Continuum Singularities of a Mean Field Theory of Collisions

Consider a complex energy $z$ for a $N$-particle Hamiltonian $H$ and let $\chi$ be any wave packet accounting for any channel flux. The time independent mean field (TIMF) approximation of the inhomogeneous, linear equation $(z-H)|\Psi>=|\chi>$ consists in replacing $\Psi$ by a product or Slater determinant $\phi$ of single particle states $\phi_i.$ This results, under the Schwinger variational principle, into self consistent TIMF equations $(\eta_i-h_i)|\phi_i>=|\chi_i>$ in single particle space. The method is a generalization of the Hartree-Fock (HF) replacement of the $N$-body homogeneous linear equation $(E-H)|\Psi>=0$ by single particle HF diagonalizations $(e_i-h_i)|\phi_i>=0.$ We show how, despite strong nonlinearities in this mean field method, threshold singularities of the {\it inhomogeneous} TIMF equations are linked to solutions of the {\it homogeneous} HF equations.Comment: 21 pages, 14 figure

Existence of a Density Functional for an Intrinsic State

A generalization of the Hohenberg-Kohn theorem proves the existence of a density functional for an intrinsic state, symmetry violating, out of which a physical state with good quantum numbers can be projected.Comment: 6 page

The quantum dynamics of atomic magnets, co-tunneling and dipolar-biased tunneling

Multi-spins tunneling cross-relaxations in an ensemble of weakly-coupled Ho$^{3+}$ ions, mediated by weak anisotropic dipolar interactions, can be evidenced by ac-susceptibility measurements in a high temperature regime. Based on a four-body representation, including the rare-earth nuclear spin, two-ions tunneling mechanisms can be attributed to both dipolar-biased tunneling and co-tunneling processes. The co-reversal involving entangled pairs of magnetic moments is discussed with a particular emphasis, giving new evidences to elucidate the many-body quantum dynamics.Comment: 4 figure

On positive functions with positive Fourier transforms

Using the basis of Hermite-Fourier functions (i.e. the quantum oscillator eigenstates) and the Sturm theorem, we derive the practical constraints for a function and its Fourier transform to be both positive. We propose a constructive method based on the algebra of Hermite polynomials. Applications are extended to the 2-dimensional case (i.e. Fourier-Bessel transforms and the algebra of Laguerre polynomials) and to adding constraints on derivatives, such as monotonicity or convexity.Comment: 12 pages, 23 figures. High definition figures can be obtained upon request at [email protected] or [email protected]

Move ordering and communities in complex networks describing the game of go

We analyze the game of go from the point of view of complex networks. We construct three different directed networks of increasing complexity, defining nodes as local patterns on plaquettes of increasing sizes, and links as actual successions of these patterns in databases of real games. We discuss the peculiarities of these networks compared to other types of networks. We explore the ranking vectors and community structure of the networks and show that this approach enables to extract groups of moves with common strategic properties. We also investigate different networks built from games with players of different levels or from different phases of the game. We discuss how the study of the community structure of these networks may help to improve the computer simulations of the game. More generally, we believe such studies may help to improve the understanding of human decision process.Comment: 14 pages, 21 figure

Distinguishing humans from computers in the game of go: a complex network approach

We compare complex networks built from the game of go and obtained from databases of human-played games with those obtained from computer-played games. Our investigations show that statistical features of the human-based networks and the computer-based networks differ, and that these differences can be statistically significant on a relatively small number of games using specific estimators. We show that the deterministic or stochastic nature of the computer algorithm playing the game can also be distinguished from these quantities. This can be seen as tool to implement a Turing-like test for go simulators.Comment: 7 pages, 6 figure