a family of numerov type exponentially fitted predictor corrector methods for the numerical integration of the radial schro dinger equation a family of predictor corrector exponential numerov type methods is developed for the numerical integration of the one dimensional schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions the new methods are very simple and integrate more exponential functions than both the well known fourth order numerov type exponentially fitted methods and the sixth algebraic order runge kutta type methods numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above

extended numerov method for computing eigenvalues of specific schrodinger equations numerov s method and an extended version of it are introduced for computing eigenvalues of schrodinger equations with potentials v x which are even functions with respect to x furthermore it is assumed that the wavefunctions tend to zero for x to or infinity the derived results are compared with previously derived numerical data and with available exact values

a spectral solution of the sturm liouville equation comparison of classical and nonclassical basis sets abstract this paper considers a spectral method of solution of the sturm liouville equation and the associated schroedinger equation the main objective is to develop a collocation method based on quadrature collocation points generated from nonclassical polynomials the polynomials and associated quadrature points are calculated with gautschi s stieltjes procedure from some specified weight function the particular spectral method used here with nonclassical basis sets is referred to as the quadrature discretization method qdm the qdm and a related weighted qdm are applied to several sturm liouville and schroedinger equations and the results are compared with the traditional spectral methods based on chebyshev and legendre quadrature points the results are also compared with the results of other workers wherever available the qdm was found to give the most rapid convergence relative to other methods for the problems studied

shooting methods for the schrodinger equation a simple stable shooting method for the schrodinger equation is described and is shown to work with various integrators including the numerov one quantum mechanical techniques are extended to apply to problems for which e is not a traditional eigenvalue and are shown to permit the extraction of useful information even from an unstable shooting method

weights of the exponential fitting multistep algorithms for first order odes abstract we describe a numerical method for the calculation of the weights of the linear multistep algorithms for solving first order differential equations the main novelties are that i we admit nonequidistant mesh points in the partition and ii the weights are determined on the basis of the exponential functions exp i x i 1 2 3 rather than on the power function set as it is done for the classical weights in this way the method allows computing not only the weights of the well established algorithms but also those of new ones another novelty consists in the construction of a general scheme for the error analysis of this kind of algorithms some relevant numerical illustrations are given

a family of trigonometrically fitted partitioned runge kutta symplectic methods abstract we are presenting a family of trigonometrically fitted partitioned runge kutta symplectic methods of fourth order with six stages the solution of the one dimensional time independent schrodinger equation is considered by trigonometrically fitted symplectic integrators the schrodinger equation is first transformed into a hamiltonian canonical equation numerical results are obtained for the one dimensional harmonic oscillator and the exponential potential

fast subset scan for spatial pattern detection summary we propose a new fast subset scan approach for accurate and computationally efficient event detection in massive data sets we treat event detection as a search over subsets of data records finding the subset which maximizes some score function we prove that many commonly used functions e g kulldorff s spatial scan statistic and extensions satisfy the linear time subset scanning property enabling exact and efficient optimization over subsets in the spatial setting we demonstrate that proximity constrained subset scans substantially improve the timeliness and accuracy of event detection detecting emerging outbreaks of disease 2 days faster than existing methods

a sixth order exponentially fitted method for the numerical solution of the radial abstract a new sixth order method of runge kutta type is developed for the numerical integration of the single channel radial schrodinger equation the formula derived contains certain free parameters which allows it to be fitted automatically to exponential functions extensive numerical testing on the resonance problem and on the bound states problem indicates that this new method is generally more efficient than other previously developed finite difference methods

numerov method maximally adapted to the schro dinger equation abstract we construct and investigate the two step scheme which is maximally specialized for the schrodinger equation the new scheme is found to preserve all attractive features of the standard numerov method ease in programming flexibility in application speed in run etc with the important advantage of being more efficient for problems involving high values of energy its efficiency is actually found to be close to that of the piecewise perturbation methods a fact which strongly recommends it for further applications

a new variable step method for the numerical integration of the one dimensional schro dinger equation the numerical integration of the one dimensional schrodinger equation is considered a new variable step method is presented within the canonical functions scheme characterized namely by i the replacement of the schrodinger equation by the equivalent integral equation ii the representation of the wavefunction by a superposition of simple integrals iii the definition of taylor series like difference equations at each interval an estimate of the step size h is determined in order to obtain the desired local accuracy for the practical case of a given potential function with polynomial interpolations the step size h is directly given by a simple formula depending on i the potential function in the interval ii the initial values of the wavefunction at the origin of the interval this variable step method is applied to the morse potential function used for the recent raptis and cash method comput phys commun 36 113 1985 for this application the present method presents some advantages over the previous one

