Based on Löwdin's projector for recovering the Strue^ 2 symmetry of an Unrestricted Hartree‐Fock Slater determinant, a robust and efficient method, which simplifies the multi‐determinantal character of the original expansion, is deduced. It has a direct application as a Projection after Variation scheme, providing the exact expected values of spin‐projected operators in polynomial time. In the presented approach the (exactly) projected Hamiltonian admits a simple explicit expression when it is written in the “Corresponding Orbitals” basis. This simplicity makes possible to optimize the projected determinant, Variation after Projection, by a direct minimization of the projected energy. Recent successful results obtained from the combination of a projected symmetry broken Slater determinant and post Hartree‐Fock methods has strengthen the attention on Projective Methods. In order to achieve a simple application of the reported formalism to the Interaction Configuration method (CI), a new rotation of the CI basis has been proposed and justified.

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The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a relationship with the cardinality of a certain class of words. By solving the counting problem for the words, and on the basis of the conjecture, several simple recurrence formulas for general cases arise. Moreover, a precise asymptotic analysis is provided. Our results coincide with all current formulas in the literature for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. We expect that the study of the relationship between the newly defined words and the networks will lead to further combinatoric characterizations of this class of phylogenetic networks.

The combinatorial study of phylogenetic networks has attracted much attention in recent times. In particular, one class of them, the so-called tree-child networks, are becoming the most prominent ones. However, their combinatorial properties are largely unknown. In this paper we address the problem of exactly counting them. We conjecture a bijection with a certain class of words, and from this assumption a simple recurrence formula arises. It is able to determine the number of all subclasses, as well as a direct formula, a simple enumeration procedure and precise asympotics. Our results coincide with all currently proved formulas for particular subclasses of tree-child networks, as well as with numerical results obtained for small networks. Since, as we will show, working with words greatly simplies the problem, we expect to contribute to further combinatoric characterizations of this class of networks.

The basic principles of self-organization of one-component charged particles, confined in disk and circular parabolic potentials, are proposed. A system of equations is derived, which allows us to determine equilibrium configurations for an arbitrary, but finite, number of charged particles that are distributed over several rings. Our approach reduces significantly the computational effort in minimizing the energy of equilibrium configurations and demonstrates a remarkable agreement with the values provided by molecular dynamics calculations. With the increase of particle number n>180 we find a steady formation of a centered hexagonal lattice that smoothly transforms to valence circular rings in the ground-state configurations for both potentials.