arXiv subjects

Prosper Rosaire Mama Assandje

Publications and source records attributed to Prosper Rosaire Mama Assandje.

10 recordsLinked to original sources

arXiv2026

Geometric Dissipation Structures and the Koszul Form

This paper investigates the canonical Koszul one-form at the interface of Jean Marie Souriau's Lie group thermodynamics and Paulette Libermann's symplectically complete foliations to describe geometric dissipation. On the Poincaré half-plane $\mathbb{H}$ under $SL(2, \mathbb{R})$, the equivariant momentum map level sets form a regular Libermann foliation framing the conservative boundaries. The Koszul-derived dissipative vector field acts strictly transverse to these leaves, driving a metriplectic flow that breaks Noether conservation laws and generates entropy. Quantum wise, we develop Berezin quantization on para-Kähler symmetric spaces, where the Koszul potential acts as a weight determining state density via the Fisher Souriau metric, and formalize Kostant's geometric quantization using the real dual polarizations of a bi-Lagrangian web.

math.DG↗

arXiv2026

A Hamiltonian driven Geometric Construction of Neural Networks via the Lognormal family, Application to Financial Fraud Detection and to Network Security

We presents a method for constructing neural networks intrinsically on statistical manifolds via the lognormal distribution. We demonstrate this approach by formulating a neural network architecture directly on statistical manifold. The construction is driven by the Hamiltonian system that is equivalent to the gradient flow on this manifold. We define the network's input values using the coordinate system of this Hamiltonian dynamics, naturally embedded in the Poincar$\acute{e}$ disk. The core of our contribution lies in the derivation of the network's components from geometric principles: the rotation component of the synaptic weight matrix is determined by the Lie group action of $SU(1,1)$ on the disk, while the activation function emerges from the symplectic structure of the system. We subsequently obtain the complete weight matrix, including its translation vector, and the resulting output values.

cs.LG↗

arXiv2026

Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3).

math.DG↗

arXiv2026

On examples of duals Saito's basis of some inhomogeneous divisors, and application

We investigate a class of non-quasi-homogeneous free divisors in the sense of Saito. These divisors are defined by equations of the form $D:= \{h=0\}$ on $\mathbb{C}^p$, where the polynomial $h$ is specific linear combination of monomials involving the product of coordinates. For this class, we explicitly construct a Saito basis for the module of logarithmic vector fields $Der(logD)$. This construction is then applied to the setting of logarithmic Poisson geometry. Focusing on the example defined by $h=xy+x^{2}y^{2}+x^3y^3$ on the Poisson algebra $(\mathcal{A}=\mathbb{C}[x,y], \{-,-\}_{h})$, where the Poisson bracket is induced by the bivector $π= h\partial x\wedge\partial y$. We define the associated Koszul bracket on the module of logarithmic 1-forms. This enables us to prove that $π$ endows the sheaf of logarithmic 1-forms $Ω^{1}(log D )$ with a Lie-Rinehart algebra structure. Furthermore, we introduce and provide explicit descriptions for the resulting cohomology theory, which we term the logarithmic Poisson cohomology $H_{log}^{\bullet} $ of $\{-,-\}_{h}$. As a related and foundational computation, we also calculate the corresponding logarithmic De Rham cohomology $H^{\bullet}_{DR}$ for the divisor $D$ and we make a generalization in dimension 2.

math.DG↗

arXiv2025

On Logit Weibull Manifold

In this work, it is shown that there is no potential function on the Weilbull statistical manifold. However, from the two-parameter Weibull model we can extract a model with a potential function called the logit model. On this logit model, there is a completely integrable Hamiltonian gradient system.

math.ST↗

arXiv2025

Deformation quantization of a hessian KV- structure on $\mathbb{R}^2$

This paper studies the quantization of the deformation of Hessian structures on a two-dimensional vector space, in the framework of Koszul-Vinberg algebras. We analyze how Hessian structures can be deformed to obtain quantum structures while preserving certain geometric and algebraic properties. The aim of the paper is to establish links between deformation theory and Hessian geometry.

math.DG↗

arXiv2025

On KV-Poisson Structure and related invariants

We propose an deepened analysis of KV-Poisson structures of on IR^2. We present their classification their properties an their possible applications in different domains. We prove that these structure give rise to a new Cohomological invariant. We explicitly compute the Cohomological groups of some of these structures.

math.DG↗

arXiv2025

Riemannian curvature of gaussian distribution in dual coordinate system

In this paper, we study the geometric nonlinearity properties, such as curvature and torsion, in a dual coordinate system of the Riemannian manifold defined by the Gaussian distribution. We also give the Amari formulas explicitly in this new coordinate system, which allows us to characterize existing geometric invariants, such as the dual potential function and the Fisher metric.

math.DG↗

arXiv2025

Completely Integrable Gradient System on the bivariate beta statistical manifold

This paper investigates the geometry of a completely integrable gradient system defined on the three parameter bivariate beta statistical manifold of the first kind. We prove that the associated vector field is Hamiltonian and admits a Lax pair representation implying complete integrability. We show that the potential function derived from exponential family structure defines a Riemannian metric equivalent to the Fisher information metric. By applying Stirling's approximation to the gamma functions involved in the potential, we obtain an explicit expression that facilitates the study of the pseudo-riemannian geometry of the manifold. Furthermore, we demonstrate that the gradient flow is linearizable in dual affine coordinates, and we identify the Hamiltonian function whose gradient defines the flow. These results highlight the deep interplay between information geometry, dynamical systems, and asymptotic analysis.

math.DG↗

arXiv2025

A Kahlerian approche to the Schrodinger equation in Siegel jacobi Space of the lognormal distribution

In this paper, we describe the evolution of spectral curves in the Siegel Jacobi space through the Schrodinger equation constructed from a Kahler geometry induced on the lognormal statistical manifold via Dombrowski's construction. We introduce new holomorphic structures and show that the Hamiltonian vector field coincides with the fundamental vector field generated by holomorphic isometries. We construct the time dependent Schrodinger equation from this geometric setting and show that the associated energy is not constant, but varies with time. This work establishes a bridge between Kahler geometry, statistical models, and the formalism of quantum mechanics.

math.DG↗