3-D and 2-D Permutohedrons in Nature


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A Centroid Computation

This is a temporary pedagogical post of an elementary computation of a centroid required in an application to a potential employer.

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A Vector Calculus Computation of the Volume of a Parallelpiped

This is a temporary  pedagogical entry of a simple vector triple product calculation required in an application to a potential employer.

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Scoot Over

Each of the two scutoids depicted below contain 2-D examples of my two favorite families of convex polytopes–the permutahedra (the hexagon) and associahedra (the pentagon, also the 2-D stellahedron, OEIS A248727), related to multiplicative (OEIS A133314) and compositional (OEIS A133437) inversion of functions (or formal generating series), respectively.

(From “Scientists Have Discovered an Entirely New Shape, And It Was Hiding in Your Cells“)

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Pre-Lie algebras, Cayley’s analytic trees, and mathemagical forests

Referring to week 299 of John Baez’s old blog or the Pre-Lie Algebra entry of nLab, a left pre-Lie algebra satisfies the associative relation, (AR),

a(bc) - (ab)c = b(ac) - (ba)c.

To see the relation to Cayley’s work of 1857 as described in my pdf Mathemagical Forests (MF), let the generators be represented by the vectors (infinitesimal Lie generators) a(x)D, b(x)D and c(x)D where D = d/dx, the derivative, and the multiplication operation by a(x)D \cdot b(x)D = a(x)b'(x)D. (Call it the canonical pre-Lie operation, CPLO.) Then both sides of the AR reduce to a(x)b(x)c''(x)D.

In MF, initially the op g(x)D is assigned to each vertex of forests of “naturally grown” rooted trees with the trees of each forest having the same number of vertices. Then starting with the leaves and working down, the CPLO is precisely the action of the resulting operator at each vertex on g(x)D of the immediate lower vertex (see also Lagrange a la Lah). For example, if three leaves are attached by edges, or branches, directly to a lower vertex, the operator generated at that lower vertex is (g(x))^3 g'''(x)D, which then operates via the CPLO on g(x)D of the next lower vertex, or node. The associated forests represent the action of powers of infinitesimal Lie generators, g(x)D, i.e., Lie vectors, and encode the repeated product differentiation rule, or Newton-Leibniz product rule, through the “natural growth” of the forests.

The resulting action of each side of the AR can be represented by a rooted tree with three nodes, or vertices, with two leaves and one root. a(x)D is assigned to one leaf, b(x)D to the other, and c(x)D to the root. The resulting operation gives a(x)b(x)c''(x)D.

(1) Arthur Cayley, On the theory of the analytical forms called trees, Phil. Mag. 13 (1857), 172-176.

(2) An N-Category Cafe posting by Baez on week 299 and pre-Lie algebras

(3) OEIS A139605 and A145271

(4) Butcher series: A story of rooted trees and numerical methods for evolution equations by McLachlan, Modin, Munthe-Kaas, and Verdier

(5) Combinatorial Hopf algebras by Loday and Ronco (Pg. 28 has a description of a pre-Lie product in terms of nonplanar unreduced trees, and pg. 18, a bijection between planar binary (PB) and planar unreduced trees (PUT) that might be useful in translating between arguments made using one formulation into the other. )

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Formal group laws and binomial Sheffer sequences

Given a compositional inverse pair f(x) and f^{-1}(x), i.e.,

f(f^{-1}(x)) = x,

with f(x) = e^{a. x} with  a_0 = 0a_1 = 1, and (a.)^n = a_n,  construct the binomial Sheffer sequence p_n(t) with the exponential generating function

e^{x p.(t)} = e^{t f^{-1}(x)}.

