analysis of hydraulic tomography using temporal moments of drawdown recovery data received 2 june 2005 revised 30 september 2005 accepted 21 october 2005 published 1 february 2006 1 transient hydraulic tomography tht is a potentially cost effective and highresolution technique for mapping spatial distributions of the hydraulic conductivity and specific storage in aquifers interpretation of abundant well hydrographs of a tht survey however is a computational challenge we take on this challenge by developing an estimation approach that utilizes the zeroth and first temporal moments of well hydrographs instead of drawdown itself the governing equations for the temporal moments are poisson s equations these equations demand less computational resources as opposed to the parabolic equation that governs drawdown evolution likewise the adjoint equations for evaluating sensitivities of the moments for parameter estimation also take the same forms therefore a temporal moment approach is expected to expedite the interpretation of tht surveys on the basis of this premise we extend our sequential successive linear estimator to use the zeroth moment and characteristic time of the drawdown recovery data generated by tht surveys we subsequently investigate computational efficiency and accuracy of the moment approach results of the investigation show that the temporal moment approach yields results similar to those from the approach that uses transient heads but at significantly less computational costs limitations using temporal moments are discussed subsequently

exact solutions of the schrodinger equation for a class of anharmonic oscillators abstract a mathematical basis for the existence of exact solutions to the time independent schrodinger equation for a class of quantum anharmonic oscillators is given a natural extension of that basis provides a broader class than that given by flessas some numerical examples of the calculation of eigenvalues are given a brief comparison is made with the earlier work of hioe macmillen and montroll

traditional analysis of aquifer tests comparing apples to oranges 1 traditional analysis of aquifer tests uses the observed drawdown at one well induced by pumping at another well for estimating the transmissivity t and storage coefficient s of an aquifer the analysis relies on theis solution or jacob s approximate solution which assumes aquifer homogeneity aquifers are inherently heterogeneous at different scales if the observation well is screened in a low permeability zone while the pumping well is located in a high permeability zone the resulting situation contradicts the homogeneity assumption in the traditional analysis as a result what does the traditional interpretation of the aquifer test tell us using numerical experiments and a first order correlation analysis we investigate this question results of the investigation suggest that the effective t and s for an equivalent homogeneous aquifer of gaussian random t and s fields vary with time as well as the principal directions of the effective t the effective t and s converge to the geometric and arithmetic means respectively at large times analysis of the estimated t and s using drawdown from a single observation well shows that at early time both estimates vary with time the estimated s stabilizes rapidly to the value dominated by the storage coefficient heterogeneity in between the pumping and the observation wells at late time the estimated t approaches but does not equal the effective t it represents an average value over the cone of depression but influenced by the location size and degree of heterogeneity as the cone of depression evolves

fourth order difference equation for the first associated of classical discrete orthogonal polynomials we derive the fourth order difference equation satisfied by the first associated of classical orthogonal polynomials of a discrete variable we give it explicitly for first associated of hahn polynomials from which can be derived by a limiting process the equation satisfied by first associated of all classical families continuous and discrete

generalized hermite polynomials associated with functions of parabolic cylinder associated hermite polynomials he n n z which generalize the usual scaled hermite polynomials he n z corresponding to n 0 are introduced for the purpose to represent the raising and lowering of the indices of functions of the parabolic cylinder d n z in finite integer steps properties of these polynomials such as recursion relations explicit representations and the differential equation are derived the generation of the associated hermite polynomials from the usual hermite polynomials by differential operators representable by means of the confluent hypergeometric function is given an application for the explicit calculation of the functions of the parabolic cylinder for negative integer indices is discussed other applications are visible for the investigation of the zeros of the functions of the parabolic cylinder

computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow finite difference and finite element methods are frequently used to study aquifer flow however additional analysis is required when model parameters and hence predicted heads are uncertain computational algorithms are presented for steady and transient models in which aquifer storage coefficients transmissivities distributed inputs and boundary values may all be simultaneously uncertain innovative aspects of these algorithms include a new form of generalized boundary condition a concise discrete derivation of the adjoint problem for transient models with variable time steps an efficient technique for calculating the approximate second derivative during line searches in weighted least squares estimation and a new efficient first order second moment algorithm for calculating the covariance of predicted heads due to a large number of uncertain parameter values the techniques are presented in matrix form and their efficiency depends on the structure of sparse matrices which occur repeatedly throughout the calculations details of matrix structures are provided for a two dimensional linear triangular finite element model

the force gradient symplectic finite difference time domain scheme for maxwell s equations in this paper a novel force gradient explicit symplectic finite difference time domain sfdtd algorithm based on the rm t rm v type hamiltonian decomposition is derived to investigate the propagation characteristics of electromagnetic waves in which maxwell s equations are numerically integrated by third order symplectic map combined with force gradient in the time direction the numerical stability and dispersion analyses are presented the numerical results effectively confirm that the force gradient sfdtd f sfdtd method is reliable for electromagnetic wave computation its accuracy is superior to the non force gradient sfdtd method and the traditional fdtd method the implementation of f sfdtd method is easily applied to compute the back scattered radar cross sections rcs using 2 d tm mode for a perfectly conducting cylinder

