The case of U(5)–Q̂(χ) · Q̂(χ) mixing in the configuration-mixed Interacting Boson Model is studied in its mean-field approximation. Phase diagrams with analytical and numerical solutions are constructed and discussed. Indications for first-order and second-order shape phase transitions can be obtained from binding energies and from critical exponents, respectively.

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) ...