Alper Ferudun

Riemann Surfaces of Genus 9 with Nilpotent Automorphism Group of the Largest Order Are Not Unique: On a Question in Farb's Problem 2.19

Problem 2.19 of Farb's list of problems on mapping class groups asks for properties of automorphism groups that determine a unique point of the moduli space M_g, and asks as an example whether there is a unique Riemann surface of genus g≥2 whose automorphism group is nilpotent an...

Containment of the Symbolic Cube in the Square for the Roulleau–Urzúa Configurations of Cubics

In an Oberwolfach report of 2015 Harbourne considered the configurations C_n of 4(n² − 3)/3 plane cubics due to Roulleau and Urzúa, defined when 3 divides n, and asked whether I^(3) ⊄ I² for the ideal I of the singular points of C_n. He reported that for C_3 the containment I^(3)...

Geometric Sampling Does Not Determine Regularized R-Diagonal Laws

For every s0>0, q>1 and finite integer K>=1, we construct a continuum of distinct affiliated R-diagonal laws having the same joint star-moments and the same first K alternating cumulants of c(1+s c*c)^(-1/2) at every scale s=s0 q^k, k in Z. All absolute moments are finite. A fini...

A Sharp Prior-Sensitivity Threshold for Fixed-Partition Support Boundaries

Consider a Poisson support-boundary experiment with intensity scale n, a known partition of [0,1] into K bins of widths Delta_j, and independent prior densities on the bin heights. For a fixed positive prior whose logarithm is locally Lipschitz, we prove that the posterior is asy...

The Entropy Cost of Infinite-Genus Weil-Petersson Local Limits

Let mu_g be normalized Weil-Petersson measure on the moduli space of closed hyperbolic surfaces of genus g. If surface laws nu_g, with arbitrary measurable rooting kernels, have a pointed smooth weak limit Q, then for every fixed h >= 1 we prove liminf H(nu_g | mu_g)/log g >= (2h...

Binary Interior Excision and Nonuniqueness of Graph Homology

We give an explicit nonuniqueness example for the (J1, categorical-product) Eilenberg-Steenrod axioms on Cech closure spaces. Applying the known clique and closed neighborhood complex constructions to the mutual singleton graph gives two integral relative homology theories, with...

Euclidean Realizability of Fundamental-Polytope Types

For a finite metric d, its fundamental polytope is the convex hull of the vectors (e_i-e_j)/d_ij. We answer the universality clause of Vershik's question about Euclidean metrics: not every abstract combinatorial type is Euclidean. The four-cycle metric gives a cube, whereas four...

Constant Normalized Area in the Planar Three-Body Problem

We prove that a collision-free noncollinear planar Newtonian three-body motion with constant triangle area divided by barycentric moment of inertia is necessarily a Lagrange homographic motion. The masses are arbitrary positive numbers, and the conclusion holds on any open time i...

Largest Acylindrical Actions for Artin Groups under a Strict Triple Condition

Let A_S be a finitely generated Artin group such that the sum of the reciprocals of the three labels on every triple of generators is strictly less than one, with 1/infinity = 0. This note records a direct consequence of the Martin-Przytycki coned-off Deligne construction and the...

Hive Polytopes with Arbitrary Integral Right-Hand Sides Have Integral Points Exactly up to Rank 5: A Counterexample to a Conjecture of De Loera and McAllister

Knutson and Tao proved that a nonempty hive polytope has an integral vertex. De Loera and McAllister conjectured that this extends to arbitrary integral right-hand sides: if B and R are the matrices of the boundary equations and of the rhombus inequalities of hive patterns of ran...

A Cayley Graph with Chromatic Number Three and No Invariant Random Three-Coloring

We construct a connected seven-regular Cayley graph of (C5 * (C2 x C2)) x C2 with chromatic number three but no translation-invariant random proper three-coloring. A central involution partitions two finite subgroup graphs into pairs. Every proper three-coloring has at least four...

Eventual Scalar Recurrences for Constrained Orthogonal Polynomials

Let S be a finite-codimensional subspace of R[x] with a positive definite moment inner product, and let q_n be its degree-ordered monic orthogonal basis. For a fixed nonzero polynomial a of degree d, we prove that an eventual scalar band recurrence for a q_n exists if and only if...

On Hayman's Problem on the Series Σ f(c_n/a_n) Formed with Second Partial Sums: A Comparison Theorem and a Characterization

Problem 7.31 of Hayman and Lingham's collection Research Problems in Function Theory, proposed by W. K. Hayman, concerns sequences with a_1 > 0 and 0 ≤ a_n ≤ n and their partial sums b_n = a_1 + … + a_n, c_n = b_1 + … + b_n. It states that Σ (a_n/c_n)^α < ∞ for α > 1/2, and asks...

Independent Arborescences in Acyclic Digraphs: The Path Condition Suffices for All Roots and All Convex Sets

Let D be a finite acyclic digraph with root-nodes r_1, ..., r_k, not necessarily distinct, and convex node sets U_1, ..., U_k with r_i ∈ U_i. The EGRES Open collection lists as open the question, proposed by S. Fujishige, N. Kamiyama and N. Katoh, whether there are independent r_...

Sharp Degree Bounds for Multipartite Lorentzian Coxeter Forms

Partition at least four vertices into at least two nonempty parts. Assign Coxeter exponent 2 within each part, a fixed exponent m >= 5 between parts, and replace selected cross-part exponents by 3. Suppose each vertex has at most one replacement neighbor in any other part. If the...

Symmetric Unimodality for Real Free Convolution Powers

We derive preservation of symmetric weak unimodality under every real free additive convolution power t >= 1, without support, moment or density assumptions. The essential input is the Cauchy-transform inequality in Anonymous's unrefereed preprint Symmetric unimodality under free...

Linear Gaussian SURE Bias for Haar Soft Thresholding under Smooth Noise

We give a fixed, smooth, sub-Gaussian noise distribution for which the bias of the Gaussian Stein unbiased risk estimate does not vanish after division by dimension, even for ordinary Haar soft thresholding. For every dyadic dimension d >= 2, threshold 1/8 and signal constant on...

Sharp Attraction Thresholds for Neutral Jordan Skew Products

For a class of holomorphic skew products tangent to the identity, we give necessary and sufficient conditions for an open attracting domain and for an open domain attracted along the base axis. The transverse leading coefficient is a fixed matrix with real spectrum and arbitrary...

Expansive Symmetry Breaking at a Five-Vertex Linkage Singularity

We give an explicit planar framework with five distinct vertices spanning the plane and six bars whose locally one-dimensional configuration set has two analytic branches at a reflection-symmetric configuration. Both outgoing branches are expansive and immediately break the refle...

On a Determinant of Teufl and Wagner Related to Set Partitions: Divisibility by the Catalan Power of det A

Teufl and Wagner posed the following problem (Problem 1 of their contribution to Oberwolfach Report 23/2018, pp. 1457–1458). Let A = (a_ij) be an n × n matrix of indeterminates, and let T = T(A) be the matrix, indexed by the set partitions of an n-set, whose entry (P, Q) is the s...