Papers by Christian Mercat
HAL (Le Centre pour la Communication Scientifique Directe), Oct 10, 2015
Outdoor and Embodied Mathematics with MathCityMap
HAL (Le Centre pour la Communication Scientifique Directe), Jun 14, 2022
PhD supervision of Frédéric Rieux on digital diffusion and convolution applied to discrete analys... more PhD supervision of Frédéric Rieux on digital diffusion and convolution applied to discrete analysis - 3 M.Sc. in computer science supervisions. -Member of the Directing Committee of the Pôle MIPS (Mathematics, Computer Sc., Physics, Mechanics) of the University Montpellier 2 -Initiator of the Inter2Geo European project -Partner of the Math-Bridge European project
Birkhäuser Boston eBooks, 2002
We derive the fusion hierarchy of functional equations for critical AD -E lattice models related ... more We derive the fusion hierarchy of functional equations for critical AD -E lattice models related to the sℓ(2) unitary minimal models, the parafermionic models and the supersymmetric models of conformal field theory and deduce the related TBA functional equations. The derivation uses fusion projectors and applies in the presence of all known integrable boundary conditions on the torus and cylinder. The resulting TBA functional equations are universal in the sense that they depend only on the Coxeter number of the AD -E graph and are independent of the particular integrable boundary conditions. We conjecture generally that TBA functional equations are universal for all integrable lattice models associated with rational CFTs and their integrable perturbations.
Journal of physics, Jul 16, 2001
We study integrable and conformal boundary conditions forŝℓ(2) Z k parafermions on a cylinder. Th... more We study integrable and conformal boundary conditions forŝℓ(2) Z k parafermions on a cylinder. These conformal field theories are realized as the continuum scaling limit of critical AD -E lattice models with negative spectral parameter. The conformal boundary conditions labelled by (a, m) ∈ (G, Z 2k) are identified with associated integrable lattice boundary conditions labelled by (r, a) ∈ (A g−2 , G) where g is the Coxeter number of the AD -E graph G. We obtain analytically the boundary free energies, present general expressions for the parafermion cylinder partition functions and confirm these results by numerical calculations.
HAL (Le Centre pour la Communication Scientifique Directe), Jun 24, 2002
We use discrete holomorphic polynomials to prove that, given a refining sequence of critical maps... more We use discrete holomorphic polynomials to prove that, given a refining sequence of critical maps of a Riemann surface, any holomorphic function can be approximated by a converging sequence of discrete holomorphic functions.
Bulletin de la Société Mathématique de France, 2004
We show that discrete exponentials form a basis of discrete holomorphic functions on a finite cri... more We show that discrete exponentials form a basis of discrete holomorphic functions on a finite critical map. On a combinatorially convex set, the discrete polynomials form a basis as well. Résumé (Les exponentielles forment une base des fonctions holomorphes discrètes sur un compact) Nous montrons que les exponentielles forment une base des fonctions holomorphes discrètes sur une carte critique compacte. Sur un convexe, les polynômes discrets formentégalement une base.
HAL (Le Centre pour la Communication Scientifique Directe), Jul 6, 2009
We describe the key achievements of Intergeo, a project co-funded within the eContentplus program... more We describe the key achievements of Intergeo, a project co-funded within the eContentplus programme of the European Union, and their impact on teacher education, based on the technical and educational concepts of the Intergeo Platform.
Springer eBooks, 2008
This paper defines a theory of conformal parametrization of digital surfaces made of surfels equi... more This paper defines a theory of conformal parametrization of digital surfaces made of surfels equipped with a normal vector. The main idea is to locally project each surfel to the tangent plane, therefore deforming its aspect-ratio. It is a generalization of the theory known for polyhedral surfaces. The main difference is that the conformal ratios that appear are no longer real in general. It yields a generalization of the standard Laplacian on weighted graphs.
Lecture Notes in Computer Science, 2010
There are a lot of transformations that can turn one raster image into a derived one in a mathema... more There are a lot of transformations that can turn one raster image into a derived one in a mathematically interesting way. This article describes a collection of such filters, implemented in OpenGL in order to use the high degree of parallelism modern GPUs provide, thereby providing performance required to process e.g. live camera images in real-time. The filters contained in this library include wallpaper groups, conformal maps described by meromorphic functions, as well as hyperbolic symmetry groups. Using examples of increasing complexity, several key implementation techniques are explained, including texture wrap configurations, user-configurable control points, and custom fragment shader programs. This work might exhibit aesthetic aspects of mathematics to the masses and provide useful building blocks for scientists as well as artists.
