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We carry out the extended symmetry analysis of an ultraparabolic Fokker–Planck equation with three independent variables, which is also called the Kolmogorov equation and is singled out within the class of such Fokker–Planck equations by its remarkable symmetry properties. In particular, its essential Lie invariance algebra is eight-dimensional, which is the maximum dimension within the above class. We compute the complete point symmetry pseudogroup of the Fokker–Planck equation using the direct method, analyse its structure and single out its essential subgroup. After listing inequivalent one- and two-dimensional subalgebras of the essential and maximal Lie invariance algebras of this equation, we exhaustively classify its Lie reductions, carry out its peculiar generalised reductions and relate the latter reductions to generating solutions with iterative action of Lie-symmetry operators. As a result, we construct wide families of exact solutions of the Fokker–Planck equation, in particular, those parameterised by an arbitrary finite number of arbitrary solutions of the (1+1)-dimensional linear heat equation. We also establish the point similarity of the Fokker–Planck equation to the (1+2)-dimensional Kramers equations whose essential Lie invariance algebras are eight-dimensional, which allows us to find wide families of exact solutions of these Kramers equations in an easy way.

According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that dimK(L)=∣L/K∣<∞, the transcendence degree trd(L/K)<∞ and trd(L/K)=∞ with difftrd(L/K)<∞, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension L=K(G); then, there is a Galois correspondence between the intermediate fields K⊂K′⊂L and the subgroups e⊂G′⊂G, provided that K′ is stable under a Lie algebra Δ of invariant derivations of L/K. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations.

In the Kolmogorov Theory of Consciousness, algorithmic agents utilize inferred compressive models to track coarse-grained data produced by simplified world models, capturing regularities that structure subjective experience and guide action planning. Here, we study the dynamical aspects of this framework by examining how the requirement of tracking natural data drives the structural and dynamical properties of the agent. We first formalize the notion of a generative model using the language of symmetry from group theory, specifically employing Lie pseudogroups to describe the continuous transformations that characterize invariance in natural data. Then, adopting a generic neural network as a proxy for the agent dynamical system and drawing parallels to Noether’s theorem in physics, we demonstrate that data tracking forces the agent to mirror the symmetry properties of the generative world model. This dual constraint on the agent’s constitutive parameters and dynamical repertoire enforces a hierarchical organization consistent with the manifold hypothesis in the neural network. Our findings bridge perspectives from algorithmic information theory (Kolmogorov complexity, compressive modeling), symmetry (group theory), and dynamics (conservation laws, reduced manifolds), offering insights into the neural correlates of agenthood and structured experience in natural systems, as well as the design of artificial intelligence and computational models of the brain.

This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics. The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth. This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.

Élie Cartan’s general equivalence problem is recast in the language of Lie algebroids. The resulting formalism, being coordinate and model-free, allows for a full geometric interpretation of Cartan’s method of equivalence via reduction and prolongation. We show how to construct certain normal forms (Cartan algebroids) for objects of finite-type, and are able to interpret these directly as ‘infinitesimal symmetries deformed by curvature’. Details are developed for transitive structures, but rudiments of the theory include intransitive structures (intransitive symmetry deformations). Detailed illustrations include subriemannian contact structures and conformal geometry.

We show that generically a pseudogroup generated by holomorphic diffeomorphisms defined about $0\\in \\mathbb{C} $ is free in the sense of pseudogroups even if the class of conjugacy of the generators is fixed. This result has a number of consequences on the topology of leaves for a (singular) holomorphic foliation defined on a neighborhood of an invariant curve. In particular, in the classical and simplest case arising from local nilpotent foliations possessing a unique separatrix which is given by a cusp of the form $\\{ {y}^{2} - {x}^{2n+ 1} = 0\\} $, our results allow us to settle the problem of showing that a generic foliation possesses only countably many non-simply connected leaves.

Consider a pseudogroup on $( \\mathbb{C} , 0)$ generated by two local diffeomorphisms having analytic conjugacy classes a priori fixed in $\\mathrm{Diff} \\hspace{0.167em} ( \\mathbb{C} , 0)$ . We show that a generic pseudogroup as above is such that every point has a (possibly trivial) cyclic stabilizer. It also follows that these generic groups possess infinitely many hyperbolic orbits. This result possesses several applications to the topology of leaves of foliations, and we shall explicitly describe the case of nilpotent foliations associated to Arnold’s singularities of type ${A}^{2n+ 1} $ .

A singular riemannian foliation F on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold E that meets every leaf of F orthogonally and whose dimension is the codimension of the regular leaves of F. We prove that the algebra of basic forms of M relative to F is isomorphic to the algebra of those differential forms on E that are invariant under the generalized Weyl pseudogroup of E. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of F coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of F are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos. [PUBLICATION ABSTRACT]