Automatized analytic continuation of Mellin-Barnes integrals

I describe a package written in MATHEMATICA that automatizes typical operations performed during evaluation of Feynman graphs with Mellin-Barnes (MB) techniques. The main procedure allows to analytically continue a MB integral in a given parameter without any intervention from the user and thus to resolve the singularity structure in this parameter. The package can also perform numerical integrations at specified kinematic points, as long as the integrands have satisfactory convergence properties. I demonstrate that, at least in the case of massive graphs in the physical region, the convergence may turn out to be poor, making naive numerical integration of MB integrals unusable. I present possible solutions to this problem, but argue that full automatization in such cases may not be achievable.Comment: 23 pages, 11 figures, numerical evaluation functionality adde

Conjugate Function Method for Numerical Conformal Mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.Comment: 23 pages, 15 figures, 5 table

Computing ODE Symmetries as Abnormal Variational Symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [Comput. Methods Appl. Math. 5 (2005), no. 4, pp. 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods

3D mappings by generalized joukowski transformations

The classical Joukowski transformation plays an important role in di erent applications of conformal mappings, in particular in the study of ows around the so-called Joukowski airfoils. In the 1980s H. Haruki and M. Barran studied generalized Joukowski transformations of higher order in the complex plane from the view point of functional equations. The aim of our contribution is to study the analogue of those generalized Joukowski transformations in Euclidean spaces of arbitrary higher dimension by methods of hypercomplex analysis. They reveal new insights in the use of generalized holomorphic functions as tools for quasi-conformal mappings. The computational experiences focus on 3D-mappings of order 2 and their properties and visualizations for di erent geometric con gurations, but our approach is not restricted neither with respect to the dimension nor to the order.Financial support from "Center for Research and Development in Mathematics and Applications" of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT), is gratefully acknowledged. The research of the first author was also supported by the FCT under the fellowship SFRH/BD/44999/2008. Moreover, the authors would like to thank the anonymous referees for their helpful comments and suggestions which improved greatly the final manuscript

Metropolis Sampling

Monte Carlo (MC) sampling methods are widely applied in Bayesian inference, system simulation and optimization problems. The Markov Chain Monte Carlo (MCMC) algorithms are a well-known class of MC methods which generate a Markov chain with the desired invariant distribution. In this document, we focus on the Metropolis-Hastings (MH) sampler, which can be considered as the atom of the MCMC techniques, introducing the basic notions and different properties. We describe in details all the elements involved in the MH algorithm and the most relevant variants. Several improvements and recent extensions proposed in the literature are also briefly discussed, providing a quick but exhaustive overview of the current Metropolis-based sampling's world.Comment: Wiley StatsRef-Statistics Reference Online, 201

What factors promote student resilience on a level 1 distance learning module?

Resilience is understood to be the ability to adapt positively in the face of adversity. In relation to new students on a distance learning module, this can mean how they adapt and make sense of the demands of their chosen study to enable them to persist in their studies. This article reports a small-scale study involving semistructured telephone interviews with students on a level 1 distance learning module at the UK Open University. Students identified the challenges they experienced such as carving out time to study alongside other commitments, as well as developing their academic writing. Students also identified factors that enabled them to adapt to these challenges and be successful in continuing to study. Students rated highly the support they received from tutors in the form of tailored, detailed feedback on their assignments. Other factors that enabled students to persist in their studies were time management, self-belief and motivation

Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement

Sensitivity Analysis and Optimization Using Energy Finite Element and Boundary Element Methods

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76597/1/AIAA-20811-196.pd

Review of Inverse Laplace Transform Algorithms for Laplace-Space Numerical Approaches

A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time