Module: Math::Algebra::Tropical

Defined in:
lib/ruuuby/math/algebra/tropical/context_numeric.rb,
lib/ruuuby/math/algebra/tropical/tropical.rb,
lib/ruuuby/math/algebra/tropical/context_matrix.rb

Overview

Note:

geometry aspects will not be primary focus in this module

“the study of polynomials and their ‘geometric properties` when addition is replaced with minimization and multiplication is replaced with ordinary addition”

‣ standard multiplication‘*` is transformed to`⨂`: `x ⨂ y = x + y` ‣ standard addition(`+`) is transformed to`⨁` w/ 2 versions:

| name               | definition            | additive identity |
| ------------------ | -------------------- | ----------------- |
| `min-plus algebra` | `x ⨁ y = min{x, y}`  | `∞`              |
| `max-plus algebra` | `x ⨁ y = max{x, y}`  | `−∞`             |

‣ “unit for ⨁ is +∞, and the unit for ⨂ is 0” ‣ ‘tropical addition` is commutative: `∀ a,b ∈ ℝ max(a, b) = max(b, a)` ‣ `tropical addition` is associative: `∀ a,b ∈ ℝ max(max(a, b), c) = max(max(a, c), b)` ‣ `tropical multiplication` is commutative and `associative ∈ ℝ` ‣ `tropical multiplication` is distributive ‣ `additive inverse` DNE as ∀ element does not necessarily have a matching element that when summed, equal the `additive identity`

‣ (as -∞ is < ∀ element other than -∞; thus, won't be returned from func{`max`})

‣ “eigenvalue-eigenvector pair ‘λ` and x for a given matrix A satisfy the equation `Ax = λx` where `λ ∈ ℝ, A ∈ ℝⁿˣⁿ`, and `x ∈ ℝⁿ`”

‣ ‘normalized weight of a cycle` = `sum of edge weights` / `number of edges`

‣ ‘tropical algebra` is a semi-ring

‣ "given by `(ℝᴹᴬˣ,⨁,⨂) where ℝᴹᴬˣ = ℝ∪{−∞}`"

See Also:

Defined Under Namespace

Modules: ContextMatrix, ContextNumeric