Lie Groups

Dynamic simulation procedures of flight vehicle (fixed-wing, rotorcraft, UAV, satellite) 3D manoeuvres need robust and efficient integration methods in order to allow for reliable, and possibly real-time, simulation missions. Since flight vehicle 3D manoeuvres necessary include complete 3D rotation domain, such procedures also require an efficiency way of dealing with large 3D rotation. Usually, in this context, the simulation procedures built around the standard numerical ordinarydifferential-equations (ODE) based on three-parameters rotation variables (such as Euler angels) have their limitations, as they impose discontinuities or even singularities in the flight vehicle attitude integral curves. Most commonly, the quaternion representation is widely used in flight simulation to overcome the mentioned deficiency [1]. However, if quaternions are used for a parameterisation of the rotation manifold, the standard model leads to integration of differential-algebraic equations (DAE) th...

One of the most important tools to study finite groups is through the use of the character
table as it reduces many pieces of information about finite groups to an invertible matrix
which is widely understood by people. Any finite group is either simple or has a non-trivial
normal subgroup and hence will be of extension type. Computation of character tables
of maximal subgroups and automorphism groups of simple groups has been the focus of
researchers for some years now. The character tables of all the maximal subgroups of the
sporadic simple groups are known, except for some maximal subgroups of the Monster M
and Baby Monster B. There are several well-developed methods for calculating the character
tables of group extensions and in particular when the kernel of the extension is an elementary
abelian group. In this dissertation we studied the method developed by Bernd Fischer and
known nowadays as the theory of Clifford-Fischer matrices which requires that we have a
normal subgroup N of group G¯ . This method derives its fundamentals from the Clifford
theory . Let G¯ = N.G, where NC G¯ and G/N ¯ ∼= G, be a group extension. For each conjugacy
class [gi
]G, we construct a non-singular matrix M(gi), called a Fischer matrix. Once we have
all the Fischer matrices together with the charater tables (ordinary or projective) and fusions
of the inertia factor groups into G, the character table of G¯ is then constructed easily. In this
dissertation we applied the coset analysis technique (this is a method to find the conjugacy
classes of group extensions) together with the theory of Clifford-Fischer matrices to calculate
the ordinary character tables of three groups of extension type, in which two are split and
one is a non-split extension. These groups are of the form 24
:(Z3:Z2), 24
:A8 and 24·A8.

Fermat's Last Theorem asserts that the Diophantine equation a n + b n = c n admits no nontrivial integer solutions for n > 2. In this note, we provide a rational reformulation of the equation and explore a constructive attempt to find rational solutions. We show that such constructions lead to algebraic contradictions when n > 2, and we outline why this heuristic obstruction is consistent with Fermat's theorem, though it does not constitute a full proof. The goal is to provide an accessible entry point into the structure of the problem and to highlight the deep nature of rational solutions in exponential Diophantine equations.

Fermat's Last Theorem asserts that the Diophantine equation a n + b n = c n admits no nontrivial integer solutions for n > 2. In this note, we provide a rational reformulation of the equation and explore a constructive attempt to find rational solutions. We show that such constructions lead to algebraic contradictions when n > 2, and we outline why this heuristic obstruction is consistent with Fermat's theorem, though it does not constitute a full proof. The goal is to provide an accessible entry point into the structure of the problem and to highlight the deep nature of rational solutions in exponential Diophantine equations.

We propose a minimal, testable theory for when geometry emerges from pregeometric tension in meaning—framing the transition as an observable event rather than a modeling assumption. We formalize the Span layer as the interference/condensation zone where folds, tension, and resonance accumulate into neighborhoods that prepare a spectrum; Span ends only when an onset triplet co-occurs: a rise in spectral gap (Δλ₂ ≥ ε), an intrinsic-dimension knee, and zigzag-persistence consolidation. On this basis we state six Emergence Conditions (E1–E6): local enabling (non-degenerate tension and directional fold-causality; resonance density; order-faithful composability; tension regularity), a global Cheeger floor for conductance, and the measurable onset triplet that licenses geometry. Together they yield a concise Geo-Bridge—P→Fold⁡σG→Spec(L,λ2,Ek)→Fisher(M,gV)\mathcal P \xrightarrow{\operatorname{Fold}_\sigma} G \xrightarrow{\text{Spec}} (L,\lambda_2,E_k) \xrightarrow{\text{Fisher}} (\mathcal M,g_V)—and delimit the post-onset window where the unique linear, local, reciprocal closure holds: B=σ∇gVSB=\sigma\nabla_{g_V}S. We corroborate the framework with Span mini-experiments: (I) resonance toggles advance onset and increase integrability, (II) Cheeger bottlenecks delay onset and destabilize topology, and (III) regularity breaks produce ill-conditioned Fisher charts despite spurious spectral hints. The work integrates an evidence-before-output gate (S⁵OS) and a reproducibility kit (seeds, thresholds, DOI’d artifacts), and clarifies minimal vs. sufficient conditions, nonlinear pre-onset dynamics, and cross-model/domain robustness for emergent discourse geometry.

