Discrete and Differential Calculus

We develop a functorial calculus for higher finite differences and
derivatives on affine Banach spaces. Cubical probes in Möbius coordinates
carry exact covariant and contravariant operators $\Delta_+$ and $\Delta^+$
for arbitrary maps; symmetric tangent and cotangent probes carry their
smooth counterparts $D_+$ and $D^+$. Composition becomes functoriality:
the coordinate formulas for $\Delta_+$ and $D_+$ are the cover- and
partition-indexed Faà di Bruno formulas. The pullbacks $\Delta^+$ and
$D^+$ are algebra morphisms and are adjoint to their respective
pushforwards. In the smooth sector, $D_+$ is the corresponding coalgebra
morphism, uniquely determined by the total differential. These
algebra-coalgebra adjunctions encode the higher product rules.

The central construction is a weighted collapse of cubical probes. Under
this collapse, covers of excess weight vanish and partitions remain. The
resulting symbol map

$\sigma_k(c) = \sum_{\pi\in{Part}(k)} \prod_{A\in\pi}c(A)$

intertwines the collapsed cubical pushforward with $D_+$. In finite
dimensions, rescaled cube measures converge weakly to point-supported
distributions; contravariantly, the collapse extracts Taylor coefficients
from cubical jets. The smooth higher chain and product rules follow from
the corresponding exact discrete identities.

Applications include vertexwise inversion of cubical pushforward, a finite
Neumann formula for inverse differential pushforward and the derivatives of
local inverses, higher-symbol expansions for collapsing stencils, and the
transport of point differential operators through arbitrary coordinates.
The latter recovers the Laplace--Beltrami and polar biharmonic operators and
produces an exact polar form of the five-point stencil. We give
self-contained proofs of the four-functor calculus and symbol map.

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Discrete Faà di Bruno via Möbius Inversion

We approach discrete and differential Faà di Bruno formulas from a Möbius inversion angle.
On the Boolean cube, Newton's discrete Taylor formula and the definition of iterated forward differences form a zeta--Möbius dual pair,
and composing two Taylor expansions and inverting once yields a closed discrete
Faà di Bruno formula at a fixed basepoint: for arbitrary maps $f, g$ between abelian groups,

$$ \Delta(f \circ g;,x;,u_1,\dots,u_k) = \sum_{H \in \mathrm{Cov}(k)} \Delta(f;,g(x);,(\Delta(g;x;u_T))_{T\in H}), $$

where $\mathrm{Cov}(k)$ denotes the coverings of $[k]$ by nonempty subsets.
Grouping repeated directions gives binomial versions on multi-index grids, and iterating gives formulas for $m$-fold composites,
with integer covering coefficients governed by explicit cross and level recursions, a discrete analogue of the Constantine--Savits formulas.

The relationship between coverings and partitions appearing in classical Faà di Bruno formulas, is exhibited in an algebraic setting.
The discrete formulas are Taylor expansions over the function algebra of the Boolean cube,
$B_k = \mathbf{k}[\delta_1,\dots,\delta_k]/(\delta_i^2-\delta_i)$, whose idempotent generators absorb overlapping products;
in the differential analogue $A_k = \mathbf{k}[\varepsilon_1,\dots,\varepsilon_k]/(\varepsilon_i^2)$,
nilpotent generators annihilate overlaps and only partitions remain.
Both algebras are fibers of the flat deformation $C_k = \mathbf{k}[t][x_1,\dots,x_k]/(x_i^2 - tx_i)$,
over which a single weighted covering formula interpolates:
its coefficients are difference quotients, non-partition coverings carry positive powers of $t$, and evaluation at $t = 0$ yields
the classical partition-indexed Faà di Bruno formula.

We demonstrate how these algebraic identities can be lifted to the analytical setting of $C^n$ maps between Banach spaces,
recovering the multivariate Faà di Bruno formula of Constantine--Savits and extending it to composites of several maps.
Boolean finite differences, binomial grid formulas, infinitesimal Taylor algebras, and Fréchet derivatives thus appear
as four realizations of one Möbius-dual Faà di Bruno formula, connected by a flat family.

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Faà di Bruno is Taylor Composition

We approach Faà di Bruno as a composition theorem for Taylor polynomials.
For $C^k$ maps $\phi: E \to F$ and $\psi: F \to G$ between Banach spaces,
let $T^k_\ast(\phi; x)$ denote the reduced Taylor polynomial of $\phi$ at $x$, obtained by removing the constant term.
We show that

$$T^k_\ast(\psi \circ \phi; x) = \pi_{\le k}(T^k_\ast(\psi; \phi(x)) \circ T^k_\ast(\phi; x)).$$

The proof is an elementary estimate of the Peano remainder and does not use partitions or combinatorial enumeration.

Expanding this composition identity recovers the classical Faà di Bruno formulas.
Polarization gives the multivariate partition formula (Lévy 2006),
while coefficient extraction gives the multi-index formula (Constantine and Savits 1996).
Our approach separates the functorial nature of Taylor approximation from the combinatorial bookkeeping of polarization and coefficient extraction.

As an application, we give a general higher-order product rule.

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A Unified Polynomial Approximation Scheme for Classical Function Spaces

We prove polynomial density theorems for a broad class of classical function spaces on domains
$\Omega \subset \IR^n$, in both local (compact-open) and global (Banach) topologies.
In the local regime we show that restrictions of polynomials are dense in $C^k(\Omega)$,
$C^\infty(\Omega)$, and $L^p_{loc}(\Omega), W^{k,p}_{loc}(\Omega)$ for arbitrary open $\Omega$.

In the global regime on bounded domains we re-prove the classical density results for
$C^0(\bar{\Omega})$ (Stone-Weierstrass) and $L^p(\Omega)$.
For higher-differentiable function spaces ($k \geq 1$), polynomial density depends on regularity of
the domain. We show density in $C^k_b(\bar{\Omega})$ for $C^k$-extension domains (e.g. smooth boundary)
and in $W^{k,p}(\Omega)$ for Sobolev-extension domains (e.g. Lipschitz boundary).

The proofs are organized around a single approximation pipeline: extension, compact truncation,
analytic regularization, and Taylor approximation. This yields straightforward, standard and unified
arguments that make explicit the domain assumptions underlying several folklore polynomial-density
statements; we also record counterexamples showing that such hypotheses are necessary for global
approximation in strong norms.

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Fundamental Theorem of Algebra

We give a very short proof of the Fundamental Theorem of Algebra using a one-parameter deformation and a discriminant-locus argument.

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The Cayley-Hamilton Theorem

We give a short real-variable proof of the Cayley--Hamilton theorem using a local-to-global rigidity argument: prove the identity on a neighbourhood of a diagonal matrix with simple spectrum and extend to all matrices using polynomiality.

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