Operators

Operators are used to transform size distributions. The operators overload the Base or LinearAlgebra methods. Operators fall into two broad categories: operators changing number concentration and spectral density fields (𝕟.N and 𝕟.S) and operators that change the sizing vector (𝕟.Dp). The former include 𝕟₁ + 𝕟₂, 𝕟₁ - 𝕟₂, 𝕟₁ ∗ 𝕟₂, 𝕟₁ / 𝕟₂, a ∗ 𝕟, T .∗ 𝕟, and A ∗ 𝕟, while the latter include a · 𝕟 and T · 𝕟.

Operator	Description
𝕟₁ + 𝕟₂	Superposition of the distributions 𝕟₁ and 𝕟₂
𝕟₁ - 𝕟₂	Superposition of the distributions 𝕟₁ and 𝕟₂
a ∗ 𝕟	Uniform scaling of the concentration fields by factor a
𝐀 ∗ 𝕟	Matrix multiplication of 𝐀 and concentration
𝕟₁ * 𝕟₂	Scaled such that total number concentration equals to N1 ∗ N2
𝕟₁ / 𝕟₂	Ratio of concentration fields of distributions 𝕟₁.N and 𝕟₂.N
a · 𝕟	Uniform scaling of the diameter field of the size distribution by factor a
T · 𝕟	Elementwise scaling the diameter field by factor T

Number Operators

Base.:+ — Method

+(𝕟₁::SizeDistribution, 𝕟₂::SizeDistribution)

Defines the sum of two size distributions. If diameter grids are not equal, then the diameter grid of n2 is interpolated onto the n1 grid prior to addition.

𝕩 = 𝕟₁ + 𝕟₂

is defined such that

𝕩.S = 𝕟₁.S + 𝕟₂.S 
𝕩.N = 𝕩.S .* 𝕟.ΔlnD

Example Usage

𝕟₁ = lognormal([[120, 90, 1.20]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝕟₂ = lognormal([[90, 140, 1.15]]; d1 = 20.0, d2 = 800.0, bins = 64)
𝕩 = 𝕟₁ + 𝕟₂

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Base.:- — Method

-(𝕟₁::SizeDistribution, 𝕟₂::SizeDistribution)

Defines the sum of two size distributions. If diameter grids are not equal, then the diameter grid of n2 is interpolated onto the n1 grid prior to addition.

𝕩 = 𝕟₁ - 𝕟₂

is defined such that

𝕩.S = 𝕟₁.S - 𝕟₂.S 
𝕩.N = 𝕩.S .* 𝕟.ΔlnD

Example Usage

𝕟₁ = lognormal([[120, 90, 1.20]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝕟₂ = lognormal([[90, 140, 1.15]]; d1 = 20.0, d2 = 800.0, bins = 64)
𝕩 = 𝕟₁ - 𝕟₂

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Base.:* — Method

*(a::Number, 𝕟::SizeDistribution)

Multiplication of scalar and size distribution. The net result is a scaling of the number concentration of the spectra by a. The function is symmetric such that a * 𝕟 == 𝕟 * a.

Let a denote a number and 𝕟 denote a size distribution. Then

𝕩 = a * 𝕟

is defined such that

𝕩.N = a * 𝕟.N
𝕩.S = a * 𝕟.S

Example Usage

𝕟 = lognormal([[120, 90, 1.20]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝕩 = 2.3 * 𝕟₂

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Base.:* — Method

*(𝐀::AbstractMatrix, 𝕟::SizeDistribution)

Multiplication of matrix and size distribution. The net result is the multiplication of the matrix with number concentration and spectral density fields.

Let 𝐀 denote an nxn matrix where n equals the number of size bins of 𝕟. Then

𝕩 = 𝐀 * 𝕟

is defined such that

𝕩.N = 𝐀 * 𝕟.N
𝕩.S = 𝐀 * 𝕟.S

𝕟 = lognormal([[100, 100, 1.1]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝐀 = rand(256,256)
𝕩 = 𝐀 * 𝕟

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Base.:* — Method

*(𝕟₁::SizeDistribution, 𝕟₂::SizeDistribution)

Multiplication of size distribution and a size distribution. The net result is a size distribution that has total number concentration square. For a probability distributions that by definition integrate to unity, this operation corresponds to the product of two random variates with distribution 1 and 2.

Let 𝕟₁ and 𝕟₂ denote a two size distribution defined on the same diameter grid. Then

𝕩 = 𝕟₁ * 𝕟₂

is defined such that

Nsq = 𝕟₁.N * 𝕟₂.N
𝕩.N = sum(𝕟₁.N) * sum(𝕟₂.N) * Nsq./sum(Nsq)
𝕩.S = N ./ 𝕟₁.ΔdlnD

Example Usage

𝕟₁ = lognormal([[120, 90, 1.20]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝕟₂ = lognormal([[90, 140, 1.15]]; d1 = 20.0, d2 = 800.0, bins = 64)
𝕩 = 𝕟₁ * 𝕟₂

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Base.:/ — Method

/(𝕟₁::SizeDistribution, 𝕟₂::SizeDistribution)

Division of size distribution and size distribution. The net result is a size distribution that is the ratio of the concentration vectors.

Let 𝕟₁ and 𝕟₂ denote a two size distribution defined on the same diameter grid. Then

𝕩 = 𝕟₁ / 𝕟₂

is defined such that

N = 𝕟₁.N ./ 𝕟₂.N
S = 𝕟₁.S ./ 𝕟₂.S

Example Usage

𝕟₁ = lognormal([[120, 90, 1.20]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝕟₂ = lognormal([[90, 140, 1.15]]; d1 = 20.0, d2 = 800.0, bins = 64)
𝕩 = 𝕟₁ / 𝕟₂

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Size Operators

LinearAlgebra.:⋅ — Method

⋅(a::Number, 𝕟::SizeDistribution)

Multiplication of a scalar and a size distribution. The net result is a uniform diameter shift of the size distribution. The function is symmetric such that a ⋅ 𝕟 == 𝕟 ⋅ a.

Let a denote a floating point scalar and 𝕟 denote a size distribution. Then

𝕩 = a ⋅ 𝕟

is defined such that

𝕩.Dp = a * 𝕟.Dp

Example Usage

a = 2.0 
𝕟 = lognormal([[300, 100, 1.3]]; d1 = 10.0, d2 = 1000.0, bins = 256)
𝕩 = a ⋅ 𝕟

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LinearAlgebra.:⋅ — Method

LinearAlgebra.:⋅(A::Vector{<:AbstractFloat}, 𝕟::SizeDistribution)

Dot product of vector and size distribution. The net result is diameter dependent shift of the size distribution. The function is symmetric such that A ⋅ 𝕟 == 𝕟 ⋅ A.

Let T denote a floating point vector with the same number of elements as the size distribution 𝕟. Then

𝕩 = T ⋅ 𝕟

is defined such that

𝕩.Dp = T .* 𝕟.dp

Example Usage

𝕟 = lognormal([[100, 100, 1.1]]; d1 = 10.0, d2 = 1000.0, bins = 256)  
μ,σ = 80.0, 2000.0
T = (1.0 .+ erf.((𝕟.Dp .- μ)./(sqrt(2σ)))) 
𝕩 = T ⋅ 𝕟

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