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Module rplus

lie_groups

Module rplus 

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ℝ⁺: The Positive Reals (Multiplicative Scaling Group)

This module implements ℝ⁺, the group of positive real numbers under multiplication. ℝ⁺ is the symmetry group for scaling transformations in applications like:

  • Volatility surface analysis (IV scaling)
  • Image processing (contrast/brightness)
  • Economic growth models (multiplicative factors)

§Mathematical Background

§Definition

ℝ⁺ = (0, ∞) with multiplication

The positive reals form a Lie group under multiplication.

§Group Structure

  • Multiplication: a · b (standard multiplication)
  • Identity: 1
  • Inverse: a⁻¹ = 1/a
  • Abelian: a · b = b · a

§Lie Algebra

Lie(ℝ⁺) ≅ ℝ (the real line with addition)
Exponential map: exp(x) = eˣ
Logarithm: log(a) = ln(a)

§Topological Properties

  • Non-compact: Unlike U(1), ℝ⁺ extends to infinity
  • Simply connected: π₁(ℝ⁺) = 0 (no winding numbers)
  • Contractible: Homotopy equivalent to a point

§Isomorphism with (ℝ, +)

The exponential map provides a Lie group isomorphism:

exp: (ℝ, +) → (ℝ⁺, ×)
log: (ℝ⁺, ×) → (ℝ, +)

§Applications

  1. Volatility surfaces: IV(K,T) = λ · IV_eq(K,T) models level shifts
  2. Scale invariance: Physical systems with no preferred scale
  3. Log-returns: r = log(S_t/S_0) lives in the Lie algebra

Structs§

RPlus
An element of ℝ⁺, the multiplicative group of positive reals
RPlusAlgebra
Lie algebra of ℝ⁺, isomorphic to (ℝ, +)