the quantum relativistic harmonic oscillator spectrum of zeros of its wave functions the distribution of zeros of the wave functions of the quantum relativistic harmonic oscillator is studied an analytic approximate expression for this distribution is derived within the framework of the wkb approximation using potential theory techniques it is proved that this wkb approach gives the right asymptotic distribution which is explicitly calculated

distributions of zeros of discrete and continuous polynomials from their recurrence relation the hypergeometric polynomials in a continuous or a discrete variable whose canonical forms are the so called classical orthogonal polynomial systems are objects which naturally appear in a broad range of physical and mathematical fields from quantum mechanics the theory of vibrating strings and the theory of group representations to numerical analysis and the theory of sturm liouville differential and difference equations often they are encountered in the form of a three term recurrence relation ttrr which connects a polynomial of a given order with the polynomial of the contiguous orders this relation can be directly found in particular by use of lanczos type methods tight binding models or the application of the conventional discretization procedures to a given differential operator here the distribution of zeros and its asymptotic limit characterized by means of its moments around the origin is described for the continuous classical hermite laguerre jacobi bessel polynomials and for the discrete classical charlier meixner kravchuk hahn polynomials by means of a general procedure which i only requires the three term recurrence relation and ii avoids the often high brow subtleties of the potential theoretic considerations used in some recent approaches let us underline that the orthogonality condition is not required in our approach the moments are given in an explicit manner which at times allow us to recognize the analytical form of the corresponding distribution

the moments of the density of zeros for the relativistic hermite and laguerre polynomials abstract the moments of the density of zeros of two new orthogonal polynomial systems called relativistic hermite polynomials h n n n 0 rhp and relativistic laguerre polynomials l n n n 0 rlp are represented by means of a cases method and of a representation formula introduced by ricci in terms of generalized lucas polynomials of the second kind by using a fortran program numerical computations are explicitly developed in some particular case

factorization of fourth order differential equations for perturbed classical orthogonal polynomials we factorize the fourth order differential equations satisfied by the laguerre hahn orthogonal polynomials obtained from some perturbations of classical orthogonal polynomials such as the rth associated for genetic r the general co recursive the general co recursive associated the general co dilated and the general co modified classical orthogonal polynomials moreover we find four linearly independent solutions of the fourth order differential equations and show that the factorization obtained for modifications of classical orthogonal polynomials is still valid with some minor changes when the polynomial family modified is semi classical finally we extend the validity of the results obtained for the associated classical orthogonal polynomials with integer order of association from integers to reals

computing the moments of the density of zeros for orthogonal polynomials abstract a representation formula by means of the generalized lucas polynomials of first kind for the moments of the density of zeros of orthogonal polynomial sets defined by a three term recurrence relation is used to derive information directly from the entries of the jacobi matrix numerical computations are explicitly developed in some particular case

resonance tongues and instability pockets in the quasi periodic hill schrodinger equation this paper concerns hill s equation with a parametric forcing that is real analytic and quasi periodic with frequency vector d and a frequency or energy parameter a and a small parameter b the 1 dimensional schrodinger equation with quasi periodic potential occurs as a particular case in the parameter plane 2 a b for small values of b we show the following the resonance tongues with rotation number frac 1 2 langle bf k omega rangle bf k in mathbb z d have c boundary curves our arguments are based on reducibility and certain properties of the schrodinger operator with quasi periodic potential analogous to the case of hill s equation with periodic forcing i e d 1 several further results are obtained with respect to the geometry of the tongues one result regards transversality of the boundaries at b 0 another result concerns the generic occurrence of instability pockets in the tongues in a reversible near mathieu case that may depend on several deformation parameters these pockets describe the generic opening and closing behaviour of spectral gaps of the schrodinger operator in dependence of the parameter b this result uses a refined averaging technique also consequences are given for the behaviour of the lyapunov exponent and rotation number in dependence of a for fixed b

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

the schrodinger equation for the x2 λx2 1 gx2 interaction certain exact odd and even wavefunctions for the potential x2 x2 1 gx2 x in are obtained the coefficients of the wavefunctions and expressions for the energies are obtained using the symbolic manipulation code reduce the chief use of these results is to provide a check for the effectiveness of numerical algorithms the results supplement those of flessas 1982 for the same potential on xe 0