explicit numerov type methods with reduced number of stages we present in this paper a new approach for the derivation of hybrid explicit numerov type methods the new methodology does not require the intermediate use of high accuracy inter polatory nodes since we only need the taylor expansion of the internal points as a consequence a sixth order method is produced at a cost of only four stages per step instead of six stages needed for the methods which have appeared in the literature until now numerical results over some well known problems in physics and mechanics indicate the superiority of the new method 2003 elsevier science ltd all rights reserved

fast generalized subset scan for anomalous pattern detection we propose fast generalized subset scan fgss a new method for detecting anomalous patterns in general categorical data sets we frame the pattern detection problem as a search over subsets of data records and attributes maximizing a nonparametric scan statistic over all such subsets we prove that the nonparametric scan statistics possess a novel property that allows for efficient optimization over the exponentially many subsets of the data without an exhaustive search enabling fgss to scale to massive and high dimensional data sets we evaluate the performance of fgss in three real world application domains customs monitoring disease surveillance and network intrusion detection and demonstrate that fgss can successfully detect and characterize relevant patterns in each domain as compared to three other recently proposed detection algorithms fgss substantially decreased run time and improved detection power for massive multivariate data sets

an exponentially fitted eighth order method for the numerical solution of the schrodinger equation an eighth algebraic order exponentially fitted method is developed for the numerical integration of the schrodinger equation the formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions an comparative error analysis is also given numerical and theoretical results indicate that the new method is much more efficient than other classical and exponentially fitted methods

an elliptic spatial scan statistic the spatial scan statistic is commonly used for geographical disease cluster detection cluster evaluation and disease surveillance the most commonly used shape of the scanning window is circular in this paper we explore an elliptic version of the spatial scan statistic using a scanning window of variable location shape eccentricity angle and size and with and without an eccentricity penalty the method is applied to breast cancer mortality data from northeastern united states and female oral cancer mortality in the united states power comparisons are made with the circular scan statistic

on computing eigenvalues of the schrodinger equation for symmetrical potentials the problem of the numerical determination of eigenvalues of the one dimensional schrodinger equation with symmetrical potential v x is considered the problem is reduced to the definition of an eigenvalue function g e well defined for the given potential v it is theoretically proved that the even parity eigenvalues are given by g e 0 and the odd parity eigenvalues are given by g e infinity the method is canonical in the sense that it is independent of the eigenfunction yet along with the eigenvalue it allows the determination of the eigenfunction initial values this method is applied to the potentials v x2 v x6 bx2 v x2 lambda x2 1 gx2 where exact eigenvalues are available the numerical results compared with the exact ones and with those of previous confirmed methods show that the present method is accurate and efficient

rational computations of the topological k theory of classifying spaces of discrete groups we compute rationally the topological complex k theory of the classifying space bg of a discrete group provided that g has a cocompact g cw model for its classifying space for proper g actions for instance word hyperbolic groups and cocompact discrete subgroups of connected lie groups satisfy this assumption the answer is given in terms of the group cohomology of g and of the centralizers of finite cyclic subgroups of prime power order we also analyze the multiplicative structure

a four step method for the numerical solution of the schro dinger equation abstract a new four step exponentially fitted method is developed in this paper the expressions for the coefficients of the method are found such as to ensure the optimal approximation to the eigenvalue schrodinger equation i e equivalent to positive energy

frequency evaluation in exponential fitting multistep algorithms for odes we consider the linear multistep algorithms for first order odes and examine the problem of how the frequencies should be tuned in order to obtain the maximal benefit from the exponential fitting versions of such algorithms we find out that the key of the answer consists in analysing the behaviour of the error on further investigating the simple case of two step bdf algorithms we produce formulae for the optimal s and show that if the optimal s are used the order of the method is increased by one unit the reported numerical illustrations suggest that further investigations along these lines deserve a real attention

cluster detection of spatial regression coefficients popular approaches to spatial cluster detection such as the spatial scan statistic are defined in terms of the responses here we consider a varying coefficient regression and spatial clusters in the regression coefficients for varying coefficient regression such as the geographically weighted regression different regression coefficients are obtained for different spatial units it is often of interest to the practitioners to identify clusters of spatial units with distinct patterns in a regression coefficient but there is no formal statistical methodology for that rather cluster identification is often ad hoc such as by eyeballing the map of fitted regression coefficients and discerning patterns in this paper we develop new methodology for spatial cluster detection in the regression setting based on hypotheses testing we evaluate our methods in terms of power and coverages for true clusters via simulation studies for illustration our methodology is applied to a cancer mortality dataset copyright 2016 john wiley sons ltd