Let G be a p-solvable group, P≤G p-subgroup and χ∈Irr(G) such that χ(1)p≥|G:P|p. We prove the restriction χP is sum of characters induced from subgroups Q≤P χ(1)p=|G:Q|p. This generalizes previous results by Giannelli–Navarro Giannelli–Sambale on number linear constituents χP. Although this statement does not hold for arbitrary groups, we conjecture weaker version which can seen as an extension...

We point out that inflaton decays can be a copious source of stable or long–lived particles χ with mass exceeding the reheat temperature TR but less than half the inflaton mass. Once higher order processes are included, this statement is true for any χ particle with renormalizable (gauge or Yukawa) interactions. This contribution to the χ density often exceeds the contribution from thermal χ pr...

Let Q(n, χ) denote the minimum clique size an n-vertex graph can have if its chromatic number is χ . Using Ramsey graphs we give an exact, albeit implicit, formula for the case χ ≥ (n + 3)/2.

Suppose that T is an acyclic r-uniform hypergraph, with r ≥ 2. We define the (t-color) chromatic Ramsey number χ(T, t) as the smallest m with the following property: if the edges of any m-chromatic r-uniform hypergraph are colored with t colors in any manner, there is a monochromatic copy of T . We observe that χ(T, t) is well defined and ⌈ R(T, t)− 1 r − 1 ⌉ + 1 ≤ χ(T, t) ≤ |E(T )| + 1 where R...

Circular-perfect graphs form a natural superclass of perfect graphs, introduced by Zhu almost 20 years ago: on the one hand due to their definition means more general coloring concept, other as an important class χ-bound with smallest non-trivial χ-binding function χ(G)≤ω(G)+1. In this paper, we survey results about circular-perfect obtained in two last decades, focus whether fascinating proper...

The RecBCD enzyme is important for both restriction of foreign DNA and recombinational DNA repair. Switching enzyme function from the destructive antiviral state to the productive recombinational state is regulated by the recombination hotspot, χ (5'-GCTGGTGG-3'). Recognition of χ is unique in that it is recognized as a specific sequence within single-stranded DNA (ssDNA) during DNA translocati...

In this chapter, we will see a proof of the analytic continuation of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) via the Hurwitz zeta function. This then gives rise to a functional equation for ζ(s) and a direct computation for the value of this function at negative integer points. 1. The Hurwitz zeta function We have already seen the definition of the Riemann zeta funct...

It is known that the problem of deciding k-colorability of a graph exhibits an easy-hard-easy pattern, with the maximal complexity being either at k = χ−1 or k = χ , where χ is the chromatic number of the graph. However, the behavior around the complexity peak is poorly understood. In this paper, we use list coloring to model coloring with a fractional number of colors between χ − 1 and χ . We ...

The hybrid χ (Chi) formalism is suited to modeling, simulation and verification of hybrid systems. It integrates concepts from dynamics and control theory with concepts from computer science, in particular from process algebra and hybrid automata. In this paper, we first provide an overview of χ. Then, the χ formalism is related to other formalisms by means of translation schemes: a translation...

We consider the regularity of measurable solutions χ to the cohomological equation φ = χ ◦ T − χ, where (T,X, μ) is a dynamical system and φ : X → R is a C valued cocycle in the setting in which T : X → X is a piecewise C Gibbs– Markov map, an affine β-transformation of the unit interval or more generally a piecewise C uniformly expanding map of an interval. We show that under mild assumptions,...