scholarly journals ENGEL RELATIONS IN 4-MANIFOLD TOPOLOGY

2016 ◽  
Vol 4 ◽  
Author(s):  
MICHAEL FREEDMAN ◽  
VYACHESLAV KRUSHKAL
Keyword(s):  
Lie Groups ◽  
Free Groups ◽  
Higher Order ◽  
Double Points

We give two applications of the 2-Engel relation, classically studied in finite and Lie groups, to the 4-dimensional (4D) topological surgery conjecture. The A–B slice problem, a reformulation of the surgery conjecture for free groups, is shown to admit a homotopy solution. We also exhibit a new collection of universal surgery problems, defined using ramifications of homotopically trivial links. More generally we show how $n$-Engel relations arise from higher-order double points of surfaces in 4-space.

scholarly journals Products of free groups in Lie groups

2021 ◽  
Author(s):  
Caterina Campagnolo ◽  
Holger Kammeyer
Keyword(s):  
Lie Groups ◽  
Free Groups

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

scholarly journals Higher Order Geometric Theory of Information and Heat Based on Poly-Symplectic Geometry of Souriau Lie Groups Thermodynamics and Their Contextures: The Bedrock for Lie Group Machine Learning

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 840 ◽  
Author(s):  
Frédéric Barbaresco
Keyword(s):  
Machine Learning ◽  
Maximum Entropy ◽  
Lie Groups ◽  
Lie Group ◽  
Data Analytics ◽  
Geometric Theory ◽  
Higher Order ◽  
Small Data ◽  
Algebra Structure ◽  
Fisher Metric

We introduce poly-symplectic extension of Souriau Lie groups thermodynamics based on higher-order model of statistical physics introduced by Ingarden. This extended model could be used for small data analytics and machine learning on Lie groups. Souriau geometric theory of heat is well adapted to describe density of probability (maximum entropy Gibbs density) of data living on groups or on homogeneous manifolds. For small data analytics (rarified gases, sparse statistical surveys, …), the density of maximum entropy should consider higher order moments constraints (Gibbs density is not only defined by first moment but fluctuations request 2nd order and higher moments) as introduced by Ingarden. We use a poly-sympletic model introduced by Christian Günther, replacing the symplectic form by a vector-valued form. The poly-symplectic approach generalizes the Noether theorem, the existence of moment mappings, the Lie algebra structure of the space of currents, the (non-)equivariant cohomology and the classification of G-homogeneous systems. The formalism is covariant, i.e., no special coordinates or coordinate systems on the parameter space are used to construct the Hamiltonian equations. We underline the contextures of these models, and the process to build these generic structures. We also introduce a more synthetic Koszul definition of Fisher Metric, based on the Souriau model, that we name Souriau-Fisher metric. This Lie groups thermodynamics is the bedrock for Lie group machine learning providing a full covariant maximum entropy Gibbs density based on representation theory (symplectic structure of coadjoint orbits for Souriau non-equivariant model associated to a class of co-homology).

scholarly journals Higher-order variational problems on lie groups and optimal control applications

2014 ◽  
Vol 6 (4) ◽  
pp. 451-478 ◽  
Author(s):  
Leonardo Colombo ◽  
◽  
David Martín de Diego ◽  
Keyword(s):  
Optimal Control ◽  
Lie Groups ◽  
Variational Problems ◽  
Higher Order ◽  
Control Applications

Kernels of higher order Cauchy differences on free groups

2015 ◽  
Vol 89 (1) ◽  
pp. 119-147 ◽  
Author(s):  
Che Tat Ng
Keyword(s):  
Free Groups ◽  
Higher Order

scholarly journals Higher Order Nonlinear Complementary Filtering on Lie Groups

2019 ◽  
Vol 64 (5) ◽  
pp. 1772-1783 ◽  
Author(s):  
David Evan Zlotnik ◽  
James Richard Forbes
Keyword(s):  
Lie Groups ◽  
Higher Order
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