Alper Ferudun

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...

Radial Barriers for Discrete Newtonian Potentials: Counterexamples and Sufficient Conditions

For a locally finite configuration of positive masses in R^n, n >= 3, we study the existence of spheres on which the Newtonian potential is uniformly small. Unit masses at the positive integers contradict the dimensional extension of an auxiliary lemma of Clunie, Eremenko and Ros...

The Elements with Property (s) of a Skew Brace Need Not Form an Ideal: On a Question of Colazzo, Ferrara and Trombetti

An element x of a skew brace B has property (s), in the sense of Colazzo, Ferrara and Trombetti, if it has only finitely many conjugates of each of the kinds x∗b, b∗x, b+x−b and b∘x∘b⁻¹. In her abstract in Oberwolfach Report 9/2023, Colazzo asks whether the elements with property...

On Reznick's Question about the Cone of Forms with a Sum-of-Squares Odd Power: Explicit Non-Convexity for Sextics in Five and Quartics in Seven Variables

For an even degree m and an odd exponent 2k+1, let Σ_{n,m}(2k+1) be the set of real forms f of degree m in n variables such that f^(2k+1) is a sum of squares of forms. B. Reznick asked (Oberwolfach Report 14/2023, p. 779) whether this set is a closed convex cone. It is a closed c...

A Negative Answer to a Question of Praeger and Seress on Minimal Subdegrees of Primitive Groups of Twisted Wreath Type

Let G be a finite primitive permutation group of twisted wreath type, with socle T^k and point stabiliser P, and let H be its primitive component: the group induced on one coordinate of the product decomposition Δ^k, where |Δ| = |T|. Praeger and Seress (Oberwolfach Report 12/2005...

Unbounded Multiplicity in the Dilatation Spectrum of Every Term of the Johnson Filtration: An Observation on a Question of Farb

Farb asked (Question 7.6 of "Some problems on mapping class groups and moduli space") whether spec(I_g(k)), the set of the logarithms of the dilatations of the pseudo-Anosov elements of the k-th term of the Johnson filtration of the mapping class group Mod_g, has bounded multipli...

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_...

Exact Degrees for a Fixed-Double-Pole Rational Hénon Family

We give an all-parameter degree calculation for the birational maps F(x,y)=(beta/x+gamma/x^2-delta*y,x) over the complex numbers, with gamma*delta nonzero. Outside the nontrivial root-of-unity resonances of -delta^3, the first dynamical degree is 2. At a resonance of order ell, t...

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...

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...

Cyclic Vertex Groups and Integral Toric Restriction

Let K be the dual boundary complex of a compact simple polytope, and let an integral matrix define a coordinate-torus quotient, with nonsingular restriction at each vertex. We show that restriction from coordinate-torus equivariant cohomology to the kernel subgroup is surjective...

On a Problem of Ferrand and Stoimenow Answered in Print by Rutherford and Ito: A Sharper Signature Bound for the HOMFLY Polynomial of Alternating Links, and a Short Proof that Quasipositive Alternating Links Are Positive

Problem 1.16 of Ohtsuki's list (E. Ferrand, A. Stoimenow) asks whether every alternating link L satisfies σ(L) ≥ min deg_l P_L ≥ min deg_a F_L(a^-1, z), where σ is the signature, P_L(l, m) the HOMFLY polynomial and F_L(a, z) the Kauffman polynomial. Both inequalities are in print...

A Sharp Three-Cycle Bound for Two-Zone Affine Systems with a Virtual Focus

We prove that a planar two-zone piecewise-affine system with a straight switching line, two strict off-line foci, and at least one virtual focus has at most three isolated crossing limit cycles. The estimate includes nonhyperbolic cycles and every switching offset. Known three-cy...

A Gaussian Obstruction to Uniform Mesh Bounds for Differential Operators

We classify the nonzero finite-order differential operators with real polynomial coefficients that preserve hyperbolicity and retain a positive uniform fraction of additive zero mesh on the whole Laguerre-Polya class: they are exactly the nonzero scalar multiples of the identity....

The Word-Search Constants C_2(ABBB) = 8/5 and C_3(ABB) = 6: On Two Questions of Schildkraut and of Halberstam and Schildkraut

For a word w and a toroidal grid of letters in dimension d, an appearance of w is a cell together with one of the 3^d − 1 directions in {−1,0,1}^d ∖ {0} along which the letters spell w, and C_d(w) is the supremum, over all grids, of the number of appearances per cell. We prove th...

A Sharp Three-Fifths Bound for Seymour Vertices in Three-Regular Oriented Graphs

Let D be a finite simple oriented graph with indegree and outdegree three at every vertex. A Seymour vertex has at least as many exact second out-neighbors as first out-neighbors. We prove that at least ceil(3|V(D)|/5) vertices are Seymour vertices. If B is the set of the remaini...

Line Configurations in P^3 Whose Symbolic Square Has Initial Degree α(I)+1 Are Coplanar or Pseudostars: A Negative Answer to a Question of Janssen

For a finite union L of lines in P^3 with homogeneous ideal I, let α(I) be the least degree of a non-zero form in I, and let I^(2) be the second symbolic power of I, the ideal of the forms that vanish to order at least two along every line of L; one always has α(I^(2)) ≥ α(I) + 1...

A Nonremovable Meromorphic Map into a Ball on a Codimension-Two Wedge

We construct a meromorphic germ in C^3 that is holomorphic from an attached wedge into the open unit ball, extends continuously to its real-algebraic generic edge with values on the unit sphere, and nevertheless has no holomorphic extension to a full neighborhood of the edge poin...

Polygons with Prescribed Turning Angles in Three Dimensions: A Counterexample to a Conjecture of Efrat, Fulek, Kobourov and Tóth, and the Corrected Criterion

A closed polygon in R^3 has at each vertex a turning angle in (0, π), and a sequence A = (α_0, ..., α_{n−1}) in (0, π)^n is called realizable if it is the sequence of turning angles of such a polygon. Efrat, Fulek, Kobourov and Tóth conjectured (Oberwolfach Rep. 17 (2020); J. Gra...

Geometric and Arithmetic Independence of Affine Chebyshev Towers

We determine joint Galois images for finite families of normalized Chebyshev maps with nonconstant affine targets over function fields. Over C(t), endpoint graphs and degree parity determine the geometric image; least common multiples merge repeated endpoint pairs, while greatest...