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One way to improve this section would be to revise the development with matrix notation and right eigenvectors, supported by appropriate literature, to better fit the example written in Matlab (which facilitates matrix operations). Presently the literature cited for the development is: Mark Newman
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In the section 'discrete heat equation' there is a mismatch between the development of the solution and the example computer code given at the end. Specifically, the solution development explicitly names the left eigenvectors defined by
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I would rather argue that there is a bunch of other applications from applied engineering to theoretical physics ;) But never mind, yes two different pages would make sense. In
English and in German, though.
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So, by far the most common use of this is the laplace filtering used for image processing. This article shows lots of nice math about this, but however completely fails at describing anything about
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should contain more on larger-stencil approximations, as in ; See in particular illustrations starting on page 3. A proper explanation should facilitate comprehension of equation 2 under section
1166:(2010). Networks: An Introduction. Oxford University Press. ISBN 978-0199206650, however sources for a matrix development are available. Are there other suggestions for improving this section?
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382:{\displaystyle \mathbf {D} _{xy}^{2}={\frac {1}{\alpha +1}}{\begin{bmatrix}\alpha &1-\alpha &\alpha \\1-\alpha &-4&1-\alpha \\\alpha &1-\alpha &\alpha \end{bmatrix}}}
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Yes, of course, e.g., P. McDonald and R. Meyers. Diffusions on graphs, Poisson problems and spectral geometry. Trans. Amer. Math. Soc., 354(12):5111–5136 (electronic), 2002.
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Everybody having a MatLab to their abuse can try this with fspecial('laplacian',alpha). (Wow, this was the most complicated maths I ever edited here.) ;-)
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MatLab itself (for probably some very good reason otherwise MatLab wouldn't do that, however, they don't cite this reason unfortunately) recommends
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The embedding of
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670:{\displaystyle \mathbf {D} _{xy}^{2}=1/2{\begin{bmatrix}0.5&1&0.5\\1&-6&1\\0.5&1&0.5\end{bmatrix}}}
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seems to be vastly superior for this (I am not very good at german though). Would say this needs a separate page at
962:{\displaystyle \mathbf {D} _{xy}^{2}=1/6{\begin{bmatrix}1&4&1\\4&-20&4\\1&4&1\end{bmatrix}}}
816:{\displaystyle \mathbf {D} _{xy}^{2}=1/3{\begin{bmatrix}1&1&1\\1&-8&1\\1&1&1\end{bmatrix}}}
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instead of the current redirect to this page. Not sure I will find much time for it, so any help is very welcome.
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517:{\displaystyle \mathbf {D} _{xy}^{2}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}}
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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help actually gives both a rule how to construct them and a recommendation which one to use.
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I always struggle with which of the 3 recommended one now to use. While the
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1098:{\textstyle L\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}}
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While the 3 filters mentioned in this article just have an
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Is any correspondence of the discrete
Laplacian for the
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this is used for image processing. To contrast, the
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845:{\displaystyle \alpha =0.2}
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170::s. --
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