Cheng Equation: A Revisit Through Symmetry Analysis

The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.Comment: 10 pages. Accepted for publication in Quaestiones Mathematicae journa

Lie symmetries of (1+2) nonautonomous evolution equations in Financial Mathematics

We analyse two classes of $(1+2)$ evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the $(1+2)$ Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a $(1+1)$ equation, the resulting equation is of maximal symmetry and so equivalent to the $(1+1)$ Classical Heat Equation.Comment: 15 pages, 1 figure, to be published in Mathematics in the Special issue "Mathematical Finance

The WMAP normalization of inflationary cosmologies

We use the three-year WMAP observations to determine the normalization of the matter power spectrum in inflationary cosmologies. In this context, the quantity of interest is not the normalization marginalized over all parameters, but rather the normalization as a function of the inflationary parameters n and r with marginalization over the remaining cosmological parameters. We compute this normalization and provide an accurate fitting function. The statistical uncertainty in the normalization is 3 percent, roughly half that achieved by COBE. We use the k-l relation for the standard cosmological model to identify the pivot scale for the WMAP normalization. We also quote the inflationary energy scale corresponding to the WMAP normalization.Comment: 4 pages RevTex4 with two figure

Coarse-grained Interaction Potentials for Anisotropic Molecules

We have proposed an efficient parameterization method for a recent variant of the Gay-Berne potential for dissimilar and biaxial particles and demonstrated it for a set of small organic molecules. Compared to the previously proposed coarse-grained models, the new potential exhibits a superior performance in close contact and large distant interactions. The repercussions of thermal vibrations and elasticity has been studied through a statistical method. The study justifies that the potential of mean force is representable with the same functional form, extending the application of this coarse-grained description to a broader range of molecules. Moreover, the advantage of employing coarse-grained models over truncated atomistic summations with large distance cutoffs has been briefly studied.Comment: 8 pages, 4 tables and 6 figures. To appear in J. Chem. Phy

Contextual Teaching with Computer-Assisted Instruction

Computer technology has made substantial contributions to education and educators are now confronted with determining how to best incorporate it as a teaching tool. Educators have also long struggled with how to make what is learned in school more useful in other contexts. This review of recent literature was undertaken in an attempt to determine if computer-assisted instruction is compatible with contextual teaching and learning approaches. The four computer-assisted assets of flexibility, format, interactivity and navigational methods were examined because they yield the most interpretive evidence of compatibility with contextual teaching and learning approaches and their characteristics. It was concluded that all four of the assets identified were compatible and should be included within contextual approaches

Computing the Effective Hamiltonian of Low-Energy Vacuum Gauge Fields

A standard approach to investigate the non-perturbative QCD dynamics is through vacuum models which emphasize the role played by specific gauge field fluctuations, such as instantons, monopoles or vortexes. The effective Hamiltonian describing the dynamics of the low-energy degrees of freedom in such approaches is usually postulated phenomenologically, or obtained through uncontrolled approximations. In a recent paper, we have shown how lattice field theory simulations can be used to rigorously compute the effective Hamiltonian of arbitrary vacuum models by stochastically performing the path integral over all the vacuum field fluctuations which are not explicitly taken into account. In this work, we present the first illustrative application of such an approach to a gauge theory and we use it to compute the instanton size distribution in SU(2) gluon-dynamics in a fully model independent and parameter-free way.Comment: 10 pages, 4 figure

Modelling the Galactic Magnetic Field on the Plane in 2D

We present a method for parametric modelling of the physical components of the Galaxy's magnetised interstellar medium, simulating the observables, and mapping out the likelihood space using a Markov Chain Monte-Carlo analysis. We then demonstrate it using total and polarised synchrotron emission data as well as rotation measures of extragalactic sources. With these three datasets, we define and study three components of the magnetic field: the large-scale coherent field, the small-scale isotropic random field, and the ordered field. In this first paper, we use only data along the Galactic plane and test a simple 2D logarithmic spiral model for the magnetic field that includes a compression and a shearing of the random component giving rise to an ordered component. We demonstrate with simulations that the method can indeed constrain multiple parameters yielding measures of, for example, the ratios of the magnetic field components. Though subject to uncertainties in thermal and cosmic ray electron densities and depending on our particular model parametrisation, our preliminary analysis shows that the coherent component is a small fraction of the total magnetic field and that an ordered component comparable in strength to the isotropic random component is required to explain the polarisation fraction of synchrotron emission. We outline further work to extend this type of analysis to study the magnetic spiral arm structure, the details of the turbulence as well as the 3D structure of the magnetic field.Comment: 18 pages, 11 figures, updated to published MNRAS versio

