Steven Finch

collaborators

11 papers

math.DG2026

Schwarz $P$ and Schwarz $D$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas

Steven Finch

Let $T$ be the tetragonal disphenoid whose four congruent faces are isosceles triangles with sides $2, 3, 3$ , so that the dihedral angles are $π /2$ along one opposite pair…

math.DG2026

An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron

Steven Finch

Smyth (1984) proved that every tetrahedron $T \subset R^{3}$ contains three embedded minimal disks meeting $\partial T$ orthogonally, obtained as conjugate surfaces of Plateau…

math.DG2026

Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface

Steven Finch

Let $Ω$ be the curvilinear square bounded by the four circular arcs $∣ ζ \mp 1∣ = 2$ , $∣ ζ \mp i ∣ = 2$ , and let $W = 1 + 14 ζ^{4} + ζ^{8}$ . Put \[ θ(ζ)=\arcsin\frac{2(1+i)ζ} {…

math.DG2026

The two cubic separable minimal surfaces: $D_{1} D_{2} D_{3} = 1$ and $C_{1} + C_{2} + C_{3} = 3 C_{1} C_{2} C_{3}$

Steven Finch

A minimal surface in $R^{3}$ is $separable$ if it is the zero set of $f (x) + g (y) + h (z)$ , and $isotropic$ if moreover $f = g = h$ . We announce that there are exac…

math.CO2026

Small Matrices with Large Inverses: Unimodular $4 \times 4$ Cases

Steven Finch

How close to singularity can an $n \times n$ unimodular matrix be? For ternary cases as $n$ increases, exact expressions are unlikely, but upon fixing $n = 4$ and assessing $(2 k + 1)$ -…

math.CO2026

Small Matrices with Small Inverses: Unimodular Zerofree Cases

Steven Finch

We consider unimodular matrices $M$ such that neither $M$ nor $M^{- 1}$ contain zero entries. Matrices typically exhibit a trade-off: small $M$ imply large $M^{- 1}$ . We investigate…

math.GM2026

$n$ th Roots of $n$ th Powers

Steven Finch

Seeking simple, efficient solutions of a matrix equation leads (quite circuitously) to optimizing unimodular zerofree matrices. Canonicalizing such matrices under signed-permutatio…

math.CA2024

Popa's "Recurrent Sequences" and Reciprocity

Steven Finch

Dumitru Popa found asymptotic expansions for certain nonlinear recurrences, but left open the numerical evaluation of associated constants. We address this issue. A change of varia…

math.CO2024

An Exceptional Convolutional Recurrence

Steven Finch

A quadratic recurrence of Faltung type, arising via ancestral path lengths of random binary trees, turns out to be related to the Painlevé I differential equation.

math.NT2024

A Deceptively Simple Quadratic Recurrence

Steven Finch

Standard techniques for treating linear recurrences no longer apply for quadratic recurrences. It is not hard to determine asymptotics for a specific parametrized model over a wide…

math.CO2024

Multisum Sets

Steven Finch

Complete infinite multisum sets are eventually linear. After 30 years of sitting in a file cabinet, the proof (thanks to James H. Schmerl) is brought from darkness into light.

Schwarz P and Schwarz D as free-boundary minimal disks in a tetragonal disphenoid: exact areas

An exact hypergeometric Weierstrass representation for Smyth's free-boundary minimal disk in the regular tetrahedron

Elimination of the parameters in an elliptic parametrization of the Schwarz primitive surface

The two cubic separable minimal surfaces: D1​D2​D3​=1 and C1​+C2​+C3​=3C1​C2​C3​

Small Matrices with Large Inverses: Unimodular 4×4 Cases

Small Matrices with Small Inverses: Unimodular Zerofree Cases

nth Roots of nth Powers

Popa's "Recurrent Sequences" and Reciprocity

An Exceptional Convolutional Recurrence

A Deceptively Simple Quadratic Recurrence

Multisum Sets

Schwarz $P$ and Schwarz $D$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas

The two cubic separable minimal surfaces: $D_{1} D_{2} D_{3} = 1$ and $C_{1} + C_{2} + C_{3} = 3 C_{1} C_{2} C_{3}$

Small Matrices with Large Inverses: Unimodular $4 \times 4$ Cases

$n$ th Roots of $n$ th Powers