Alper Ferudun

The Schrijver-Seymour Bound for Four Weight Classes

Let a finite matroid be labeled by an arbitrary abelian group, and let S be the set of its basis-weight sums. We prove the Schrijver-Seymour stabilizer bound whenever the nonloop labels occupy at most four cosets modulo the stabilizer of S. There is no restriction on the rank, gr...

Essential Periods and Monochromatic Triangles in Measurable Layered Colorings

Let f: R -> {-1,1} be measurable and color the plane by chi(x,y)=f(y). Call an equilateral side length essentially avoided when, for each fixed orientation, the set of monochromatic translations has planar measure zero. We prove that essential avoidance of side s forces the essen...

Paneitz Resonances and Index Growth on a Sphere Times a Hyperbolic Surface

For a fixed closed connected hyperbolic surface (Sigma,h) of curvature -1, we determine the full Paneitz spectrum for g_t = g_S2 plus t^(-1)h, where the round sphere has curvature 1 and t>0. The exceptional scales with nonconstant kernel form a locally finite set, with explicit r...

The Scaling-Rotation Chain Metric for Positive-Definite Two by Two Matrices

The scaling-rotation distance between positive-definite matrices minimizes the distance between their eigendecomposition fibers. Its associated chain metric also allows changes of eigenframe at repeated eigenvalues. For real two by two matrices we compute that chain metric explic...

Central Obstructions and Block-Complement Categories for Schur Covers

We study explicit ways to replace the symmetric-group automorphisms of the category of finite sets and injections by their Schur double covers. If the central involution acts freely on a finite lift fibre of an injection of codimension d, then the fibre cardinality is divisible b...

Sharp Risk Bounds and a Shrinking-Variance Transition for Normal and Gamma Fits

Suppose lognormal data are fitted by an ordinary normal model and a shape-scale gamma model. If s > 0 is the log variance, the difference Delta(s) between their optimized forward Kullback-Leibler risks satisfies 2s/3 < Delta(s) < s and 2/3 < Delta'(s) < 1; both constants are opti...

A Proper Subclass of the Separable Permutations Whose Generating Function Lies Outside Q(z, √(1−4z), √(1−6z+5z²))

Question 5 of R. Brignall's abstract (joint work with M. Bóna, C. Defant, M. Opler and V. Vatter) in Oberwolfach Report 6/2024, p. 292, asks whether every proper subclass of the separable permutations Av(2413, 3142) has a generating function in K = Q(z, √(1−4z), √(1−6z+5z²)). The...

Pedal Areas of Periodic Elliptic Billiard Trajectories with Respect to an Arbitrary Point

Reznik, Garcia and Koiller listed many quantities that are experimentally constant along the families of N-periodic billiard trajectories in an ellipse whose caustic is a confocal ellipse. We treat three of their rows on the signed areas of pedal polygons with respect to an arbit...

Antipedal Areas of N-Periodics in the Elliptic Billiard: Row k401 Fails at the Centre and Holds at the Foci

Row k401 of Reznik, Garcia and Koiller's list of experimentally found invariants of N-periodic billiard trajectories in an ellipse (record AMR-050-0018 of the UnsolvedMath collection) states that for N ≡ 2 (mod 4) and every point M the product A'A*_M is constant along the family;...

A Rank-One Obstruction for Odd-Primary Wood Equivalences

Let p be odd, h=(p-1)p^(r-1)m with r>=1 and m>1, and H=C_(p^r) a subgroup of the height-h Morava stabilizer. For Honda Morava E-theory over F_(p^h), put A=E^(hH). We prove that no finite p-local spectrum F satisfies A smash F equivalent to E as A-modules. More generally, if A sma...

Minimum Support Augmentation in Chordal Graphical Models

Given an observed support in a finite Cartesian table, we minimize the number of distinct new cells needed to make every maximal-clique projection of a chordal graphical model surjective. For a clique path, the missing clique configurations form a partially ordered set, and the m...

Half-Angle Sines, Steiner Pedal Areas and Evolute Areas of N-Periodics in the Elliptic Billiard

Reznik, Garcia and Koiller listed many quantities that they found experimentally to be constant along the families of N-periodic billiard trajectories in an ellipse with a confocal elliptic caustic. We treat four rows of their tables (records AMR-050-0002, -0028, -0039 and -0032...

Exact Positive-Focal Certificates for Fundamental-Matrix Pencils

For an independent rational 3-by-3 matrix pencil, we give an exact procedure that counts its real projective rank-two members compatible with two positive finite focal lengths, or reports that there are infinitely many. Principal points are known, skew is zero and aspect ratio is...

An Exact Singular-Locus Formula for Rooted-Tree Compartmental Models

We compute the coefficient-map Jacobian determinant of every bidirected-tree linear compartmental model whose sole input, output and leak are at one root. The determinant is a product of explicit edge powers and squared resultants of characteristic polynomials of sibling branches...

Brauer-Picard Groups and Module Orbits for Twisted Cyclic Categories

For every cyclic group C_n and every associator class omega_t, this preprint explicitly computes the Brauer-Picard group of Vec_(C_n)^(omega_t) and its action on all indecomposable module categories. Module labels are the divisors of gcd(n,t). For p^k exactly dividing n and s=min...

Fare Zone Assignment with Hard Cut Budgets under Bounded Congestion

At the Oberwolfach workshop on Combinatorial Optimization in November 2024, Peis, with Pabst, Kauther and Fiorini, posed the following problem (Oberwolfach Report 50/2024, p. 2992). Commodities on a tree have paths, weights and budgets. A set of tree edges, the zone borders, is t...

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)...

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...

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...

On Conjecture 1 of Aldous for a Spatial Model of City Growth: A Proof of Part (a), and Counterexamples to Parts (b) and (c) for 2α/√3 < β < 2α

In rough notes of 2007 on a spatial model of city growth, D. J. Aldous stated a three-part conjecture about the unbalanced regime. Immigrants arrive one at a time at uniform random points of the unit square; a city of population n exerts the influence c0 n^α r^(−β) at distance r;...