53 Results Found

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th...

Given a compact connected Lie group G with an $S^1$ -module structure and a maximal compact torus T of $G^{S^1}$ , we define twisted Weyl group $W(G,S^1,T)$ of G associated to $S^1$ -module and show that two elements of...

In this paper, we study left-invariant Einstein-like metrics on the compact Lie group G. Assume that there exist two subgroups, $H\subset K\subset G$, such that $G/K$ is a compact, connected, irreducible, symmetric space, and the isotropy representation of $G/H$...

This is mainly a review of an intense 15-year long collaboration between the authors on explicit realisations of compact Lie groups and their applications. Starting with an elementary example, we will illustrate the main idea at the foundation of the...

We consider the phenomenon of concentration of measures, which is restricted to the case of families of compact connected Lie groups. While in the literature, powerful general results regarding the existence of concentration and its relations to extr...

A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a compl...

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we pr...

In the present paper, we review the progress of the project of the classification and construction of invariant differential operators for non-compact, semisimple Lie groups. Our starting point is the class of algebras which we called earlier ‘...

The scale of an endomorphism of a totally disconnected, locally compact group G is defined and an example is presented which shows that the scale function is not always continuous with respect to the Braconnier topology on the automorphism group of G...

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces $\mathrm{U}/\mathrm{H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduc...

An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum $S+T$ of two (L)-semigroups S and T having boundary B is the quotient space obtained from t...

In this paper, new types of associated curves, which are defined as rectifying-direction, osculating-direction, and normal-direction, in a three-dimensional Lie group G are achieved by using the general definition of the associated curve, and some ch...

In computational anatomy, organ’s shapes are often modeled as deformations of a reference shape, i.e., as elements of a Lie group. To analyze the variability of the human anatomy in this framework, we need to perform statistics on Lie groups. A Lie g...

Let $G/H$ be a homogeneous space of a compact simple classical Lie group G. Assume that the maximal torus $T_H$ of H is conjugate to a torus $T_{\beta }$ whose Lie algebra ${\mathfrak{t}}_{\beta }$ is the kernel of the maximal root $\beta$...

This is a survey on harmonic maps and biharmonic maps into (1) Riemannian manifolds of non-positive curvature, (2) compact Lie groups or (3) compact symmetric spaces, based mainly on my recent works on these topics.

This paper establishes a functorial algebraic isomorphism between the moduli space ${\mathcal{B}}_{\mathcal{C}}^{\mathrm{ps}}(\Sigma ,G)$ of polystable principal G-bundles with prescribed monodromy on a punctured Riemann surface $\Sigma$ of genus $g\ge 2$, for a complex reductive Lie group G...

Thestability of a control system is essential for its effective operation. Stability implies that small changes in input, initial conditions, or parameters do not lead to significant fluctuations in output. Various stability properties, such as inner...

The authors wish to make the following corrections to their paper [...]

Let G be a compact Lie group acting smoothly on a manifold M. The local structure of such actions is described by the slice theorem, which reduces the study of G-spaces to smaller isotropy group actions. In this paper, we investigate the equivariant...

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically...

In this paper, we study the extended Hamilton–Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group G on a manifold M and a G-invariant vector field X on M, we construct complete solutions of the Hamilton–J...

This article presents a 2D pose estimation method for an omnidirectional mobile robot with Mecanum wheels, using an extended Kalman filter (EKF) formulated on the Lie group $\mathrm{SO}\left(2\right)$. The purpose is estimate the robot’s position and orientation by...

Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces...

Motivated by known results in locally conformal symplectic geometry, we study different classes of G ${}_2$ -structures defined by a locally conformal closed 3-form. In particular, we provide a complete characterization of invariant exact locally c...

We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic sphe...

This study considers a left-invariant Riemannian metric g on the Lie group $Ni{l}^4$. We introduce a Ricci solitons’ classification on $(Ni{l}^4,g)$. These are expansive non-gradient Ricci solitons. We examine the existence of harmonic maps into $(Ni{l}^4,g)$...

Consider the Euclidean space ${\mathbb{R}}^n$ with the orthogonal action of a compact Lie group G. We prove that a locally Lipschitz G-invariant mapping f from ${\mathbb{R}}^n$ to $\mathbb{R}$ can be uniformly approximated by G-invariant smooth mappings g in such a way that the gradient o...

Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters $n\ge r\ge s$ on dimensions. We discuss the symmetry on th...

