multi indexed q racah polynomials as the second stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with real shifts in one dimension the multi indexed q racah polynomials they are obtained from the q racah polynomials by the multiple application of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state vectors in a similar way to the multi indexed laguerre and jacobi polynomials reported earlier the virtual state vectors are the solutions of the matrix schrodinger equation with negative eigenvalues except for one of the two boundary points

a new fast physical optics for smooth surfaces by means of a numerical theory of diffraction a new technique to compute the physical optics po integral is presented the technique consists of a blind code that computes the different contributions stationary phase points end points etc numerically this technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plane this algorithm permits a fast and accurate evaluation of the po integral for smooth large surfaces the cpu time is almost independent of frequency

machine based methods in parameterized complexity theory we give machine characterizations of most parameterized complexity classes in particular of w p of the classes of the w hierarchy and of the a hierarchy for example we characterize w p as the class of all parameterized problems decidable by a nondeterministic fixed parameter tractable algorithm whose number of nondeterministic steps is bounded in terms of the parameter the machine characterizations suggest the introduction of a hierarchy wfunc between the w and the a hierarchy we study the basic properties of this hierarchy

linear operators with invariant polynomial space and graded algebra the irreducible finite dimensional representations of the graded algebras osp j 2 j 1 2 3 are expressed in terms of differential operators some quantum deformations of these algebras are shown to admit similar kinds of representations these are formulated in terms of finite difference operators the results are discussed in the framework of the quasi exactly solvable equations

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

recurrence relations of the multi indexed orthogonal polynomials v racah and q racah types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson and askey wilson types in this paper we explore those of the racah and q racah types for the m indexed q racah polynomials we derive 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation and annihilation operators of the quantum mechanical systems described by the multi indexed q racah polynomials are obtained in appendix b and appendix c we present a proof and some data of the recurrence relations with constant coefficients for the multi indexed wilson and askey wilson polynomials

the parametrized complexity of some fundamental problems in coding theory the parametrized complexity of a number of fundamental problems in the theory of linear codes and integer lattices is explored concerning codes the main results are that maximum likelihood decoding and weight distrubution are hard for the parametrized complexity class w 1 the np completeness of these two problems was established by berlekamp mceliece and van tilborg in 1978 using by means of a reduction from three dimensional matching on the other hand our proof of hardness for w 1 is based on a parametric polynomial time transformation from perfect code in graphs an immediate consequence of our results is that bounded distance decoding is likely to be hard for binary linear codes concerning lattices we address the theta series problem of determining for an integer lattice l given by a set of generators and a positive integer k whether there is a vector x in l of euclidean norm k we prove here for the first time that theta series is np complete and show that it is also hard for w 1 furthermore we prove that the nearest vector problem for integer lattices is hard for w 1 these problems are the counterparts of weight distrubution and maximum likelihood decoding for lattices relations between all these problems and combinatorial problems in graphs are discussed

quasi exactly solvable extensions of the lame equation quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors for a few of these equations such as the lame equation the number of algebraic solutions is larger than predicted by the general theory a class of quasi exactly solvable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed families of equations which by themselves are not quasi exactly solvable but can be transformed into systems of coupled quasi exactly solvable equations are discussed these results suggest a scheme for the classification of quasi exactly solvable systems in one variable

equilibrium positions shape invariance and askey wilson polynomials we show that the equilibrium positions of the ruijsenaars schneider van diejen systems with the trigonometric potential are given by the zeros of the askey wilson polynomials with five parameters the corresponding single particle quantum version which is a typical example of discrete quantum mechanical systems with a q shift type kinetic term is shape invariant and the eigenfunctions are the askey wilson polynomials this is an extension of our previous study 1 2 which established the discrete analogue of the well known fact the equilibrium positions of the calogero systems are described by the hermite and laguerre polynomials whereas the corresponding single particle quantum versions are shape invariant and the eigenfunctions are the hermite and laguerre polynomials

procedures for computing one and two dimensional integrals of functions with rapid irregular oscillations a collocation procedure for efficient integration of rapidly oscillatory functions is presented the integration problem is transformed into a certain o d e problem and this is solved by a collocation technique the method is also extended to two dimensional integration and some numerical results are appended showing the efficiency of the method in handling difficult cases of rapid irregular oscillations 1 the procedure for one dimensional integrals we consider integrals of the form a 1 1 1 f x e iq x dx where f is smooth and nonoscillatory and iq x i b a 1 two practical methods for evaluating rapidly oscillatory integrals are described in 1 the use of approximation as in filon s method 3 4 and the speedup method of longman 5 formally both methods are applicable to any integral of the form 1 1 but their best performance is for the case of a constant frequency q w in this note we present an efficient method which is applicable for cases of varying frequency q using only a small number of values of f and q in a b and the values q a and q b the proposed method follows the spirit of filon s method it is based upon the fact that if f were of the form 1 2 f x iq x p x p x l l1p x a x b then the integral could be evaluated directly as 1 3 j b iq x p x pt x eiq x dx f d p x eiq x dx b e qbp a eiqa equation 1 2 can be considered as a differential equation for p x and any solution of this equation can be used in 1 3 for evaluating i the general solution of this equation is 1 4 p x e iq x f t eiq t dt cj received march 24 1981 revised july 29 1981 1980 mathematics subject classification primary 65d30 secondary 65d32 1982 american mathematical society 0025 5718 8 1 ooo 0 1 23 03 25 531 this content downloaded from 157 55 39 104 on sun 19 jun 2016 06 36 01 utc all use subject to http about jstor org terms