Vous trouverez dans ce texte le recueil de témoignages des acteurs d'une histoire qui commence en 1931 avec un article de von Neumann sur les représentations irréductibles du groupe de Heisenberg, les travaux de jacques Dixmier sur les représentations unitaires des groupes de Lie nilpotents et les algèbres de von Neumann, la bombe lancée par Alexandre A. Kirillov dans les années 50 avec sa thèse sur la « méthode des orbites » qui déclare à Alain Guichardet à Moscou « j'ai dépassé Dixmier ». Les travaux parallèles de Valentine Bargmann, étudiant d'Eugène Wigner, qui s'intéresse aux représentations unitaires des groupes non-compacts et en particulier au groupe homogène de Lorentz. Les travaux de la fin des années 60 de Bertam Kostant et Jean-Marie Souriau qui inventent en parallèle l'application moment (géométrisation du théorème de Noether) et le fait que l'on peut associer aux orbites coadjointes une structure de variété symplectique via la 2 forme KKS (Kirillov-Kostant-Souriau). Enfin, les

In the rapidly advancing field of robotics, mathematical tools have become essential for addressing the complex challenges associated with motion analysis and optimization [1-9]. This paper explores the hyper-state of a rigid body in motion, focusing on its position, velocity vector field, and acceleration vector field. It proposes hypercomplex entities representing the hyper-state of motion for rigid bodies and multibody systems. This trident algebra extends dual algebra applied in rigid body kinematics. A differential transform, a symbolic representation that links a real-time function with a trident time function, is the cornerstone of this approach. The analysis employs automatic differentiation through nilpotent trident algebra, eliminating the need for further differentiation concerning time. A proper orthogonal trident tensor and trident unit quaternions encompasses the hyper-state of the rigid body and is an isomorphic homogenous trident matrix. A hyper-state analysis of the general cylindrical spatial chain is presented. The POE quaternionic formula for the hyper-state of lower-pair kinematics chains is demonstrated in compact representation. In the end an application for the hyper-state of a four-degrees-of-freedom general serial 3C manipulator by trident unit quaternions is presented. Moreover, the significant impact of nilpotent algebra extends to automatic differentiation [5], [10], a critical method for achieving accurate computations. This capability is essential for skillfully and precisely navigating complex optimization landscapes.

This paper introduces LUMU-Λ, a new recursive algebra generated by the Law of Universal Mathematical Unity (LUMU), and applies it to the largest exceptional Lie group, E₈. We define a transformation Φ(λᵢ) = gᵢ + εᵢ, embedding LUMU-Λ into E₈ via a direct sum Ξ = E₈ ⊕ LUMU-Λ. Computing the self-interaction operator Ω(x) = Ξ(x) ∘ Ξ⁻¹(x), we find ǁΩ(x)-xǁ > ε over a connected region of the group manifold-a real fault line in E₈. This fault demonstrates that consequence exceeds symmetry, a new principle in mathematical physics. The result is not a numerical artifact but a structural consequence of recursion, validated via consistency with the Fibonacci-Xibof cycle and the inertial field of past mass. This work establishes LUMU as a meta-framework capable of generating new mathematics and altering foundational symmetry.

Résumé – Nous exposons ici le modèle géométrique de l'Information Quantique, tel qu'introduit par Jean-Marie  Souriau. Ce dernier a élaboré le concept de quantification géométrique en introduisant la notion de variété quantique fibrée en cercle au-dessus d’une variété symplectique, laquelle est une orbite coadjointe. Il devient ainsi possible d’étendre la définition de la fonction hamiltonienne, permettant de caractériser les « états quantiques » comme des fonctions complexes définies sur le groupe 𝐺, et satisfaisant à deux inégalités précises. Pour ce faire, la construction de Gelfand Naimark-Segal est utilisée pour établir un lien entre l’état quantique, la représentation unitaire du groupe,  et le vecteur unitaire dans l’espace de Hilbert. Ces axiomes garantissent l’interprétation probabiliste de la mécanique  quantique, où l’état 𝑚 associe une loi de probabilité à toute « observable » du groupe 𝐺. Cela permet, dans le cadre  linéaire, de garantir les relations d'incertitude énoncées par Heisenberg. La convexité de l'espace des états produit des états quantiques dits « mixtes », dont l'existence s'avère indispensable pour la thermodynamique quantique, notamment  en ce qui concerne les états de Gibbs.
Abstract – We present here the geometric model of Quantum Information, as introduced by Jean-Marie Souriau. He developed the concept of geometric quantisation by introducing the notion of a quantum fibre bundle over a symplectic  manifold, which is a coadjoint orbit. This allows for the extension of the definition of the Hamiltonian function,  enabling the characterisation of "quantum states" as complex functions defined on the group 𝐺, satisfying two specific  inequalities. To achieve this, the Gelfand-Naimark-Segal construction is employed to establish a connection between  the quantum state, the unitary representation of the group, and the unit vector in the Hilbert space. These axioms ensure  the probabilistic interpretation of quantum mechanics, where the state 𝑚 associates a probability measure with each  "observable" of the group 𝐺. In the linear case, this guarantees the Heisenberg uncertainty relations. The convexity of  the state space produces so-called "mixed" quantum states, whose existence is essential for quantum thermodynamics,  particularly concerning Gibbs states.

