[HN Gopher] Algebraic graph calculus (2021) ___________________________________________________________________ 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). ___________________________________________________________________ (page generated 2023-04-15 23:00 UTC)