Related papers: Counterexample to a conjecture about dihedral quan…

A conjecture of Verstra\"ete states that for any fixed $\ell < k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\ell}$-free subgraph with at least $c |E(G)|$ edges. For $\ell = 2$, this…

We show that there is an unique connected quandle of order twice an odd prime number greater than 3. It has order 10 and is isomorphic to the conjugacy class of transpositions in the symmetric group of degree 5. This establishes a…

We use a weight-preserving, sign-reversing involution to find a combinatorial expansion of $\Delta_{e_k} e_n$ at $q=1$ in terms of the elementary symmetric function basis. We then use a weight-preserving bijection to prove the Delta…

Problem 4.19 in Ziegler's "Lectures on Polytopes" asserts that every simple $3$-dimensional polytope has the property that its dual can be constructed as the convex hull of a subset of the vertices of the original simple polytope. In this…

We prove that at least $\left( \dfrac{(1+\epsilon)2m}{N-1}+1+\epsilon \right)^N$, where $0\leqslant \epsilon <1$, many general points, satisfy Demailly's conjecture. Previously, it was known to be true for at least $(2m+2)^N$ many general…

A conjecture of Le says that the Deligne polytope $\Delta_d$ is generically ordinary if $p\equiv 1\ (\!\!\bmod\ D(\Delta_d))$, where $D(\Delta_d)$ is a combinatorial constant determined by $\Delta_d$. In this paper a counterexample is given…

The Descartes-Frenicle-Sorli conjecture predicts that $k=1$ if $q^k n^2$ is an odd perfect number with Euler prime $q$. In this note, we present some conditions equivalent to this conjecture.

We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$…

In this paper, we extend the rectangular side of the shuffle conjecture by stating a rectangular analogue of the square paths conjecture. In addition, we describe a set of combinatorial objects and one statistic that are a first step…

Let $\sigma(x)$ be the sum of the divisors of $x$. If $N$ is odd and $\sigma(N) = 2N$, then the odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and…

The Legendre conjecture has resisted analysis over a century, even under assumption of the Riemann Hypothesis. We present, a significant improvement on previous results by greatly reducing the assumption to a more modest statement called…

We formulate a generalization of the volume conjecture for planar graphs. Denoting by <G, c> the Kauffman bracket of the graph G whose edges are decorated by real "colors" c, the conjecture states that, under suitable conditions, certain…

We establish sharp upper and lower estimates of the Dunkl kernel in the case of dihedral groups.

A graph is said to be diameter-$k$-critical if its diameter is $k$ and removal of any of its edges increases its diameter. A beautiful conjecture by Murty and Simon, says that every diameter-2-critical graph of order $n$ has at most…

We give an explicit counterexample to an entanglement inequality suggested in a recent paper [quant-ph/0005126] by Benatti and Narnhofer. The inequality would have had far-reaching consequences, including the additivity of the entanglement…

We show that the "double circle" order type and some of its generalizations have a compatible triangulation with any other order types with the same number of points and number of edges on convex hull, thus proving another special case of…

Let $I$ be an ideal of height $d$ in a regular local ring $(R,m,k=R/m)$ of dimension $n$ and let $\Omega$ denote the canonical module of $R/I$. In this paper we first prove the equivalence of the following: the non-vanishing of the edge…

We construct an explicit diagonal \Delta_P on the permutahedra P. Related diagonals on the multiplihedra J and the associahedra K are induced by Tonks' projection P --> K and its factorization through J. We introduce the notion of a…

For $\delta>0$ sufficiently small and $A\subset \mathbb{Z}^k$ with $|A+A|\le (2^k+\delta)|A|$, we show either $A$ is covered by $m_k(\delta)$ parallel hyperplanes, or satisfies $|\widehat{\operatorname{co}}(A)\setminus A|\le c_k\delta |A|$,…

Another counter-example to Dirac's Conjecture is presented, which resembles the Cawley model but is so modified as to include second class constraints. The arbitrary function in the general solution to the defining equations of momenta…