Syneos Health (South Korea)

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...

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...

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...

A Schwarzian inequality for the Riemann zeta function and its real dynamics

We prove a two-sided moment inequality for the Riemann zeta function: for $A_j(s)=\sum_{n\ge2}(\log n)^j n^{-s}$, one has $A_2(s)^2< A_1(s)A_3(s)<\tfrac32 A_2(s)^2$ for every $s>1$. The upper bound is equivalent to negativity of the Schwarzian derivative, whose endpoint limits ar...

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...

The number of regions determined by all (d-1)-simplices spanned by a point set in R^d

Let S be a set of n points in strong general position in R^d, d ≥ 2, and let Σ_d be the union of the C(n,d) closed (d−1)-simplices spanned by S. We study R_d(S), the number of connected components of int conv S \ Σ_d, extending the exact theory available for d = 3. The cells of t...

Attracting fixed points of Dedekind zeta functions

Every Dedekind zeta function has a unique fixed point on (1, infinity), but this point need not be repelling. We give certified attracting examples in a real quadratic field and four abelian fields, each with a certified repelling real two-cycle. Two arithmetic limits explain the...

The number of regions determined by all (d-1)-simplices spanned by a point set in R^d

Let S be a set of n points in strong general position in R^d, d ≥ 2, and let Σ_d be the union of the C(n,d) closed (d−1)-simplices spanned by S. We study R_d(S), the number of connected components of int conv S \ Σ_d, extending the exact theory available for d = 3. The cells of t...

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...

A Schwarzian inequality for the Riemann zeta function and its real dynamics

We prove a two-sided moment inequality for the Riemann zeta function: for $A_j(s)=\sum_{n\ge2}(\log n)^j n^{-s}$, one has $A_2(s)^2< A_1(s)A_3(s)<\tfrac32 A_2(s)^2$ for every $s>1$. The upper bound is equivalent to negativity of the Schwarzian derivative, whose endpoint limits ar...