B. G. Giraud

Diffusion over a saddle with a Langevin equation

The diffusion problem over a saddle is studied using a multidimensional Langevin equation. An analytical solution is derived for a quadratic potential and the probability to pass over the barrier deduced. A very simple solution is given for the one-dimensional problem and a general scheme is shown for higher dimensions.

Fusion Dynamics of Massive Heavy-Ion Systems

A two-step model is proposed for the fusion mechanism of massive heavy-ion systems which is the most unknown part in the reaction theory for the synthesis of the superheavy elements. It consists of the approaching phase of incident ions and the dynamical shape evolution of the amalgamated system toward the spherical compound nucleus. Preliminary results are presented.

Lorentzian neural nets

Abstract We consider neural units whose response functions are Lorentzians rather than the usual sigmoids or steps. This consideration is justified by the fact that neurons can be paired and that a suitable difference of the sigmoids of the paired neurons can create a window response function. Lorentzians are special cases of such windows and we take advantage of their simplicity to generate polynomial equations for several problems such as: i) fixed points of completely connected net, ii) classification of operational modes, iii) training of a feedforward net, iv) process signals represented by complex numbers.

Optimal approximation of square integrable functions by a flexible one-hidden-layer neural network of excitatory and inhibitory neuron pairs

Abstract We present here a new type of feedforward neural network, the basic element of which is a pair of excitatory/inhibitory formal neurons. Indeed, a difference of sigmoidal functions can be used to generate a basis of band-like functions for the reconstruction of any plane wave with any momentum. With an infinite number of neurons, a one-hidden-layer neural network can thus implement any square integrable response function on the available subspace of elementary response functions. With a finite number of neurons, the accuracy of the reconstruction of the response function can be optimized. Our explicit calculation of synaptic weights is based on a projector algebra and provides convergence, in a single step, towards the absolute minimum of an explicit cost function. Our numerical examples illustrate the flexibility and robustness of such a network for the implementation of both continuous and discontinuous response functions.

Effects of collateral inhibition in a model of the immature rat cerebellar cortex: multineuron correlations

A model of the immature rat cerebellar cortex is used to simulate the effect of the inhibitory recurrent collateral axons of the Purkinje cells on the spike trains in the network. Inhibition induces an important overall change in the statistical characteristics of individual spike trains. It is also instrumental in producing a strong cooperativity between the different neurons. Moreover, a functional spatial anisotropy appears. A specific entropy index is used to analyze levels of information transfer between clustered and faraway neurons in the network. The formatting effect of recurrent collateral inhibition on spike trains and on network functional dynamics is studied by means of a model of the newborn rat cerebellar cortex. This immature structure has simpler morphological characteristics and fewer physiological parameters than the adult one. It is thus a good candidate for the comparison between experimental and theoretical data. The model network is made of 256 formal neurons (FN), arranged in a square lattice. Each neuron is coupled to its eight nearest neighbors by inhibitory links. All the parameters of the different elements of the model--in particular integration of inhibitory and excitatory inputs--are given anatomical and physiological values derived from biological data. Activities of single FNs and correlations between spatially distant ones are analyzed with classical statistical techniques as well as with a specific informational entropy method we introduce. Simulation results indicate that inhibition is instrumental in: (1) the transformation of the spike train characteristics. This includes a lengthening of the mean interspike interval as well as an overall change in the statistical distribution of intervals, with an emergence of long-lasting ones; (2) the functional structuration of the network. Inhibitory connections between nearest neighbors induce a strong cooperativity between FNs. Furthermore a clear spatial anisotropy occurs in the functioning of the network, with inhibitory effects extending beyond local connectivity in preferential directions. We propose an interpretation of this functional structuration in terms of the various routes followed by the inhibition, including relay effects. The parameters of the model (levels of activities, inhibition rules and connectivities) were varied in order to test the robustness of the above results. Finally, the results are compared with those obtained in an experimental situation.

Constrained orthogonal polynomials

We define sets of orthogonal polynomials satisfying the additional constraint of a vanishing average. These are of interest, for example, for the study of the Hohenberg-Kohn functional for electronic or nucleonic densities and for the study of density fluctuations in centrifuges. We give explicit properties of such polynomial sets, generalizing Laguerre and Legendre polynomials. The nature of the dimension 1 subspace completing such sets is described. A numerical example illustrates the use of such polynomials.

Complex scaled spectrum completeness for coupled channels

The Complex Scaling Method (CSM) provides scattering wave functions which regularize resonances and suggest a resolution of the identity in terms of such resonances, completed by the bound states and a smoothed continuum. But, in the case of inelastic scattering with many channels, the existence of such a resolution under complex scaling is still debated. Taking advantage of results obtained earlier for the two channel case, this paper proposes a representation in which the convergence of a resolution of the identity can be more easily tested. The representation is valid for any finite number of coupled channels for inelastic scattering without rearrangement.

Scalar nature of the nuclear density functional

We prove the existence theorem for a scalar density functional (DF) for nuclei. The theorem is a direct consequence of the rotational invariance of the nuclear Hamiltonian. Since the DF depends only on scalar densities, practical predictions of ground state (g.s.) energies reduce to one-dimensional, radial calculations.

Beyond the time independent mean field theory for nuclear and atomic reactions: Inclusion of particle-hole correlations in a generalized random phase approximation.

The time independent mean field method for scattering defines biorthonormal sets of single-particle wave functions and corresponding creation and annihilation operators. Two-particle--two-hole (2[ital p]-2[ital h]) correlations can be introduced through a generalized random phase approximation; 1[ital p]-1[ital h] contributions vanish (Brillouin theorem). While the genreal variational method for scattering by Giraud and Nagarajan solves inhomogeneous Euler equations by inversion of the standard, Hermitean Hamiltonian, the present approach diagonalizes a non-Hermitean Hamiltonian, which carries the information about entrance and exit channels.

Di-nucleus dynamics towards fusion of heavy nuclei

The Two-Step Model for fusion of massive systems is briefly recapitulated, which clarifies the mechanism of so-called fusion hindrance. Since the neck changes the potential landscape, especially the height of the conditional saddle point, time evolution of the neck degree of freedom plays a crucial role in fusion. We analytically solve time-evolution of nuclear shape of the composite system from di-nucleus to mono-nucleus. The time-dependent distribution function of the neck is obtained, which elucidates dynamics of fusion processes in general, and thus, is useful for theoretical predictions on synthesis of the superheavy elements with various combinations of incident heavy ions.