'Twisted duality' for Clifford Algebras

Viewing the complex Clifford algebra $C(V)$ of a real inner product space $V$ as a superalgebra, we offer several proofs of the fact that if $W$ is a subspace of the complexification of $V$ then the supercommutant of the Clifford algebra $C(W)$ is precisely the Clifford algebra $C(W^{\perp})$.Comment: 8 page

A tangential approach to trigonometry

We construct the complex tangent as a meromorphic function in the plane, using an approach developed by Weierstrass in his characterization of analytic functions that satisfy algebraic addition theorems

Creator-annihilator domains and the number operator

We show that for the bosonic Fock representation in infinite dimensions, the maximal common domain of all creators and annihilators properly contains the domain of the square-root of the number operator.Comment: 7 pages, no figure

The triple-zero Painlev\'e I transcendent

We offer elementary proofs for fundamental properties of the unique triple-zero solution to the first Painlev\'e equation

Distributive Quotients

We note that each lattice $L$ has a unique largest distributive quotient, of which every distributive quotient of $L$ is itself a quotient.Comment: 3 page

Remarks on trigonometric functions after Eisenstein

We modify the Whittaker-Watson account of the Eisenstein approach to the trigonometric functions, basing these functions independently on the Eisenstein function $\varepsilon_2$

L1\mathbf L^1 completeness for Fourier series

We note that the Fubini theorem may be used to prove that an $L^1$ function is determined by its Fourier coefficients

Implicational Completeness

We present a proof of completeness for the implicational propositional calculus, based on a variant of the Lindenbaum procedure

Conservative inclusion of N\"orlund methods

We offer practical necessary and sufficient conditions in order that every sequence convergent relative to the N\"orlund summation method $(N, p)$ be convergent relative to the N\"orlund summation method $(N, q)$ without the requirement that limits be preserved

Automorphisms of Quadratic Liouville Structures

We examine the diffeomorphisms of a symplectic vector space that preserve a chosen symplectic potential. Our examination yields an explicit description of these diffeomorphisms when the chosen potential differs from the canonical potential by the differential of a homogeneous quadratic in one of three broad classes.Comment: 8 page