It has been shown[19] that in a loop of length u to which a single stub of length v is attached (fig. 1 in ref. 19), the parity effect is completely destroyed when v/u> 2. It was also shown that such minute topological defects (v/u<< 1) act as singular perturbations. However ref. 19 studies the effect of a single topological defect and says that for v/u<< 1 parity effect is not violated in the ...

This conjecture can be traced to Chowla ([1], p. 96). It is closely related to the Bunyakovsky– Schinzel conjecture on primes represented by irreducible polynomials. The one-variable analogue of (1.1) is classical for deg f = 1 and quite hopeless for deg f > 1. We know (1.1) itself when deg f ≤ 2. (The main ideas of the proof go back to de la Vallée-Poussin ([3], [4]); see [11], §3.3, for an ex...

In 2007, Andrews and Paule introduced the family of functions [Formula: see text] which enumerate the number of broken k-diamond partitions for a fixed positive integer k. Since then, numerous mathematicians have considered partitions congruences satisfied by [Formula: see text] for small values of k. In this work, we provide an extensive analysis of the parity of the function [Formula: see tex...

In this paper, we present a conjecture on the degree of unipotent characters in the cohomology of particular Deligne–Lusztig varieties for groups of Lie type, and derive consequences of it. These degrees are a crucial piece of data in the geometric version of Broué’s abelian defect group conjecture, and can be used to verify this geometric conjecture in new cases. The geometric version of Broué...

An edge coloring of a tournament T with colors 1, 2, . . . , k is called ktransitive if the digraph T (i) defined by the edges of color i is transitively oriented for each 1 ≤ i ≤ k. We explore a conjecture of the first author: For each positive integer k there exists a (least) p(k) such that every k-transitive tournament has a dominating set of at most p(k) vertices. We show how this conjectur...

In this paper, we give a new and direct proof for the recently proved conjecture raised in Soltani and Roozegar (2012). The conjecture can be proved in a few lines via the integral representation of the Gauss-hypergeometric function unlike the long proof in Roozegar and Soltani (2013).

We prove that the v-decomposition number dλμ(v) is an even or odd polynomial according to whether the partitions λ and μ have the same relative sign (or parity) or not. We then use this result to verify Martin’s conjecture for weight 3 blocks of symmetric group algebras — that these blocks have the property that their projective (indecomposable) modules have a common radical length 7.

We study the family of elliptic curvesy2 = x3−t2x+1, both overQ(t) and over Q. In the former case, all integral solutions are determined; in the latter case, computation in the range 1 ≤ t ≤ 999 shows large ranks are common, giving a particularly simple example of curves which (admittedly over a small range) apparently contradict the once held belief that the rank under specialization will tend...

We analyse the capacity of several two-dimensional constraint families — the exclusion, colouring, parity and charge model families. Using Baxter’s corner transfer matrix formalism combined with the corner transfer matrix renormalisation group method of Nishino and Okunishi, we calculate very tight lower bounds and estimates on the growth rates of these models. Our results strongly improve prev...

A parity walk in an edge-coloring of a graph is a walk traversing each color an even number of times. We introduce two parameters. Let p(G) be the least number of colors in a parity edge-coloring of G (a coloring having no parity path). Let b p(G) be the least number of colors in a strong parity edge-coloring of G (a coloring having no open parity walk). Note that b p(G) ≥ p(G) ≥ χ′(G). The val...