General SG programs:
This is a function to calculate derivatives of the map f: R^m1 -> R^m2 defined by restricting an (m1)st level complex of delta circuits to an (m2)nd level one. We approximate derivatives with respect to perturbing each edge, and for each edge in the restriction, using difference quotients with a variety of sizes of perturbation |
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Computes the energy inner product of two functions on SG |
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Computes energy measure for a given harmonic function and cell |
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Computes energy of a function |
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A contractive similarity of the plane centered at 0 of dilation factor 1/2 |
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A contractive similarity of the plane centered at the point qi of dilation factor 1/2 |
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Returns relative coordinates to use when printing out a table of values at vertex points on SG |
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Plots a function defined on the level m vertices of SG, with edges connecting adjacent vertices |
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Computes the harmonic extension from V_0 to V_1 of a harmonic function. |
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Computes the harmonic extension (given boundary points) to a particular level |
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Calculates the Kusuoka measure of a cell |
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Plots the values of an energy measure |
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Plots the Kusuoka measure |
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Calculates the Radon-Nikodym derivative of an energy measure with respect to the Kusuoka measure |
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Applies the function fi to a column of coordinate pairs |
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A reference file with some important matrices for SG |
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Computes the coordinates of the midpoints of all level m edges of SG |
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Prints the values of a function on the vertices of SG |
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Performs a sequence of contractions on a set of vectors (in a matrix) in $R^{2}$ given a word w |
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Performs a sequence of contractions on a vector given a word w |
Indexing:
Gives a matrix of all words of length m consisting of 0,1,2 |
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Returns a vector listing the address of a given index k in a level m SG-type ordering (TLR ordering) |
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Finds the alternative address for a given point in an SG ordering |
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Finds the alternative index for a given point in TLR ordering on SG. |
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Finds the index in TLR ordering corresponding to a given address |
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This function finds the addresses of those edges which are incident to a given vertex |
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A headache-saving formula for the jth group of 3^m consecutive indices |
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Accepts a vector $y$ of size 3^m1, for some m1, and a permutation $indices$ of size 3^m2, for m2 <= m1, and scrambles the 3^m2 subvectors of y (each of size 3^(m1-m2)) according to the permutation |
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Composes a function f:SG->R with a permutation of the cells of SG |
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Just a simple function to reorder a vector from largest to smallest element |
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Solves the problem of performing calculations such as computing the spectrum of the Laplacian on SG when our chosen data storage scheme is redundant |
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Finds the addresses of those edges which are incident to a given vertex |
Graph restrictions and their relation to the Kusuoka measure:
Plots derivatives of resistances of restriction of graph after perturbing an edge |
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Computes derivative measures |
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Computes measures on all cells of a level m graph |
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Compute the derivatives of the resistances of the restricted graph with respect to perturbing each edge individually by a small t (see inside comments on how we use 3^m by 3 matrices in describing Y-networks |
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Performs Delta-Y transformation on resistances of a Delta graph. |
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Produces coordinates of edges of Y-network of level m. |
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A level 1 Y-network of resistors to a single Y-circuit |
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Gives a matrix with zeros everywhere except at one specified entry |
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Plots derivatives of resistances of Y graph after perturbing edges of an m-cell of the level m Y-network. |
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Plots derivatives when only perturbing an edge. |
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Calculates radon-nikodym derivative of the derivative measures with respect to the Kusuoka measure. |
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Produces a concatenation of each step of the restriction of a system of resistors on SG (in Y-circuits) |
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This function computes the restriction of an (m1)st level network of resistors to an (m2)nd level such network, where a network at the mth level is the stacking of three (m-1)st level networks, and a 1st level network is a Y shaped network. |
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Computes the restriction of a level m Y-network of resistors on SG to a single Y-circuit on the boundary points of SG. |
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Stores the intermediate computations involved in restricting a function between two levels of a Y-circuit. (Note: The point of this is that if you want to perform a restriction of a function that is perturbed on only one cell, you can save computation by simply executing this function ahead of time and, instead of doing redundant calculations, simply reading the restriction data for all cells not containing the perturbed cell since those remain unchanged). |
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This is a computation-saving program for calculating restrictions of Y-networks of resistors. Performs the restriction of an m1-level SG-circuit to an m2-level circuit. |
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Finds the delta-circuit of resistors on three vertices which is equivalent to an inputed y-circuit on the same three vertices with a central fourth vertex added. |
Laplacians on the Sierpinski Gasket
Calculates the Laplacian of 1 (the constant function). |
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Defines a step function on a level m graph approximation of SG. |
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Restricts a Delta network of resistors |
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Computes a Y network equivalent to a (single) given Delta network of resistors. |
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Computes a difference quotient approximation to a directional derivative of the map f: R^{m1} -> R^{m2} which takes a level m1 Delta network of resistors and restricts to a level m2 network. |
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This is a computation-saving version of diffquot.m (runs much faster for m1 >= 7 or so). |
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Makes a plot of the Weyl ratio for a computed spectrum, which is assumed to be stored as the variable “cspec”. |
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Once you have computed the spectrum of a conformal Laplacian, doplot allows you to create a plot of the Weyl file. |
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Script to print out pictures of several Weyl ratio plots. |
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Script to calculate the spectrum for the Dirichlet Laplacian, assuming a conformal factor g and level m are given. |
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This is what I used to make conformal factors to put on SG. I tried to come up with some wavy sinusoidal thing that wouldn’t have any symmetry. |
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Returns the Dirichlet spectrum of the Laplacian for the conformal factor g using level m approximation. |
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Allows one to define an edge function on SG by inputting a function on R^2, and calculating the value of the function at the midpoint of each edge, where we use embedding of SG into R^2 such that the vertices form equilateral triangles. |
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Creates a graphic plot of a level m edge function on SG. |
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This function takes an inline function of the plane and computes its values at level m vertex points on a certain embedding of SG in the plane (as in an equilateral triangle with boundary points (0,0), (1/2,sqrt(3)/2),and (sqrt(3)/2,1/2)), then takes the level m laplacian of the resulting function and produces a plot of it. |
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Labels the current Weyl ratio plot with ‘log(x)’ on the x-axis and ‘N(x)/x^\beta’ on the y-axis. |
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Changes a decimal number into a standard format to append to a file name. |
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This function allows the inputing of an R^2 function whose value to find on the junction points of a certain embedding of SG into the plane. |
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This function accepts a function $u$ stored as a vector of length $3^(m+1)$ and a conformal factor $g$ of the same length, and computes the g-laplacian of u. |
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Script to make a plot of the Weyl ratio for a conformal Laplacian once the spectrum has been computed |
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Calculates the matrix corresponding to the Weyl ratio |
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This function computes the Neumann spectrum of the conformal laplacian with factor g size of g = 3^(m+1) |
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Script to print the current Weyl ratio plot under a coherent naming scheme |
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Calculates the laplacian of a function at a single point |
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Calculates the gaps between subsequent eigenvalues of the Laplacian. |
Miscellaneous:
Plots Vicek graphs |
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computes log base 3. |
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computes p dimension of an energy measure. |
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does color point plot (Matlab only does color plot in the meshplot, as far as I know). |
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reorders a vector of values. |
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computes energy of functions on the tetrahedral Sierpinski Gasket. |
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plots tetrahedral Sierpinski Gasket. |
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allows printing of a table of values into an SG-type arrangement. |
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computes L_p dimensions. |