scholarly journals HYPERKÄHLER ARNOLD CONJECTURE AND ITS GENERALIZATIONS

2012 ◽  
Vol 23 (08) ◽  
pp. 1250077 ◽  
Author(s):  
VIKTOR L. GINZBURG ◽  
DORIS HEIN
Keyword(s):  
Generating Functions ◽  
Dimensional Reduction ◽  
Morse Theory ◽  
Three Dimensional ◽  
Degenerate Case ◽  
Floer Theory ◽  
Arnold Conjecture ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Degenerate Version

We generalize and refine the hyperkähler Arnold conjecture, which was originally established, in the non-degenerate case, for three-dimensional time by Hohloch, Noetzel and Salamon by means of hyperkähler Floer theory. In particular, we prove the conjecture in the case where the time manifold is a multidimensional torus and also establish the degenerate version of the conjecture. Our method relies on Morse theory for generating functions and a finite-dimensional reduction along the lines of the Conley–Zehnder proof of the Arnold conjecture for the torus.

Wavelet-like collocation method for finite-dimensional reduction of distributed systems

2000 ◽  
Vol 24 (12) ◽  
pp. 2687-2703 ◽  
Author(s):  
A. Adrover ◽  
G. Continillo ◽  
S. Crescitelli ◽  
M. Giona ◽  
L. Russo
Keyword(s):  
Distributed Systems ◽  
Collocation Method ◽  
Dimensional Reduction ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional

scholarly journals A note on the Dancer–Fučík spectra of the fractional p-Laplacian and Laplacian operators

2015 ◽  
Vol 4 (1) ◽  
pp. 13-23 ◽  
Author(s):  
Kanishka Perera ◽  
Marco Squassina ◽  
Yang Yang
Keyword(s):  
Dimensional Reduction ◽  
Simple Eigenvalue ◽  
Critical Groups ◽  
Local Analysis ◽  
Laplacian Operator ◽  
Sufficient Condition ◽  
Nontrivial Solutions ◽  
Finite Dimensional Reduction ◽  
Finite Dimensional ◽  
Laplacian Operators

AbstractWe study the Dancer–Fučík spectrum of the fractional p-Laplacian operator. We construct an unbounded sequence of decreasing curves in the spectrum using a suitable minimax scheme. For p = 2, we present a very accurate local analysis. We construct the minimal and maximal curves of the spectrum locally near the points where it intersects the main diagonal of the plane. We give a sufficient condition for the region between them to be nonempty and show that it is free of the spectrum in the case of a simple eigenvalue. Finally, we compute the critical groups in various regions separated by these curves. We compute them precisely in certain regions and prove a shifting theorem that gives a finite-dimensional reduction in certain other regions. This allows us to obtain nontrivial solutions of perturbed problems with nonlinearities crossing a curve of the spectrum via a comparison of the critical groups at zero and infinity.

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