Galpern—Sobolev Type Equations with Non-constant Coefficients

2004 ◽  
pp. 301-310
Author(s):  
K. Gürlebeck ◽  
W. Sprößig
Keyword(s):  
Sobolev Type ◽  
Constant Coefficients

scholarly journals GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES

2014 ◽  
Vol 2 ◽  
Author(s):  
MARTINS BRUVERIS ◽  
PETER W. MICHOR ◽  
DAVID MUMFORD
Keyword(s):  
Initial Data ◽  
Geodesic Distance ◽  
Geodesic Equation ◽  
Plane Curves ◽  
Sobolev Type ◽  
Geodesic Completeness ◽  
Constant Coefficients ◽  
Geodesically Complete ◽  
Well Posed ◽  
Closed Plane

AbstractWe study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

Differential p-forms and q-vector fields with constant coefficients

2020 ◽  
pp. 1
Author(s):  
Jaime Muñoz Masqué ◽  
Luis M. Pozo Coronado ◽  
M. Eugenia Rosado
Keyword(s):  
Vector Fields ◽  
Constant Coefficients ◽  
Q Vector

scholarly journals Coarse cohomology with twisted coefficients

2020 ◽  
Vol 70 (6) ◽  
pp. 1413-1444
Author(s):  
Elisa Hartmann
Keyword(s):  
Sheaf Cohomology ◽  
Grothendieck Topology ◽  
Coarse Structure ◽  
Relative Cohomology ◽  
Number Of Ends ◽  
Constant Coefficients ◽  
Coarse Space ◽  
Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

scholarly journals B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

2020 ◽  
Vol 18 (1) ◽  
pp. 715-730
Author(s):  
Javanshir J. Hasanov ◽  
Rabil Ayazoglu ◽  
Simten Bayrakci
Keyword(s):  
Differential Operator ◽  
Singular Integral ◽  
Riesz Potential ◽  
Type Theorem ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Sobolev Type ◽  
Riesz Potentials ◽  
Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

scholarly journals Adaptive Control Structure with Neural Data Processing Applied for Electrical Drive with Elastic Shaft

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3389
Author(s):  
Marcin Kamiński ◽  
Krzysztof Szabat
Keyword(s):  
Neural Network ◽  
Adaptive Control ◽  
Dynamic Properties ◽  
Experimental Studies ◽  
Drive System ◽  
Particle Swarm Optimizer ◽  
The Neural Network ◽  
Control Concept ◽  
Constant Coefficients ◽  
Selection Of

This paper presents issues related to the adaptive control of the drive system with an elastic clutch connecting the main motor and the load machine. Firstly, the problems and the main algorithms often implemented for the mentioned object are analyzed. Then, the control concept based on the RNN (recurrent neural network) for the drive system with the flexible coupling is thoroughly described. For this purpose, an adaptive model inspired by the Elman model is selected, which is related to internal feedback in the neural network. The indicated feature improves the processing of dynamic signals. During the design process, for the selection of constant coefficients of the controller, the PSO (particle swarm optimizer) is applied. Moreover, in order to obtain better dynamic properties and improve work in real conditions, one model based on the ADALINE (adaptive linear neuron) is introduced into the structure. Details of the algorithm used for the weights’ adaptation are presented (including stability analysis) to perform the shaft torque signal filtering. The effectiveness of the proposed approach is examined through simulation and experimental studies.

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