Brooks’ Theorem implies that if a graph has ∆ ≥ 3 and and χ > ∆, then ω = ∆+1. Borodin and Kostochka conjectured that if ∆ ≥ 9 and χ ≥ ∆, then ω ≥ ∆. We show that if ∆ ≥ 13 and χ ≥ ∆, then ω ≥ ∆ − 3. For a graph G, let H(G) denote the subgraph of G induced by vertices of degree ∆. We also show that if χ ≥ ∆, then ω ≥ ∆ or ω(H(G)) ≥ ∆− 5.

Let ρ(x) = x − [x], χ = χ (0,1). In L2(0, ∞) consider the subspace B generated by {ρa|a ≥ 1} where ρa(x) := ρ 1 ax. By the Nyman-Beurling criterion the Riemann hypothesis is equivalent to the statement χ ∈ B. For some time it has been conjectured, and proved in this paper, that the Riemann hypothesis is equivalent to the stronger statement that χ ∈ B nat where B nat is the much smaller subspace...

Recently, many authors have embraced the study of certain properties modules such as projectivity, injectivity and flatness from an alternative point view. Rather than saying a module has property or not, each is assigned relative domain which, somehow, measures to which extent it this particular property. In work, we introduce new fresh perspective on modules. However, will first investigate m...

In this note we relate Emerton’s Jacquet functor JP to his ordinary parts functor OrdP , by computing the χ-eigenspaces Ord χ P for central characters χ. This fills a small gap in the literature. One consequence is a weak adjunction property for unitary characters χ appearing in JP , with potential applications to local-global compatibility in the p-adic Langlands program in the ordinary case.

Let K be a totally imaginary quadratic extension of a totally real number field F , and assume that F has narrow ideal class number 1. Let χ be a character of the ideal class group CL(K) of K, and let LK(χ, s) be its associated L–function. In this paper we prove that for all > 0, #{χ ∈ CL(K)∧ : LK(χ, 1 2 ) 6= 0} ,F d 1 100− K as the absolute discriminant dK →∞.

We discuss the light variability of the Mira variable χ Cygni, present a homogeneous set of times of minimum and maximum light since its discovery date in 1686, and analyse the variations of the pulsation period. There is a substantial cycleto-cycle variation in the pulsations (standard error ∼ 6. d5) with quite strong evidence for a linearly increasing period. Quasicyclic patterns in the O−C d...

We establish the equation of motion of pseudoscalar particles coupled to an electromagnetic field in a classical gravitational background through the use of conformal time and flat geometry. We show that the expansion of the universe can lead to larger mixing than in a stationary universe. We also show that for a broad range of parameters, one can obtain resonant mixing, i.e. a region in which ...

Total Variation (TV) is an effective method of removing noise in digital image processing while preserving edges [27]. The choice of scaling or regularization parameter in the TV process defines the amount of denoising, with value of zero giving a result equivalent to the input signal. Here we explore three algorithms for specifying this parameter based on the statistics of the signal in the to...