scholarly journals Generalized spinning particles on $${\mathcal {S}}^2$$ in accord with the Bianchi classification

2021 ◽  
Vol 81 (3) ◽  
Author(s):  
Anton Galajinsky
Keyword(s):  
External Field ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Degrees Of Freedom ◽  
Scalar Potential ◽  
Three Dimensional ◽  
Dirac Monopole ◽  
Spinning Particles ◽  
Group Manifold ◽  
Internal Degrees Of Freedom

AbstractMotivated by recent studies of superconformal mechanics extended by spin degrees of freedom, we construct minimally superintegrable models of generalized spinning particles on $${\mathcal {S}}^2$$ S 2 , the internal degrees of freedom of which are represented by a 3-vector obeying the structure relations of a three-dimensional real Lie algebra. Extensions involving an external field of the Dirac monopole, or the motion on the group manifold of SU(2), or a scalar potential giving rise to two quadratic constants of the motion are discussed. A procedure how to build similar models, which rely upon real Lie algebras with dimensions $$d=4,5,6$$ d = 4 , 5 , 6 , is elucidated.

Helical Axis Data Visualization and Analysis of the Knee Joint Articulation

2016 ◽  
Vol 138 (9) ◽  
Author(s):  
Ricardo Manuel Millán Vaquero ◽  
Alexander Vais ◽  
Sean Dean Lynch ◽  
Jan Rzepecki ◽  
Karl-Ingo Friese ◽  
...  
Keyword(s):  
Knee Joint ◽  
Lie Algebra ◽  
Degrees Of Freedom ◽  
Spatial Scales ◽  
Three Dimensional ◽  
Patient Specific ◽  
Helical Axis ◽  
Accurate Analysis ◽  
Related Data ◽  
Flexion Extension

We present processing methods and visualization techniques for accurately characterizing and interpreting kinematical data of flexion–extension motion of the knee joint based on helical axes. We make use of the Lie group of rigid body motions and particularly its Lie algebra for a natural representation of motion sequences. This allows to analyze and compute the finite helical axis (FHA) and instantaneous helical axis (IHA) in a unified way without redundant degrees of freedom or singularities. A polynomial fitting based on Legendre polynomials within the Lie algebra is applied to provide a smooth description of a given discrete knee motion sequence which is essential for obtaining stable instantaneous helical axes for further analysis. Moreover, this allows for an efficient overall similarity comparison across several motion sequences in order to differentiate among several cases. Our approach combines a specifically designed patient-specific three-dimensional visualization basing on the processed helical axes information and incorporating computed tomography (CT) scans for an intuitive interpretation of the axes and their geometrical relation with respect to the knee joint anatomy. In addition, in the context of the study of diseases affecting the musculoskeletal articulation, we propose to integrate the above tools into a multiscale framework for exploring related data sets distributed across multiple spatial scales. We demonstrate the utility of our methods, exemplarily processing a collection of motion sequences acquired from experimental data involving several surgery techniques. Our approach enables an accurate analysis, visualization and comparison of knee joint articulation, contributing to the evaluation and diagnosis in medical applications.

scholarly journals Post-Lie algebra structures for nilpotent Lie algebras

2018 ◽  
Vol 28 (05) ◽  
pp. 915-933
Author(s):  
Dietrich Burde ◽  
Christof Ender ◽  
Wolfgang Alexander Moens
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Nilpotent Lie Algebras ◽  
Dimension Formula ◽  
Necessary And Sufficient

We study post-Lie algebra structures on [Formula: see text] for nilpotent Lie algebras. First, we show that if [Formula: see text] is nilpotent such that [Formula: see text], then also [Formula: see text] must be nilpotent, of bounded class. For post-Lie algebra structures [Formula: see text] on pairs of [Formula: see text]-step nilpotent Lie algebras [Formula: see text] we give necessary and sufficient conditions such that [Formula: see text] defines a CPA-structure on [Formula: see text], or on [Formula: see text]. As a corollary, we obtain that every LR-structure on a Heisenberg Lie algebra of dimension [Formula: see text] is complete. Finally, we classify all post-Lie algebra structures on [Formula: see text] for [Formula: see text], where [Formula: see text] is the three-dimensional Heisenberg Lie algebra.