Let G be a finite, nontrivial group. In a paper in 1994, Cohn defined the covering number of a finite group, denoted as σ(G), as the minimum number of proper subgroups whose union is equal to the whole group. This concept has received considerable attention lately, mainly due to the importance of recent discoveries. In this thesis we study a dual concept to the covering number. We define the intersection number of a finite group, denoted by ι(G), as the minimum number of maximal subgroups whose intersection is equal to the Frattini subgroup. Similarly, we define the inconjugate intersection number of a finite group, ι(G) as the minimum number of inconjugate maximal subgroups whose intersection is equal to the Frattini subgroup. We study basic properties for these intersection numbers and determine its values for certain types of finite groups. Then we single out some similarities and differences between the covering number and the intersection number in specific types of groups. Finally we point to some directions of future developments and research.

In the present study, a two-dimensional (2D), nonlinear, and pseudo-homogeneous mathematical model of a fixed-bed catalytic reactor with an integrated membrane for the methane steam reforming over a nickelbased catalyst is developed. A permselective Pd based membrane is used in order to remove hydrogen from the reaction zone and shift the equilibrium towards hydrogen production thus enabling the achievement of a high methane conversion. The nickel-based catalyst allows for significantly low operating temperatures (less than 550 o C) than conventional methane reforming. The necessary heat for the initiation and preservation of the reactions is supplied to the reactor through an external source. The mathematical model is based on rigorous mass, energy, and momentum balances, where both axial and radial gradients of mass and temperature are fully considered. Hydrogen flux through the membrane is calculated by Sieverts law where the driving force is the hydrogen partial pressure difference between the two sides of the membrane. Results referring to the distribution of species and methane conversion along the reactor and temperature and hydrogen flowrate along the reactor for different radial positions are obtained and analyzed. Furthermore, sensitivity analysis on the effect of different wall temperatures (500 o C, and 550 o C) and operating pressures (1, 5, and 10 atm) show that in a membrane reactor methane conversion (60.24 % at 10 atm) can reach similar values to that in a traditional reactor (61.21 % at 10 atm and 700 o C) at significantly lower temperatures (550 o C).

Let M be a compact manifold with a Hamiltonian T action and moment map Φ. The restriction map in rational equivariant cohomology from M to a level set Φ -1 (p) is a surjection, and we denote the kernel by I p . When T has isolated fixed points, we show that I p distinguishes the chambers of the moment polytope for M . In particular, counting the number of distinct ideals I p as p varies over different chambers is equivalent to counting the number of chambers.

This work presents a quaternion-based formulation of vector analysis, with particular emphasis on the angular part of the Laplacian operator and its spectral decomposition. By leveraging the algebraic structure of quaternions and the Lie group of rotations SO(3), the author derives a simplified expression for the angular component of the Laplacian, The spectral analysis reveals that the eigenfunctions of
L are vector spherical harmonics, constructed from classical scalar spherical harmonics.

The paper further develops a basis of vector functions on the sphere using these harmonics and applies the formalism to quantum wave equations. Specifically, it incorporates electromagnetic corrections from Pauli and Dirac into the Klein-Gordon equation using the van der Waerden–Kramers method, with a return to two-component wave functions as suggested by Weyl. The wave functions are valued in the quaternionic space and are represented via 2×2 complex matrices.

Finally, the study explores the fine structure of the quantum system by solving the second-order wave equation under the influence of electromagnetic potentials. The spectral properties of the associated operators are analyzed, leading to a refined understanding of the energy levels and their dependence on angular momentum and spin interactions.

Homosexuality has become a global trend both in developing and developed countries. While it has been legalized in some countries, it's still illegal in others. In this paper, a comprehensive mathematical model of sexual orientations with recovery centers was developed using a set of ordinary differential equations and solved using Wolfram Mathematica and the fourth-order Runge-Kutta method. The population was divided into ten compartments; Males (M), Females (F), gays (G), lesbians (L), heterosexual males (Hm), heterosexual females (Hf), bisexual males (Bm), bisexual females (Bf), recovery centers for males (Rm) and recovery centers for females (Rf). Positivity, boundedness, the homosexuality epidemic threshold, and equilibria for the model were determined, and stabilities were investigated. Moreover, control reproduction number and bifurcation analysis were also studied. The sensitivity of the parameters was investigated using the partial rank correlation coefficient (PRCC) method and Latin hypercube sampling (LHS). Numerical simulation was done using MATLAB 45-ODE Solver and showed that increasing homosexuality recruitment rates increased homosexuality in the population, and vice versa. It was also established numerically that increasing transfer rates from the bisexual classes to recovery centers increased the heterosexual populations. In conclusion, the effective transfer to recovery centers of the homosexual populations and low contact rates are most significant in reducing the spread of homosexuality in the community.