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

eigenvalues from power series expansions an alternative approach an appropriate rational approximation to the eigenfunction of the schrodinger equation for anharmonic oscillators enables one to obtain the eigenvalue accurately as the limit of a sequence of roots of hankel determinants the convergence rate of this approach is greater than that for a well established method based on power series expansions weighted by a gaussian factor with an adjustable parameter the so called hill determinant method

rational jacobi matrices and certain quantum mechanical problems in several branches of physics the hamiltonian of a many body system can be reduced to a rational jacobi matrix i e a jacobi matrix whose elements are rational functions of the suffix by means of a method of the lanczos type here it is shown that one can calculate in a simple analytical way the asymptotic eigenvalue density of all these matrices by means of its moments without solving the corresponding eigenvalue problem the method is applied to a large class of quantum mechanical models of hamiltonians

definite integrals as solutions for the x2 λx2 1 gx2 potential presents exact solutions and eigenvalues for the schrodinger equation with the x2 x2 1 gx2 interaction on the half axis x or 0 the solutions are given in the form of definite integrals and the eigenvalues by means of a well defined limiting procedure

applications of the monotonicity of extremal zeros of orthogonal polynomials in interlacing and optimization problems abstract we investigate monotonicity properties of extremal zeros of orthogonal polynomials depending on a parameter using a functional analysis method we prove the monotonicity of extreme zeros of associated jacobi associated gegenbauer and q meixner pollaczek polynomials we show how these results can be applied to prove interlacing of zeros of orthogonal polynomials with shifted parameters and to determine optimally localized polynomials on the unit ball

exact solutions for nonpolynomial potentials in n space dimensions using a factorization method and supersymmetry a supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of a broad class of nonpolynomial potentials in an arbitrary space dimension under suitable constraints on the potential parameters for the specific nonpolynomial oscillator potential v x x2 x2 1 gx2 previous analytic results including algebraic and integral type in one and three dimensions are easily obtainable from these compact expressions interesting aspects of the existence of pairs of solutions for certain combinations of and g are also discussed

further polynomial type eigenfunctions the method developed previously by the author for obtaining eigenfunctions in the form polynomial exponential polynomial for a linear second order differential equation in normal form is extended to embrace the general case when the interaction potential or square of the refractive index profile has the form polynomial polynomial various situations arise in the development of the theory some of which can be concluded algebraically but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters

theory of recursive generation of systems of orthogonal polynomials an illustrative example abstract pending its publication in full detail the theory of recursive generation of systems of orthogonal polynomials is verbally described and illustrated by means of a typical example of practical application for this purpose the theory is applied to the polynomials of legendre chosen as initially given orthogonal sequence

on the zeros of eigenfunctions of polynomial differential operators it is shown that for polynomial eigenfunctions of an ordinary polynomial differential operator with coefficients depending only on the independent variable it is possible to determine the density of nodes around the mean without solving the corresponding eigenvalue problem this is done by means of the first few moments which can be directly expressed in terms of the above mentioned coefficients also very simple expressions for the asymptotic values i e when the degree of the polynomial becomes very large of these quantities are found for illustration these results are applied to various orthogonal polynomials which satisfy ordinary differential equations of second fourth and or sixth order

use of power averaging for quantifying the influence of structure organization on permeability upscaling in on lattice networks under mean parallel flow with the local permeability distribution variance s 2 is nonnegligible but remains small it is equal to 0 09 for sparse networks and 0 14 for dense networks representing 4 5 and 7 respectively of the full possible range of w values power averaging is not strictly valid but gives an estimate of upscaling at a few percent here w depends slightly on the local permeability distribution shape beyond its variance but mostly on the morphological network structures most of the morphological control on upscaling for on lattice structures is local and topological and can be explained by the dependence on the average number of neighbor by points effective coordination number within the flowing structure backbone

spectral properties of solutions of hypergeometric type differential equations abstract the second order differential equation x y x y y 0 is usually called equation of hypergeometric type provided that are polynomials of degree not higher than two and one respectively and is a constant their solutions are commonly known as hypergeometric type functions htfs in this work a study of the spectrum of zeros of those htfs for which 1 2 1 v r and are independent of is done within the so called semiclassical or wkb approximation specifically the semiclassical or wkb density of zeros of the htfs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy applications to the gaussian and confluent hypergeometric functions as well as to hermite functions are shown

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

relativistic orthogonal polynomials are jacobi polynomials we identify the recently studied relativistic orthogonal polynomials in terms of jacobi polynomials so all their properties follow from the corresponding properties of jacobi polynomials through a change of variable

wkb approach to zero distribution of solutions of linear second order differential equations given a second order linear differential equation y z s z y z 0 the distribution of zeros of its solutions is defined by v y z 0 z where z stands for the dirac delta at the point z some techniques of approximation of the restriction of v to r directly from s z are considered in particular for the wkb method error bounds are provided and some related results established in the second part formulas for the appropriate scaling in the holonomic case are given as an illustration we obtain the asymptotic distribution of the real zeros of some families of polynomials