Calculating potentials of mean force and diffusion coefficients from nonequilibirum processes without Jarzynski's equality

In general, the direct application of the Jarzynski equality (JE) to reconstruct potentials of mean force (PMFs) from a small number of nonequilibrium unidirectional steered molecular dynamics (SMD) paths is hindered by the lack of sampling of extremely rare paths with negative dissipative work. Such trajectories, that transiently violate the second law, are crucial for the validity of JE. As a solution to this daunting problem, we propose a simple and efficient method, referred to as the FR method, for calculating simultaneously both the PMF U(z) and the corresponding diffusion coefficient D(z) along a reaction coordinate z for a classical many particle system by employing a small number of fast SMD pullings in both forward (F) and time reverse (R) directions, without invoking JE. By employing Crook's transient fluctuation theorem (that is more general than JE) and the stiff spring approximation, we show that: (i) the mean dissipative work W_d in the F and R pullings are equal, (ii) both U(z) and W_d can be expressed in terms of the easily calculable mean work of the F and R processes, and (iii) D(z) can be expressed in terms of the slope of W_d. To test its viability, the FR method is applied to determine U(z) and D(z) of single-file water molecules in single-walled carbon nanotubes (SWNTs). The obtained U(z) is found to be in very good agreement with the results from other PMF calculation methods, e.g., umbrella sampling. Finally, U(z) and D(z) are used as input in a stochastic model, based on the Fokker-Planck equation, for describing water transport through SWNTs on a mesoscopic time scale that in general is inaccessible to MD simulations.Comment: ReVTeX4, 13 pages, 6 EPS figures, Submitted to Journal of Chemical Physic

Analytic Behaviour of Competition among Three Species

We analyse the classical model of competition between three species studied by May and Leonard ({\it SIAM J Appl Math} \textbf{29} (1975) 243-256) with the approaches of singularity analysis and symmetry analysis to identify values of the parameters for which the system is integrable. We observe some striking relations between critical values arising from the approach of dynamical systems and the singularity and symmetry analyses.Comment: 14 pages, to appear in Journal of Nonlinear Mathematical Physic

Development of probabilistic models for quantitative pathway analysis of plant pest introduction for the EU territory

This report demonstrates a probabilistic quantitative pathway analysis model that can be used in risk assessment for plant pest introduction into EU territory on a range of edible commodities (apples, oranges, stone fruits and wheat). Two types of model were developed: a general commodity model that simulates distribution of an imported infested/infected commodity to and within the EU from source countries by month; and a consignment model that simulates the movement and distribution of individual consignments from source countries to destinations in the EU. The general pathway model has two modules. Module 1 is a trade pathway model, with a Eurostat database of five years of monthly trade volumes for each specific commodity into the EU28 from all source countries and territories. Infestation levels based on interception records, commercial quality standards or other information determine volume of infested commodity entering and transhipped within the EU. Module 2 allocates commodity volumes to processing, retail use and waste streams and overlays the distribution onto EU NUTS2 regions based on population densities and processing unit locations. Transfer potential to domestic host crops is a function of distribution of imported infested product and area of domestic production in NUTS2 regions, pest dispersal potential, and phenology of susceptibility in domestic crops. The consignment model covers the several routes on supply chains for processing and retail use. The output of the general pathway model is a distribution of estimated volumes of infested produce by NUTS2 region across the EU28, by month or annually; this is then related to the accessible susceptible domestic crop. Risk is expressed as a potential volume of infested fruit in potential contact with an area of susceptible domestic host crop. The output of the consignment model is a volume of infested produce retained at each stage along the specific consignment trade chain