Y. Nakamura established that gradient systems defined on specific statistical manifolds, such as those associated with Gaussian and multinomial distributions, satisfy the conditions of Liouville complete integrability. Furthermore, he demonstrated that gradient flows on statistical manifolds may be linearised through the application of dual coordinates from information geometry, in a manner analogous to the action-angle coordinates employed in Hamiltonian mechanics to characterise integrable systems. We extend this line of inquiry to Souriau’s symplectic model of Information Geometry for Lie groups. Subsequently, we examine algebraic complete integrability in the sense of Adler and van Moerbeke, as well as the symplectic structure underlying Lax pairs, which can be formulated in terms of algebraic-geometric structures. This approach underlines the interplay between the analytical and group-theoretical methodologies in the study of integrable systems. Finally, we study the Adler-Kostant-Symes theorem as a principal tool for the construction of integrable systems, leveraging its capacity to establish algebraic integrability, and its link to Souriau fundamental equation of Lie Groups Thermodynamics ⟨Q,[ β ,Z]⟩ + Θ̃( β ,Z) = 0 .
Nonholonomic mechanics has received considerable attention in dynamics and control area. However, due to a wide range of fluctuations in the physical world, the ideal mathematical models of mechanical systems with nonholonomic constraints suffer from issues of ignoring the real-world perturbations and physically difficult to realize. Motivated by recent developments in stochastic and constrained mechanics, here we present a stochastic variational formulation for mechanical systems with or without stochastic nonholonomic constraints. We give stochastic variational principles for both stochastically unconstrained and nonholonomic cases under the same framework by deriving the stochastic implicit Hamel equations. Moreover, an interesting example of the stochastic rolling disk is provided to illustrate the proposed method.
The tangent space at a KMS state (Kubo Martin Schwinger state) can be decomposed into two subspaces in such a way that the time evolution, which is described by the modular automorphism group, satisfies the KMS condition w.r.t. each of these two subspaces. The tangent space is said to be two-typed because each tangent vector is the sum of two vectors, one in each subspace. The type of the tangent vector is conserved under time evolution. An explicit construction of the pair of subspaces is presented. The starting point is an orthonormal basis diagonalizing the modular operator. The example of the quantum spin-1/2 is worked out in detail. The constructed subspaces are of dimension 3, respectively 1. The paper is restricted to the finite-dimensional case. In this way the technicalities of handling unbounded operators are not exposed.
The Bayesian Cramér-Rao Bound (BCRB) is generally attributed to Van Trees who published it in 1968. According to Stigler’s law of eponymy, no scientific discovery is named after its first discoverer. This is the case not only for the Cramér-Rao bound itself—due in particular to the French mathematicians Fréchet and Darmois—but also for the van Trees inequality: The French physician, geneticist, epidemiologist and mathematician Marcel-Paul (Marco) Schützenberger, in a paper of just fifteen lines written in 1956—more than a decade before van Trees—had not only derived the BCRB but, as a close examination of his proof shows, used a very original approach based on the Weyl-Heisenberg uncertainty principle on the square root of the posterior distribution. This work reviews and extends Schützenberger’s approach to Fisher information matrices, which opens up new perspectives.
We propose a natural intrinsic extension of the ridge regression from Euclidean spaces to general manifolds, which relies on Riemannian least-squares fitting, empirical covariance, and Mahalanobis distance. We utilize it for time-series prediction and apply the approach to forecast hurricane tracks and their intensities (maximum wind speeds).
Building on stochastic geometric mechanics on Riemannian manifolds, we shall focus on extensions of the classical Maupertuiss variational principle to a class of diffusion processes as extremals of a stochastic action functional preserving the expectation of energy. We shall also mention a recent and related stochastic Jacobi integration theorem, whose consequence will be analyzed elsewhere.
Because of the strict separation of mass and energy in Galilei physics, a Galilei-invariant tensor formalism is most at home in a 5-dimensional extended spacetime associated with the Bargmann-Galilei (traditionally ‘Bargmann’) group, a central extension of the Galilei group that explicitly exhibits the transformation properties of kinetic energy. While not necessary for a tensor formalism fully embodying Poincaré physics, a similar central extension of the Poincaré group to the Bargmann-Poincaré group may illuminate a path towards a strong-field ‘Galilei general relativity’. Here the Bargmann metric is generalized to curved spacetime by extending the usual 1+3 (traditionally ‘3+1’) formalism of general relativity on 4-dimensional spacetime to a 1+3+1 formalism, whose spacetime kinematics is shown to be consistent with that of the usual 1+3 formalism. On Bargmann spacetime, tensor laws governing the motion of an elementary classical material particle and the dynamics of a simple fluid reference the foliation of spacetime in a manner that partially reverts the Einstein perspective (accelerated fiducial observers, and geodesic material particles and fluid elements) to a Newton-like perspective (geodesic fiducial observers, and accelerated material particles and fluid elements subject to a gravitational force).
In Bayesian statistics, the selection of noninformative priors is a crucial issue. There have been discussions on theoretical justification and problems for the Jeffreys prior, as well as alternative objective priors. Among them, we will focus on the two types of matching priors consistent with frequency theory: the probability matching priors and the moment matching priors. In particular, there is no clear relationship between these two matching priors on non-regular statistical models, even though they have similar objectives. Considering information geometry on a one-sided truncated exponential family, a typical example of non-regular statistical models, we obtain the result that the Lie derivative along one vector field provides the conditions for the probability and moment matching priors. Note that this Lie derivative does not appear in regular models. This result promotes a unified understanding of probability and moment matching priors on non-regular models. Further, we discuss the relationship between the probability and moment matching priors and the α -parallel priors.