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

newton sum rules of zeros of semiclassical orthogonal polynomials abstract the distribution of zeros of the semiclassical orthogonal polynomials with weights w x x w c x wc x denoting a classical weight function and x being equal to x c or x2 c2 is investigated via the newton sum rules of zeros i e the sums of the rth powers of zeros recursion relations satisfied by these sum rules for semiclassical legendre laguerre and hermite polynomials are explicitly given extensions to other semiclassical polynomials are indicated

direct expectation value calculations a new application of the hellmann feynman theorem is shown to allow accurate calculation of expectation values along with energies for the schrodinger equation for variable boundary position and boundary conditions power series and finite difference versions of the method are developed and applied to perturbed oscillator and perturbed hydrogen atom problems

quasi orthogonal polynomials quadrature and interpolation abstract we investigate the quasi orthogonal polynomials by expressing them as characteristic polynomials of symmetric matrices many properties of zeros of quasi orthogonal polynomials are derived from this formulation we derive new formulae for quadrature and interpolation based on the zeros of quasi orthogonal polynomials and prove the mean convergence of lagrange interpolation

the schrodinger equation for the interaction potential x2 λx2 1 gx2 and the first heun confluent equation the authors present systematically the solutions and the quasi polynomial solutions for the schrodinger equation with the x2 lambda x2 1 gx2 interaction potential with the help of the first heun confluent equation or spheroidal heun equation

potential r2 λr2 1 gr2 and the analytic continued fractions the author constructs the exact wavefunctions and an analytic continued fractional green function for the one and three dimensional schrodinger equation with the potential v r h r2 r2 lambda r2 1 gr2 h 1 4 lambda g 0 in the numerical application of the formulae the energy and norm of the boundary state psi may be approximated by both their lower and upper estimates

fourth order differential equation satisfied by the associated of any order of all classical orthogonal polynomials a study of their distribution of zeros abstract the first associated numerator polynomials of all classical orthogonal polynomials satisfy one fourth order differential equation valid for the four classical families but for the associated of arbitrary order the differential equations are only known separately in this work we introduce a program built in mathematica symbolic language which is able to construct the unique differential equation satisfied by the associated of any order of the classical class then we use this differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of the kth power of zeros which are closely related with the moments of such a distribution

on two sets of orthogonal polynomial systems encountered in nonlinear physics two sets with an infinite number of new systems of orthogonal polynomials have recently been discovered by smith in connection with some non linear physical problems e g the dispersion of a buoyant contaminant in a fluid they appear as solutions of non linear differential equations let p x m k s and q x m k with n 0 k m k 2m 2m k denote a generic system of each set the positive integers k and m are restricted by k 1 k although the orthogonality interval of these polynomials is real their zeros are generally complex here the sum rules y x x r 1 2 for the zeros x i 1 2 n of the nth degree polynomials of these sets are studied it is found that all these quantities vanish except for r pm p being an arbitrary positive integer simple recurrent expressions for ypn are given

an evaluation of the bouwer and rice method of slug test analysis the method of bouwer and rice 1976 for analyzing slug test data is widely used to estimate hydraulic conductivity k based on steady state flow assumptions this method is specifically intended to be applicable to unconfined aquifers therefore it is of practical value to investigate the limits of accuracy of the k estimates obtained with this method accordingly using a numerical model for transient flow we evaluate the method from two perspectives first we apply the method to synthetic slug test data and study the error in estimated values of k second we analyze the logical basis of the method parametric studies helped assess the role of the effective radius parameter specific storage screen length and well radius on the estimated values of k the difference between unconfined and confined systems was studied via conditions on the upper boundary of the flow domain for the cases studied the bouwer and rice analysis was found to give good estimates of k with errors ranging from 10 to 100 we found that the estimates of k were consistently superior to those obtained with hvorslev s 1951 basic time lag method in general the bouwer and rice method tends to underestimate k the greatest errors occurring in the presence of a damaged zone around the well or when the top of the screen is close to the water table when the top of the screen is far removed from the upper boundary of the system no difference is manifest between confined and unconfined conditions it is reasonable to infer from the simulated results that when the screen is close to the upper boundary the results of the bouwer and rice method agree more closely with a confined idealization than an unconfined idealization in effect this method treats the aquifer system as an equivalent radial flow permeameter with an effective radius re which is a function of the flow geometry our transient simulations suggest that re varies with time and specific storage thus the effective radius may be reasonably viewed as a time averaged mean value the fact that the method provides reasonable estimates of hydraulic conductivity suggests that the empirical electric analog experiments of bouwer and rice have yielded shape factors that are better than the shape factors implicit in the hvorslev method

fourth order differential equations satisfied by the generalized co recursive of all classical orthogonal polynomials a study of their distribution of zeros the unique fourth order differential equation satisfied by the generalized co recursive of all classical orthogonal polynomials is given for any but fixed level of recursivity up to now these differential equations were known only for each classical family separately and also for a specific recursivity level moreover we use this unique fourth order differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of powers of their zeros which are closely related with the moments of such distribution both results are obtained with the help of two programs built in mathematica symbolic language

