'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

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

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

Bessel function asymptotics: a relation due to Lommel

We show that the Bessel function asymptotic relation $J_{\nu}^2(z) + J_{\nu + 1}^2 (z) \sim 2/(\pi z)$ of Lommel is valid when $z$ is real but can fail otherwise

Cayley-Sylvester invariants and the Hamilton equations

We note implications of the Cayley-Sylvester theory of invariants and covariants for the Hamilton equations generated by cubic and quartic Hamiltonian functions

Beyond Quartic Hamiltonians

We investigate the Hamiltonian system generated by a homogeneous binary polynomial $U$ of order greater than four. In particular, we study the circumstances under which the associated Hessian is a Weierstrass $\wp$ function and find that the vanishing of two covariants of $U$ is involved

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

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