How was this infinite sequence of numbers found? (non-commutative geometry )

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    Laplace equation
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SUMMARY

This discussion centers on the exploration of an infinite sequence of numbers related to non-commutative geometry, specifically referencing Alain Connes' work. The sequence includes values such as 5/4, 2, 5/2, and 13/4, which are identified as solutions to the Laplacian equation associated with the frequencies of vibrating drums. The discussion highlights resources for understanding the derivation of these frequencies, particularly for square and triangular drums, and provides links to relevant academic materials.

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  • Understanding of non-commutative geometry concepts
  • Familiarity with the Laplacian equation in mathematical physics
  • Knowledge of isospectral drums and their properties
  • Basic principles of differential equations and vibrations of membranes
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  • Study Alain Connes' work on non-commutative geometry
  • Learn about the derivation of frequencies for triangular drums
  • Explore the mathematical principles behind isospectrality
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Mathematicians, physicists, and researchers interested in the intersections of geometry and spectral theory, particularly those focusing on the vibrations of membranes and non-commutative geometry.

Heidi
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Hi Pfs,
I read these slides:
https://indico.math.cnrs.fr/event/782/attachments/1851/1997/Connes.pdf
It is about non commutative geometry (Alain Connes)
After Shapes II, you see a the plots of the square roots of a sequence of numbers given below:
5/4, 2, 5/2, 13/4 ....
I think that they are the solutions of laplacian equation related to the shape of the "drums" above. that is to say the frequencies one can get when hitting this drums.
How to retrieve this infinite sequence?
It is at page 50
thanks
 
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In this article Connes gives an example of two isospectral drums (which are non connex).
One drum with a triangle and a quare and the other with a rectangle and a different triangle.
I found this
https://math.libretexts.org/Bookshe...ons/6.01:_Vibrations_of_Rectangular_Membranes
explaining how to get the possible frequencies emitted by a square or rectangular drum.
Do you know what are the frequencies ommitted by a trangular drum (half a square)?
 
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