Then the associated formal group law (FGL) may be expressed as

FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)] = e^{a.[f^{-1}(x)+f^{-1}(y)]}

= e^{a. D_t}|_{t=0} \;\; e^{t[f^{-1}(x)+f^{-1}(y)]}

= f(D_t)|_{t=0} \;\; e^{t[f^{-1}(x)+f^{-1}(y)]}

= f(D_t)|_{t=0} \;\; e^{t f^{-1}(x)} e^{tf^{-1}(y)}

= \sum_{j \geq 0 } \; \; \sum_{k \geq 0 } \; \; \frac{x^j}{j!} \frac{y^k}{k!} \; f(D_t)|_{t=0} \;\; p_j(t)p_k(t).

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An lnfinite Wronskian Matrix, Binomial Sheffer Polynomials, and the Lagrange Reversion Theorem

Form the infinite Wronskian matrix W(x,y) with elements

W_{j,k} = D_x^{j-1}\frac{[y \cdot h(x)]^k}{k!}.

A generating function for this matrix is

e^{\alpha D_x} e^{\beta y h(x)} = e^{\beta y h(\alpha+x)}= G

with k! \; M_{j,k} = D_\alpha^{j-1} \; D_\beta^k \; G \; |_{\alpha=\beta=0}.

If h(0) = 0 = h^{(-1)}(0), then also

G = e^{(\alpha+x)p.(\beta y)},

where (p.(y))^n = p_n(y) = \sum_{m=0}^n \; p_{n,m} \; y^m is a binomial Sheffer sequence of polynomials.

Then in this particular case,

\; W_{j,k} = \sum_{m \ge 0} \frac{x^{m-j+1}}{(m-j+1)!} \; p_{m,k} \; y^k

and so is the product of an upper triangular Toeplitz matrix of divided-powers in x, whose rows are the shifted summands of the Taylor series for e^x , and the Sheffer polynomial summand matrix in y. For example, these are the 4 by 4 submatrices:

\begin{bmatrix} 1 & x & x^2/2!& x^3/3! \\ 0 & 1 & x & x^2/2! \\ 0 & 0 & 1 & x\\ 0 & 0 & 0 & 1 \end{bmatrix}

\begin{bmatrix} p_{0,0} & 0 & 0 & 0 \\  p_{1,0} & y \;p_{1,1} & & 0 \\ p_{2,0} & y \; p_{2,1} & y^2 \; p_{2,2} & 0\\ p_{3,0} & y \; p_{3,1} & y^2 \; p_{3,2} & y^3 \; p_{3,3}\end{bmatrix} .

By inspection,

W_{j,k} = D_x^{j-1} \; y^k \; C_k(x)

where C_k(x)= \sum_{n. \ge 0} \; p_{n,k} \; x^n/n! = (h(x))^k/k!, the e.g.f. for the k-th column of the Sheffer matrix.

Revisiting the Lie infinigens of previous posts, we have, for u=h(x) and g(u)=1/(h^{(-1)}(u))^{'},

W_{j,k} = (g(u)D_u)^{j-1}\frac{[y \cdot u]^k}{k!} |_{u=h(x)},

and, consistently,

G = e^{\alpha g(u) D_u}\; e^{\beta y u} \; |_{u=h(x)} = e^{\beta y h(\alpha + h^{-1}(u))} \; |_{u=h(x)}.

The trace for the general matrix,

Tr[W] = \sum_{n \ge 0} W_{n,n} = \sum_{n \ge 0} \; D_x^{n-1}\frac{[y \cdot h(x)]^n}{n!}

with D^{-1} 1 = x , appears in several guises (see the earlier post The Lagrange Reversion Theorem and the Lagrange Inversion Formula), changing colors but maintaining the same basic form, in related but distinct formulations for compositional inversion and, therefore, pops up in the analyses of formal group laws; antipodes for Hopf algebras; combinatorics of forests of tree graphs; convex polytopes; moduli spaces of marked discs and punctured Riemann spheres; Feynman graphs for quantum fields; solutions of nonlinear PDEs, such as the inviscid Burgers’ equation; Hirzebruch genera; and umbral, or finite operator, calculus.

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