Maxwell's equations have two solutions, retarded potential and the other is advanced potential. Physics and electronic industry only accept the retarded wave. Advanced waves are not acceptable. However, a group of scientists believes the advanced wave is an objective existence. These scientists include Wheeler, Feynman, Cramer, etc. The author establishes the mutual energy flow theory of electromagnetic field, which supports the objective existence of advanced wave. This paper reviews the existing advanced wave theory and clarifies the expression form of advanced wave. The retarded wave and advanced wave can be expressed by the same symbol. For example, in quantum mechanics, if the wave function is used to represent the retarded wave, this wave function can also represent the advanced wave, and it is wrong to use the conjugate complex of the wave function to represent the advanced wave. In addition, advanced wave and retarded wave are inseparable. For example, the wave generated by the current element in the waveguide is a Leftward wave and a right wave. The Leftward wave starts as an advanced wave and moves towards the current element. When the Leftward wave crosses the current, it becomes a retarded wave. The rightward wave is the same.

We shall elucidate the foliation structures, namely, the 3-web and the bi-Lagrangian (Künneth) structure, that were jointly employed by the physicist and mathematician Jean-Marie Souriau in his Lie Groups Thermodynamics, extended to include dissipation models, and by Gérard Debreu, the Nobel Laureate in Economics, within the context of preferences and utility theory. Debreu examined the conditions under which a web is considered trivial, that is, whether there exists a change of coordinates rendering the web equivalent to a standard orthogonal grid, integrable into a potential function. He employed the Frobenius theorem and Pfaffian forms to investigate whether certain distributions, in the sense of foliations, satisfy the necessary conditions for integrability. Souriau developed a symplectic model of thermodynamics based on symplectic foliation, wherein dissipation dynamics are defined on a transverse Riemannian foliation, thereby inducing a web structure linked to a bi-Lagrangian manifold. A bi-Lagrangian manifold may be endowed with a metric 3-web structure via a Riemannian metric. For every 3-web, one can associate a canonical Chern connection, whose flatness guarantees additive separability. This connection, originally introduced by Hess for Lagrangian 2-webs, also known as bipolarised symplectic manifolds, is particularly employed in the study of bi-Lagrangian manifolds.

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.

Let x : M m → M , m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifold M with sectional curvature satisfying -k 2 ≤ K M ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L m -norm. If K M ≡ 0, assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L 2 harmonic 1-forms on M has finite dimension. Moreover there exists a constant Λ > 0, explicitly computed, such that if the total curvature is bounded from above by Λ then there is no nontrivial L 2 -harmonic 1-forms on M .

We give a new construction of the outer automorphism of the symmetric group on six points. Our construction features a complex Hadamard matrix of order six containing third roots of unity and the algebra of split quaternions over the real numbers.

In this paper we relate the study of actions of discrete groups over connected manifolds to that of their orbit spaces seen as differentiable stacks. We show that the orbit stack of a discrete dynamical system on a simply connected manifold encodes the dynamics up to conjugation and inversion. We also prove a generalization of this result for arbitrary discrete groups and non-simply connected manifolds, and relate it to the covering theory of stacks. As applications, we obtain a geometric version of Rieffel's theorem on irrational rotations of the circle, we compute the stack-theoretic fundamental group of hyperbolic toral automorphisms, and we revisit the classification of lens spaces.

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We consider quantum transition amplitudes, partition functions and observables for 3D spin foam models within SU (2) quantum group deformation symmetry, where the deformation parameter is a complex fifth root of unity. By considering "fermionic" cycles through the foam we couple this SU (2) quantum group with the same deformation of SU (3), so that we have quantum numbers linked with spacetime symmetry and charge gauge symmetry in the computation of observables. The generalization to higher-dimensional Lie groups SU (N ), G 2 and E 8 is suggested. On this basis we discuss a unifying framework for quantum gravity. Inside the transition amplitude or partition function for geometries, we have the quantum numbers of particles and fields interacting in the form of a spin foam network -in the framework of state sum models, we have a sum over quantum computations driven by the interplay between aperiodic order and topological order.

Considering the predictions from the standard model of particle physics coupled with experimental results from particle accelerators, we discuss a scenario in which from the infinite possibilities in the Lie groups we use to describe particle physics, nature needs only the lower dimensional representations…

The goal of this work is to elaborate on new geometric methods of constructing exact and parametric quasiperiodic solutions for anamorphic cosmology models in modified gravity theories, MGTs, and general relativity, GR. There exist previously studied generic off-diagonal and diagonalizable cosmological metrics encoding gravitational and matter fields with quasicrystal like structures, QC, and holonomy corrections from loop quantum gravity, LQG. We apply the anholonomic frame deformation method, AFDM, in order to decouple the (modified) gravitational and matter field equations in general form. This allows us to find integral varieties of cosmological solutions determined by generating functions, effective sources, integration functions and constants. The coefficients of metrics and connections for such cosmological configurations depend, in general, on all spacetime coordinates and can be chosen to generate observable (quasi)-periodic/ aperiodic/ fractal / stochastic / (super) cluster / filament / polymer like (continuous, stochastic, fractal and/or discrete structures) in MGTs and/or GR. In this work, we study new classes of solutions for anamorphic cosmology with LQG holonomy corrections. Such solutions are characterized by nonlinear symmetries of generating functions for generic off-diagonal cosmological metrics and generalized connections, with possible nonholonomic constraints to Levi-Civita configurations and diagonalizable metrics depending only on a time like coordinate. We argue that anamorphic quasiperiodic cosmological models integrate the concept of quantum discrete spacetime, with certain gravitational QC-like vacuum and nonvacuum structures. And, that of a contracting universe that homogenizes, isotropizes and flattens without introducing initial conditions or multiverse problems.

