Mielnik–Plebański–Strichartz Formula

• Mielnik–Plebański–Strichartz Formula is a closed-form expression for the Magnus logarithm that handles linear initial-value problems with non-commuting operators.
• It unifies the continuous Magnus and Baker–Campbell–Hausdorff expansions with a discrete analogue via summations over permutations and surjections.
• The formula leverages combinatorial structures like planar trees and algebraic frameworks such as dendriform and tridendriform algebras to derive explicit coefficient identities.

Searching arXiv for the cited papers and closely related records to ground the article. The Mielnik–Plebański–Strichartz formula is a closed form for the Magnus logarithm associated with linear initial-value problems for non-commuting operators. In the continuous setting, it expresses the Magnus expansion, and hence the continuous Baker–Campbell–Hausdorff series, as a sum over permutations weighted by the descent number and a binomial coefficient. In the discrete setting, an analogous formula expresses the logarithm of the solution of a linear first-order finite-difference equation as a sum over surjections and partial diagonals. In both cases, the formula is derived from combinatorial and algebraic structures built from planar trees, dendriform or tridendriform algebras, and Rota–Baxter operators (Ebrahimi-Fard et al., 2012, Ebrahimi-Fard et al., 2013).

1. Continuous formulation in the Magnus and BCH setting

The continuous point of departure is the linear initial-value problem

$$\frac{d}{dt}\,X(t)\;=\;A(t)\,X(t)\,,\qquad X(0)=X_0\,.$$

Formally, one writes the Dyson–Chen series

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$

but in the non-commutative setting it is more natural to seek a single logarithm,

$X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$

where $\Omega(t)$ is a Lie-series in $A$. This $\Omega(t)$ is called the Magnus expansion (Ebrahimi-Fard et al., 2012).

In the formulation singled out by Ebrahimi-Fard and Manchon, the Mielnik–Plebański–Strichartz formula gives $\Omega(t)$ directly as

$$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$

where $d(\sigma)$ is the number of descents of the permutation $\sigma$ (Ebrahimi-Fard et al., 2012).

The same result admits a Lie-theoretic reformulation. Since $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$0 is a Lie-element, each integrand may be rewritten as a nested commutator by the Dynkin–Specht–Wever trick: $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$1 This places the formula simultaneously in the Magnus framework and in the continuous BCH framework.

2. Dendriform and tridendriform algebraic framework

A decisive structural feature is that the logarithm is not taken with respect to ordinary multiplication, but with respect to an associative product obtained by splitting it into finer operations. In the continuous case, one works in a dendriform algebra $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$2, where

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$3

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$4

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$5

and the sum $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$6 is associative. Every dendriform algebra carries two pre-Lie products,

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$7

(Ebrahimi-Fard et al., 2012).

In the discrete case, the relevant structure is tridendriform. Let $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$8 be a unital non-commutative algebra over a field of characteristic zero, and let $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$9 be a coefficient sequence. The forward difference operator $X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$0 and its right-inverse summation operator $X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$1 are defined on sequences $X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$2 by

$X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$3

Then $X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$4 is a Rota–Baxter operator of weight $X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$5, satisfying

$X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$6

The linear finite-difference initial-value problem

$X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$7

is equivalent to

$X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$8

where $X(t)\;=\;\exp\!\bigl(\Omega(t)\bigr)\;X_0,$9 and

$\Omega(t)$0

Since $\Omega(t)$1 is tridendriform, one defines the associative star-product

$\Omega(t)$2

and the corresponding exponential and logarithm

$\Omega(t)$3

The pre-Lie product is

$\Omega(t)$4

and the discrete Magnus expansion is obtained in this pre-Lie algebra (Ebrahimi-Fard et al., 2013).

A common simplification is to identify the Mielnik–Plebański–Strichartz formula only with the continuous integral formula over permutations. The discrete theory shows that the same logarithmic mechanism persists for finite differences, but with tridendriform rather than dendriform splitting and with surjections rather than permutations (Ebrahimi-Fard et al., 2013).

