Approximate solutions of Fredholm integral equations of the second kind

This note is concerned with the problem of determining approximate solutions of Fredholm integral equations of the second kind. Approximating the solution of a given integral equation by means of a polynomial, an over-determined system of linear algebraic equations is obtained involving the unknown coefficients, which is finally solved by using the least-squares method. Several examples are examined in detail. (c) 2009 Elsevier Inc. All rights reserved.

A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Transient state work fluctuation theorem for a classical harmonic oscillator linearly coupled to a harmonic bath

Based on a Hamiltonian description we present a rigorous derivation of the transient state work fluctuation theorem and the Jarzynski equality for a classical harmonic oscillator linearly coupled to a harmonic heat bath, which is dragged by an external agent. Coupling with the bath makes the dynamics dissipative. Since we do not assume anything about the spectral nature of the harmonic bath the derivation is not restricted only to the Ohmic bath, rather it is more general, for a non-Ohmic bath. We also derive expressions of the average work done and the variance of the work done in terms of the two-time correlation function of the fluctuations of the position of the harmonic oscillator. In the case of an Ohmic bath, we use these relations to evaluate the average work done and the variance of the work done analytically and verify the transient state work fluctuation theorem quantitatively. Actually these relations have far-reaching consequences. They can be used to numerically evaluate the average work done and the variance of the work done in the case of a non-Ohmic bath when analytical evaluation is not possible.

Classical and Cosserat plasticity and viscoplasticity models for slow granular flow

We consider models for the rheology of dense, slowly deforming granular materials based of classical and Cosserat plasticity, and their viscoplastic extensions that account for small but finite particle inertia. We determine the scale for the viscosity by expanding the stress in a dimensionless parameter that is a measure of the particle inertia. We write the constitutive relations for classical and Cosserat plasticity in stress-explicit form. The viscoplastic extensions are made by adding a rate-dependent viscous stress to the plasticity stress. We apply the models to plane Couette flow, and show that the classical plasticity and viscoplasticity models have features that depart from experimental observations; the prediction of the Cosserat viscoplasticity model is qualitatively similar to that of Cosserat plasticity, but the viscosities modulate the thickness of the shear layer.

NMR relaxation study of disorder in the mixed system BP x GPI(1-x) and the low temperature transition from classical to quantum rotation

1H NMR spin-lattice relaxation time (T1) studies have been carried out in the temperature range 100 K to 4 K, at two Larmor frequencies 11.4 and 23.3 MHz, in the mixed system of betaine phosphate and glycine phosphite (BPxGPI(1-x)), to study the effects of disorder on the proton group dynamics. Analysis of T1 data indicates the presence of a number of inequivalent methyl groups and a gradual transition from classical reorientations to quantum tunneling rotations. At lower temperatures, microstructural disorder in the local environments of the methyl groups, result in a distribution in the activation energy (Ea) and the torsional energy gap (E01). For certain values of x, the magnetisation recovery shows biexponential behaviour at lower temperatures.

Classical Langevin dynamics of a charged particle moving on a sphere and diamagnetism: A surprise

It is generally known that the orbital diamagnetism of a classical system of charged particles in thermal equilibrium is identically zero —the Bohr-van Leeuwen theorem. Physically, this null result derives from the exact cancellation of the orbital diamagnetic moment associated with the complete cyclotron orbits of the charged particles by the paramagnetic moment subtended by the incomplete orbits skipping the boundary in the opposite sense. Motivated by this crucial but subtle role of the boundary, we have simulated here the case of a finite but unbounded system, namely that of a charged particle moving on the surface of a sphere in the presence of an externally applied uniform magnetic field. Following a real space-time approach based on the classical Langevin equation, we have computed the orbital magnetic moment that now indeed turns out to be non-zero and has the diamagnetic sign. To the best of our knowledge, this is the first report of the possibility of finite classical diamagnetism in principle, and it is due to the avoided cancellation.

On states, on a lattice, with half-integral charge

In certain molecular models, and related one-dimensional field theories, localized objects appear with half-integral expectation values of charge. We consider whether these states are eigenstates of charge, with half-integral eigenvalue. We find that it is indeed so for a suitably diffuse definition of the charge operator in question. This diffuse charge operator has a spectrum which approaches a continuum. The analysis is made on a lattice, to avoid divergence ambiguities, and on a finite length, which is only subsequently made large. The half-integral charge phenomenon is not tied to solitons, but can also arise as an end effect.

Simplified algorithms based on Haar transforms for signal recognition in protective relays

This letter presents the development of simplified algorithms based on Haar functions for signal extraction in relaying signals. These algorithms, being computationally simple, are better suited for microprocessor-based power system protection relaying. They provide accurate estimates of the signal amplitude and phase.

Derivation of the solution of certain singular integral equations

It is shown that the a;P?lication of the Poincare-Bertrand fcm~ulaw hen made in a suitable manner produces the s~lutiano f certain singular integral equations very quickly, thc method of arriving at which, otherwise, is too complicaled. Two singular integral equations are considered. One of these quaiions is with a Cauchy-tyge kcrnel arid the other is an equalion which appears in the a a w guide theory and the theory of dishcations. Adifferent approach i? alw made here to solve the singular integralquation> of the waveguide theor? ind this i ~ v o l v eth~e use of the inversion formula of the Cauchy-type singular integral equahn and dudion to a system of TIilberl problems for two unknowns which can be dwupled wry easily to obi& tbe closed form solutim of the irilegral equatlou at band. The methods of the prescnt paper avoid all the complicaled approaches of solving the singular integral equaticn of the waveguide theory knowr todate.

A cryptographic system based on finite field transforms

In this paper, we develop a cipher system based on finite field transforms. In this system, blocks of the input character-string are enciphered using congruence or modular transformations with respect to either primes or irreducible polynomials over a finite field. The polynomial system is shown to be clearly superior to the prime system for conventional cryptographic work.

Metal-organic framework structures - how closely are they related to classical inorganic structures?

Metal-organic frameworks (MOFs) have emerged as an important family of compounds for which new properties are increasingly being found. The potential for such compounds appears to be immense, especially in catalysis, sorption and separation processes. In order to appreciate the properties and to design newer frameworks it is necessary to understand the structures from a fundamental perspective. The use of node, net and vertex symbols has helped in simplifying some of the complex MOF structures. Many MOF structures are beginning to be described as derived from inorganic structures. In this tutorial review, we have provided the basics of the node, the net and the vertex symbols and have explained some of the MOF structures. In addition, we have also attempted to provide some leads towards designing newer structures/topologies.

Spectral analysis of finite section normal integral operators

Some continuity and differentiability properties of the eigenvalues and eigenfunctions of finite section normal integral operators are proved. These are the extension of corresponding results for symmetric operators ([4.], 554–566; K. B. Athreya and R. Vittal Rao, to appear; [10.], 463–471.

Kac-Akhiezer formula for normal integral operators

The Kac-Akhiezer formula for finite section normal Wiener-Hopf integral operators is proved. This is an extension of the corresponding result for symmetric operator [2, 3, 4, 5, 6, 7].

On solitons with half integral charge

Can certain soliton states, with half integral expectation value of charge, be also eigenstates of charge X with half integral eigenvalue? It can be so only with a somewhat sophisticated definition of charge.