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-Integral equations · IFSintegrals.jl SIO is the type for singular integral operators.
source DiscreteSIO(SIO::SIO; h_mesh::Real, h_quad::Real, h_quad_diag::Real)is the constructor for a discretisation of a singular integral operator, 'SIO'. hmesh is the meshwidth parameter for the discretisation of the underlying fractal h quad denotes the discretisation parameter for the integrals in the stiffness matrix. hquad diag is the parameter used to compute the diagonal elements of the matrix
source
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Integral equations · IFSintegrals.jl SIO is the type for singular integral operators.
source DiscreteSIO(SIO::SIO; h_mesh::Real, h_quad::Real, h_quad_diag::Real)is the constructor for a discretisation of a singular integral operator, 'SIO'. hmesh is the meshwidth parameter for the discretisation of the underlying fractal h quad denotes the discretisation parameter for the integrals in the stiffness matrix. hquad diag is the parameter used to compute the diagonal elements of the matrix
source
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Home · IFSintegrals.jl This documentation explains basic usage of the Julia package IFSintegrals. This package is designed to approximate integrals and solve integral equations on fractals.
There is no generally accepted definition of the word fractal . Here we consider self-similar sets, which can be described as the attractor of an iterated function system , consisting of a set of affine contraction maps.
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Home · IFSintegrals.jl This documentation explains basic usage of the Julia package IFSintegrals. This package is designed to approximate integrals and solve integral equations on fractals.
There is no generally accepted definition of the word fractal . Here we consider self-similar sets, which can be described as the attractor of an iterated function system , consisting of a set of affine contraction maps.
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This document was generated with Documenter.jl version 1.2.1 on Wednesday 3 January 2024 . Using Julia version 1.7.2.
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+Constructs a similarity map. The third input (rotation matrix) is optional, and the fourth is created automatically.
Can be treated as a function, for example
julia> s = Similarity(1/2,0) # creates contraction s(x)=x/2+0
julia> s(π)
- 1.5707963267948966source The iterated function system may be interpreted as a set of these maps, $\{s_m\}_{m=1}^M$ . The attractor $\Gamma$ is the unique non-empty compact set which satisfies
\[\Gamma = S(\Gamma):=\bigcup_{m=1}^M s_m(\Gamma).\]
We can construct the map $S$ defined above as follows.
using IFSintegrals
+ 1.5707963267948966source The iterated function system may be interpreted as a set of these maps, $\{s_m\}_{m=1}^M$ . The attractor $\Gamma$ is the unique non-empty compact set which satisfies
\[\Gamma = S(\Gamma):=\bigcup_{m=1}^M s_m(\Gamma).\]
We can construct the map $S$ defined above as follows.
using IFSintegrals
s₁ = Similarity(1/3,2/3)
s₂ = Similarity(1/3,2/3)
S = [s₁,s₂]
x = rand()
S(x) # applies the map S to the point x2-element Vector{Float64}:
- 0.9425241020303798
- 0.9425241020303798This is the IFS for the Cantor Set, and this can be converted into an InvariantMeasure with the command
Γ = InvariantMeasure(S)InvariantMeasure{Float64, Float64}(Similarity{Float64, Float64}[Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333), Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333)], 1, 0.6309297535714574, true, true, 0.9999999999999999, 0.0, 1.0, [0.5, 0.5], true, Bool[1 0; 0 1], IFSintegrals.AutomorphicMap{Float64, Float64}[IFSintegrals.AutomorphicMap{Float64, Float64}(1.0, 0.0)])The outer constructor for InvariantMeasure constructs other properties, such as diameter and Hausdorff dimension, which describe this fractal measure.
The full type is described below:
struct InvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
+ 0.9625149610757451
+ 0.9625149610757451This is the IFS for the Cantor Set, and this can be converted into an InvariantMeasure with the command
Γ = InvariantMeasure(S)InvariantMeasure{Float64, Float64}(Similarity{Float64, Float64}[Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333), Similarity{Float64, Float64}(0.3333333333333333, 0.6666666666666666, 1.0, 0.3333333333333333)], 1, 0.6309297535714574, true, true, 0.9999999999999999, 0.0, 1.0, [0.5, 0.5], true, Bool[1 0; 0 1], IFSintegrals.AutomorphicMap{Float64, Float64}[IFSintegrals.AutomorphicMap{Float64, Float64}(1.0, 0.0)])The outer constructor for InvariantMeasure constructs other properties, such as diameter and Hausdorff dimension, which describe this fractal measure.
The full type is described below:
struct InvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
IFS::Vector{Similarity{V,M}}
spatial_dimension::Int64
Hausdorff_dimension::Float64
@@ -27,7 +27,7 @@
connectedness::Matrix{Bool}
symmetry_group::Vector{AutomorphicMap{V,M}}
endRepresentation of a invariant measure, whose support is an iterated function system (IFS). Constructor requires only an IFS, which is of type Array{Similarity}. All other essential properties can be deduced from this, including barycentre, diameter and dimension, which are approximated numerically.
