Conjecture Of Hill Research Articles

Operadic lifts of the algebra of indexing systems

Abstract For a given finite group G , the homotopy category of N ∞ G -operads is equivalent to a finite lattice , and G varies, there are various image constructions between these lattices. In this paper, we explain how to lift this algebraic structure back to the operad level. We show that lattice joins and meets correspond to N ∞ coproducts and products, and we show that the image constructions correspond to N ∞ induction, restriction, and coinduction constructions, at least when taken along an injective homomorphism. We also prove that a N ∞ variant of the Boardman-Vogt tensor product lifts the join. Our result does not resolve Blumberg and Hill's conjecture that the ordinary tensor product of suitably cofibrant N ∞ operads models the join, but it does imply a closely related result. If O and P are operads, then an algebra over the Boardman-Vogt tensor product O ⊗ P is equipped with a pair of interchanging O and P -actions. We prove that under mild hypotheses on a N ∞ operad O , every orthogonal O -ring spectrum is weakly equivalent to a spectrum over an operad O ′ ≃ O that interchanges with itself.

Drawings of complete graphs in the projective plane

AbstractHill's Conjecture states that the crossing number of the complete graph in the plane (equivalently, the sphere) is . Moon proved that the expected number of crossings in a spherical drawing in which the points are randomly distributed and joined by geodesics is precisely , thus matching asymptotically the conjectured value of . Let denote the crossing number of a graph in the projective plane. Recently, Elkies proved that the expected number of crossings in a naturally defined random projective plane drawing of is . In analogy with the relation of Moon's result to Hill's conjecture, Elkies asked if . We construct drawings of in the projective plane that disprove this.

The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

A bipartite-cylindrical drawing of the complete bipartite graph $$ K_{m,n} $$ is a drawing on the surface of a cylinder, where the vertices are placed on the boundaries of the cylinder, one vertex-partition per boundary, and the edges do not cross the boundaries. The bipartite-cylindrical crossing number of $$ K_{m,n}$$, denoted by $$ cr_\circledcirc (K_{m,n}) $$, is the minimum number of crossings among all bipartite-cylindrical drawings of $$ K_{m,n} $$. This problem is equivalent to minimizing the number of crossings in drawings of $$ K_{m,n} $$ in the plane where each of the vertex classes in the bipartition is placed on a circle and the edges do not cross the two circles. We determine $$ cr_\circledcirc (K_{m,n}) $$ and give explicit constructions that achieve this number. This in turn gives an alternative proof to the Harary–Hill Conjecture restricted to a subclass of cylindrical drawings of the complete graph.

Closing in on Hill's Conjecture

Borrowing L\'aszl\'o Sz\'ekely's lively expression, we show that Hill's conjecture is "asymptotically at least 98.5% true". This long-standing conjecture states that the crossing number cr($K_n$) of the complete graph $K_n$ is $H(n) := \frac{1}{4}\lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor \lfloor \frac{n-2}{2}\rfloor \lfloor\frac{n-3}{2}\rfloor$, for all $n\ge 3$. This has been verified only for $n\le 12$. Using flag algebras, Norin and Zwols obtained the best known asymptotic lower bound for the crossing number of complete bipartite graphs, from which it follows that for every sufficiently large $n$, cr$(K_n) > 0.905\, H(n)$. Also using flag algebras, we prove that asymptotically cr$(K_n)$ is at least $0.985\, H(n)$. We also show that the spherical geodesic crossing number of $K_n$ is asymptotically at least $0.996\, H(n)$.

The Crossing Number of the Cone of a Graph

Motivated by a problem asked by Richter and by the long standing Harary--Hill conjecture, we study the relation between the crossing number of a graph $G$ and the crossing number of its cone $CG$, the graph obtained from $G$ by adding a new vertex adjacent to all the vertices in $G$. Simple examples show that the difference $cr(CG)-cr(G)$ can be arbitrarily large for any fixed $k=cr(G)$. In this work, we are interested in finding the smallest possible difference; that is, for each nonnegative integer $k$, find the smallest $f(k)$ for which there exists a graph with crossing number at least $k$ and cone with crossing number $f(k)$. For small values of $k$, we give exact values of $f(k)$ when the problem is restricted to simple graphs and show that $f(k)=k+\Theta (\sqrt {k})$ when multiple edges are allowed.

Bishellable drawings of $K_n$

The Harary--Hill conjecture, still open after more than 50 years, asserts that the crossing number of the complete graph $K_n$ is \(H(n) := \frac 1 4 lfloor\fracn2\rfloor lfloor\fracn-12\rfloor lfl...