the relativistic hermite polynomial is a gegenbauer polynomial it is shown that the polynomials introduced recently by aldaya bisquert and navarro salas phys lett a 156 381 1991 in connection with a relativistic generalization of the quantum harmonic oscillator can be expressed in terms of gegenbauer polynomials this fact is useful in the investigation of the properties of the corresponding wave function some examples are given in particular related to the asymptotic behavior and to the distribution of zeros of the polynomials for large quantum numbers

geometry imbedded darcy s law and transient subsurface flow the traditional interpretation of darcy s experiment views it as a valuable means for setting up the partial differential equation of transient or steady state subsurface fluid flow in the present work darcy s observations are viewed from a different perspective enabling the statement of transient subsurface fluid flow in terms of an equation defined over finite domains of space and time two new notions namely geometry imbedding and location of average are introduced the equation describes transient flow along a flow tube with arbitrarily varying cross section consisting of materials with properties dependent on fluid potential this equation based on its own postulates is fully consistent within itself and exists independently of the classical partial differential equation this technical note presents preliminary ideas on what appears to be a promising new line of inquiry that departs from the traditional approach based on continuum mechanics further work is in progress

polynomial type eigenfunctions the differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial the problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form properties of the eigenvalues of this matrix are investigated since they are functions of a parameter occurring in the positive definite quadratic form the asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter

fourth order differential equations for numerator polynomials the author gives explicitly the fourth order differential equation satisfied by the numerator polynomials associated polynomials of the classical orthogonal polynomials the coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau and their derivatives defined via the relation sigma rho te this differential equation is therefore valid for the jacobi laguerre hermite and bessel associated polynomials

quasi orthogonality and differential equations abstract the following theorem is proved quasi orthogonal polynomials corresponding to the semi classical or laguerre hahn families and the orthogonal polynomials themselves verify differential equations of same order reducible to two or four by the hahn theorem

sum rules for zeros of polynomials and generalized lucas polynomials a representation formula in terms of generalized lucas polynomials of the second kind see formula 4 3 for the sum rules js i introduced by case j math phys 21 702 1980 and studied by dehesa et al j math phys 26 1547 1985 26 2729 1985 in order to obtain information about the zeros distribution of eigenfunctions of a class of ordinary polynomial differential operators is derived

shifted 1 n expansion approach to the interaction v r r2 λr2 1 gr2 the authors use the shifted 1 n expansion method to calculate the energy eigenvalues of the non polynomial oscillator v r r2 r2 1 gr2 the results thus obtained are in excellent agreement with exact numerical results it is also shown that the non polynomial interaction can be made supersymmetric and in this case also the results obtained by the shifted 1 n method matches extremely well with the exact analytic results given by supersymmetry

double exponential sinc collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function we show that the double exponential sinc collocation method provides an efficient uniformly accurate solution to the one dimensional time independent schrodinger equation for a general class of rational potentials of the form v x p x q x the derived algorithm is based on the discretization of the hamiltonian of the schrodinger equation using sinc expansions this discretization results in a generalized eigenvalue problem the eigenvalues of which correspond to approximations of the energy values of the starting hamiltonian a systematic numerical study is conducted beginning with test potentials with known eigenvalues and moving to rational potentials of increasing degree

spectral properties of the biconfluent heun differential equation abstract the spectrum of zeros of the polynomial solutions of the biconfluent heun differential equation is investigated by two different methods first the spectral newton sums i e the sums of the r th powers of the zeros are given in a rigorous and recurrent way second the density of zeros i e the number of zeros per unit interval is calculated in an explicit way within the so called semiclassical or bkw approximation this is done by using a general theorem which applies to a linear second order differential equation under certain conditions applications to the radial schrodinger equations associated to some quantum mechanical systems rotating harmonic oscillator confinement potentials as well as to the bessel differential equation are also shown

survey on symplectic finite difference time domain schemes for maxwell s equations to discretize maxwell s equations a variety of high order symplectic finite difference time domain schemes which use th order symplectic integration time stepping and th order staggered space differencing are surveyed first the order conditions for the symplectic integrators are derived second the comparisons of numerical stability dispersion and energy conservation are provided between the high order symplectic schemes and other high order time approaches finally these symplectic schemes are studied by using different space and time strategies according to our survey high order time schemes for matching high order space schemes are required for optimum electromagnetic simulation numerical experiments have been conducted on radiation of electric dipole and wideband s parameter extraction of dielectric filled waveguide the results demonstrate that the high order symplectic scheme can obtain satisfying numerical solutions under high courant friedrichs levy number and coarse grid conditions

