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Commutative formal series and formal vector fields

Lax825748.FormalSeries · concepts/Lax825748/FormalSeries.lean · lax-825748

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    Natural Language Statement

    Definition

    The state algebra in dimension dd is R[[q1,…,qd]]\mathbb R[[q_1,\ldots,q_d]]. Its elements are arbitrary commutative formal power series.

    A formal vector field with coefficients θi\theta_i acts on ff by ∑iθi∂if\sum_i\theta_i\partial_i f. Formal continuity means continuity for the topology of agreement of all coefficients below a fixed total degree.

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    Lean source view on GitHub

    1import Lax825748.Monomials
    2import Mathlib.RingTheory.MvPowerSeries.Derivative
    3
    4/-!
    5---
    6title: Commutative formal series and formal vector fields
    7type: definition
    8---
    9The state algebra in dimension dd is R[[q1,…,qd]]\mathbb R[[q_1,\ldots,q_d]].
    10Its elements are arbitrary commutative formal power series.
    11
    12A formal vector field with coefficients θi\theta_i acts on ff by
    13∑iθi∂if\sum_i\theta_i\partial_i f. Formal continuity means continuity for the topology of
    14agreement of all coefficients below a fixed total degree.
    15-/
    16
    17noncomputable section
    18
    19namespace Lax825748.FormalSeries
    20
    21open Monomials
    22open scoped BigOperators
    23
    24abbrev StateSeries (d : ℕ) := MvPowerSeries (Fin d) ℝ
    25
    26def AgreeBelow {d : ℕ} (N : ℕ) (f g : StateSeries d) : Prop :=
    27 ∀ I : Fin d →₀ ℕ, totalDegree I < N →
    28 MvPowerSeries.coeff I f = MvPowerSeries.coeff I g
    29
    30def FormallyContinuous {d e : ℕ} (F : StateSeries d → StateSeries e) : Prop :=
    31 ∀ N, ∃ M, ∀ f g, AgreeBelow M f g → AgreeBelow N (F f) (F g)
    32
    33abbrev VectorField (d : ℕ) := Fin d → StateSeries d
    34
    35noncomputable def applyVectorField {d : ℕ} (A : VectorField d)
    36 (f : StateSeries d) : StateSeries d :=
    37 ∑ i, A i * MvPowerSeries.pderiv ℝ i f
    38
    39end Lax825748.FormalSeries
    40
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