Insight into the fractional calculus via a spreadsheet

Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

Secular series and renormalization group for amplitude equations

We have developed a technique that circumvents the process of elimination of secular terms and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems. © 2008 The American Physical Society.

GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion

The efficient computation of matrix function vector products has become an important area of research in recent times, driven in particular by two important applications: the numerical solution of fractional partial differential equations and the integration of large systems of ordinary differential equations. In this work we consider a problem that combines these two applications, in the form of a numerical solution algorithm for fractional reaction diffusion equations that after spatial discretisation, is advanced in time using the exponential Euler method. We focus on the efficient implementation of the algorithm on Graphics Processing Units (GPU), as we wish to make use of the increased computational power available with this hardware. We compute the matrix function vector products using the contour integration method in [N. Hale, N. Higham, and L. Trefethen. Computing Aα, log(A), and related matrix functions by contour integrals. SIAM J. Numer. Anal., 46(5):2505–2523, 2008]. Multiple levels of preconditioning are applied to reduce the GPU memory footprint and to further accelerate convergence. We also derive an error bound for the convergence of the contour integral method that allows us to pre-determine the appropriate number of quadrature points. Results are presented that demonstrate the effectiveness of the method for large two-dimensional problems, showing a speedup of more than an order of magnitude compared to a CPU-only implementation.

Developments of the total entropy utility function for the dual purpose of model discrimination and parameter estimation in Bayesian design

The total entropy utility function is considered for the dual purpose of Bayesian design for model discrimination and parameter estimation. A sequential design setting is proposed where it is shown how to efficiently estimate the total entropy utility for a wide variety of data types. Utility estimation relies on forming particle approximations to a number of intractable integrals which is afforded by the use of the sequential Monte Carlo algorithm for Bayesian inference. A number of motivating examples are considered for demonstrating the performance of total entropy in comparison to utilities for model discrimination and parameter estimation. The results suggest that the total entropy utility selects designs which are efficient under both experimental goals with little compromise in achieving either goal. As such, the total entropy utility is advocated as a general utility for Bayesian design in the presence of model uncertainty.

Comparison of two pressurisers for cementation of the proximal femur

PURPOSE: To compare pressures generated by 2 different cement pressurisers at various locations in the proximal femur. METHODS: Two groups of 5 synthetic femurs were used, and 6 pressure sensors were placed in the femur at 20-mm intervals proximally to distally. Cement was filled into the femoral canal retrogradely using a cement gun with either the half-moon pressuriser or the femoral canal pressuriser. Maximum pressures and pressure time integrals (cumulative pressure over time) of the 2 pressurisers were compared. RESULTS: At all sensors, the half-moon pressuriser produced higher maximum pressures and pressure time integrals than the femoral canal pressuriser, but the difference was significant only at sensor 1 (proximal femur). This may result in reduced cement interdigitation in the proximal femur. CONCLUSION: The half-moon pressuriser produced higher maximum cementation pressures and pressure time integrals than the femoral canal pressuriser in the proximal femur region, which is critical for rotational stability of the implant and prevention of implant fracture. KEYWORDS: arthroplasty, replacement, hip; bone cements; femur

A New Mechanism For Ferromagnetism In C-60-TDAE

Based on the topology of C-60 and the resulting non-disjoint nature of the lowest unoccupied molecular orbitals, Ne propose a new model for ferromagnetic exchange in C-60-TDAE. Within the Hubbard model, we find that the ferromagnetic exchange integral is stabilized to first order in the inter-ball transfer integral, while the antiferromagnetic coupling is stabilized only to second order. This difference is adequate to counter the larger phase space available for stabilizing the antiferromagnetic state. Thus, the ground state is found to be ferromagnetic for reasonable inter-ball transfer integrals.

Universal crack closure integral for SIF estimation

Numerical analysis of cracked structures often involves numerical estimation of stress intensity factors (SIFs) at a crack tip/front. A newly developed formulation called universal crack closure integral (UCCI) for the evaluation of potential energy release rates (PERRs) and the corresponding SIFs is presented in this paper. Unlike the existing element dedicated forms of crack closure integrals (MCCI, VCCI) with application limited to finite element analysis, this new numerical SIF/PERR estimation technique is independent of the basic stress analysis procedure, making it universally applicable. The second merit of this procedure is that it avoids the generally error-producing zones close to the crack tip/front singularity. The UCCI procedure, based on Irwin's original CCI, is formulated and explored using a simple 2D problem of a straight crack in an infinite sheet. It is then applied to some three-dimensional crack geometries with the stresses and displacements obtained from a boundary element program.