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

parameterized circuit complexity and the w hierarchy abstract a parameterized problem l k belongs to w t if there exists k computed from k such that l k reduces to the weight k satisfiability problem for weft t circuits we relate the fundamental question of whether the w t hierarchy is proper to parameterized problems for constant depth circuits we define classes g t as the analogues of ac 0 depth t for parameterized problems and n t by weight k existential quantification on g t by analogy with np p we prove that for each t w t equals the closure under fixed parameter reductions of n t then we prove using sipser s results on the ac 0 depth t hierarchy that both the g t and the n t hierarchies are proper if this separation holds up under parameterized reductions then the w t hierarchy is proper we also investigate the hierarchy h t defined by alternating quantification over g t by trading weft for quantifiers we show that h t coincides with h 1 we also consider the complexity of unique solutions and show a randomized reduction from w t to unique w t

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

analysis of antennas on board arbitrary structures modeled by nurbs surfaces an accurate and efficient numerical scheme has been developed for predicting high frequency radiation patterns of antennas mounted on arbitrary structures modeled by parametric surfaces the method is based on geometric optics go and the uniform theory of diffraction utd nonuniform rational b splines nurbs surfaces have been used to describe the geometry of the structure as most of the computer aided geometric design cagd tools available in the industry are biased on nurbs the scheme can perform the electromagnetic analysis without any new or additional remeshing of the geometrical model a special ray tracing technique that combines go and utd with nurbs has been developed this technique uses some selective criteria in order to identify rapidly the nurbs where a ray impact may occur impact point coordinates are obtained by means of an optimization procedure based on the conjugate gradient method cgm the accuracy and efficiency of the approach are shown comparing it with other methods

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

multi indexed wilson and askey wilson polynomials as the third stage of the project multi indexed orthogonal polynomials we present in the framework of discrete quantum mechanics with pure imaginary shifts in one dimension the multi indexed wilson and askey wilson polynomials they are obtained from the original wilson and askey wilson polynomials by multiple applications of the discrete analogue of the darboux transformations or the crum krein adler deletion of virtual state solutions of types i and ii in a similar way to the multi indexed laguerre jacobi and q racah polynomials reported earlier

shape reconstruction of three dimensional conducting curved plates using physical optics nurbs modeling and genetic algorithm a microwave inverse scattering problem including a method for shape reconstruction of three dimensional electrically large conducting patches with simple geometries using genetic algorithm is presented unknown shape reconstruction algorithm starts from the knowledge of the simulated radar cross section rcs data through back scattering far field computation using physical optics approximation the forward problem involves the computation of the multiple frequency and multiple direction rcs of three dimensional large conducting patches modeled by nonuniform rational b spline nurbs surfaces the control points of nurbs are the geometrical parameters which are optimized for the shape reconstruction procedure the extended stationary phase method and critical cases which occur in physical optics computations in the forward problem are also discussed noise effect and the influence of increment in the number of control points of a nurbs over the inversion algorithm are investigated as well numerical results are presented to verify the operation of the proposed algorithm

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

efficient algorithm for the evaluation of the physical optics scattering by nurbs surfaces with relatively general boundary condition an adaptive integration algorithm is presented for the computation of the physical optics po electric and magnetic field scattered by electrically large objects modeled by non uniform rational b splines nurbs the algorithm is the customization of a more general purpose result that has been recently published by using a unique formulation both impenetrable e g impedance surfaces coated conductors as well as transparent thin sheet materials e g thin dielectric panels or frequency selective surfaces are treated via their fresnel reflection and transmission coefficients the po radiation integral is evaluated over the nurbs parametric domain since most of the computer aided geometric design cagd tools are based on nurbs the proposed algorithm allows a straightforward electromagnetic analysis of the structures by exploiting the standard available geometrical description with no need of generating new geometrical models furthermore the proposed adaptive sampling requires a number of integration points that is found to be drastically smaller than that resulting from standard nyquist based sampling integration algorithms such reduction of the sampling points is achieved by resorting to high frequency technique concepts and allows a significant reduction of the cpu computational burden therefore the algorithm is efficient and particulary suitable for the electromagnetic characterization of real life electrically large objects

exactly solvable birth and death processes many examples of exactly solvable birth and death processes a typical stationary markov chain are presented together with the explicit expressions of the transition probabilities they are derived by similarity transforming exactly solvable matrix quantum mechanics which is recently proposed by odake and the author s odake and r sasaki j math phys 49 053503 2008 the q askey scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role the most generic solvable birth death rates are rational functions of qx with x being the population corresponding to the q racah polynomial

