arXiv subjects

S. Blake Allan

Publications and source records attributed to S. Blake Allan.

4 recordsLinked to original sources

arXiv2025

Algebra: The Eighth Liberal Art?

What is the role of algebra in classical mathematics education? How does it relate to the four quadrivial arts? These questions have troubled the mathematical community since the introduction of algebra into the Renaissance academy by men like François Viète, Guillame Gosselin, and René Descartes. Their challenge is perhaps most starkly articulated at the conclusion of Viète's Introduction to the Analytic Art, where he claims that his algebra "appropriates to itself by right the proud problem of problems, which is: [sic] TO LEAVE NO PROBLEM UNSOLVED". Some contemporary educators respond by eschewing these methods to avoid the excessive formalization often accompanying algebra, and to give a central place to the geometrical tradition of Euclid's Elements. Others embrace the rise of algebra in the curriculum, focusing on contemporary techniques and priorities. This paper seeks to reconcile these perspectives by clarifying the way in which algebra participates in the quadrivial arts. Based on testimony from both the origins of algebra and its contemporary practitioners, I argue that algebra is not so much an eighth liberal art as an arithmetical language of form; an actualized potential in arithmetic. I conclude by offering curricular recommendations which provide glimpses of the practical insights available from this vantage.

math.HO↗

arXiv2021

On essential self-adjointness of singular Sturm-Liouville operators

Considering singular Sturm--Liouville differential expressions of the type \[ τ_α = -(d/dx)x^α(d/dx) + q(x), \quad x \in (0,b), \; α\in \mathbb{R}, \] we employ some Sturm comparison-type results in the spirit of Kurss to derive criteria for $τ_α$ to be in the limit point and limit circle case at $x=0$. More precisely, if $α\in \mathbb{R}$ and for $0 < x$ sufficiently small, \[ q(x) \geq [(3/4)-(α/2)]x^{α-2}, \] or, if $α\in (-\infty,2)$ and there exist $N\in\mathbb{N}$, and $\varepsilon>0$ such that for $0<x$ sufficiently small, \begin{align*} &q(x)\geq[(3/4)-(α/2)]x^{α-2} - (1/2) (2 - α) x^{α-2} \sum_{j=1}^{N}\prod_{\ell=1}^{j}[\ln_{\ell}(x)]^{-1} \\ &\quad\quad\quad +[(3/4)+\varepsilon] x^{α-2}[\ln_{1}(x)]^{-2}. \end{align*} then $τ_α$ is nonoscillatory and in the limit point case at $x=0$. Here iterated logarithms for $0 < x$ sufficiently small are of the form, \[ \ln_1(x) = |\ln(x)| = \ln(1/x), \quad \ln_{j+1}(x) = \ln(\ln_j(x)), \quad j \in \mathbb{N}. \] Analogous results are derived for $τ_α$ to be in the limit circle case at $x=0$. We also discuss a multi-dimensional application to partial differential expressions of the type \[ - {\rm div} |x|^α \nabla + q(|x|), \quad α\in \mathbb{R}, \; x \in B_n(0;R)\backslash\{0\}, \] with $B_n(0;R)$ the open ball in $\mathbb{R}^n$, $n\in \mathbb{N}$, $n \geq 2$, centered at $x=0$ of radius $R \in (0, \infty)$.

math.CA↗

arXiv2021

On critical dipoles in dimensions $n\geq 3$

We reconsider generalizations of Hardy's inequality corresponding to the case of (point) dipole potentials $V_γ(x) = γ(u, x) |x|^{-3}$, $x \in \mathbb{R}^n \backslash \{0\}$, $γ\in [0,\infty)$, $u \in \mathbb{R}^n$, $|u|=1$, $n \in \mathbb{N}$, $n \geq 3$. More precisely, for $n \geq 3$, we provide an alternative proof of the existence of a critical dipole coupling constant $γ_{c,n} > 0$, such that \begin{align*} &\text{for all $γ\in [0,γ_{c,n}]$, and all $u \in \mathbb{R}^n$, $|u|=1$,} \\ &\quad \int_{\mathbb{R}^n} d^n x \, |(\nabla f)(x)|^2 \geq \pm γ\int_{\mathbb{R}^n} d^n x \, (u, x) |x|^{-3} |f(x)|^2, \quad f \in D^1(\mathbb{R}^n). \end{align*} with $D^1(\mathbb{R}^n)$ denoting the completion of $C_0^{\infty}(\mathbb{R}^n)$ with respect to the norm induced by the gradient. Here $γ_{c,n}$ is sharp, that is, the largest possible such constant, and we discuss a numerical scheme for its computation. Moreover, we discuss upper and lower bounds for $γ_{c,n} > 0$. We also consider the case of multicenter dipole interactions with dipoles centered on an infinite discrete set.

math.AP↗

arXiv2019

Explicit Krein Resolvent Identities for Singular Sturm-Liouville Operators with Applications to Bessel Operators

We derive explicit Krein resolvent identities for generally singular Sturm-Liouville operators in terms of boundary condition bases and the Lagrange bracket. As an application of the resolvent identities obtained, we compute the trace of the resolvent difference of a pair of self-adjoint realizations of the Bessel expression $-d^2/dx^2+(ν^2-(1/4))x^{-2}$ on $(0,\infty)$ for values of the parameter $ν\in[0,1)$ and use the resulting trace formula to explicitly determine the spectral shift function for the pair.

math.SP↗