[HN Gopher] Algebraic graph calculus (2021)
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       Algebraic graph calculus (2021)
        
       Author : 082349872349872
       Score  : 54 points
       Date   : 2023-04-15 20:22 UTC (2 hours ago)
        
 (HTM) web link (gabarro.org)
 (TXT) w3m dump (gabarro.org)
        
       | marosgrego wrote:
       | Neat. I wish someone wrote this down in the language of
       | differential forms.
        
         | enriquto wrote:
         | This is called "discrete exterior calculus" and you can find it
         | easily.
         | 
         | The idea is that k-forms are functions defined on the k-cliques
         | of the graph. TFA is just the 1-dimensional case of this.
         | 
         | The 2-dimensional case would be:
         | 
         | 0-forms: functions defined on vertices
         | 
         | 1-forms: functions defined on edges
         | 
         | 2-forms: functions defined on triangles
         | 
         | The exterior derivative is defined in a natural way, by taking
         | differences along signed boundaries.
         | 
         | In the case of a triangulated surface, the Hodge dual has a
         | nice interpretation via the dual triangulation.
        
       | just_a_quack wrote:
       | One cool application of the discrete laplace operator is that we
       | can use it to calculate the number of spanning trees of the graph
       | by considering the determinants of the submatrices we get my
       | removing 1 row and 1 column from the matrix!
       | 
       | https://en.wikipedia.org/wiki/Kirchhoff%27s_theorem
        
       | molticrystal wrote:
       | We have also covered two other forms of calculus over the the
       | last week:
       | 
       | Alien calculus [0][1] Matrix Calculus [2][3]
       | 
       | Might as well throw in some other generalizations of the
       | derivative [4] , it is amazing how these concepts apply to all
       | sorts of mathematical structures once you generalize them, whlie
       | you are probably fairly interested in math if you are reading the
       | comments here, but on the offi chance you have not done so, look
       | up measure theory, it is a fun concept that allows you to
       | generalize the integral in neat ways that you'll notice has a lot
       | of application to computer science especially vision related
       | applications.
       | 
       | [0] https://news.ycombinator.com/item?id=35476236 [1]
       | https://www.quantamagazine.org/alien-calculus-could-save-par...
       | [2] https://news.ycombinator.com/item?id=35568311 [3]
       | https://www.matrixcalculus.org/ [4]
       | https://math.stackexchange.com/a/1209684
        
         | quickthrower2 wrote:
         | Of course LLMs probably have sparked an interest in calculus. I
         | have dusted of decades old knowledge, relearning the difference
         | between a d and a d, or what is df/dx ln(x).
        
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