The relation between Souriaus Lie group thermodynamics and statistical transformation models is investigated. Particularly, the Fisher distributions on the 2-sphere is discussed in detail and the Fisher-Rao metric is found explicitly. The geodesic flow on SO(3) associated with the statistical transformation model of the Fisher distributions on the 2-sphere is integrable and provides an interesting example of dynamical systems on Gibbs sets.
In this paper we propose a novel methodology that extends Linear Discriminant Analysis (LDA) to Kendall's shape space to classify 3D shapes and analyze which features most influence class differentiation. Our approach adapts LDA to the non-Euclidean geometry of the shape space, generalizing assumptions about the probability distribution of data in Euclidean spaces and incorporating parallel transport to improve the estimation of shape variability between clusters. A simulation study is performed to show the effectiveness of the proposed methodology.
In a previous paper, we proposed a symplectic version of the Brezis-Ekeland-Nayroles principle. We applied it to the standard plasticity. The object of this work is to extend the previous formalism to non associated laws. For this aim, we introduce the concept of symplectic bipotential which extends that of bipotential to dynamical systems. We present a method to build it from a bipotential. Next, we generalize the symplectic Brezis-Ekeland-Nayroles principle to the non associated dissipative laws. As example, we apply it to the unilateral contact law with Coulombs dry friction.
RNA residues come in a plethora of geometric conformational shapes, some of which are very common and some of which are rather rare. In this contribution we extend the previously developed clustering algorithm MINT-AGE in order to find within a large database rare conformational clusters of very small sizes. To this end in the MINT step we replace the previous nonparametric circular mode hunting by a parametric version. In validation, this allows to identify conformational classes of sizes ≥ 2 via statistical unsupervised learning on the gold standard database, hand curated by the Richardson Laboratory.
This note presents a notion of generalised metriplectic systems which includes not necessarily holonomic constraints using methods from port-Hamiltonian systems theory. Metriplectic systems can be written as particular dissipative Hamiltonian systems with the exergy as Hamiltonian. Within the generalised dissipative Hamiltonian systems, a generalisation of metriplectic systems is identified.
Morphoelastic bodies are elastic materials that undergo complex shape changes due to intrinsic growth, remodelling, or active internal processes. In a theoretical context, these materials are typically represented by non-Euclidean material manifolds characterized by an evolving metric structure. In this work, we formulate remodelling on such manifolds through a gradient system, where the inertialess dynamics of the system are driven by the steepest descent of an energy functional within an appropriate metric space. We obtain a gradient flow equation by combining isotropic non-linear elasticity with growth-induced dissipative mechanisms and illustrate the formalism through numerical simulations of stress relaxation at fixed strain.
A triplet (ω , ℱ_1,ℱ_2) is a bilagrangian structure on a manifold M, if ω is a 2-form, closed and non-degenerate (called symplectic form) on M, and (ℱ_1,ℱ_2) is a pair of transversal Lagrangian foliations on the symplectic manifold (M,ω ) . The quadruplet (M, ω , ℱ_1,ℱ_2) is called a bilagrangian manifold. We prolong a bi-Lagrangian structure on M on its cotangent bundle T^*M in different ways. As a consequence, some dynamics on the set of bi-Lagrangian structures of M can be prolonged as dynamics on the set of the bi-Lagrangian structures of TM and T^*M .
Gaussian Mixture Models (GMMs) are important tools for modeling complex data in machine learning tasks and computer vision applications. However, computing f-divergences between GMMs remains challenging due to the absence of a closed-form expression, which led to expensive numerical approximations that limit practical applications. In this paper, we give an efficient f-divergence approximation through the embedding of GMMs into the symmetric positive definite (SPD) matrices. Our main result is that for any compact set of non-degenerate GMM parameters, the f-divergence between two GMMs and the computationally efficient f-divergence between their corresponding centered multivariate normal distributions in the SPD space are uniformly equivalent. Our approach preserves the geometric structure of GMMs while enabling closed-form computation. As an instance the proposed framework is applied on the UIUC texture recognition datasets.
In this paper, we propose a Hamilton-Dirac formulation for non-simple nonequilibrium thermodynamic systems with chemical reactions and diffusion. A key feature of these systems is the degeneracy of their associated Lagrangian function. To address this, we build upon Diracs theory of constraints for degenerate Lagrangians and develop a Hamiltonian variational formulation for nonholonomic systems with nonlinear thermodynamic constraints, as well as primary constraints arising from the degeneracy. We introduce the constrained Hamiltonian on the primary constraint set and clarify the underlying geometric structure using Dirac structures. Finally, we illustrate our Hamilton-Dirac formulation with an example of a membrane undergoing matter reaction and diffusion.
In this paper, we propose a physics-informed machine learning method called Deep Dirac Neural Networks, based on the framework of Dirac dynamical system [10]. This method is a powerful tool for learning a generalized energy function in mechanical systems with constraints. Specifically, we focus on mechanical systems with holonomic constraints. Our approach enables the learning of not only dynamic entities such as the generalized energy but also kinematic entities such as Lagrange multipliers and holonomic constraint functions, directly from training and target data, without requiring any prior knowledge of the constraint conditions. This is achieved by training three separate modular models in the framework of the Dirac dynamical systems, where a loss function is designed to enforce the holonomic constraints. Unlike conventional approaches, our method does not rely on predefined constraint conditions, offering greater flexibility in modeling. Finally, we demonstrate the effectiveness of the proposed method through numerical experiments using a double pendulum as an illustrative example.
