A graph is said to be exceptional if it is connected, has least eigenvalue greater than or equal to −2, and is not a generalized line graph. Such graphs are known to be representable in the exceptional root system E 8. The 473 maximal exceptional graphs have been found by computer, and the 467 with maximal degree 28 have been characterized. Here we construct the remaining six maximal exceptiona...

Abstract We relate quantum degree cones, parametrizing PBW degenerations of quantized enveloping algebras, to (negative tight monomial) cones introduced by Lusztig in the study monomials canonical bases, K -theoretic for quiver representations, and some maximal prime tropical flag varieties.

A clique-transversal set in a graph is a subset of the vertices that meets all maximal complete subgraphs on at least two vertices. We prove that every connected graph of order n and maximum degree three has a clique-transversal set of size b19n/30 + 2/15c. This bound is tight, since 19n/30− 1/15 is a lower bound for infinitely many values of n. We also prove that the vertex set of any connecte...

We show that the top of any diamond with bottom 0 in the r.e. degrees is also the top of a stack of n diamonds with bottom 0. Let R be the upper semilattice of the recursively enumerable degrees. A minimal pair consists of two incomparable r.e. degrees with innmum equal to the recursive degree 0. An r.e. degree is cappable if it is one half of a minimal pair. An r.e. degree a is the top of a di...

(0.0) For projective curves,there exists a fundamental dichotomy between curves of genus 0 or 1 on one side, and curves of genus 2 or more on the other side.This dichotomy appears at many levels, such as: Kodaira dimension, topology (fundamental group), hyperbolicity properties (as expressed by the Kobayashi pseudo-metric), and arithmetic geometry (see [La 1,2] and section 7 below). The objecti...

where f (x, y) and g(x, y) are real polynomials of x and y with degree not greater than n, 1h is an oval lying on real algebraic curve H(x, y)=h, deg H(x, y)=m (H(x, y) are called Hamiltonians), and 7 is a maximal interval of existence of 1h . This question is called the weakened Hilbert 16th problem, posed by V.I. Arnold in [1, 2]. The general result of solving the weakened Hilbert 16th proble...

In this paper, we are interested in the question of separating two characters of the absolute Galois group of a number field K, by the Frobenius of a prime ideal p of OK . We first recall an upper bound for the norm N(p) of the smallest such prime p, depending on the conductors and on the degrees. Then we give two applications: (i) find a prime number p for which P (mod p) has a certain type of...

Using the Hatcher–Oertel algorithm for finding boundary slopes of Montesinos knots, we prove the Slope Conjecture and the Strong Slope Conjecture for a family of 3-tangle pretzel knots. More precisely, we prove that the maximal degrees of the colored Jones polynomial of such a knot determine a boundary slope as predicted by the Slope Conjecture, and that the linear terms in the degrees correspo...

We give new descriptions of lattice SU(N) Yang-Mills theory in terms of new lattice variables. The validity of such descriptions has already been demonstrated in the SU(2) Yang-Mills theory by our previous works from the point of view of defining and extracting topological degrees of freedom such as gaugeinvariant magnetic monopoles and vortices which play the dominant role in quark confinement...