Sungsoo Na

Coalition wins in multiplayer k-in-a-row

In cyclic k-in-a-row on Z2 with p ≥ 3 players, a player has a winning strategy against all the others exactly when it moves first and either k = 1, or k = 2 and p ≤ 8. Against every other player, the target, the others force a win of one of them. When 2p > k they win in round k,...

When does a random disk first overlap an earlier one? An expansion of the geometric birthday problem

The expected index of the first disk that overlaps an earlier one, when disks of diameter σ are placed independently and uniformly with centres in a disk of radius R, has the formal expansion √(π/(2q)) + 2/3 + √3/(4π) + X√q + c2q in the pair-overlap probability q, whose constant...

The fifth virial coefficient of hard disks

We determine the fifth virial coefficient of the hard-disk fluid. We prove that B5/B24 = 140/81 − 101√3/(9π) + 1169/(18π2) − 71√3/(2π3) + 4P1/π3, where P1 is a two-dimensional integral of squared triple-overlap areas of unit disks, and that B5/B24 is an explicit expression in two...

The fifth virial coefficient of hard disks

We determine the fifth virial coefficient of the hard-disk fluid. We prove that B5/B24 = 140/81 − 101√3/(9π) + 1169/(18π2) − 71√3/(2π3) + 4P1/π3, where P1 is a two-dimensional integral of squared triple-overlap areas of unit disks, and that B5/B24 is an explicit expression in two...

Equidistribution of quadratic roots in ray classes and Crocker's irreducible primes

Let O = Z[ω], ω = (−1 + i√3)/2. For a prime ideal 𝔭 not dividing 6, of residue degree one and norm q, let t𝔭 ∈ [0, q) be the root of 36T2 + 6T + 1 modulo q determined by 6t𝔭 ≡ ω (mod 𝔭). We prove that for any fixed nonzero integral ideal 𝔪 of O, any finite-order ray class charact...

The least self-power root of a prime

For a prime q let f(q) = min{n ≥ 2 : q ∣ nn − 1} be its least self-power root. Writing V(X) for the number of its values in [2, X], we prove an explicit inequality giving V(X) ≥ X1/2−ε and, from the Baker–Harman theorem on shifted primes, V(X) ≥ X0.677−o(1). We explain why the fi...

Exact self-tilings of convex polygons: constructions and angle obstructions

Every convex trapezoid and every convex quadrilateral with two opposite right angles admits an exact countably infinite tiling by strictly smaller similar closed copies. In contrast, quadrilaterals whose cyclic angles have only the total-angle integer relation admit no finite or...

Finite metric fibres for exact self-tilings of convex pentagons

For every realizable cyclic pentagon angle tuple admitting no nonnegative integer combination equal to π, we prove that only finitely many similarity classes satisfy the side-length relations forced by exact self-tiling. The relations arise from full vertex stars in a locally fin...

Maximal triangles in space-form balls and simplices in Euclidean balls with prescribed vertex centroid

We determine maximum-volume simplices with prescribed interior vertex centroid in Euclidean balls and maximum-area triangles in geodesic balls of simply connected space forms. In full Euclidean dimension at least two, the maximizers are kites: a regular (n−1)-simplex base with an...

Length profiles of graphs in expanding squares: an exact realization theorem

Which functions describe the Euclidean length of a continuous graph inside every square centered at the origin? We characterize the finite continuous profiles that vanish at zero and are realized by continuous graphs on the real line through the origin: subtracting the square's s...

Maximal triangles in space-form balls and simplices in Euclidean balls with prescribed vertex centroid

We determine maximum-volume simplices with prescribed interior vertex centroid in Euclidean balls and maximum-area triangles in geodesic balls of simply connected space forms. In full Euclidean dimension at least two, the maximizers are kites: a regular (n−1)-simplex base with an...

Translation-invariant Fourier covariance and path-correlated codeword search

Translation-invariant second moments on a finite abelian group have diagonal Fourier spectrum. We observe that this standard diagonalization equates the expected coherent bad mass after subgroup addition with the ordinary bad mass, for any fixed subgroup and bad-error set. The id...

Translation-invariant Fourier covariance and path-correlated codeword search

Translation-invariant second moments on a finite abelian group have diagonal Fourier spectrum. We observe that this standard diagonalization equates the expected coherent bad mass after subgroup addition with the ordinary bad mass, for any fixed subgroup and bad-error set. The id...

Sierksma's conjecture for nine points in R³

We prove Sierksma's conjecture for nine points in R³: every nine points of R³, not necessarily distinct, have at least eight Tverberg partitions into three parts, to our knowledge the first case in dimension three. The proof combines equivariant topology (Dold's theorem on twiste...

Sierksma's conjecture for nine points in R³

We prove Sierksma's conjecture for nine points in R³: every nine points of R³ have at least eight Tverberg partitions into three parts, to our knowledge the first case in dimension three. The proof combines equivariant topology (Dold's theorem on twisted constraint complexes), a...

A finite-data criterion for Feigenbaum convergence in real-analytic unimodal families

We give a finite-data criterion for Feigenbaum parameter scaling in real-analytic unimodal families with a quadratic critical point. Implementation requires supplied holomorphic domains, outward evaluators, mixed jets, and Cauchy majorants, not just four callbacks. The hypotheses...

A finite-data criterion for Feigenbaum convergence in real-analytic unimodal families

We give a finite-data criterion for Feigenbaum parameter scaling in real-analytic unimodal families with a quadratic critical point. Implementation requires supplied holomorphic domains, outward evaluators, mixed jets, and Cauchy majorants, not just four callbacks. The hypotheses...

Sierksma's conjecture for nine points in R³

We prove Sierksma's conjecture for nine points in R³: every nine points of R³ have at least eight Tverberg partitions into three parts, to our knowledge the first case in dimension three. The proof combines equivariant topology (Dold's theorem on twisted pair-constraint complexes...

Attracting fixed points of Dedekind zeta functions

Every Dedekind zeta function has a unique fixed point on $(1,\infty)$, but its stability and real two-cycles depend on arithmetic data. We give certified attracting examples in quadratic, cyclic cubic, cyclic quartic and cyclotomic fields. In contrast, infinitely many real and im...

Attracting fixed points of Dedekind zeta functions

Every Dedekind zeta function has a unique fixed point on $(1,\infty)$, but its stability and real two-cycles depend on arithmetic data. We give certified attracting examples in quadratic, cyclic cubic, cyclic quartic and cyclotomic fields. In contrast, infinitely many real and im...