3. Tree combinatorics and free algebra constructions

The closed form is governed by planar trees. In the continuous approach, the relevant objects are planar binary trees and planar rooted trees. A planar binary tree is a rooted tree in which every internal vertex has exactly two incoming edges and one outgoing edge; a planar rooted tree allows any number of children. Knuth’s rotation correspondence

$\Omega(t)$5

is the unique bijection satisfying

$\Omega(t)$6

where $\Omega(t)$7 is binary grafting and $\Omega(t)$8 is the left Butcher product on planar rooted trees (Ebrahimi-Fard et al., 2012).

The free unital dendriform algebra on one generator $\Omega(t)$9 is naturally spanned by planar binary trees. The grafting $A$0 and the dendriform operations are related by

$A$1

This free setting is the natural domain in which the logarithm $A$2 can be computed in closed form (Ebrahimi-Fard et al., 2012).

In the discrete theory, the corresponding combinatorial objects are planar reduced trees. A planar reduced tree is a finite oriented tree embedded in the plane, with one outgoing edge, $A$3 incoming edges at each internal vertex, and external leaves. Writing $A$4 for the vector space spanned by all such trees, including the trivial tree $A$5, one has the grafting operation

$A$6

Loday–Ronco showed that $A$7 is the free tridendriform algebra on one generator $A$8, with products recursively defined by, for $A$9, $\Omega(t)$0,

$\Omega(t)$1

$\Omega(t)$2

$\Omega(t)$3

where $\Omega(t)$4 (Ebrahimi-Fard et al., 2013).

These free algebra models encode the combinatorics of the coefficients in the logarithm. A plausible implication is that the formula is best understood not as an isolated summation identity, but as the image of a universal tree-level statement under suitable algebra morphisms.

4. Closed-form expressions over trees, permutations, and surjections

In the free unital dendriform algebra generated by $\Omega(t)$5, the solution of

$\Omega(t)$6

has logarithm

$\Omega(t)$7

where $\Omega(t)$8 is the number of edges of the planar rooted tree $\Omega(t)$9 and $\Omega(t)$0 its number of leaves. Pulling back along Knuth’s rotation gives an equivalent sum over planar binary trees of degree $\Omega(t)$1, with a factor involving the number of descents $\Omega(t)$2 of the tree $\Omega(t)$3 (Ebrahimi-Fard et al., 2012).

The discrete analogue takes the same closure phenomenon into the tridendriform setting. The tree-indexed pre-Lie Magnus formula can be summed over all trees of a given order with explicit coefficients: $\Omega(t)$4 satisfies

$\Omega(t)$5

where $\Omega(t)$6 is the number of leaves minus one and $\Omega(t)$7 is the number of strict descents of $\Omega(t)$8. Applying the tridendriform morphism $\Omega(t)$9 yields

$$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$0

(Ebrahimi-Fard et al., 2013).

The surjection form is the discrete Mielnik–Plebański–Strichartz formula: $$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$1 Here $$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$2 is the set of surjections $$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$3,

$$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$4

is the strict descent number, and

$$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$5

is the partial diagonal associated to $$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$6 (Ebrahimi-Fard et al., 2013).

The continuous and discrete formulas differ in the indexing sets and coefficients, but the parallel is exact at the structural level: the logarithm is assembled from descent statistics, universal tree expansions, and a morphism from free splitting algebras to concrete iterated integrals or iterated sums.