In this paper we introduce Morse Lie groupoid morphisms and study their main properties. We show that this notion is Morita invariant which gives rise to a well defined notion of Morse function on differentiable stacks. We show a groupoid version of the Morse lemma which is used to describe the topological behavior of the critical subgroupoid levels of a Morse Lie groupoid morphism around its nondegenerate critical orbits. We also prove Morse type inequalities for certain separated differentiable stacks and construct a Morse double complex whose total cohomology is isomorphic to the Bott-Shulman-Stasheff cohomology of the underlying Lie groupoid. We provide several examples and applications.

VB-groupoids can be thought of as vector bundle objects in the category of Lie groupoids. Just as Lie algebroids are the infinitesimal counterparts of Lie groupoids, VB-algebroids correspond to the infinitesimal version of VB-groupoids. In this work we address the problem of the existence of a VB-groupoid admitting a given VB-algebroid as its infinitesimal data. Our main result is an explicit characterization of the obstructions appearing in this integrability problem as the vanishing of the spherical periods of certain cohomology classes. Along the way, we illustrate our result in concrete examples. Finally, as a corollary, we obtain computable obstructions for a $2$-term representation up to homotopy of Lie algebroid to arise as the infinitesimal counterpart of a smooth such representation of a Lie groupoid.

VB-groupoids are vector bundle objects in the category of Lie groupoids: the total and the base spaces of the vector bundle are Lie groupoids and the vector bundle structure maps are required to define Lie groupoid morphisms. The infinitesimal version of VBgroupoids are VB-algebroids, namely, vector bundle objects in the category of Lie algebroids. Following recent developments in the area, we show that a VB-algebroid is integrable to a VB-groupoid if and only if its base algebroid is integrable and the spherical periods of certain underlying cohomology classes vanish identically. We illustrate our results in concrete examples. Finally, we obtain as a corollary computable obstructions for a 2-term representation up to homotopy of Lie algebroid to arise as the infinitesimal counterpart of a smooth such representation of a Lie groupoid. 1 Introduction 439 2 Background material about VB-groupoids and VB-algebroids 443 3 Integrability of VB-algebroids 456 4 Integrating 2-term representations up to homotopy 472 References 480 Obstructions to the integrability of VB-algebroids 441 define VB-groupoids which integrate the above VB-algebroids (see e.g. ). The concept of VB-algebroid was introduced by Pradines [32] and it has been further studied by Mackenzie [27], Gracia-Saz and Mehta , among other authors (see for their relation to supergeometry). VB-groupoids were also studied in . Lie theory for VB-algebroids and VB-groupoids was systematically studied in . VB-algebroids and VB-groupoids have shown to be specially important in the infinitesimal description of Lie groupoids equipped with multiplicative geometric structures, namely, in the study of generalizations of the symplectic/Poisson correspondence mentioned earlier. For example, multiplicative differential forms turn out to be equivalent to cocycles involving the VB-groupoid structure on T G while multiplicative multivector fields to cocycles involving T * G. Their infinitesimal counterpart then becomes clear: they correspond to cocycles for VB-algebroid structures arising from T A and T * A (see and references therein). More general VB-groupoids and VB-algebroids appear when considering other multiplicative geometries like multiplicative foliations or, more generally, multiplicative Dirac structures . The VB-structure plays a fundamental role in all these cases and hints that Lie theory in the VB-category is an interesting object to study within this area. The aim of this work is to characterize the obstructions appearing in the integrability problem for VB-algebroids, namely, the obstructions for the existence of a VB-groupoid (H, G, E, M ) such that Lie(H, G, E, M ) is isomorphic to a given VB-algebroid. A key result on that matter was shown in , stating that a VB-algebroid (D, A, E, M ) is integrable iff D → E is integrable as a Lie algebroid. This means that if the total space D → E of a VB-algebroid is integrable then so is A → M and, furthermore, the extra vector bundle structure integrates automatically at the level of the (s.s.c.) Lie groupoids. For this reason we only need to study the integrability of D. Our main result is given by Theorem 3.22, which states that a VBalgebroid (D, A, E, M ) is integrable iff A is integrable and the spherical periods of certain cohomology classes associated to (D, A, E, M ) vanish. Assuming that A is integrable, we first show that the obstructions to the integrability lie in N (D) ∩ ker(p : D → A) and then express the elements in this set as spherical integrals of an underlying 2-cochain on A. The first part comes from the general theory of obstructions of adapted to the VBcase and the second from the description of the structure of a VB-algebroid in terms of connections and cochains given in . The vanishing of these

In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that these categories carry natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid, and we prove that the 1-truncation of this 2-groupoid is isomorphic to the Weinstein groupoid of the VB-algebroid associated with a representation up to homotopy. Contents 1. Introduction 503 2. Generalities 509 3. VB-algebroids as Lie algebroid fibrations 514 4. Strict 2-groupoids and 2-representations 519 5. Application to the integration of VB-algebroids 528 Appendix A. Proof of Theorem 4.19 535 Appendix B. A geometric description of the semidirect product 539 Acknowledgments 541 References 542

We study vector bundles over Lie groupoids, known as VB-groupoids, and their induced geometric objects over differentiable stacks. We establish a fundamental theorem that characterizes VB-Morita maps in terms of fiber and basic data, and use it to prove the Morita invariance of VB-cohomology, with implications to deformation cohomology of Lie groupoids and of classic geometries. We discuss applications of our theory to Poisson geometry, providing a new insight over Marsden–Weinstein reduction and the integration of Dirac structures. We conclude by proving that the derived category of VB-groupoids is a Morita invariant, which leads to a notion of VB-stacks, and solves (an instance of) an open question on representations up to homotopy.