MATHEMATICA™ PACKAGES FOR COMPUTING PRINCIPAL DECOMPOSITIONS OF SIMPLE LIE ALGEBRAS AND APPLICATIONS IN EXTENDED CONFORMAL FIELD THEORIES

2003 ◽  
Vol 14 (01) ◽  
pp. 1-27 ◽  
Author(s):  
DANIELA GĂRĂJEU ◽  
MIHAIL GĂRĂJEU
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Adjoint Representation ◽  
Cartan Matrix ◽  
Field Theories ◽  
Conformal Field Theories ◽  
Killing Form ◽  
Simple Lie Algebras ◽  
Conformal Field

In this article, we propose two Mathematica™ packages for doing calculations in the domain of classical simple Lie algebras. The main goal of the first package, [Formula: see text], is to determine the principal three-dimensional subalgebra of a simple Lie algebra. The package provides several functions which give some elements related to simple Lie algebras (generators in fundamental and adjoint representation, roots, Killing form, Cartan matrix, etc.). The second package, [Formula: see text], concerns the principal decomposition of a Lie algebra with respect to the principal three-dimensional embedding. These packages have important applications in extended two-dimensional conformal field theories. As an example, we present an application in the context of the theory of W-gravity.

The classification of three-dimensional Lie algeb­ras on complex field

2020 ◽  
pp. 143-155
Author(s):  
E. R. Shamardina
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Three Dimensional ◽  
Structural Equations ◽  
Complex Numbers ◽  
Complex Field ◽  
Low Dimensional

In this paper, we study the classification of three-dimensional Lie al­gebras over a field of complex numbers up to isomorphism. The proposed classification is based on the consideration of objects invariant with re­spect to isomorphism, namely such quantities as the derivative of a subal­gebra and the center of a Lie algebra. The above classification is distin­guished from others by a more detailed and simple presentation. Any two abelian Lie algebras of the same dimension over the same field are isomorphic, so we understand them completely, and from now on we shall only consider non-abelian Lie algebras. Six classes of three-dimensional Lie algebras not isomorphic to each other over a field of complex numbers are presented. In each of the classes, its properties are described, as well as structural equations defining each of the Lie alge­bras. One of the reasons for considering these low dimensional Lie alge­bras that they often occur as subalgebras of large Lie algebras

scholarly journals Control of connectivity and rigidity in prismatic assemblies

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200485
Author(s):  
Gary P. T. Choi ◽  
Siheng Chen ◽  
Lakshminarayanan Mahadevan
Keyword(s):  
Number Theory ◽  
Degrees Of Freedom ◽  
Three Dimensional ◽  
Connected Components ◽  
Elementary Number Theory ◽  
Percolation Transition ◽  
Elementary Number ◽  
Topological Connectivity ◽  
Internal Degrees Of Freedom ◽  
Deterministic Setting

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

scholarly journals Three-Dimensional Electromagnetic Metamaterials with Non-Maxwellian Effective Fields

2007 ◽  
Vol 1014 ◽  
Author(s):  
Jonghwa Shin ◽  
Jung-Tsung Shen ◽  
Shanhui Fan
Keyword(s):  
Degrees Of Freedom ◽  
Effective Permeability ◽  
Three Dimensional ◽  
Homogeneous Medium ◽  
Effective Field ◽  
Long Wavelength ◽  
Wavelength Limit ◽  
New Class ◽  
Electromagnetic Metamaterials ◽  
Internal Degrees Of Freedom

