Kriebel Alec

A telescoping verification of a focal-inversion area invariant

For a proper elliptical billiard with a fixed strictly nested nondegenerate confocal elliptical caustic, the original signed shoelace area times the signed area of the straight polygon obtained by unit inversion of its vertices about either original focus is a positive phase-inde...

A simple brace outside asymmetric products of two trivial braces

A full negative answer to Cedo Problem 9 in Oberwolfach Report 51/2019, printed p.3228 (problem 30004311; OWR-17294-011). An explicit nontrivial simple left brace of order 20,000 has additive group C5^4 x C2^5 and metabelian multiplicative group (C5^3 semidirect C2) x (C2^4 semid...

A negative-curvature normal barrier on an algebraic cone that is not a hyperbolicity cone

An explicit affirmative answer to Tunçel's 2017 existential problem: the regular algebraic convex cone K={x,y,z≥0:xy²≥z³} has the 100-normal barrier F=-25 log(z(xy²-z³)) with D³F(a)[h,u,u]≤0 for every interior base point a, every cone direction h and every real direction u, but K...

A three-point counterexample to a random lattice discrepancy comparison

A full negative answer to the universal expected p-th-power comparison posed by Novak, reporting joint work with Hinrichs, in Oberwolfach Report 13/2004, printed p.698 (problem 30000052; OWR-723-005). For independent continuous uniform generator and shift, three one-dimensional i...

A two-tail obstruction to ambient conjugation in Dlab groups

We give an affirmative answer to the corrected Kourovka Notebook Problem 21.149. The compact-support Dlab group with slopes in the powers of two has an order automorphism that cannot be implemented by conjugation after any embedding into any standard real Dlab group. The ambient...

Vanishing of the cellularization map for non-p-nilpotent component groups

For every prime p, every field k of characteristic p, and every compact Lie group G whose component group is not p-nilpotent, the canonical k-cellularization map of C_*(Omega((BG)^p-completed);k) induces zero on every homotopy group. This gives an affirmative answer to Greenlees'...

Negative answers to the printed totally mixed face questions

A full-dimensional lattice-polytopal counterexample to both unrestricted assertions in Sam Payne's printed 2019 totally mixed face-poset question. For the standard four-simplex D and I=[0,1], the Minkowski sum of D×I and (-D)×I has a positive-dimensional, dimension-additive face...

A polynomial versus superpolynomial gap in vertex-distribution-free graph testing

A full negative answer to Gishboliner–Shapira's comparison question in Oberwolfach Report 19/2019, Problem 5 (printed p.1139), and arXiv:1905.09903v5, Problem 5.6. The single fixed hereditary, extendable property consists of K4-free graphs becoming 3-colorable after deletion of a...

Verification of the odd-period focal inversion area product

Attributed verification/research note giving a self-contained flag-curve proof of the historical odd-period focal inversion signed-area product: k903,a in Reznik--Garcia--Koiller arXiv:2004.12497v11 and Conjecture 1 of Reznik--Garcia--Helman arXiv:2012.03020v2. Alper Ferudun, Pro...

An effective criterion for prescribed block-code extensions

For a specified onto block self-map f of a nonempty two-sided irreducible sofic shift T, this note gives a complete finite computable criterion for extending that same map into T from a containing shift of finite type. It answers the specified-map question in Boyle's Problem 16.3...

A projective-line quotient of the Rogalski–Sierra algebra

For the exact six-relation presentation R(alpha,beta) printed in the 2010 Oberwolfach report and 2012 Rogalski–Sierra article, the linear ideal I=(x3-x1,x4-x2) is nonzero, homogeneous and completely prime, strictly below the augmentation ideal. The exact quotient is the projectiv...

A flag-curve proof of an original-side pedal-area invariant

For a noncircular ellipse and a proper nested confocal elliptical caustic whose primitive least billiard period is divisible by four, the signed original trajectory area times the signed area of its original-side-line pedal polygon is independent of phase for every fixed finite r...

A parity-selected endpoint theorem for the regular dodecahedron

This note gives two precise answers associated with Fuchs Conjecture 3.2. The imported bare endpoint-angle condition is false: exactly the two proper face diagonals per initial face and vertex are additional paths and have graph distance two. An explicitly adopted parity-selected...

An interior second-order defect from a Hele-Shaw contact arc

A detailed attributed proof/application of Ross–Witt Nyström's smooth Hele-Shaw contact-arc construction. The associated bounded weak homogeneous complex Monge–Ampère solution on P1 times the disc lacks a second-order real Taylor expansion at a genuine interior point, despite smo...

A cone-angle proof of Fuchs's labelled length-ratio conjecture

This research note gives a cone-angle and classical Veech-cusp proof of Fuchs's Conjecture 2.7: the lengths of source-parallel first-vertex billiard trajectories in every regular n-gon, n≥3, have the canonical labelled sine ratio. The proof includes sides, odd and even double pol...

A counterexample to ideal closure of property-(s) elements in skew braces

This concise research note gives an explicit counterexample to Colazzo's unrestricted question asking whether the elements with property (s) in a skew left brace form an ideal. On the abelian additive group Z x Z/4Z, a standard semidirect multiplication gives exactly S = Z x {0,1...

Nonunique nearest stable complex quadratics

For every fixed positive-definite real coefficient inner product, unstable complex quadratics with at least two distinct attained global nearest closed-left-half-plane stable quadratics occur arbitrarily close to z^2, both in monic coefficient space and in the full exact-degree c...

Radial orthogonality does not imply dual 1-conformal flatness

We give an explicit negative answer to Takashi Kurose's Question 3(e) in Furuhata, Matsuzoe and Urakawa, Open Problems in Affine Differential Geometry and Related Topics (1998), p.126, indexed as AMR-059-0011 / 6000011. On the product S^2(1) x R with its product metric and Levi-C...

Radial orthogonality does not imply dual 1-conformal flatness

We give an explicit negative answer to Takashi Kurose's Question 3(e) in Furuhata, Matsuzoe and Urakawa, Open Problems in Affine Differential Geometry and Related Topics (1998), p.126, indexed as AMR-059-0011 / 6000011. On the product S^2(1) x R with its product metric and Levi-C...