VB-groupoids are vector bundle objects in the category of Lie groupoids: the total and the base spaces of the vector bundle are Lie groupoids and the vector bundle structure maps are required to define Lie groupoid morphisms. The infinitesimal version of VB-groupoids are VB-algebroids, namely, vector bundle objects in the category of Lie algebroids. Following recent developments in the area, we show that a VB-algebroid is integrable to a VBgroupoid if and only if its base algebroid is integrable and the spherical periods of certain underlying cohomology classes vanish identically. We illustrate our results in concrete examples. Finally, we obtain as a corollary computable obstructions for a 2-term representation up to homotopy of Lie algebroid to arise as the infinitesimal counterpart of a smooth such representation of a Lie groupoid. Our main result is given by Theorem 3.22, which states that a VB-algebroid (D, A, E, M ) is integrable iff A is integrable and the spherical periods of certain cohomology classes associated to (D, A, E, M ) vanish. Assuming that A is integrable, we first show that the obstructions to the integrability lie in N (D)∩ker(p : D → A) and then express the elements in this set as spherical integrals of an underlying 2cochain on A. The first part comes from the general theory of obstructions of adapted to the VB-case and the second from the description of the structure of a VB-algebroid in terms of connections and cochains given in . The vanishing of these integrals can also be seen as the condition for certain algebroid cohomology classes to be in the image of the Van Est map ([13]). We end this introduction with a second 'extrinsic' approach to VB-algebroids and VB-groupoids. Here, the key point stems from the work of Gracia-Saz and Mehta, [21] and , where they showed that, upon non-canonical splittings, a VB-algebroid (resp. VB-groupoid) structure boils down to a representation up to homotopy ([3, 4]) of A (resp. of G) on a 2-term complex coming from the fibers of the horizontal structures. Ordinary representations of A and G define particular cases of the 'up to homotopy' ones (and hence of VB's), but these last are strictly more general. For example, the adjoint and coadjoint representations of A, which are well known to be only 'up to homotopy' for a general A, arise from the intrinsic VB-algebroid structure of T A and T * A after suitable splitting. As an application of our main result and building on the above correspondences, we obtain a notion of integrability of a 2-term representation up to homotopy of a Lie algebroid A and characterize the underlying obstructions by translating our main result from the world of VB-algebroids to that of representations. (This notion of integrability coincides with the one stemming from .) This paper is organized as follows. In section 2 we present background material on VB-groupoids and VB-algebroids, including the main examples and properties. We also recall the notion of splitting of a VB-algebroid and its relation to representations up to homotopy. In section 3, we briefly review the general theory of obstructions from [15] and then prove Theorem 3.4, Proposition 3.17 and Theorem 3.22, which are the main results of this work. In Section 4 we apply our results to obtain Corollary 4.10 which provides general criteria for the integrability of twoterm representations up to homotopy. Lastly, we mention the role of integrability in the simplicial formalism for integration of representations up to homotopy given in . Acknowledgements. The authors would like to thank the PPGMA at UFPR, Curitiba, as well as IMPA, Rio de Janeiro, for supporting several visits while part of this work was carried out. Also, the authors thank Henrique Bursztyn, Matias del Hoyo, Thiago Drummond and Rajan Mehta, for interesting discussions and useful comments and suggestions that have improved this paper. Brahic was supported by CNPq (Programa Ciência sem Fronteiras 401253/2012-0). Cabrera also thanks CNPq (Projeto Universal 471864/2012-9) for support. Ortiz thanks CNPq (Projeto Universal 482796/2013-8) and CAPES-COFECUB (grant 763/13) for supporting this work. The authors are very grateful to the anonymous referees for all comments and suggestions which have improved this manuscript significantly.

We show in this paper that the correspondence between 2-term representations up to homotopy and VB-algebroids, established in , holds also at the level of morphisms. This correspondence is hence an equivalence of categories. As an application, we study foliations and distributions on a Lie algebroid, that are compatible both with the linear structure and the Lie algebroid structure. In particular, we show how infinitesimal ideal systems in a Lie algebroid A are related with subrepresentations of the adjoint representation of A.

We study multiplicative Dirac structures on Lie groups. We show that the characteristic foliation of a multiplicative Dirac structure is given by the cosets of a normal Lie subgroup and, whenever this subgroup is closed, the leaf space inherits the structure of a Poisson-Lie group. We also describe multiplicative Dirac structures on Lie groups infinitesimally.

