175 research outputs found
Symplectic Reduction for Semidirect Products and Central Extensions
This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980's by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle.
We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein, 1974) and present a new proof and approach to semidirect product theory. We shall then give an idea of how the more general theory of group extensions proceeds (the details of which are given in Marsden, Misiołek, Perlmutter and Ratiu, 1998). The case of central extensions is illustrated in this paper with the example of the Heisenberg group. The theory, however, applies to many other interesting examples such as the Bott-Virasoro group and the KdV equation
On a two-component -Camassa--Holm system
A novel -Camassa--Holm system is studied as a geodesic flow on a
semidirect product obtained from the diffeomorphism group of the circle. We
present the corresponding details of the geometric formalism for metric Euler
equations on infinite-dimensional Lie groups and compare our results to what
has already been obtained for the usual two-component Camassa--Holm equation.
Our approach results in well-posedness theorems and explicit computations of
the sectional curvature.Comment: 12 page
The Camassa-Holm Equation: Conserved Quantities and the Initial Value Problem
Using a Miura-Gardner-Kruskal type construction, we show that the
Camassa-Holm equation has an infinite number of local conserved quantities. We
explore the implications of these conserved quantities for global
well-posedness.Comment: 8 pages, LaTe
Determinants of corporate governance in banks
In most cases, banks are large and organizationally complex corporations. Effective corporate governance in banks is the basis for achieving and maintaining public confidence in the banking system and constitutes a critical factor for the proper functioning of individual banks as well as the entire economic system.Udostępnienie publikacji Wydawnictwa Uniwersytetu Łódzkiego finansowane w ramach projektu „Doskonałość naukowa kluczem do doskonałości kształcenia”. Projekt realizowany jest ze środków Europejskiego Funduszu Społecznego w ramach Programu Operacyjnego Wiedza Edukacja Rozwój; nr umowy: POWER.03.05.00-00-Z092/17-00
Generalized Euler-Poincar\'e equations on Lie groups and homogeneous spaces, orbit invariants and applications
We develop the necessary tools, including a notion of logarithmic derivative
for curves in homogeneous spaces, for deriving a general class of equations
including Euler-Poincar\'e equations on Lie groups and homogeneous spaces.
Orbit invariants play an important role in this context and we use these
invariants to prove global existence and uniqueness results for a class of PDE.
This class includes Euler-Poincar\'e equations that have not yet been
considered in the literature as well as integrable equations like Camassa-Holm,
Degasperis-Procesi, CH and DP equations, and the geodesic equations
with respect to right invariant Sobolev metrics on the group of diffeomorphisms
of the circle
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