Analysis of a finite composite plate with smooth rigid pin

A continuum method of analysis is presented in this paper for the problem of a smooth rigid pin in a finite composite plate subjected to uniaxial loading. The pin could be of interference, push or clearance fit. The plate is idealized to an orthotropic sheet. As the load on the plate is progressively increased, the contact along the pin-hole interface is partial above certain load levels in all three types of fit. In misfit pins (interference or clearance), such situations result in mixed boundary value problems with moving boundaries and in all of them the arc of contact and the stress and displacement fields vary nonlinearly with the applied load. In infinite domains similar problems were analysed earlier by ‘inverse formulation’ and, now, the same approach is selected for finite plates. Finite outer domains introduce analytical complexities in the satisfaction of boundary conditions. These problems are circumvented by adopting a method in which the successive integrals of boundary error functions are equated to zero. Numerical results are presented which bring out the effects of the rectangular geometry and the orthotropic property of the plate. The present solutions are the first step towards the development of special finite elements for fastener joints.

Cresols - An investigation into their electronic structure and spectra

The three isomeric cresols were subjected to the all-valence-electron CNDO/2 andPPP-CI calculations. Results from this study were used: (i) to compare the electronic structures of these isomers vis-Ã-vis parent compounds-phenol and toluene, (ii) to obtain a quantitative picture of their chemical reactivities and electronic absorption spectra. Using the sgr-core charges derived from CNDO/2 calculations and subsequently revising the valence-state ionisation potential and one-center-two-electron repulsion integrals, thePPP-CI calculations were performed on the title compounds according toNishimoto andForster scheme. In these calculations the pseudo-unsaturated nature of the methyl group has been given due consideration. In spectral assignment, compared to the conventionalPPP approach, the CNDO/2-basedPPP-CI method gave a better agreement with the experimental data.

On the breakdown of the most probable distribution for mayer clusters

We present an analysis of the breakdown of the most probable approximation to the Mayer cluster size distribution for clusters of size comparable to the size of the system. This failure is illustrated by considering an ideal Bose gas for which exact volume dependent reducible cluster integrals are available.

A PPP AND CNDO Study of Dithiocarbonic Antonic Derivatives

The application of the CNDO and PPP-CI methods to N,N-dimethyl dithiocarbamate, O-methyl dithiocarbonate (methyl xanthate) and methyl trithiocarbonate ions for the elucidation of electronic structure and electronic spectra is described. The CNDO/2 calculations have been used to obtain the one centre core integrals of the ionic compounds required in calculating the pi electronic spectra of these molecules using the PPP method. The calculated spectra are in good agreement with the experiment. The atomic charge densities determined for alkyl xanthate, dithiocarbamate and trithiocarbonate ions support the earlier qualitative predictions regarding electronic structure from spectroscopic and other studies.

Study of turbid media with light: Recovery of mechanical and optical properties from boundary measurement of intensity autocorrelation of light

We discuss the inverse problem associated with the propagation of the field autocorrelation of light through a highly scattering object like tissue. In the first part of the work, we reconstructed the optical absorption coefficient mu(u) and particle diffusion coefficient D-B from simulated measurements which are integrals of a quantity computed from the measured intensity and intensity autocorrelation g(2)(tau) at the boundary. In the second part we recover the mean square displacement (MSD) distribution of particles in an inhomogeneous object from the sampled g(2)(tau) measure on the boundary. From the MSD, we compute the storage and loss moduli distributions in the object. We have devised computationally easy methods to construct the sensitivity matrices which are used in the iterative reconstruction algorithms for recovering these parameters from the measurements. The results of the reconstruction of mu(a), D-B, MSD and the viscoelastic parameters, which are presented, show reasonable good position and quantitative accuracy.

Representation and properties of para-Bose oscillator operators. II. Coherent states and the minimum uncertainty states

The energy, position, and momentum eigenstates of a para-Bose oscillator system were considered in paper I. Here we consider the Bargmann or the analytic function description of the para-Bose system. This brings in, in a natural way, the coherent states ||z;alpha> defined as the eigenstates of the annihilation operator ?. The transformation functions relating this description to the energy, position, and momentum eigenstates are explicitly obtained. Possible resolution of the identity operator using coherent states is examined. A particular resolution contains two integrals, one containing the diagonal basis ||z;alpha><−z;alpha||. We briefly consider the normal and antinormal ordering of the operators and their diagonal and discrete diagonal coherent state approximations. The problem of constructing states with a minimum value of the product of the position and momentum uncertainties and the possible alpha dependence of this minimum value is considered. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Charged-soliton excitations in statistical mechanics

A method is developed for demonstrating how solitons with some internal periodic motion may emerge as elementary excitations in the statistical mechanics of field systems. The procedure is demonstrated in the context of complex scalar fields which can, for appropriate choices of the Lagrangian, yield charge-carrying solitons with such internal motion. The derivation uses the techniques of the steepest-descent method for functional integrals. It is shown that, despite the constraint of some fixed total charge, a gaslike excitation of such charged solitons does emerge.

Water-wave scattering by two submerged plane vertical barriers-Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207-215, 1971) who employed a completely different approach involving a Schwarz-Christoffel transformation of complex-variable theory to solve the problem.