Nous exposons ici le modèle géométrique de l'Information Quantique, tel qu'introduit par Jean-Marie Souriau. Ce dernier a élaboré le concept de quantification géométrique en introduisant la notion de variété quantique fibrée en cercle au-dessus d'une variété symplectique, laquelle est une orbite coadjointe. Il devient ainsi possible d'étendre la définition de la fonction hamiltonienne, permettant de caractériser les « états quantiques » comme des fonctions complexes définies sur le groupe 𝐺, et satisfaisant à deux inégalités précises. Pour ce faire, la construction de Gelfand-Naimark-Segal est utilisée pour établir un lien entre l'état quantique, la représentation unitaire du groupe, et le vecteur unitaire dans l'espace de Hilbert. Ces axiomes garantissent l'interprétation probabiliste de la mécanique quantique, où l'état 𝑚 associe une loi de probabilité à toute « observable » du groupe 𝐺. Cela permet, dans le cadre linéaire, de garantir les relations d'incertitude énoncées par Heisenberg. La convexité de l'espace des états produit des états quantiques dits « mixtes », dont l'existence s'avère indispensable pour la thermodynamique quantique, notamment en ce qui concerne les états de Gibbs.

We show that a single finite field, built on an odd prime 𝑝, contains the entire scope of algebraic machinery to support smooth geometry, differential calculus and continuous harmonic analysis. By arranging the field's basic arithmetic moves in a 4-dimensional "symmetry cube", we obtain a finite lattice that has the combinatorial shape of a 2-sphere. Completing the field via an internally defined infinitesimal extension turns this lattice into a genuinely smooth surface with constant curvature. The field itself provides finite versions of the familiar constants 𝑖, 𝜋 and 𝑒, identified by their structural roles. Using these constants we build a Fourier kernel that works simultaneously in the finite, discrete and continuous settings, merging the conventional and the finite harmonic analysis into one algebraic framework. Finally, exact Lie groups are systematically derived inside the resultant finite ring continuum. The proposed construct provides a common foundation for discrete mathematics, classical analysis, and physical modelling within a single, finite relational universe.

We present a formal reconstruction of the conventional number systems, including integers, rationals, reals, and complex numbers, based on the principle of relational finitude over a finite field F 𝑝. Rather than assuming actual infinity, we define arithmetic and algebra as observer-dependent constructs grounded in finite field symmetries. Conventional number classes are then reinterpreted as pseudo-numbers, expressed relationally with respect to a chosen reference frame. We define explicit mappings for each number class, preserving their algebraic and computational properties while eliminating ontological dependence on infinite structures. The resultant framework-that we denote as Finite Ring Continuum-establishes a coherent foundation for mathematics, physics and formal logic in ontologically finite paradox-free informational universe.

-Dès 1993, A. Fukjiwara et Y. Nakamura ont développé des liens étroits entre la géométrie de l'information et les systèmes intégrables en étudiant les systèmes dynamiques sur des modèles statistiques et les systèmes à gradients complètement intégrables sur les variétés de distributions gaussiennes et multinomiales. Parallèlement, dans le cadre des systèmes dynamiques intégrables et de la méthode de déformation isospectrale de Moser, issue des travaux de P. Lax, AM Perelomov a appliqué des modèles pour l'espace des matrices hermitiennes définies positives d'ordre 2, en considérant cet espace comme un espace homogène. Nous concluons par des liens entre les paires de Lax et l'équation de Souriau pour le calcul des coefficients du polynôme caractéristique d'une matrice. L'objectif principal de cet article est de réaliser une synthèse de ces recherches sur l'intégrabilité et les paires de Lax et leurs liens avec la géométrie de l'information, afin de stimuler de nouveaux développements à l'interface de ces disciplines. Abstract-As early as 1993, A. Fukjiwara and Y. Nakamura developed close links between information geometry and integrable systems by studying dynamical systems on statistical models and gradient systems that are completely integrable on manifolds of Gaussian and multinomial distributions. At the same time, within the framework of integrable dynamical systems and Moser's isospectral deformation method, derived from the work of P. Lax, A. M. Perelomov applied models for the space of second-order positive definite Hermitian matrices, considering this space as a homogeneous space. We conclude with links between Lax pairs and the Souriau equation for calculating the coefficients of the characteristic polynomial of a matrix. The main objective of this article is to synthesize this research on integrability and Lax pairs and their links with information geometry, in order to stimulate new developments at the interface of these disciplines.

La thermodynamique est une expression de la physique à un niveau épistémique élevé. À ce titre, son potentiel en tant que biais inductif pour aider les procédures d'apprentissage automatique à obtenir des prédictions précises et crédibles a été récemment reconnu dans de nombreux domaines. La thermodynamique fournit des informations utiles dans le processus es phénomènes dissipatifs. Nous allons expliquer le lien entre les TINN (Réseaux de Neurones Informés par la Thermodynamique), le flot métriplectique et la thermodynamique des groupes de Lie et comment ces concepts cherchent collectivement à intégrer les principes de la thermodynamique et des symétries dans la modélisation mathématique, l'apprentissage automatique et les systèmes physiques.