We derive expressions for the invariant length element and measure for the simple compact Lie group SU(4) in a coordinate system particularly suitable for treating entanglement in quantum information processing. Using this metric, we compute the inva...

It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive...

The widespread adoption of autonomous vehicle technologies requires accurate predictions of coordinated multi-agent motion. While predicting such motion under arbitrary control mechanisms is generally intractable, this paper focuses on certain classe...

We studied the random variable $V_t={\mathrm{vol}}_{S^2}(g_{t}B\cap B)$, where B is a disc on the sphere $S^2$ centered at the north pole and ${\left(g_t\right)}_{t\ge 0}$ is the Brownian motion on the special orthogonal group $SO\left(3\right)$ starting at the identity. We applied the results of the theo...

Let G be a compact connected Lie group, $(X,\omega ,\mu )$ a Hamiltonian G-manifold with moment map $\mu$, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices $(A,u)$ near Z, where A is...

Karhunen–Loève expansions play a key role in mathematical statistics. Two kernels associated with the well-known Anderson–Darling statistic admit such developments involving Legendre polynomials and their derivatives. We give an in...

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the f...

Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various...

This paper reports an application of the Lie algebra $se\left(3\right)$ of the Euclidean group $SE\left(3\right)$, which is isomorphic to the theory of screws in the velocity and acceleration analyses of serial manipulators. The symbolic computation of the infinitesimal kinem...

We present a comparative exposition of Riemannian and Lorentzian geometries, organized through successive levels of increasing structure. Beginning with scalar products on vector spaces, we proceed through smooth manifolds, left-invariant metrics on...

We develop a collection of nontrivial examples that illustrate and test recent stability results for linear control systems ($LCS$) on Lie groups. We treat the main structural classes: Abelian (${\mathbb{R}}^n$), nilpotent (Heisenberg), solvable non-nilpotent (rigid...

Let X be a compact Riemann surface of genus $g\ge 2$ and $M\left(G_2\right)$ be the moduli space of polystable principal bundles over X, the structure group of which is the simple complex Lie group of exceptional type $G_2$. In this work, it is proved that the only aut...

Let X be a compact Riemann surface of genus $g\ge 2$, G be a complex semisimple Lie group, and ${\mathcal{M}}_G\left(X\right)$ be the moduli space of stable principal G-bundles. This paper studies the fixed point set of involutions on ${\mathcal{M}}_G\left(X\right)$ induced by an anti-holomorphic involu...

Deception detection with low-channel wearable EEG requires protocols that generalize across people while remaining practical for portable devices. Using the public LieWaves dataset (27 subjects recorded with a five-channel Emotiv Insight headset), we...

The review is devoted to nonassociative algebras, rings and modules over them. The main actual and recent trends in this area are described. Works are reviewed on radicals in nonassociative rings, nonassociative algebras related with skew polynomials...

Let X be a compact Riemann surface of genus $g\ge 2$. An octonionic bundle over X is a fiber bundle whose fiber is the non-associative algebra of complex octonions, equivalently a principal $G_2\left(\mathbb{C}\right)$-bundle, where $G_2\left(\mathbb{C}\right)$ is the exceptional Lie group of auto...

Symmetry (and group theory) is a fundamental principle of theoretical physics. Finite symmetries, continuous symmetries of compact groups, and infinite-dimensional representations of noncompact Lie groups are at the core of solid physics, particle ph...

Let X be a compact Riemann surface of genus $g\ge 2$, ${\sigma }_X\in Aut\left(X\right)$ (the group of biholomorphic self-maps of X) an involution, and $F_4\left(\mathbb{C}\right)$ (one of the five exceptional complex simple Lie groups) the fixed-point subgroup of the outer involution $\mathrm{\&si}$...

In his 1842 lectures on dynamics C.G. Jacobi summarized difficulties with differential equations by saying that the main problem in the integration of differential equations appears in the choice of right variables. Since there is no general rule for...

The Hodgkin–Huxley equations have successfully described neuronal excitability for over seventy years, yet their mathematical structure remains empirically justified rather than theoretically explained. Why are gating variables bounded between...

In this paper, exact solutions with a linear velocity field are sought for the gas dynamics equations in the case of the special state equation and the state equation of a monatomic gas. These state equations extend the transformation group admitted...

Stable attitude control of unmanned or autonomous operations of vehicles moving in three spatial dimensions is essential for safe and reliable operations. Rigid body attitude control is inherently a nonlinear control problem, as the Lie group of rigi...