\leq i<j\leq n} b_ib_jd_{ij}\leq 0 with b_i\in\Z and \sum_{i=1}^{n}b_i=1$. A Delaunay polytope of a lattice is called extremal if the only affine bijective transformations of it into a Delaunay polytope, are the homotheties; there is a correspondance between such Delaunay polytopes and extreme rays of $HYP_n$. We show that unique Delaunay polytopes of root lattice $A_1$ and $E_6$ are the only extreme Delaunay polytopes of dimension at most 6. We describe also the skeletons and adjacency properties of $HYP_7$ and of its dual.

math.MG↗

arXiv2000

Bifurcation vers l'etat d'Abrikosov et diagramme des phases

Nous étudions dans cette thèse la fonctionnelle de Ginzburg-Landau dans $\R^3$ sur des couples de fonctions $(ϕ, \overrightarrow{A})$ qui vérifient des conditions de périodicité de jauge en $x_3$ et selon un réseau discret de $(x_1,x_2)$. Nous montrons que le problème variationnel est équivalent au problème de la minimisation d'une autre fonctionnelle sur un tore. Dans le cadre de la démonstration, un fibré vectoriel non trivial apparaît. On se limite alors pour la suite à une quantification de 1. On montre ensuite que la fonctionnelle admet un minimum sur l'espace fonctionnel $H^{1}$ qui vérifie un système d'équations aux dérivées partielles appelé système de Ginzburg-Landau. Le minimum est $C^{\infty}$ par l'ellipticité du système d'équations de Ginzburg-Landau. On montre qu'il y a une bifurcation du couple $(0,0)$ pour le champ critique $H_{ext}=k$ où $k$ est un paramètre caractéristique du système. On étudie alors la stabilité de la solution bifurquée. On étudie la dépendance de l'énergie minimale à l'égard de la géométrie du tore. Enfin nous décrivons toutes les solutions du système d'équations de Ginzburg-Landau dans la limite $k$ tend vers l'infini. Dans le dernier chapitre, nous donnons pour notre modèle la structure du diagramme des phases en précisant quelles régions sont normales, supraconductrices pure, mixte.

math-ph↗

arXiv2000

$L^p$ version of a result by Rankin

We extend a classical result by Rankin. We consider the following question: given $n$ vectors $v_i$ in the ball of radius $R$ of an infinite dimensional Banach space ${\cal B}$ with $d(v_i,v_j)\geq 1$, can we bound the number $n$?

math.CA↗