Robert Brouzet - Academia.edu (original) (raw)

Papers by Robert Brouzet

The College Mathematics Journal, 2015

Symmetry, Integrability and Geometry: Methods and Applications

We state and prove that a certain class of smooth functions said to be BH-separable is a meagre s... more We state and prove that a certain class of smooth functions said to be BH-separable is a meagre subset for the Fréchet topology. Because these functions are the only admissible Hamiltonians for Arnold-Liouville systems admitting a bi-Hamiltonian structure, we get that, generically, ArnoldLiouville systems cannot be bi-Hamiltonian. At the end of the paper, we determine, both as a concrete representation of our general result and as an illustrative list, which polynomial Hamiltonians H of the form H(x, y) = xy + ax + bxy + cxy + dy are BH-separable. AMS classification (2020): 26A21, 26B35, 26B40, 37J35, 37J39, 58K15, 70H06 Key-words : Completely integrable Hamiltonian system, Arnold-Liouville theorem, action-angle coordinates, bi-Hamiltonian system, separability of functions, change of coordinates, Fréchet topology, meagre set.

In this paper, we present a new method for sharing images between two parties exploiting homomorp... more In this paper, we present a new method for sharing images between two parties exploiting homomorphic property of public key cryptosystem. With our method, we show that it is possible to multiply two encrypted images, to decrypt the resulted image and after to extract and reconstruct one of the two original images if the second original image is available. Indeed, extraction and reconstruction of original image at the receiving end is done with the help of carrier image. Experimental results and security analysis show the effectiveness of the proposed scheme.

Comptes Rendus De L Academie Des Sciences Serie I-mathematique, 1998

Nous étudions un certain type de fibrations lagrangiennes dont l'espace total est muni d'une pseu... more Nous étudions un certain type de fibrations lagrangiennes dont l'espace total est muni d'une pseudo-métrique compatible avec la forme symplectique. Nous montrons que, sous certaines conditions géométriques, la base de telles fibrations hérite d'un certain nombre d'invariants permettant d'en donner une classification.We study a kind of Lagrangian fibrations whose total space is endowed with a compatible pseudo-metric. Under suitable geometrical conditions we construct, on the basis, a family of invariants which classify them.

Physics Letters A, 2006

We explain why only the Pfaffian case appears in the study of quasi-bi-Hamiltonian systems.

Journal of Geometry and Physics, 2006

In this paper, after some recalls about Poisson cohomology, we first study what the general metho... more In this paper, after some recalls about Poisson cohomology, we first study what the general method is in order to obtain a bi-Hamiltonian formulation of a given Hamiltonian system by means of a deformation. Then we show that the bi-Hamiltonian formulation which results from the deformation of a Poisson structure by means of a suitable non-Noether symmetry cannot explain the complete integrability for a large class of Arnold–Liouville integrable systems; next we prove that the deformation must be made in this context by a suitable mastersymmetry. At last, we give several examples.

Differential Geometry and Its Applications, 2008

We study completely integrable quasi-bi-Hamiltonian systems whose common level surfaces are compa... more We study completely integrable quasi-bi-Hamiltonian systems whose common level surfaces are compact and prove in particular their separability in the sense of Falqui and Pedroni.

In this paper, the main objective is to establish an existence result of a variational model in i... more In this paper, the main objective is to establish an existence result of a variational model in image segmentation constrained by a given vector field. In the one dimensional case, we give a discrete version converging in a variational way to the continuous model. We finally describe the numerical analysis of this model with application in image segmentation.

The main objective of this paper is to introduce and illustrate a new tool stemming from Young me... more The main objective of this paper is to introduce and illustrate a new tool stemming from Young measure theory in order to capture concentrations, jump sign and gradient oscillations of sequences of SBV-functions. We show how this notion of measure can be applied for the analysis of approximating solutions of Mumford-Shah type energy functionals in the one dimensional case.

Optics Communications, 2011

In this paper, we present a new approach for sharing images between l players by exploiting the a... more In this paper, we present a new approach for sharing images between l players by exploiting the additive and multiplicative homomorphic properties of two well-known public key cryptosystems, i.e. RSA and Paillier. Contrary to the traditional schemes, the proposed approach employs secret sharing in a way that limits the influence of the dealer over the protocol and allows each player to participate with the help of his key-image. With the proposed approach, during the encryption step, each player encrypts his own key-image using the dealer's public key. The dealer encrypts the secret-to-be-shared image with the same public key and then, the l encrypted key-images plus the encrypted to-be shared image are multiplied homomorphically to get another encrypted image. After this step, the dealer can safely get a scrambled image which corresponds to the addition or multiplication of the l + 1 original images (l key-images plus the secret image) because of the additive homomorphic property of the Paillier algorithm or multiplicative homomorphic property of the RSA algorithm. When the l players want to extract the secret image, they do not need to use keys and the dealer has no role. Indeed, with our approach, to extract the secret image, the l players need only to subtract their own key-image with no specific order from the scrambled image. Thus, the proposed approach provides an opportunity to use operators like multiplication on encrypted images for the development of a secure privacy preserving protocol in the image domain. We show that it is still possible to extract a visible version of the secret image with only l-1 key-images (when one key-image is missing) or when the l key-images used for the extraction are different from the l original key-images due to a lossy compression for example. Experimental results and security analysis verify and prove that the proposed approach is secure from cryptographic viewpoint.

Secret sharing between two or more parties, exploiting multiplicative homomorphic properties of R... more Secret sharing between two or more parties, exploiting multiplicative homomorphic properties of RSA, may result in false data blocks during the extraction of message. This paper investigates factors that lead to such false blocks and suggest a controlled solution by analyzing their probability distribution. In this paper we prove that false blocks do exist and show that larger sizes of the prime product reduce the error probability of false blocks in extracted messages.

Expositiones Mathematicae, 2004

Ce texte a pour but de décrire les contextes mathématiques qui permirent l'émergence d'un des gro... more Ce texte a pour but de décrire les contextes mathématiques qui permirent l'émergence d'un des groupes dits classiques, à savoir le groupe symplectique, par lequel le terme même de “symplectique”, fit son apparition en mathématiques sous la plume d'Hermann Weyl. Nous montrerons plus précisément, que ce groupe possède une double origine, l'une dans le cadre de la géométrie projective avec l'étude des complexes de droites, et l'autre dans des questions relatives aux transformations des intégrales abéliennes.The goal of this paper is the description of the mathematical contexts which allowed the emergence of one of the so called classical groups, namely the symplectic group, by which the word “symplectic” appeared in mathematics. We show that it has a double origin, one in the line complexes of the projective geometry and the other one, in some questions about transformations of abelian integrals.