Three new approximate symmetry theories are proposed. The approximate symmetries are contrasted with each other and with the exact symmetries. The theories are applied to nonlinear ordinary differential equations for which exact solutions are available. It is shown that from the symmetries, approximate solutions as well as exact solutions in some restricted cases can be retrievable. Depending on the specific approximate theory and the equations considered, the approximate symmetries may expand the Lie Algebra of the exact symmetries, may be a perturbed form of the exact symmetries or may be a subalgebra of the exact symmetries. Exact and approximate solutions are retrieved using the symmetries.

This paper extends previous work on approximation of loops to the case of special orthogonal groups SO(N ), N ≥ 3. We prove that the best approximation of an SO(N ) loop Q(t) belonging to a Hölder class Lip α , α > 1, by a polynomial SO(N ) loop of degree ≤n is of order O(n -α+ ) for n ≥ k, where k = k(Q) is determined by topological properties of the loop and > 0 is arbitrarily small. The convergence rate is therefore -close to the optimal achievable rate of approximation. The construction of polynomial loops involves higher-order splitting methods for the matrix exponential. A novelty in this work is the factorization technique for SO(N ) loops which incorporates the loops' topological aspects.

We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green's functions which we derive in explicit form. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant basis of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green's functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, second order stochastic evolution equations. To probe an image at every location x ∈ R 2 and in every direction e iθ ∈ T, one translates and rotates an anisotropic wavelet ψ by means of a representation g → U g of the Euclidean motion group U g ψ(y) = ψ(R -1 θ (yx)), g = (x, e iθ ). The result of such an image sampling is a function U f ∈ L 2 (G) on the Euclidean motion group manifold G = R 2 T, which is

Let x : M m → M , m ≥ 3, be an isometric immersion of a complete noncompact manifold M in a complete simply-connected manifold M with sectional curvature satisfying -k 2 ≤ K M ≤ 0, for some constant k. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite L m -norm. If K M ≡ 0, assume further that the first eigenvalue of the Laplacian of M is bounded from below by a suitable constant. We prove that the space of the L 2 harmonic 1-forms on M has finite dimension. Moreover there exists a constant Λ > 0, explicitly computed, such that if the total curvature is bounded from above by Λ then there is no nontrivial L 2 -harmonic 1-forms on M .

Shuffling-type gradient method is a popular machine learning algorithm that solves finite-sum optimization problems by randomly shuffling samples during iterations. In this paper, we explore the convergence properties of shuffling-type gradient method under mild assumptions. Specifically, we employ the bandwidth-based step size strategy that covers both monotonic and non-monotonic step sizes, thereby providing a unified convergence guarantee in terms of step size. Additionally, we replace the lower bound assumption of the objective function with that of the loss function, thereby eliminating the restrictions on the variance and the second-order moment of stochastic gradient that are difficult to verify in practice. For non-convex objectives, we recover the last iteration convergence of shuffling-type gradient algorithm with a less cumbersome proof. Meanwhile, we also establish the convergence rate for the minimum iteration of gradient norms. Under the Polyak-Łojasiewicz (PL) condition, we prove that the function value of last iteration converges to the lower bound of the objective function. By selecting appropriate boundary functions, we further improve the previous sublinear convergence rate results. Overall, this paper contributes to the understanding of shufflingtype gradient method and its convergence properties, providing insights for optimizing finite-sum problems in machine learning. Finally, numerical experiments demonstrate the efficiency of shuffling-type gradient method with bandwidth-based step size and validate our theoretical results.

The document titled "De Natura Rerum" (reference to Lucretius' "On the Nature of Things") is a mathematical and physical treatise, by French mathematician Guy Duhaut (1948-2024), focusing on the group SO(3) and its applications.

It is a mathematically rich and pedagogically structured treatise that explores the group SO(3) and its connections to geometry, algebra, and physics. Drawing from his university lectures in the early 1980s, Duhaut presents a unified framework that integrates Clifford algebras, Lie groups, differential operators, and representation theory. The work emphasizes the role of symmetry and invariance in mathematical physics, particularly in the context of spin, quantum mechanics, and integration on manifolds. Rather than offering a purely theoretical exposition, the text is designed to be accessible and applicable, with a focus on concrete operators (like Dirac and Laplacian), invariant measures (like Haar), and the structure of SO(3) as a model for physical systems with rotational symmetry.

A theoretical model for converging cylindrical and spherical shock waves in non-ideal gas characterized by condensed matter equation of state (EOS) of Mie-Gruneisen type is investigated. The governing equations considered are non-linear system of one-dimensional partial differential equations in Eulerian hydrodynamics. We analyze the system for possible analytical and numerical solutions. The analytical solution for the system is attempted using Lie group of point transformations.The arbitrary constants that appear in the expressions of generators of the Lie group transformations provide possibility of cases with and without artificial viscosity. The symmetry variables which lead the governing system of non-linear partial differential equations to a reduced system of ordinary differential equations are determined using symmetry generators. Numerical calculations have been performed to obtain the similarity exponent and the profiles of the flow variables using the forward difference and Lax-Friedrichs schemes. The effect of artificial viscosity on the flow variables, the process of stability and consistency along with the Courant-Friedrichs Lewy condition is investigated. The numerical solution by Lax-Friedrichs scheme found to be highly suitable for the system since the effect of flow variables, velocity and pressure in the presence of artificial viscosity parameter is negligible which agrees with the analytical solution obtained by Lie symmetry method.