hill determinants and 1 n theory a computationally efficient form of the hill determinant method is devised giving both e and x2 values and is used to test the accuracy of that method in the light of recent criticisms l 1 and n 1 asymptotic theories are used to check the numerical results

a comparison of methods to determine measurement support volumes we examine methods that can be used to identify the spatial filter function associated with the measurement of quantities on two scales we divide the methods into two general classes deconvolution methods and sensitivity based methods we briefly describe the two classes of methods provide examples of each from the literature and introduce a new sensitivity based method inspired by backus gilbert model resolution ideas we compare the approaches by examining the filter function that relates smaller scale grid block parameters to parameters determined by a theis model interpretation of drawdown data in general we find that sensitivity approaches are easier to implement numerically and are less subject to the problems of identifiability stability and uniqueness associated with deconvolution methods

study of a class of non polynomial oscillator potentials we develop a variational method to obtain accurate bounds for the eigenenergies of h v in arbitrary dimensions n 1 where v r is the nonpolynomial oscillator potential the variational bounds are compared with results previously obtained in the literature an infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues

the weight functions generating functions and miscellaneous properties of the sequences of orthogonal polynomials of the second kind associated with the jacobi and the gegenbauer polynomials abstract in the introduction the main object of the paper namely the calculation and study of the weight functions and orthogonality relations corresponding to the sequences of polynomials of the second kind associated with the jacoby and the gegenbauer polynomials is briefly discussed section 2 contains the underlying theory restricted to the citation of a few general formulas regarding the orthogonality property of associated polynomials of the second kind section 3 which deals with the application to the case of the jacobi polynomials p n z n n starts with the construction of the orthogonality relation and the weight function 1 of the associated polynomials for arbitrary gt 1 the denominator of the fraction representing 1 comprises the principal value of an integral which is studied in detail in order to bring it in an explicit form most suited for practical use attention is also paid to the associated functions of the second kind and some of their properties next in a subsection 1 is expressed solely in terms of elementary functions in a number of special cases involving subsets of pairs when 1 1 n gt 1 1 1 gt0 and 1 0 gt 1 in another subsection the generating function for the associated polynomials is calculated and the analysis of the result comprises some related formulas in the final subsection the moments of the weight function 1 with respect to 0 1 and 1 are expressed in terms of the corresponding moments of the weight function of the jacobi polynomials both in finite determinantal form and by means of a recurrence relation in section 4 which concerns the application to the case of the gegenbauer polynomials c v n z n n the advantages of a new definition of the associated polynomials of the second kind are at first expounded then follow the orthogonality relation and the weight function 1 for arbitrary v gt 1 2 next 1 is calculated in terms of elementary functions for all integer and half odd integer values of v exceeding 1 2 in the former case a remarkable simplification occurs in the denominator of the fraction representing 1 after that follow three subsections devoted respectively to the functions of the second kind associated with the gegenbauer polynomials two different generating functions for the associated polynomials and related formulas and the moments of the weight function 1 the paper ends with two appendices in which certain types of sums are reduced to a one term expression

fourth order difference equation for the associated classical discrete orthogonal polynomials we derive the fourth order dierence equation satised by the associated order r of classical orthogonal polynomials of a discrete variable the coecients of this equation are given in terms of the polynomials and which appear in the discrete pearson equation dening the weight x of the classical discrete orthogonal polynomials c 1998 elsevier science b v all rights reserved ams classication 33c25

exact analytical eigenfunctions for the x2 λ x2 1 gx2 interaction the author discusses an ansatz for the eigenfunctions of the x2 x2 1 gx2 interaction introduced by blecher and leach 1987 the author shows that the ansatz provides pairs of solutions in five cases having and g connected by 6g2 4g and conjectures the possible existence of an infinite number of such solution pairs

on the x2 λx2 1 gx2 interaction presents an infinite set of exact solutions of odd parity for the one dimensional schrodinger equation with the interaction x2 lambda x2 1 gx2 these complement the exact solutions of even parity that have been obtained recently by flessas phys lett a vol 84 p 121 1981

computation of newton sum rules for associated and co recursive classical orthogonal polynomials abstract by using some preceding results of buendia et al in alfaro et al eds orthogonal polynomials and their applications lecture notes in mathematics vol 1329 springer berlin 1986 pp 222 235 ricci j math phys 34 1993 4884 4891 natalini calcolo 31 1994 127 144 and differential equations of associated see belmedhi ronveaux rend mat 11 1991 313 326 zarzo et al j comput appl math 49 1993 349 359 and co recursive see ronveaux marcellan j comput appl math 25 1989 105 109 ronveaux et al j comput appl 59 1995 295 328 orthogonal polynomials we obtain numerical results for newton sum rules of associated and co recursive laguerre hermite and jacobi polynomials

