Convergent Asymptotic Expansions of Charlier, Laguerre and Jacobi Polynomials

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.Comment: 20 pages, 5 figures. Keywords: Charlier polynomials, Laguerre polynomials, Jacobi polynomials, asymptotic expansions, saddle point methods, two-points Taylor expansion

Multi-point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to several points, allowing confluence of any of them. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in several points as well as Taylor-Laurent expansions.Comment: 20 pages, 7 figures. Keywords: multi-point Taylor expansions, Cauchy's theorem, analytic functions, multi-point Laurent expansions, uniform asymptotic expansions of integral

Two-point Taylor Expansions of Analytic Functions

Taylor expansions of analytic functions are considered with respect to two points. Cauchy-type formulas are given for coefficients and remainders in the expansions, and the regions of convergence are indicated. It is explained how these expansions can be used in deriving uniform asymptotic expansions of integrals. The method is also used for obtaining Laurent expansions in two points.Comment: 14 pages, 10 figure

Computer based human-centered display system

A human centered informational display is disclosed that can be used with vehicles (e.g. aircraft) and in other operational environments where rapid human centered comprehension of an operational environment is required. The informational display integrates all cockpit information into a single display in such a way that the pilot can clearly understand with a glance, his or her spatial orientation, flight performance, engine status and power management issues, radio aids, and the location of other air traffic, runways, weather, and terrain features. With OZ the information is presented as an integrated whole, the pilot instantaneously recognizes flight path deviations, and is instinctively drawn to the corrective maneuvers. Our laboratory studies indicate that OZ transfers to the pilot all of the integrated display information in less than 200 milliseconds. The reacquisition of scan can be accomplished just as quickly. Thus, the time constants for forming a mental model are near instantaneous. The pilot's ability to keep up with rapidly changing and threatening environments is tremendously enhanced. OZ is most easily compatible with aircraft that has flight path information coded electronically. With the correct sensors (which are currently available) OZ can be installed in essentially all current aircraft

The Bayesian Mutation Sampler Explains Distributions of Causal Judgments

One consistent finding in the causal reasoning literature is that causal judgments are rather variable. In particular, distributions of probabilistic causal judgments tend not to be normal and are often not centered on the normative response. As an explanation for these response distributions, we propose that people engage in ‘mutation sampling’ when confronted with a causal query and integrate this information with prior information about that query. The Mutation Sampler model (Davis &amp; Rehder, 2020) posits that we approximate probabilities using a sampling process, explaining the average responses of participants on a wide variety of tasks. Careful analysis, however, shows that its predicted response distributions do not match empirical distributions. We develop the Bayesian Mutation Sampler (BMS) which extends the original model by incorporating the use of generic prior distributions. We fit the BMS to experimental data and find that, in addition to average responses, the BMS explains multiple distributional phenomena including the moderate conservatism of the bulk of responses, the lack of extreme responses, and spikes of responses at 50%.</p

Asymptotic behaviour of three-dimensional singularly perturbed convection–diffusion problems with discontinuous data

AbstractWe consider three singularly perturbed convection–diffusion problems defined in three-dimensional domains: (i) a parabolic problem −ϵ(uxx+uyy)+ut+v1ux+v2uy=0 in an octant, (ii) an elliptic problem −ϵ(uxx+uyy+uzz)+v1ux+v2uy+v3uz=0 in an octant and (iii) the same elliptic problem in a half-space. We consider for all of these problems discontinuous boundary conditions at certain regions of the boundaries of the domains. For each problem, an asymptotic approximation of the solution is obtained from an integral representation when the singular parameter ϵ→0+. The solution is approximated by a product of two error functions, and this approximation characterizes the effect of the discontinuities on the small ϵ− behaviour of the solution and its derivatives in the boundary layers or the internal layers

Dressed test particles, oscillation centres and pseudo-orbits

A general semi-analytical method for accurate and efficient numerical calculation of the dielectrically screened ("dressed") potential around a non-relativistic test particle moving in an isotropic, collisionless, unmagnetised plasma is presented. The method requires no approximations and is illustrated using results calculated for two cases taken from the MSc thesis of the first author: test particles with velocities above and below the ion sound speed in plasmas with Maxwellian ions and warm electrons. The idea that the fluctuation spectrum of a plasma can be described as a superposition of the fields around \emph{non-interacting} dressed test particles is an expression of the quasiparticle concept, which has also been expressed in the development of the oscillation-centre and pseudo-orbit formalisms.Comment: 14 pages to Plasma Physics and Controlled Fusion for publication with a cluster of papers associated with workshop Stability and Nonlinear Dynamics of Plasmas, October 31, 2009 Atlanta, GA on occasion of the 65th birthday of R.L. Dewar. Version 2: Reference [27] added in Sec. 5. Version 3: Revised in response to referee

The Schrodinger equation with Hulthen potential plus ring-shaped potential

We present the solutions of the Schr$\ddot{o}$dinger equation with the Hulth$\acute{e}$n potential plus ring-shape potential for $\ell\neq 0$ states within the framework of an exponential approximation of the centrifugal potential.Solutions to the corresponding angular and radial equations are obtained in terms of special functions using the conventional Nikiforov-Uvarov method. The normalization constant for the Hulth$\acute{e}$n potential is also computed.Comment: Typed with LateX,12 Pages, Typos correcte

Power Utility Maximization in Discrete-Time and Continuous-Time Exponential Levy Models

Consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Levy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.Comment: 18 pages, to appear in Mathematical Methods of Operations Research. The final publication is available at springerlink.co

Reproducibility and validity of a diet quality index for children assessed using a FFQ

The diet quality index (DQI) for preschool children is a new index developed to reflect compliance with four main food-based dietary guidelines for preschool children in Flanders. The present study investigates: (1) the validity of this index by comparing DQI scores for preschool children with nutrient intakes, both of which were derived from 3d estimated diet records; (2) the reproducibility of the DQI for preschoolers based on a parentally reported forty-seven-item FFQ DQI, which was repeated after 5 weeks; (3) the relative validity of the FFQ DQI with 3d record DQI scores as reference. The study sample included 510 and 58 preschoolers (2-5-6.5 years) for validity and reproducibility analyses, respectively. Increasing 3d record DQI scores were associated with decreasing consumption of added sugars, and increasing intakes of fibre, water, Ca and many micronutrients. Mean FFQ DQI test-retest scores were not significantly different: 72 (so 11) v. 71 (Si) 10) (P-=0-218) out of a maximum of 100. Mean 3d record DQI score (66 (so 10)) was significantly lower than mean FFQ DQI (71 (so 10);