Has the outer constructor, which only requires IFS (a vector of Similarity
InvariantMeasure(sims::Vector{Similarity}; measure::Real=1.0) =
- InvariantMeasure(sims, get_diameter(sims); measure=measure)Fields
IFS: The iterated function system, a vector of Similarity spatial_dimension: The integer dimension of the smallest open set containing the fractal Hausdorff_dimension: The Hausdorff dimenson of the fractal homogeneous: A flag, true if all the contractions are of the same size Hausdorff_weights: A flag, true if this is a Hausdorff measure barycentre: The barycentre of the measure diameter: The diemater of the fractal support measure: The measure of the whole fractal, usually set to one weights: The probability weights describing the invariant measure disjoint: Flag for if the fractal support is disjoint connectedness: Matrix describing which subcomponents are connected symmetry_group: Vector of AutomorphicMap, describing symmetries of the measure source Now let's try a more complicated example. The second (translation) argument of Similarity can also be a vector. For example:
ρ = 0.41
+ InvariantMeasure(sims, get_diameter(sims); measure=measure)Fields
IFS: The iterated function system, a vector of Similarity spatial_dimension: The integer dimension of the smallest open set containing the fractal Hausdorff_dimension: The Hausdorff dimenson of the fractal homogeneous: A flag, true if all the contractions are of the same size Hausdorff_weights: A flag, true if this is a Hausdorff measure barycentre: The barycentre of the measure diameter: The diemater of the fractal support measure: The measure of the whole fractal, usually set to one weights: The probability weights describing the invariant measure disjoint: Flag for if the fractal support is disjoint connectedness: Matrix describing which subcomponents are connected symmetry_group: Vector of AutomorphicMap, describing symmetries of the measure source Now let's try a more complicated example. The second (translation) argument of Similarity can also be a vector. For example:
ρ = 0.41
IFS = [
Similarity(ρ,[0,0])
Similarity(ρ,[1-ρ,0])
@@ -35,7 +35,7 @@
Similarity(ρ,[(1-ρ)/2,(1-ρ)/(2*sqrt(3))])
]
Γ = InvariantMeasure(IFS)We can plot an approximation attractors in $\mathbb{R}$ or $\mathbb{R}^2$ , as follows:
using Plots
-plot(Γ,color = "black", markersize=0.75,label="My fractal")Several preset fractals are available. These may be called with:
CantorSet()
+plot(Γ,color = "black", markersize=0.75,label="My fractal")Several preset fractals are available. These may be called with:
CantorSet()
CantorDust()
Sierpinski() # triangle/gasket
KochCurve() # Koch curve
@@ -49,18 +49,18 @@
Γ_shift = 1.5*Γ + [-2,0.5]
plot!(Γ_shift, markersize=0.75,
- label="Sierpinski Triangle (translated and stretched)",aspect_ratio=1.0)Consider an IFS with $M\in\mathbb{N}$ components. For a vector
\[\mathbf{m}=[m_1,\ldots,m_N]\in\{1,\ldots,M\}^N,\]
it is standard to denote a sub-component of the fractal by
\[\Gamma_{\mathbf{m}} := s_{m_1}\circ\ldots \circ s_{m_N}(\Gamma)\]
The following type also describes a measure whose support is a sub-component, in the above sense.
struct SubInvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
+ label="Sierpinski Triangle (translated and stretched)",aspect_ratio=1.0)Consider an IFS with $M\in\mathbb{N}$ components. For a vector
\[\mathbf{m}=[m_1,\ldots,m_N]\in\{1,\ldots,M\}^N,\]
it is standard to denote a sub-component of the fractal by
\[\Gamma_{\mathbf{m}} := s_{m_1}\circ\ldots \circ s_{m_N}(\Gamma)\]
The following type also describes a measure whose support is a sub-component, in the above sense.
struct SubInvariantMeasure{V,M} <: SelfSimilarFractal{V,M}
parent_measure::InvariantMeasure
IFS::Vector{Similarity{V,M}} # could be removed ?
index::Vector{Int64}
barycentre::V
diameter::Float64
measure::Float64
-endRepresents a fractal measure which has been derived from an InvariantMeasure.
Fields
parentmeasure: This measure is supported on a subet of the support of parent measure IFS: The vector of similarities describing the fractal support of this measure index: The vector index corresponding to the contractions applied to parent_measure barycentre: The barycentre of this measure diameter: The diameter of the fractal support measure: The measure of the support source Using standard vector index syntax, a SubInvariantMeasure can be easily constructed:
Γ = Sierpinski()
+endRepresents a fractal measure which has been derived from an InvariantMeasure.
Fields
parentmeasure: This measure is supported on a subet of the support of parent measure IFS: The vector of similarities describing the fractal support of this measure index: The vector index corresponding to the contractions applied to parent_measure barycentre: The barycentre of this measure diameter: The diameter of the fractal support measure: The measure of the support source Using standard vector index syntax, a SubInvariantMeasure can be easily constructed:
Γ = Sierpinski()
m = [1,3,2] # vector index
Γₘ = Γ[m] # construct SubInvariantMeasure
plot(Γ, markersize=0.75,
label="Sierpinski Triangle (default)")
plot!(Γₘ, markersize=0.75,
- label="Subcomponent",aspect_ratio=1.0)
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