Convolution Square Roots of B-Series Characters #
This file constructs the convolution square root of a B-series character,
following Shmelev, Ebrahimi-Fard, Tapia & Salvi, Explicit and Effectively
Symmetric Runge-Kutta Methods (arXiv:2507.21006), Theorem 6.1 with q = 1/2:
whenever 2 is invertible, every character ψ has a unique convolution
square root normalized to 1 on the empty forest, given by the recursion
2 σ(φ) = ψ(φ) - Σ_{proper cuts} σ(P^c) σ(R^c).
The paper uses this to define the symmetric component of a B-series method
via (ψ⁻)² = ψ* ψ (Section 6), which is Character.symmetricPart below.
Main definitions #
RootedForest.sqrtCoeff- the square-root coefficientsForestAlgebra.Character.sqrtFunctional- the square root functionalForestAlgebra.Character.convolution_sqrtFunctional-σ σ = ψForestAlgebra.Character.sqrtFunctional_unique- uniquenessForestAlgebra.Character.symmetricPart- the symmetric component of a B-series method,(ψ* ψ)^{1/2}
The coefficients of the convolution square root of a character, defined
by the recursion 2 σ(φ) = ψ(φ) - Σ_{proper cuts} σ(P^c) σ(R^c)
(arXiv:2507.21006, Theorem 6.1).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The square-root recursion, solved for the character value.
Coefficients of the left convolution inverse of a normalized linear
functional, by the recursion g(φ) = -f(φ) - Σ_{proper cuts} g(P^c) f(R^c).
Equations
- One or more equations did not get rendered due to their size.
Instances For
Coefficients of the right convolution inverse of a normalized linear
functional, by the recursion g(φ) = -f(φ) - Σ_{proper cuts} f(P^c) g(R^c).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The square-root coefficients depend only on the character's values on forests up to the given order.
The coproduct terms of the empty forest.
The convolution square root of a character, as a linear functional.
Equations
Instances For
The square root squares to the character:
σ σ = ψ in the convolution algebra (arXiv:2507.21006, Theorem 6.1).
Uniqueness of the normalized convolution square root, on forest monomials.
The symmetric component of a B-series method: the convolution square root of
the canonical symmetric composition ψ* ψ, so that (ψ⁻)² = ψ* ψ
(arXiv:2507.21006, Section 6).
Equations
Instances For
The defining property of the symmetric component:
(ψ⁻)² = ψ* ψ (arXiv:2507.21006, Section 6).
The left convolution inverse of a normalized linear functional.
Equations
Instances For
The right convolution inverse of a normalized linear functional.
Equations
Instances For
Left and right convolution inverses coincide.
The left inverse is a two-sided inverse.
The counit is fixed by the grading involution.
Uniqueness of the normalized convolution square root.
The symmetric component of a B-series method is symmetric: the involution of
ψ⁻ = (ψ* ψ)^{1/2} is its convolution inverse (arXiv:2507.21006, Section 6;
the case q = 1/2 of Proposition 6.2).
The symmetric component is invertible with inverse its involution.
The antisymmetric component of a B-series method: ψ⁺ := ψ (ψ⁻)⁻¹, so that
ψ = ψ⁺ ψ⁻ (arXiv:2507.21006, Corollary 5.3 via the Section 6
construction).
Equations
- One or more equations did not get rendered due to their size.
Instances For
The symmetric decomposition of a B-series method: ψ = ψ⁺ ψ⁻ with ψ⁻
symmetric and ψ⁺ antisymmetric (arXiv:2507.21006, Corollary 5.3).
The antisymmetric component is even: it is fixed by the canonical involution (arXiv:2507.21006, Corollary 5.3).
Uniqueness of the symmetric factor: if ψ = e ⋆ o with e even, o odd and
both normalized, then o is the symmetric component
(arXiv:2507.21006, Theorem 5.2, uniqueness).
Uniqueness of the antisymmetric factor: if ψ = e ⋆ o with e even, o odd
and both normalized, then e is the antisymmetric component
(arXiv:2507.21006, Theorem 5.2, uniqueness).
Two characters are S-equivalent if they have the same symmetric component (arXiv:2507.21006, Section 7).
Equations
Instances For
S-equivalence is precisely the kernel of the map ψ ↦ ψ* ψ on
characters (arXiv:2507.21006, Section 7).
An odd (symmetric) character is its own symmetric component.
Every S-equivalence class contains at most one odd character: the symmetric method is the unique symmetric element of its class (arXiv:2507.21006, Section 7).
A character is S-equivalent to its own symmetric decomposition data:
ψ ~ ζ whenever ζ is odd with ψ = e ⋆ ζ for an even e.
Splitting a sum over all cuts into the two boundary terms and the proper part.
The convolution square root of a character is multiplicative:
σ(φ₁ φ₂) = σ(φ₁) σ(φ₂), so the square root of a character on the BCK Hopf
algebra is again a character (arXiv:2507.21006, Theorem 6.1).
Proof: compare the double cut sum Σ_{x,y} σ(x₁+y₁)σ(x₂+y₂) (which equals
ψ(φ₁)ψ(φ₂) via Δ(φ₁φ₂) = Δφ₁·Δφ₂ and σ⋆σ = ψ) with its factored form
Σ_{x,y} σ(x₁)σ(x₂)·σ(y₁)σ(y₂) (which also equals ψ(φ₁)ψ(φ₂)); by strong
induction on the total order, the summands agree except at the two
double-boundary pairs, which contribute 2σ(φ₁+φ₂) versus 2σ(φ₁)σ(φ₂).
The square-root coefficients as a monoid homomorphism on forest
monomials, by sqrtCoeff_add.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The convolution square root of a character, as a character: the square root of a character on the BCK Hopf algebra is again a character (arXiv:2507.21006, Theorem 6.1).
Equations
Instances For
The character square root has the square-root functional as its underlying linear functional.
√ψ ⋆ √ψ = ψ at the level of characters.
Odd roots are odd: the square root of an odd (symmetric) character is again odd (arXiv:2507.21006, Proposition 6.2).
A B-series method is symmetric iff it factors as Ω* ⋆ Ω
(arXiv:2507.21006, Theorem thm:main): a character is odd precisely when
it is the convolution of some character's adjoint with that character.
The symmetric component of a B-series method, as a character:
ψ⁻ := (ψ*⋆ψ)^{1/2} is itself the character of a (symmetric) B-series
method (arXiv:2507.21006, Corollary 5.3 with Theorem 6.1).
Equations
Instances For
The character-level symmetric part has the symmetric-part functional as its underlying linear functional.
The symmetric component is a symmetric method: ψ⁻ is odd
(arXiv:2507.21006, Proposition 6.2 with Corollary 5.3).
The inverse of the symmetric component is its involution (it is odd).
The antisymmetric component of a B-series method, as a character:
ψ⁺ := ψ ⋆ (ψ⁻)⁻¹ (arXiv:2507.21006, Corollary 5.3).
Equations
Instances For
The character-level antisymmetric part has the antisymmetric-part functional as its underlying linear functional.