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

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 $\varepsilon_2$

The bosonic Fock representation and a generalized Shale theorem

We detail a new approach to the bosonic Fock representation of a complex Hilbert space V: our account places the bosonic Fock space S[V] between the symmetric algebra SV and its full antidual SV'; in addition to providing a context in which arbitrary (not necessarily restricted) real symplectic automorphisms of V are implemented, it offers simplified proofs of many standard results of the theory.Comment: LaTeX, 24 pages, no figure

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

'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

Index theory for partial-bijections

We offer streamlined proofs of fundamental theorems regarding the index theory for partial self-maps of an infinite set that are bijective between cofinite subsets.Comment: 5 page

Neville's primitive elliptic functions: the case g3=0{\bf g_3 = 0}

The vanishing of the invariant $g_3$ attached to a lattice $\Lambda$ singles out a midpoint lattice and yields a square-root of the associated Weierstrass function $\wp_{\Lambda}$

The Clifford algebra and its antidual

We analyze the purely algebraic antidual $C'(V)$ of the complex Clifford algebra $C(V)$ over a real inner product space $V$. In particular, we introduce a partially defined product in $C'(V)$ and study its properties

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