the optimal force gradient symplectic finite difference time domain scheme for electromagnetic wave propagation in this communication we design an optimal force gradient symplectic finite difference time domain sfdtd scheme for numerically solving maxwell s equations based on the additional constraint of the minimized norm of fourth order truncation errors the numerical dissipation can be minimized and the optimal time coefficients for the third order force gradient symplectic method can be obtained due to the second order accurate adopted in space domain in general the proposed optimal sfdtd method is still second order accurate the numerical stability and numerical dispersion analysis indicate that the optimal force gradient sfdtd method has higher accuracy and stability compared with the nonoptimal force gradient sfdtd method moreover in order to verify the validity of the presented method the scattering parameters of a line fed rectangular microstrip antenna are calculated

electrostatic models for zeros of polynomials old new and some open problems we give a survey concerning both very classical and recent results on the electrostatic interpretation of the zeros of some well known families of polynomials and the interplay between these models and the asymptotic distribution of their zeros when the degree of the polynomials tends to infinity the leading role is played by the differential equation satisfied by these polynomials some new developments applications and open problems are presented

what does a slug test measure an investigation of instrument response and the effects of heterogeneity 1 we consider parameters determined by the inversion of slug test head recovery data with the homogeneous parameter model of cooper et al 1967 to be weighted spatial averages of transmissivity and storage defined at a smaller scale we quantify the spatial averaging using a power average spatial filter expression we determine the form of the filter function and the power exponent using numerically simulated slug test data the filter function that describes how smaller scale transmissivity is weighted by slug tests displays an approximate 1 r2 behavior with r the radial distance from the well the radius of the cylinderical volume that is averaged by the slug test is inversely proportional to the square root of the storage coefficient larger averaging volume with smaller storage the power exponent grows from 0 19 to 0 345 as the ratio of the characteristic scale of the heterogeneity to the characteristic scale of the averaging volume grows although a power exponent of zero corresponding to geometric averaging provides good results for most simulations our results show that while slug tests are useful to estimate transmissivity they have dubious value for estimating storage coefficients we find that the transmissivity estimate is unbiased and does not appear to be strongly influenced by storage properties the storage coefficient estimate is however strongly influenced by the transmissivity and is biased we investigate the interaction between storage coefficient and transmissivity by examining an analytical slug test model that contains two annular zones each with distinct transmissivity and storage coefficient

the three term recurrence relation and the differentiation formulas for hypergeometric type functions abstract the functions of hypergeometric type are the solutions y y x of the differential equation z y z y y 0 where are polynomials of degrees not higher than 2 and 1 respectively and is a constant here we consider a class of functions of hypergeometric type with the additional condition that 1 2 1 0 being a complex number in general moreover we assume that the coefficients of the polynomials and have no dependence on to this class of functions belong gauss kummer and hermite functions the classical orthogonal polynomials and many other functions encountered in linear and non linear physics we obtain two important structural properties of these functions i the so called three term recurrence relation which correlates three functions of successive orders and ii the differentiation formulas also called ladder or structure relations or even differential recurrence relations which relate the first derivative y z with the functions y z and y 1 z or y 1 z finally these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y where is a positive integer number and the associated contour is closed for completeness the explicit expressions corresponding to all classical orthogonal polynomials jacobi laguerre hermite and bessel are tabulated

algebraic and spectral properties of some quasiorthogonal polynomials encountered in quantum radiation the nodal structure of the wavefunctions of a large class of quantum mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials it is known that these polynomials i may be described by an arbitrary linear combination of two orthogonal polynomials pn x and ii have real and simple zeros here the three term recurrence relation the second order differential equation and the distribution of zeros of quasiorthogonal polynomials of the classical class i e when pn x is a jacobi laguerre or hermite polynomial are derived and analyzed specifically the exact values of the newton sum rules and the wkb density of zeros of these polynomials are found

spatial averaging of transmissivity in heterogeneous fields with flow toward a well steady state nonuniform flow in a heterogeneous medium is investigated using a combined numerical empirical approach the objective of the study is to determine the effective transmissivity of a square field with a constant head external boundary and a constant rate well at its center transmissivity at the point scale is modeled as an isotropic and multivariate lognormal spatial random function block transmissivities are defined using an empirical scaling up of point support values within the field the scaling process is a weighted spatial geometric averaging where log transmissivities are weighted by the inverse square of their distance from the well block transmissivities thus obtained and true effective values calculated using a numerical flow model are compared for realizations a finely discretized transmissivity field the principal finding of this study is that effective transmissivities obtained from the numerical flow model and block values obtained by spatial averaging are in excellent agreement for low to moderate variances of log transmissivity and moderately eccentric field geometries the geostatistical model for point scale transmissivity and the deterministic spatial averaging law are combined in order to create a geostatistical model for transmissivity at the block scale exact expressions are derived for the ensemble moments of block averaged transmissivities and well drawdowns in both the nonconditional case and the conditional case when transmissivity at the well bore is known the ensemble mean of block averaged transmissivity is found to decrease from the ensemble arithmetic mean toward the ensemble geometric mean as the field size becomes large compared to the integral range of spatial correlation this result contradicts earlier published work numerical results presented in this study also disagree with the analytical results of a recent stochastic analysis of specific discharge in radial systems

