The prevalence of three symmetries, reverse, complement, and inverse (or reverse-complement), in about 800 complete genomic sequences extant in GenBank is studied. Each symmetry is measured by an index, χ, where χ=1 indicates lack of symmetry and χ=0, perfect symmetry. All random sequences have χ∼1. For reverse and complement symmetries it is found that χ∼1 in all complete sequences. In sharp c...

Pseudo-Riemannian geometry and Hilbert–Schmidt norms are two important fields of research in applied mathematics. One the main goals this paper will be to find a link between these fields. In respect, present paper, we introduce analyze quantities pseudo-Riemannian geometry, namely H-distorsion and, respectively, Hessian χ-quotient. This second quantity investigated using Frobenius (Hilbert–Sch...

Let χ be a primitive, real and even Dirichlet character with conductor q, and let s be a positive real number. An old result of H. Davenport is that the cycle sums Sν(s, χ) = ∑(ν+1)q−1 n=νq+1 χ(n) ns , ν = 0, 1, 2, . . . , are all positive at s = 1, and this has the immediate important consequence of the positivity of L(1, χ). We extend Davenport’s idea to show that in fact for ν ≥ 1, Sν(s, χ) ...

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, ...