arXiv subjects
Publications and source records attributed to P. L. Robinson.
We consider the differential equations $s' = c^5$ and $c' = - s^5$ with initial conditions $s(0) = 0$ and $c(0) = 1$. We show that $s^2 c^2$ extends to the reciprocal of the Weierstrass elliptic function with invariants $g_2 = 0$ and $g_3 = 16$. We also show that powers of $s^2 c^2$ are the only functions of the form $s^m c^n$ that extend meromorphically in the plane.
We investigate the first-order system `$s\,' = c^3, \, c\,' = - s^3; \, s(0) = 0, \, c(0) = 1$'. Its solutions have the property that $s \, c$, $s^2$ and $c^2$ extend to simply-poled elliptic functions, which we explicitly identify in terms of the lemniscatic Weierstrass $\wp$ function.
We develop the theory of the lemniscatic functions sl and cl from their definition as solutions to an initial value problem.
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.
We offer a careful development of the Dixonian elliptic functions with parameter $α= 0$ from the initial value problem of which they are solutions.
We show that the Bessel function asymptotic relation $J_ν^2(z) + J_{ν+ 1}^2 (z) \sim 2/(πz)$ of Lommel is valid when $z$ is real but can fail otherwise.
We offer a simple direct proof of the unitarity of the Julia operator associated to a contraction $A$, from which follow the intertwining identity $(I - A A^*)^{1/2} A = A (I - A^* A)^{1/2}$ and the unitarity of Halmos dilations.
We offer practical necessary and sufficient conditions in order that every sequence convergent relative to the Nörlund summation method $(N, p)$ be convergent relative to the Nörlund summation method $(N, q)$ without the requirement that limits be preserved.
We note that the necessary and sufficient conditions established by Marcel Riesz for the inclusion of regular Nörlund summation methods are in fact applicable quite generally.
We draw attention to simplifications in the theory of a Nörlund summation method $(N, p)$ that arise when the series $\sum_{n \geqslant 0} p_n$ is convergent.
We extend the inner product from bosonic Fock space to a pairing between suitable antifunctionals on the symmetric algebra. Our account is illustrated by the (Gaussian) half-form pairing between positive polarizations in the form needed for geometric quantization.
We note that the Fubini theorem may be used to prove that an $L^1$ function is determined by its Fourier coefficients.
We show that in finite dimensions, the Berezin expectation on an exterior algebra corresponds to the trace on a Clifford algebra, the correspondence being mediated by what is effectively a version of antinormal ordering.
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.
We discuss extensions of an inner product from a vector space to its full antidual. None of these extensions is weakly continuous, but partial extensions recapture some familiar structure including the Hilbert space completion and the antiduality pairing.
We present a manifestly geometrically self-dual version of the Fano harmonicity axiom for the projective plane.
For an axiomatization of three-dimensional projective space based on points and planes, we discuss appropriate versions of the harmonicity axiom and the projectivity axiom, showing that each axiom is equivalent to its spatial dual.
Within an axiomatic framework for three-dimensional projective space based on lines alone, we explore the Fano axiom of harmonicity according to which the diagonal lines of a complete quadrilateral are not concurrent.