AbstractIt is commonly assumed that the long-wavelength limit of a metamaterial can always be described in terms of effective permeability and permittivity tensors. Here we report that this assumption is not necessary–there exists a new class of metamaterial consisting of several interlocking disconnected metal networks, for which the effective long-wavelength theory is local, but the effective field is non-Maxwellian, and possesses much more internal degrees of freedom than effective Maxwellian fields in a local homogeneous medium.

scholarly journals BOSONIZATION OF NON-RELATIVISTIC FERMIONS AND W-INFINITY ALGEBRA

1992 ◽  
Vol 07 (01) ◽  
pp. 71-83 ◽  
Author(s):  
SUMIT R. DAS ◽  
AVINASH DHAR ◽  
GAUTAM MANDAL ◽  
SPENTA R. WADIA
Keyword(s):  
Magnetic Field ◽  
External Field ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Cartan Subalgebra ◽  
Space Dimension ◽  
Three Dimensional ◽  
Classical Action ◽  
Yang Mills ◽  
The One

We discuss the bosonization of non-relativistic fermions in one-space dimension in terms of bilocal operators which are naturally related to the generators of W-infinity algebra. The resulting system is analogous to the problem of a spin in a magnetic field for the group W-infinity. The new dynamical variables turn out to be W-infinity group elements valued in the coset W-infinity/H where H is a Cartan subalgebra. A classical action with an H gauge invariance is presented. This action is three-dimensional. It turns out to be similar to the action that describes the color degrees of freedom of a Yang–Mills particle in a fixed external field. We also discuss the relation of this action with the one recently arrived at in the Euclidean continuation of the theory using different coordinates.

BIANCHI IX GROUP-MANIFOLD REDUCTIONS OF GRAVITY

2005 ◽  
Vol 20 (40) ◽  
pp. 3115-3125
Author(s):  
ROMAN LINARES
Keyword(s):  
Lie Algebras ◽  
Three Dimensional ◽  
Einstein Gravity ◽  
Spin Connection ◽  
Higher Dimension ◽  
Dimensional Theory ◽  
Bianchi Ix ◽  
Group Manifold ◽  
Manifold Reduction ◽  
Kaluza Klein

We exhibit a new way to perform the group-manifold reduction of pure Einstein gravity in the vielbein formulation when the compactification group manifold is S3. The new Bianchi IX group-manifold reduction is obtained by exploiting the two three-dimensional Lie algebras that the S3 group manifold admits. As an application of the new reduction we show that there exists a domain wall solution to the lower-dimensional theory which upon uplifting to the higher-dimension turns out to be the self-dual (in the nonvanishing components of both curvature and spin connection) Kaluza–Klein monopole.

Reversibly Sampling Conformations and Binding Modes Using Molecular Darting

2020 ◽  
Author(s):  
Samuel C. Gill ◽  
David Mobley
Keyword(s):  
Monte Carlo ◽  
Ligand Binding ◽  
Binding Sites ◽  
Degrees Of Freedom ◽  
Dynamics Simulation ◽  
Hiv Integrase ◽  
Binding Modes ◽  
System A ◽  
Multiple Binding ◽  
Internal Degrees Of Freedom

<div>Sampling multiple binding modes of a ligand in a single molecular dynamics simulation is difficult. A given ligand may have many internal degrees of freedom, along with many different ways it might orient itself a binding site or across several binding sites, all of which might be separated by large energy barriers. We have developed a novel Monte Carlo move called Molecular Darting (MolDarting) to reversibly sample between predefined binding modes of a ligand. Here, we couple this with nonequilibrium candidate Monte Carlo (NCMC) to improve acceptance of moves.</div><div>We apply this technique to a simple dipeptide system, a ligand binding to T4 Lysozyme L99A, and ligand binding to HIV integrase in order to test this new method. We observe significant increases in acceptance compared to uniformly sampling the internal, and rotational/translational degrees of freedom in these systems.</div>

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