quasi orthogonality and zeros of some 2ϕ2 and 3ϕ2 polynomials abstract we state and prove the q extension of a result due to johnston and jordaan cf johnston and jordaan 2015 and make use of this result the orthogonality of q laguerre little q jacobi q meixner and al salam carlitz i polynomials as well as contiguous relations satisfied by the polynomials to establish the quasi orthogonality of certain 2 2 and 3 2 polynomials the location and interlacing properties of the real zeros of these quasi orthogonal polynomials are studied interlacing properties of the zeros of q laguerre quasi orthogonal polynomials l n z q when 2 1 with those of l n 1 1 z q and l n 1 z q are also considered

beyond thiem a new method for interpreting large scale pumping tests in heterogeneous aquifers 1 the heterogeneous character of aquifers prevents a consistent interpretation of pumping tests based on thiem s formula which has been developed for homogeneous aquifers we present a new mathematical method for interpreting an ensemble and single large scale pumping tests in heterogeneous confined isotropic aquifers by combining a multiscale technique called coarse graining with renormalization group analysis we derive an explicit formula for the coarse grained aquifer transmissivity the coarse grained transmissivity is dependent on the radial distance from the well close to the well it equals the local transmissivity in single pumping tests and the harmonic mean of the heterogeneous transmissivity field in ensemble pumping tests in the far field of the well the coarse grained transmissivity approaches the geometric mean the transition between both regimes is mainly determined by the transmissivities correlation length this relation is used for an inversely estimate of the statistical properties of the transmissivity such as its geometric mean in addition this method allows for determining aquifer structures characterized by a correlation length from a single pumping test while the variance can be estimated from pumping tests with ensemble averaged hydraulic heads

on the polynomial solutions of ordinary differential equations of the fourth order the density of zeros of polynomial solutions of ordinary differential equations of the fourth order with coefficients depending only on the independent variable is analyzed the first four moments of such a density are given directly in terms of the coefficients which characterize the differential operator application to the nonclassical orthogonal polynomials corresponding to the names krall legendre krall laguerre and krall jacobi is done global asymptotic properties of the zeros of these polynomials are also obtained

a study of the orthogonal polynomials associated with the quantum harmonic oscillator on constant curvature spaces recently carinena et al ann phys 322 434 2007 introduced a new family of orthogonal polynomials that appear in the wave functions of the quantum harmonic oscillator in two dimensional constant curvature spaces they are a generalization of the hermite polynomials and will be called curved hermite polynomials in the following we show that these polynomials are naturally related to the relativistic hermite polynomials introduced by aldaya et al phys lett a 156 381 1991 and thus are jacobi polynomials moreover we exhibit a natural bijection between the solutions of the quantum harmonic oscillator on negative curvature spaces and on positive curvature spaces at last we show a maximum entropy property for the ground states of these oscillators

on quasi orthogonal polynomials their differential equations discriminants and electrostatics abstract in this paper we develop a general theory of quasi orthogonal polynomials we first derive three term recurrence relation and second order differential equations for quasi orthogonal polynomials we also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials in addition we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the quasi orthogonal polynomials the examples of the freud weight w x e x 4 2 t x 2 and the jacobi weight w x 1 x 1 x are discussed in some detail finally we consider the nonlinear orthogonality preserving transformation and related matrix problem

second order difference equations for certain families of discrete polynomials abstract in this paper we will consider two algorithms which allow us to obtain second order linear difference equations for certain families of polynomials the corresponding algorithms can be implemented in any computer algebra system in order to obtain explicit expressions of the coefficients of the difference equations

shape factors for constant head double packer permeameters 1 local estimates of hydraulic conductivity are often obtained from constant head double packer tests the tests yield ratios of flow against head difference these are converted to values of hydraulic conductivity by dividing by a shape factor previously used shape factors have generally assumed that the packers are infinitely large such shape factors lead to overestimates of hydraulic conductivity as they ignore the effect of water flow around the packers a semianalytical solution is derived for a constant head double packer permeameter test the solution is compared with existing approximate techniques including the spheroidal equipotential solution the narrow packer approximation of rehbinder 1996 and the long packer approximation of rehbinder 2005 generally it is shown that ignoring the length of the packer and assuming a fully cased well lead to an overestimate of hydraulic conductivity the significance of this reduces with increasing well screen aspect ratio evaluation of the solution is nontrivial for this reason a polynomial relationship between shape factor well screen aspect ratio and packer length obtained by curve fitting with the semianalytical solution is provided