Magnetic properties and Spin dynamics in magnetic Molecular Rings

Molecular nanomagnets are spin clusters whose topology and magnetic interactions can be modulated at the level of the chemical synthesis. They are formed by a small number of transition metal ions coupled by the Heisenberg's exchange interactions. Each cluster is magnetically isolated from its neighbors by organic ligands, making each unit not interacting with the others. Therefore, we can investigate the magnetic properties of an isolated molecular nanomagnet by bulk measurements. The present thesis has been mostly devoted to the experimental investigation of the magnetic properties and spin dynamics of different classes of antiferromagnetic (AF) molecular rings. This study has been exploiting various techniques of investigations, such as Nuclear Magnetic Resonance (NMR), muon spin relaxation (muSR) and SQUiD magnetometry. We investigate the magnetic properties and the phonon-induced relaxation dynamics of the first regular Cr9 antiferromagnetic (AF) ring, which represents a prototype frustrated AF ring. The magnetically-open AF rings like Cr8Cd are model systems for the study of the microscopic magnetic behaviour of finite AF Heisenberg chains. In this type of system the different magnetic behaviour depends length and on the parity of the chain (odd or even). In order to study the local spin densities on the Cr sites, the Cr-NMR spectra was collected at low temperature. The experimental result confirm the theoretical predictions for the spin configuration. Finally, the study of Dy6, the first rare-earth based ring that has been ever synthesized, has been performed by AC-SQuID and muSR measurements. We found that the dynamics is characterized by more than one characteristic correlation time, whose values depend strongly on the applied field.

Noisy fitness evaluation in genetic algorithms and the dynamics of learning

A theoretical model is presented which describes selection in a genetic algorithm (GA) under a stochastic fitness measure and correctly accounts for finite population effects. Although this model describes a number of selection schemes, we only consider Boltzmann selection in detail here as results for this form of selection are particularly transparent when fitness is corrupted by additive Gaussian noise. Finite population effects are shown to be of fundamental importance in this case, as the noise has no effect in the infinite population limit. In the limit of weak selection we show how the effects of any Gaussian noise can be removed by increasing the population size appropriately. The theory is tested on two closely related problems: the one-max problem corrupted by Gaussian noise and generalization in a perceptron with binary weights. The averaged dynamics can be accurately modelled for both problems using a formalism which describes the dynamics of the GA using methods from statistical mechanics. The second problem is a simple example of a learning problem and by considering this problem we show how the accurate characterization of noise in the fitness evaluation may be relevant in machine learning. The training error (negative fitness) is the number of misclassified training examples in a batch and can be considered as a noisy version of the generalization error if an independent batch is used for each evaluation. The noise is due to the finite batch size and in the limit of large problem size and weak selection we show how the effect of this noise can be removed by increasing the population size. This allows the optimal batch size to be determined, which minimizes computation time as well as the total number of training examples required.