Dynamics of learning and generalization in perceptrons with constraints

Depending on the kind of constraints imposed on the weights of a perceptron, learning can be a combinatorially hard problem. As an example of this type, I discuss a perception with binary weights comparing results obtained from replica theory, dynamic mean field theory and simulated annealing. Contrary to the replica calculation, dynamics yields information about the performance of a polynomial algorithm in a situation where the best solution cannot be found in polynomial time. I also discuss improved learning algorithms and results for finite size perceptrons.     

Dynamics of the finite SK spin-glass

The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of response and correlation function on time scales related to those free energy barriers which diverge with system sizeN. For the largest of these barriers we find a behavior proportional toN with =1/3.     

Kinetics of oxygen exchange between the two isomers of bisulfite ion,disulfite Ion (S(2)O(5)(2-)), and water as studied by oxygen-17 nuclear magnetic resonance spectroscopy

The nuclear magnetic transverse relaxation time of oxygen-17 in aqueous sodium bisulfite solutions in the pH range from 2.5 to 5 was measured over a range of temperatures, pH, and S(IV) concentrations at an ionic strength of 1.0 m. From these data the rate law for oxygen exchange between bisulfite ion and water was determined and found to be consistent with oxygen exchange occurring via the reactions SHO(3)(-) + H(+) SO(2) + H(2)O, SO(3)H(-) + SHO(3-) SO(3)(2-) + SO(2) + H(2)O, and SO(3)H(-) + SHO(3-) S(2)O(5)(2-) + H(2)O, where the symbol SHO(3-) refers to both isomeric forms of bisulfite ion, one in which the hydrogen is bonded to the sulfur (denoted HSO(3-)) and another in which the hydrogen is bonded to an oxygen atom (denoted SO(3)H(-)). The SO(3)H(-) isomer exchanges oxygen atoms with water much more rapidly than does the HSO(3-) isomer. The value of k(-1) was determined and is in essential agreement with the results of a previous determination by relaxation measurements. The value of k(16a) + k(16b) was also found, and k(16b) is at least as large as k(16a). The rate and mechanism of oxygen exchange between the two bisulfite ion environments were studied by analyzing the broadening of the HSO(3-) resonance. Oxygen exchange occurs through isomerization caused by proton transfers.     

Three- and four-coordinate gold(I) complexes of 3,6-bis(diphenylphosphino)pyridazine: monomers,polymers, and a metallocryptand cage

The slightly yellow polymeric complexes [Au(2)Cl(2)(P(2)pz)(3)](n), 1 x 6CHCl(3), (P(2)pz is 3,6-bis(diphenylphosphino)pyridazine) and [[Au(2)(P(2)pz)(3)](PF(6))(2)](n), 2, are prepared by the stoichiometric reaction of AuCl(tht) (tht is tetrahydrothiophene) and P(2)pz in either dichloromethane or dichloromethane/methanol, respectively. Addition of 2 equiv of AuCl(tht) to a dichloromethane solution of 1 equiv of P(2)pz generates the simple (AuCl)(2)(P(2)pz) compound, 3. Compound 3 contains nearly linear P-Au-Cl units with intermolecular Au.Au separations of 3.570 A. Au(2)I(2)(P(2)pz)(3), 4, is prepared by reacting excess NaI with 2 in a dichloromethane/methanol mixture. Characterization of 1, 2, and 4 by X-ray crystallography confirms the 2:3 gold/ligand ratio of all three complexes. The coordination polymer 1 maintains a high degree of solvation in the solid-state with three chloroform adducts hydrogen-bonded to the chloride ligand on each gold atom. These chloroform molecules are sandwiched between the two-dimensional polymeric sheets of 1. The crystal structure of 4 reveals an empty, iodide-capped metallocryptand cage with the tetrahedrally distorted gold atoms and the nitrogen atoms on the pyridazine rings directed away from the center of the cavity. No metal ion encapsulation was observed for complex 4. Complex 2 forms one-dimensional arrays of [Au(2)(P(2)pz)(2)](2+) metallomacrocycles connected to each other by a third P(2)pz ligand. The electronic absorption spectra (CH(2)Cl(2)) of 1-4 show broad, nearly featureless absorption bands that tail into the visible with pi-pi bands at 296 nm and discernible shoulders at 314 nm for 2 and 334 nm for 3. Excitation into the low energy band of 2 produces only a modest emission in solution at 540 nm (lambda(ex) 468 nm) and 493 nm (lambda(ex) 403 nm). Under identical conditions, the P(2)pz ligand also emits at 540 and 493 nm.     

Renormalization group equations with several length scales and crossover scaling functions for axial anisotropy

The critical behaviour of axially anisotropicn-vector models is characterized by two distinct length scales, the correlation lengths and for the easy and hard axes. In order to handle the full range of anisotropics from to partial differential renormalization group equations are derived, depending on and . The anisotropicX-Y model is studied in detail near four dimensions. The crossover scaling functions for the susceptibilities are calculated to first order in=4–d. Two distinct crossover regions are found for weak and dominant anisotropy, respectively.