The training and generalization errors of three well-known learning algorithms are calculated using methods of statistical physics. We focus in particular on inconsistent algorithms that are unable to perfectly classify the training examples, and show that the asymptotic behavior of these algorithms is different from the case of consistent algorithms. Our results are in agreement with bounds derived by computational learning theorists. We further find that the replica-symmetric theory is stable everywhere for two of the algorithms studied, which leads us to conjecture that it is the exact solution in these cases. We also demonstrate that one of the algorithms studied performs almost indistinguishably from the Bayes learning algorithm, while having the advantage of being implementable in a single-layer network.

We present a learning-parameter adjustment algorithm, valid for a large class of learning rules in neural-network literature. The algorithm follows directly from a consideration of the statistics of the weights in the network. The characteristic behavior of the algorithm is calculated, both in a fixed and a changing environment. A simple example, Widrow-Hoff learning for statistical classification, serves as an illustration.

We present an alternative approach to efficient Monte Carlo simulations of biological molecules. By relaxing the usual restriction to Markov processes, we are able to optimize performance while dealing directly with the inhomogeneity and anisotropy inherent in these systems. This approach allows us to sample configurational space more efficiently than with either standard Monte Carlo or molecular-dynamics methods.

We used a pseudorandom-walk representation in a four-dimensional embedding to estimate the global fractal dimension D of 164 sequences from GenBank and generated length-matched control sequences of three types: random, matched in base content, and matched in dimer content. The mean D of the sequences was 1.631±0.137. This D was significantly lower than the D’s for all three control types, indicating the presence of significant information content in DNA sequences not explained by base or dimer frequencies. This variation was due largely to nonuniform distribution of bases and dimers within DNA sequences. The D of genomic DNA sequences was different from the D of messenger RNA sequences.

Quantum and thermal effects on the lifetime of the Davydov soliton are studied. Results of a quantum-mechanical perturbation treatment of the lifetime are presented for a wide range of parameter values. We derive a simple condition on the parameter values that must be satisfied for the soliton to be a viable mechanism for localized energy transport in the Davydov model for the α-helix region of protein. Our conclusion is that the Davydov soliton as originally proposed is not a likely candidate for this mechanism, but that a single soliton with a very large excitation number could provide the mechanism.

We study multilayer networks which implement a fixed Boolean function from the hidden layer to the output, and have a fully connected architecture from the input to the hidden layer. We analyze the organization of the first layer of weights and the capacity of the network by a statistical mechanical approach. The mean-field equations that govern the behavior of a network at saturation are derived assuming a replica symmetric solution. The theory is applied for a detailed analysis of the learning ability of an AND machine. We calculate the maximal capacity, and the overlap between the subnetworks. Attention is paid to the organization of internal representations at saturation. The results are compared with detailed numerical simulations, and with bounds on the capacity. Good agreement is found. The behavior of the system depends on the ratio between the number of patterns with a (+) and (-) output. We compare the fully connected architecture with networks that have nonoverlapping receptive fields from the input to the hidden layer. The mean-field equations for this architecture are also presented.

We study the properties of ‘‘quantum’’ neural networks based on a two-state model of a neuron, in which the two neural states correspond to an ‘‘excited’’ and a ‘‘deexcited’’ state. Transitions from the excited to the deexcited state are associated with emission of spikes, in analogy to photon-emission models in quantum optics. We consider strongly diluted networks with low firing rates and sparse encoding. We present simple analytic estimations of critical storage capacity of quantum neural networks.

Biochemical binding systems present many-body problems, even after integrating over solvent and polymer configurational degrees of freedom, because biological molecules are dense in cellular compartments. Furthermore, the resulting effective potentials are likely to include substantial n-body contributions with n>2. The binding of oxygen to hemoglobin is studied as a prototype problem of this kind. (i) The Hill coefficient (simply related to the change in the fractional binding with a change in the ligand chemical potential) has proven to be a useful, model-independent descriptor of ligand binding behavior. An obvious generalization of this approach is given and the resulting coefficients are shown to describe stably the binding of oxygen to hemoglobin over a broad range of free-ligand concentrations. (ii) An expansion of the standard Helmholtz free energy in the local binding density is used to find the excess standard Helmholtz free energy for saturation of hemoglobin with oxygen. This requires equations to change from local potential field variables to density variables that are equivalent to the two-, three-, and four-point Ornstein-Zernike equations. The method permits exact inclusion of two-, three-, and four-body potentials in the limit of weak interaction potentials, but is very poor for strong interactions. (iii) An expansion of the two-body direct correlation function with r≠0 and the core condition are used in the compressibility equation to find the excess pressure as a function of the fractional saturation, and thus also to find the standard free-energy change upon saturating the protein with ligand.

