We observe that ω(G) + χ(S( G)) = n = ω(S( G)) + χ(G) for any graph G with n vertices, where G is any acyclic orientation of G and where S( G) is the (complement of the) auxiliary line graph introduced in [1]. (Where as usual, ω and χ denote the clique number and the chromatic number.) It follows that, for any graph parameter β(G) sandwiched between ω(G) and χ(G), then Φβ( G) := n−β(S( G)) is s...

Let q ≥ 2 and N ≥ 1 be integers. W. Zhang recently proved that for any fixed ε > 0 and q ≤ N ≤ q1/2−ε , ∑ χ 6=χ0 ∣∣∣∣ N ∑ n=1 χ(n) ∣∣∣∣2|L(1, χ)|2 = (1+ o(1))αq q N , where the sum is taken over all nonprincipal characters χ modulo q, L(1, χ) denotes the L-functions corresponding to χ , and αq = qo(1) is some explicit function of q. Here we improve this result and show that the same asymptotic ...

When G has size n and g ∈ G, for any character χ of G we have χ(g)n = χ(gn) = χ(1) = 1, so the values of χ lie among the nth roots of unity in S1. More precisely, the order of χ(g) divides the order of g (which divides #G). Characters on finite abelian groups were first studied in number theory, since number theory is a source of many interesting finite abelian groups. For instance, Dirichlet u...

We study the conjecture that ∑ n≤x χ(n) = o(x) for any primitive Dirichlet character χ (mod q) with x ≥ q , which is known to be true if the Riemann Hypothesis holds for L(s, χ). We show that it holds under the weaker assumption that ‘100%’ of the zeros of L(s, χ) up to height 14 lie on the critical line. We also establish various other consequences of having large character sums; for example, ...

1. χtrivial(x) = 1 for every x ∈ G is a character. It is called the trivial character. 2. If χ1 and χ2 are characters of G then so is χ1 · χ2 (where χ1 · χ2(x) = χ1(x)χ2(x)). 3. For every character χ of G and x ∈ G, |χ(x)| = 1 (the absolute norm is of course in C). In particular, if we define χ1(x) = χ1(x), then χ1 is also a character and χ · χ = χtrivial. 4. This implies that Ĝ = {χ ∈ C[G] | χ...

We summerize the status of factorization hypothesis in the color-suppressed B meson decays : B → J/ψK (*). We present the general formalism for decay rates and polarization fractions with considering all possible non-factorizable contributions which also include the color-octet contributions, and a new factorization scheme comes appear when the universality is exist : χ F 1 = χ A1 = χ A2 = χ V ...

Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set A of arcs pairwise intersecting at most once, which start at p and end in Q, is |χ|(|χ| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise intersecting at most twice is |χ|(|χ|+ 1)(|χ|+ 2).

This paper is a contribution to the theory of substructural logics. We introduce the notions of (MP)-based and almost (MP)-based logics (w.r.t. a special set of formulae D), which leads to an alternative proof of the well-known forms of the local deduction theorems for prominent substructural logics (FL, FLe, FLew , etc.). Roughly speaking, we decompose the proof of the local deduction theorem ...