5. Morphisms, descent combinatorics, and the origin of the coefficients

The continuous construction uses two dendriform-algebra morphisms in succession. First, there is a canonical unital dendriform-algebra monomorphism

$$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$7

where $$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$8. Combinatorially, $$\Omega(t) \;=\; \sum_{n=1}^\infty \;\sum_{\sigma\in S_n} \frac{(-1)^{d(\sigma)}}{n\,\binom{n-1}{d(\sigma)}} \int_{0<u_n<\cdots<u_1<t} A\bigl(u_{\sigma(1)}\bigr)\,\cdots\,A\bigl(u_{\sigma(n)}\bigr)\; du_1\cdots du_n,$$9 takes each planar binary tree $d(\sigma)$0 to the sum of all permutations whose descent set corresponds to the left-pointing leaves of $d(\sigma)$1. Second, there is a morphism $d(\sigma)$2 sending $d(\sigma)$3 to the $d(\sigma)$4-fold iterated integral

$d(\sigma)$5

Applying the Rota–Baxter integral $d(\sigma)$6 to $d(\sigma)$7 then yields the continuous Magnus logarithm (Ebrahimi-Fard et al., 2012).

The discrete derivation proceeds analogously, but with a tridendriform Hopf-algebraic embedding into WQSym, the tridendriform Hopf algebra of word quasi-symmetric functions, whose basis is indexed by surjections $d(\sigma)$8. The tree-basis expansion becomes the surjection expansion through the identity on partial diagonals

$d(\sigma)$9

which splits according to whether $\sigma$0, $\sigma$1, or $\sigma$2, matching the tridendriform splitting of the product of two basis elements in WQSym (Ebrahimi-Fard et al., 2013).

The coefficients themselves are combinatorial. In the continuous case, the binomial factor $\sigma$3 arises from the bijection between planar binary trees of degree $\sigma$4 and permutations in $\sigma$5, together with the fact that a tree with $\sigma$6 left-pointing leaves corresponds to exactly those permutations with $\sigma$7 descents. The sign $\sigma$8 appears in the free dendriform logarithm $\sigma$9, and the factor $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$00 is characteristic of passing from the associative logarithm $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$01 to the actual Magnus logarithm $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$02 (Ebrahimi-Fard et al., 2012).

In the discrete case, the correction terms in the logarithm combine into the single coefficient

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$03

for each tree $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$04, proved by induction on the number of leaves within the pre-Lie Magnus recursion (Ebrahimi-Fard et al., 2013).

6. Examples, continuous limit, and relation to BCH

The discrete formula admits explicit low-order illustrations. For order $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$05, there is only one tree $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$06, with no descents, and one obtains

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$07

For order $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$08, the surjections are $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$09, with $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$10 and $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$11. The $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$12 contribution is

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$13

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$14

This matches the naive discrete commutator correction needed to rewrite the second-order sum $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$15 in strictly decreasing order of discrete time (Ebrahimi-Fard et al., 2013).

The discrete formula has the continuous Mielnik–Plebański–Strichartz formula as its limit. In the continuous limit, with step $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$16,

$$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$17

This identifies the discrete theory as a direct analogue rather than a separate construction (Ebrahimi-Fard et al., 2013).

The relation to the classical BCH series is also explicit. When $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$18 and $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$19 are constant operators and one studies $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$20, the discrete BCH formula may be recovered by restricting the continuous formula to delta-functions or by a short-time expansion; conversely, setting $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$21, $$X(t)\;=\;1\;+\;\int_0^t A(s)\,ds \;+\;\int_{0<s_2<s_1<t} A(s_1)A(s_2)\,ds_2\,ds_1 \;+\;\cdots$$22 in the integral formula reproduces the classical Baker–Campbell–Hausdorff series in nested commutators with Bernoulli-number coefficients (Ebrahimi-Fard et al., 2012).

Taken together, these results place the Mielnik–Plebański–Strichartz formula within a unified algebraic-combinatorial scheme: continuous Magnus and BCH expansions are controlled by planar rooted or binary trees and dendriform algebras, while the discrete finite-difference analogue is controlled by planar reduced trees, WQSym, and tridendriform algebras. This suggests that the formula is best viewed as the closed logarithmic expression naturally induced by splitting associativity and transporting universal tree identities into concrete integral or summation models.