At first order, this procedure provides a model for the binding of oxygen to hemoglobin that does not require the assumption of pair decomposability of the interaction potentials or explicit calculation of a partition function, and yet is competitive with models that require both. The model is accurate for a wide range of interaction potentials. The result is a one-parameter equation of state that does not exhibit a phase transition for a finite system, but may exhibit a phase transition in the limit of infinite system size (depending on the magnitude of the cooperativeness parameter).

Recently the classical electrodynamic zero-point spectrum was derived as being the appropriate spectrum for the thermodynamic definition of absolute-zero temperature to hold for classical, electric dipole harmonic oscillators [Cole, Phys. Rev. A 42, 1847 (1990)]. This analysis involved the lengthy calculation of nonperturbative, retarded van der Waals thermodynamic functions. Here this derivation is shown to hold for the much simpler unretarded, resonance situation. The connection to quantum mechanics is then more immediate.

It is shown that stationarity and causality alone completely determine the second moments of the random forces in a system of coupled nonlinear Langevin equations, and lead to white noise as well as to the appropriate fluctuation dissipation theorems.

The optical-wavelet transform (OWT) is used to measure the generalized fractal dimensions ${\mathit{D}}_{\mathit{q}}$ and the related f(α) spectrum of singularities of fractal aggregates. Its ability to characterize geometrical multifractality is demonstrated on deterministic fractals. Application of the OWT to experimental copper electrodeposition clusters in the limit of small ionic concentration and small voltage is reported. The statistical self-similarity of these experimental clusters and Witten and Sander diffusion-limited aggregates is confirmed.

We introduce the correlation function defined by fractional cumulants and apply it for a model having violent spatial fluctuations with diverging average and variance. The weak correlation, which has never been observed by conventional methods, is observed numerically.

We provide a path-integral Monte Carlo (PIMC) technique for two-dimensional quantum rotators in static potentials. In our PIMC scheme we perform the summations over the different homotopy classes on the circle. In the case of a homonuclear diatomic molecule the coupling of nuclear spin and rotations due to the symmetry requirement of the total wave function is also taken into account in the PIMC technique. We introduce different experimentally relevant PIMC averaging procedures, revealing the quantum effects due to discrete level spacing and exchange, and compare with analytical results.

The well-known acoustoelectric coupling effect in semiconductors, whereby a sound wave is amplified if a charge carrier’s drift velocity exceeds the speed of sound, also exists for ions and electrons in a gaseous medium, but theoretical analysis so far has been limited to simplified collisional-exchange models [G. M. Sessler, Phys. Fluids 7, 90 (1964); U. Ingard and M. Schultz, Phys. Rev. 158, 106 (1967); T. D. Mantei and M. Fitaire, in 2 Proceedings of the 10th International Conference on Phenomena in Ionized Gases (Oxford University Press, Oxford, 1971), p. 309]. The present paper is based upon more accurate considerations of collisional phenomena, leading to more realistic predictions of the qualitative and quantitative nature of the effect.

A dynamic analysis of the growth of a gas bubble system in a lipid monolayer is presented. The bubbles are separated by thin liquid films. The two-dimensional bubble pattern has, independently of the initial gas phase covering the surface, two different regimes of growth with time. Using simple arguments, we find a differential equation which links the radius of the bubble with the width of the liquid film. According to the mechanism of thinning this film, the model predicts two different power laws for the areal growth with time. If drainage is due to gravity, the power-growth exponent is α=1/2 (the experimental value is ${\mathrm{\alpha }}_{\mathit{e}\mathit{x}\mathit{p}\mathit{t}}$=0.59±0.11). When the film reaches the equilibrium thickness before rupture, the theoretical exponent α=1 is in agreement with the experiment.

In this paper we study the linear Langevin equation related to interface-growth phenomena and to diffusion-limited reactions. Using a discretized scheme and the Chapman-Kolmogorov relation, we derive an expression for the time evolution of the mean-squared fluctuations, an expression valid both for Euclidean and fractal lattices. The theoretical analysis is corroborated by numerical investigations carried out for Euclidean lattices and fractals.

The regeneration and stopping of (αμ${)}^{+}$ following dt fusion is computed within the environment of a dense degenerate plasma. It is found that the final sticking fraction can be reduced to ${10}^{\mathrm{-}5}$ of the final sticking fraction in molecular hydrogen.

Dielectronic recombination cross sections between the hyperfine levels of the ground state of ${\mathrm{Bi}}^{82+}$ are calculated in a hydrogenic approximation. The radiative decay rates from the doubly excited resonances 1s(F=5)nl are found to be orders of magnitude stronger than the autoionizing decay rates to the 1s(F=4) continuum. The associated dielectronic recombination cross sections are strongly peaked towards zero energy, but their overall magnitudes are extremely small compared to the radiative recombination background.