Shellable Drawings and the Cylindrical Crossing Number of $$K_{n}$$ K n

The Harary---Hill Conjecture states that the number of crossings in any drawing of the complete graph $$K_n$$Kn in the plane is at least $$Z(n):=\frac{1}{4}\left\lfloor \frac{n}{2}\right\rfloor \left\lfloor \frac{n-1}{2}\right\rfloor \left\lfloor \frac{n-2}{2}\right\rfloor \left\lfloor \frac{n-3}{2}\right\rfloor $$Z(n):=14n2n-12n-22n-32. In this paper, we settle the Harary---Hill conjecture for shellable drawings. We say that a drawing $$D$$D of $$K_n$$Kn is $$s$$s-shellable if there exist a subset $$S = \{v_1,v_2,\ldots , v_ s\}$$S={v1,v2,?,vs} of the vertices and a region $$R$$R of $$D$$D with the following property: For all $$1 \le i < j \le s$$1≤i

On the slice spectral sequence

We introduce a variant of the slice spectral sequence which uses only regular slice cells, and state the precise relationship between the two spectral sequences. We analyze how the slice filtration of an equivariant spectrum that is concentrated over a normal subgroup is related to the slice filtration of its geometric fixed points, and use this to prove a conjecture of Hill on the slice filtration of an Eilenberg MacLane spectrum. We also show how the (co)connectivity of a spectrum results in the (co)connectivity of its slice tower, demonstrating the "efficiency" of the slice spectral sequence.

Cartan invariants of symmetric groups and Iwahori-Hecke algebras

Külshammer, Olsson and Robinson conjectured that a certain set of numbers determined the invariant factors of the ℓ-Cartan matrix for Sn (equivalently, the invariant factors of the Cartan matrix for the Iwahori–Hecke algebra ℋn(q), where q is a primitive ℓth root of unity). We call these invariant factors Cartan invariants. In a previous paper, the second author calculated these Cartan invariants when ℓ=pr, p is prime and r⩽p, and went on to conjecture that the formulae should hold for all r. Another result was obtained, which is surprising and counterintuitive from a block theoretic point of view. Namely, given the prime decomposition ℓ = p 1 r 1 ⋅ … ⋅ p k r k , the Cartan matrix of an ℓ-block of Sn is a product of Cartan matrices associated to p i r i -blocks of Sn. In particular, the invariant factors of the Cartan matrix associated to an ℓ-block of Sn can be recovered from the Cartan matrices associated to the p i r i -blocks. In this paper, we formulate an explicit combinatorial determination of the Cartan invariants of Sn, not only for the full Cartan matrix, but also for an individual block. We collect evidence for this conjecture by showing that the formulae predict the correct determinant of the ℓ-Cartan matrix. We then go on to show that Hill's conjecture implies the conjecture of Külshammer, Olsson and Robinson.

Minimax Risk Inequalities for the Location–Parameter Classification Problem

Minimax risk inequalities are obtained for the location-parameter classification problem. For the classical single observation case with continuous distributions, best possible bounds are given in terms of their Lévy concentration, establishing a conjecture of Hill and Tong (1989). In addition, sharp bounds for the minimax risk are derived for the multiple (i.i.d.) observations case, based on the tail concentration and the Lévy concentration. Some fairly sharp bounds for discontinuous distributions are also obtained.

The gliding humps technique for 𝐹𝐾-spaces

The gliging humps technique has been used by various authors to establish the existence of bounded divergent sequences in certain summability domains. The purpose of this paper is to extend these results and to obtain analogous ones for sequence spaces other than c and m. This serves to unify and improve many known results and to obtain several new ones—applications include extensions to theorems of Dawson, Lorentz-Zeller, Snyder-Wilansky and Yurimyae. Improving another result of Wilansky allows us to consider countable collections of sequence spaces—applications including the proof of a conjecture of Hill and Sledd and extensions to theorems of Berg and Brudno. A related result of Petersen is also considered and a simple proof using the Baire category theorem is given.

A Note on Convergence Fields

The purpose of this note is to show that the (bounded) convergence field of a conservative matrix is closed under a certain diagonalization procedure.As an application of the above result we establish a conjecture of Hill and Sledd in (1) and obtain a result of Lorentz originally proved in (2).First we introduce some notation and definitions, most of which are standard. Let l∞ denote the Banach space of bounded sequences with the supremum norm and let c denote the closed subspace of l∞ consisting of convergent sequences.