• Volume 38, Issue 3

March 1992,   pages  219-333

• First passage time on a multifurcating hierarchical structure

Asymptotic behaviour of the moments of the first passage time (FPT) on a one-dimensional lattice holding a multifurcating hierarchy of teeth is studied. There is a transition from ordinary to anomalous diffusion when the parameter controlling the relative sizes of the teeth, is varied with respect to the furcating number of the hierarchy. The scaling behaviour of the moments of FPT with the linear dimensions of the lattice segment indicates that in the anomalcus phase the probability density of the FPT is multifractal.

• Bivariate averaging functions, translation and scale autocorrelations, Fourier and Mellin transforms, the Wiener-Khinchine theorem and their inter-relationships

We first draw attention to the fact that the position operator,$$\hat X$$, its translation generator,$$\hat P$$, and its scale generator,$$\hat D$$, form an important group of triplet of operators that appear in the Heisenberg uncertainty relation stated in its most general form. The pair$$(\hat X,\hat P)$$ forms the phase-space and they have led to Fourier transform pair, the autocorrelation function, the Wiener-Khinchine theorem, and the Wigner function with many different applications to wave phenomena. The importance of the pairs$$(\hat X,\hat D)$$ and$$(\hat P,\hat D)$$ has been pointed out by Moses and Quesada (1972, 1973, 1974) who showed that we must then consider a Mellin transform pair, a scale autocorrelation function, and a corresponding Wiener-Khinchine theorem. In the present paper, we define and explore properties of a bivariate averaging function defined in a new “phase-space” involving the Mellin transform variable and its partner which can either be the position or momentum, analogous to the Wigner function. The not-necessarily positive feature of the bivariate averaging functions is traced to the general Heisenberg uncertainty mentioned above. The properties and their inter-relationships among the averaging functions are given. We hope this will be of use in discussing physical phenomena involving fractals, turbulence, and near phase transitions where the scaling properties are of importance.

• Slowly rotating white holes

A model of gravitationally anticollapsing objects, white holes, is constructed on the basis of the Kerr metric in the limit of small rotation with a corresponding interior metric. The extended space-time manifold is considered and the spectral shift of radiation from the point of view of a remote observer is calculated for different parameters of such white holes.

• Self-segregation in chemical reactions, diffusion in a catalytic environment and an ideal polymer near a defect

We study a family of equivalent continuum models in one dimension. All these models map onto a single equation and include simple chemical reactions, diffusion in presence of a trap or a source and an ideal polymer chain near an attractive or repulsive site. We have obtained analytical results for the survival probability, total growth rate, statistical properties of nearest-neighbour distribution between a trap and unreacted particle and mean-squared displacement of the polymer chain. Our results are compared with the known asymptotic results in the theory of discrete random walks on a lattice in presence of a defect.

• State generators and complex neural memories

The mechanism of self-indexing for feedback neural networks that generates memories from short subsequences is generalized so that a single bit together with an appropriate update order suffices for each memory. This mechanism explains how stimulating an appropriate neuron can recall a memory. Although information is distributed in this model, yet our self-indexing mechanism makes it appear localized. Also a new complex valued neuron model is presented to generalize McCulloch-Pitts neurons.

• Investigation of alpha particle induced reactions on thulium

Alpha particle induced reactions on the target element thulium were investigated up to 75 MeV, using foil-stack activation technique and Ge(Li) gamma ray spectroscopy method. Excitation functions for eight reactions of the type169Tm(α, xn),x=1 − 4;169Tm(α, pxn),x=3; and169Tm(α, αxn),x=1, 2, 4 were investigated. Of these, four reactions169Tm(α, p3n),169Tm(α, αn),169Tm(α, α2n)169Tm(α, α4n), were studied for the first time and in the remaining four reactions, some 19 new energy-point cross-sections were measured for the first time. The experimental cross-sections were compared with the predictions of pre-equilibrium hybrid model, as well as the more recent index model, using the initial excition number,n0=4 (4p0h). Both the models show better agreement in respect of (α, xnyp) type of reactions. However they are equally bad for (α, αxn) type of reactions which involve theα-particle in the exit channels, and for which some direct reaction contributions are indicated.

• Yields of evaporation residues and average angular momentum in heavy ion induced fusion reactions leading to compound nucleus96Ru

Cross-sections for production of evaporation residues from the compound nucleus96Ru* formed by fusion reactions28Si+68Zn,32S+64Ni,37Cl+59Co and45Sc+51V have been obtained from the yields of their characteristicγ-rays. The measurements span an excitation energy range of 55 MeV to 70 MeV of the compound nucleus. The evaporation residue (ER) cross-sections have been analysed in terms of statistical model for the decay of the compound nucleus. A good agreement is found between statistical model calculation and the experimental evaporation residue cross-sections in all the four cases. It is shown that the average angular momentum$$\bar \ell$$ of the compound nucleus can be deduced from a comparison of the experimentally measured and the statistical model predictions for the ER cross-sections. The validity of this method of deriving$$\bar \ell$$ has been discussed for the case of16O+154Sm system.

• An analysis of quantum chromodynamic structure function beyond leading order

We obtain approximate solutions of Altarelli-Parisi equations beyond leading logarithmic approximation. Our results suggest quantitative utility of higher order terms in the structure function analysis of deep inelastic scattering.

• Relativistic remnants in the reduction of the Bethe-Salpeter equation to the Schrödinger equation

Following Salpeter, the Bethe-Salpeter equation for the bound system of two oppositely charged particles is reduced to a Schrödinger equation for each of the following cases: (a) both particles are spin 1/2 particles, (b) one particle is a spinor while the other is spinless, and (c) both particles are spinless. It is shown that ife is the magnitude of charge carried by each of the particles whose masses are set equal to the electron and proton masses then, strictly speaking, only in case (a) do we obtain the familiar Schrödinger equation for the hydrogen atom. The latter equation is recovered in the other two cases only if relativistic remnants—terms of the order of 10−5 and smaller—are neglected in comparison with unity. Attention is drawn to a situation where such remnants may not be negligibly small, viz. the problem of confinement of quarks.

• Correlation-polarization effects in electron-atom scattering

The correlation-polarization (CP) model-potential, given by O’Connell and Lane (1983) as well as Padial and Norcross (1984) is examined for the elastic scattering of electrons by oxygen atoms, at intermediate energies. The correlation potential is found to be stronger than necessary at low energies. Two model polarization potentials based on the CP model, are suggested and employed to thee — O scattering at 8.7, 30, 50 and 100 eV energies. The calculated total cross-sections agree in general with the other data at these energies.

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