2,504 research outputs found
Characterization of Sobolev-Slobodeckij spaces using curvature energies
We give a new characterization of Sobolev-Slobodeckij spaces W^{1+s,p} for
n/p<1+s, where n is the dimension of the domain. To achieve this we introduce a
family of curvature energies inspired by the classical concept of integral
Menger curvature. We prove that a function belongs to a Sobolev-Slobodeckij
space if and only if it is in L^p and the appropriate energy is finite.Comment: 11 pages; the definition of curvature energies was slightly changed,
which allowed to extend the result; some proofs got simplified, mainly thanks
to [Dor85
Hopf fibration and monopole connection over the contact quantum spheres
Noncommutative geometry of quantised contact spheres introduced by Omori,
Maeda, Miyazaki and Yoshioka is studied. In particular it is proven that these
spheres form a noncommutative Hopf fibration in the sense of Hopf-Galois
extensions. The monopole (strong) connection is constructed, and projectors
describing projective modules of all monopole charges are computed.Comment: 15 pages, LaTe
Twisted Dirac Operators over Quantum Spheres
We construct new families of spectral triples over quantum spheres, with a
particular attention focused on the standard Podles quantum sphere and twisted
Dirac operators.Comment: 17 page
Basic Study of Synchronization in a Two Degree of Freedom, Duffing Based System
The issue of synchronization has been analyzed from many points of view since it was first
described in 17th century. Nowadays the theory that stands behind it is so developed,
it might be difficult to understand where it all comes from. This paper reminds the
key concepts of synchronization by examination of two coupled Duffing oscillators. It
describes also the origins of Duffing equation and proves its ability to behave chaotically
Implementation of the New Control Methods in Simplification of a Multidimensional Control and Optimization of a Control System Parameters.
The main purpose of this text is to present application of the Largest Lyapunov Exponent (LLE) as a criterion for optimization of the new type of simple controller parameters. Investigated controller is the part of numerically simulated control system. The calculation of LLE was done with a new method [2]. Introduction contains reference to previous publications on inverted pendulum control and Lyapunov stability. Application of the new simple formula for LLE estimation in control systems is discussed. In the next part simulated dynamical system is described and new type of simple controller allowing to control multidimensional system is introduced. In the last part results of the simulation are shown along with conclusions to whole dynamics analysis. Comparison of the proposed regulator with the linearquadratic regulator (LQR) was verified and its better effectiveness with respect to LQR was proved
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