Lecture Notes in Mathematics, Dec 21, 2010
EMS Press eBooks, May 25, 2007
HAL (Le Centre pour la Communication Scientifique Directe), Jan 28, 2009
In this overview article we describe the manifold achievements and challenges of Intergeo 1 , a p... more In this overview article we describe the manifold achievements and challenges of Intergeo 1 , a project co-funded within the eContentplus programme 2 of the European Union.
Physics Letters B, Oct 1, 2001
We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) u... more We construct integrable lattice realizations of conformal twisted boundary conditions for s (2) unitary minimal models on a torus. These conformal field theories are realized as the continuum scaling limit of critical AD -E lattice models with positive spectral parameter. The integrable seam boundary conditions are labelled by (r, s, ζ) ∈ (A g−2 , A g−1 , Γ) where Γ is the group of automorphisms of the graph G and g is the Coxeter number of G = A, D, E. Taking symmetries into account, these are identified with conformal twisted boundary conditions of Petkova and Zuber labelled by (a, b, γ) ∈ (A g−2 ⊗ G, A g−2 ⊗ G, Z 2) and associated with nodes of the minimal analog of the Ocneanu quantum graph. Our results are illustrated using the Ising (A 2 , A 3) and 3-state Potts (A 4 , D 4) models.
HAL (Le Centre pour la Communication Scientifique Directe), Dec 12, 2022
-Dans cet article nous présentons un projet d'étude basé sur l'Analyse de interactions élèves/ens... more -Dans cet article nous présentons un projet d'étude basé sur l'Analyse de interactions élèves/enseignant présentée dans (Kiwan-Zacka & Roditi, 2019). Cette étude portera sur les ateliers récréatifs tels que présentés au Mathscope et aura trois buts principaux : mieux comprendre le rôle joué par les médiateurs lors de l'activité, faire prendre conscience des pratiques et ainsi mieux former les médiateurs et enfin comprendre les différences entre les divers ateliers du Mathscope.
The Finnish higher education system consists of two complementary sectors: polytechnics and unive... more The Finnish higher education system consists of two complementary sectors: polytechnics and universities. The mission of universities is to conduct scientific research and provide undergraduate and postgraduate education based on it. Polytechnics train professionals in response to labor market needs and conduct R&D which supports instruction and promotes regional development in particular. Universities promote free research and scientific and artistic education, provide higher education based on research, and educate students to serve their country and humanity. In carrying out this mission, universities must interact with the surrounding society and strengthen the impact of research findings and artistic activities on society. Under the new Universities Act, which was passed by Parliament in June 2009, Finnish universities are independent corporations under public law or foundations under private law (Foundations Act). The universities have operated in their new form from 1 January 2010 onwards. Their operation is built on the freedom of education and research and university autonomy. Universities confer Bachelor's and Master's degrees, and postgraduate Licentiate and Doctoral degrees. Universities work in cooperation with the surrounding society and promote the social impact of research findings. The higher education
Springer eBooks, 2011
We introduce a new method to compute conformal parameterizations using a recent definition of dis... more We introduce a new method to compute conformal parameterizations using a recent definition of discrete conformity, and establish a discrete version of the Riemann mapping theorem. Our algorithm can parameterize triangular, quadrangular and digital meshes. It can be adapted to preserve metric properties. To demonstrate the efficiency of our method, many examples are shown in the experiment section.
Springer eBooks, 2018
Kazan National Research Technical University named after A.N. Tupolev-KAI (KNRTU-KAI) was establi... more Kazan National Research Technical University named after A.N. Tupolev-KAI (KNRTU-KAI) was established in 1932. The history of the University is closely related to the progress of Russian aeronautics. Fundamental education and profound scientific research are the distinguishing features of the university, which make it very attractive for a great number of young people. Until recent times, it was known as Kazan Aviation Institute (KAI). In 1973, the Institute was named after Andrey N. Tupolev, the prominent aircraft designer. In 1992, it obtained the status of State Technical University. In 2009, KNRTU-KAI became 1 of 12 universities selected among all the Russian universities (from about 900 state HE institutions and 2000 private ones) which was awarded the prestigious title of the "National Research University". KNRTU-KAI is a member of the European Universities Association EUA (2008) and of the European Association of Aerospace Universities PEGASUS (2009).
Journal of Mathematical Imaging and Vision, Jun 28, 2012
HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific re... more HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
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Papers by Christian Mercat