fixed parameter tractability and completeness i basic results for many fixed parameter problems that are trivially soluable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as k feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is soluable in time bounded by a polynomial of degree c where c is a constant independent of k we establish the main results of a completeness program which addresses the apparent fixed parameter intractability of many parameterized problems in particular we define a hierarchy of classes of parameterized problems fpt subseteq w 1 subseteq w 2 subseteq cdots subseteq w sat subseteq w p and identify natural complete problems for w t for t geq 2 in other papers we have shown many problems complete for w 1 dominating set is shown to be complete for w 2 and thus is not fixed parameter tractable unless independent set clique irredundant set and many other natural problems in w 2 are also fixed parameter tractable we also give a compendium of currently known hardness results as an appendix

delaminating quadrature method for multi dimensional highly oscillatory integrals a delaminating quadrature method for multi dimensional highly oscillatory integrals is put forward in comparison with the levin method the new method has the benefits of being numerically stable and having a high computational speed numerical experiments testified the advantages of this method

failure of the hill determinant method for the sextic anharmonic oscillator an extended analysis of eigenvalues and wavefunctions resulting from the hill determinant method is carried out for the one dimensional sextic anharmonic oscillator it is shown that in spite of a seemingly natural ansatz of the method irrelevant wavefunctions may appear leading to incorrect eigenvalues the domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented

threshold dominating sets and an improved characterization of w 2 the threshold dominating set problem is that of determining for a graph g v e whether there is a subset v v of size k such that for each vertex v v there are at least r elements of the closed neighborhood n v that belong to v we consider the complexity of the problem parameterized by the pair k r it is trivial to observe that this is hard for w 2 it can also be easily shown to belong to a natural extension w 2 of w 2 defined in terms of circuit families of depth bounded by a function of the parameter we prove membership in w 2 and thus w 2 completeness using this as a starting point we prove that w 2 w 2

on an improved levin oscillatory quadrature method abstract performance of an improved levin quadrature method for oscillatory integrals is studied in the study the behavior of the target system of linear equations is analyzed and an error reduction factor is proposed to measure the behavior s impact on the integral result numerical investigations show that the error reduction factor is extremely small for ill conditioned case and the ill conditioning has little impact on the final integral result therefore the concerned quadrature method is numerically very stable and it has addressed the levin method s problem of being susceptible to the ill conditioning

finite dimensional irreducible modules of the universal askey wilson algebra since the introduction of askey wilson algebras by zhedanov in 1991 the classification of the finite dimensional irreducible modules of askey wilson algebras remains open a universal analog triangle q of the askey wilson algebras was recently studied in this paper we consider a family of infinite dimensional triangle q modules by the universal property of these triangle q modules we classify the finite dimensional irreducible triangle q modules when q is not a root of unity

application of physical optics to the rcs computation of bodies modeled with nurbs surfaces the paper presents a method for the computation of the monostatic radar cross section rcs of electrically large conducting objects modeled by nonuniform rational b spline nurbs surfaces using the physical optic po technique the nurbs surfaces are expanded in terms of rational bezier patches by applying the cox de boor transform algorithm this transformation is justified because bezier patches are numerically more stable than nurbs surfaces the po integral is evaluated over the parametric space of the bezier surfaces using asymptotic integration the scattering field contribution of each bezier patch is expressed in terms of its geometric parameters excellent agreement with po predictions is obtained the method is quite efficient because it makes use of a small number of patches to model complex bodies so it requires very little memory and computing time

on the evaluation of highly oscillatory integrals by analytic continuation we consider the integration of one dimensional highly oscillatory functions based on analytic continuation rapidly converging quadrature rules are derived for a general class of oscillatory integrals with an analytic integrand the accuracy of the quadrature increases both for the case of a fixed number of points and increasing frequency and for the case of an increasing number of points and fixed frequency these results are then used to obtain quadrature rules for more general oscillatory integrals i e for functions that exhibit some smoothness but that are not analytic the approach described in this paper is related to the steepest descent method but it does not employ asymptotic expansions it can be used for small or moderate frequencies as well as for very high frequencies the approach is compared with the oscillatory integration techniques recently developed by iserles and norsett

a new family of shape invariantly deformed darboux poschl teller potentials with continuous l we present a new family of shape invariant potentials which could be called a continuous version of the potentials corresponding to the exceptional x j1 jacobi polynomials constructed recently by the present authors in a certain limit it reduces to a continuous family of shape invariant potentials related to the exceptional x l1 laguerre polynomials the latter was known as one example of the conditionally exactly solvable potentials on a half line

evaluation of physical optics integrals from b spline surfaces by means of a fast locating algorithm of stationary points a novel modeling method based on the parametric domain decomposition is proposed for the b spline surfaces this method combines the advantages of conventional b spline and bezier surfaces moreover a fast algorithm based on the new modeling method is presented to locate the stationary points including inner and boundary stationary points of physical optics po integrals from targets modeled with double quadratic b spline surfaces in this algorithm the locating problem is reduced to a numerical or analytical solution no approximation is involved in the proposed modeling method and locating algorithm the stationary phase method by means of the fast locating algorithm is employed to evaluate po integrals the cpu time is independent of the incident frequency and physical dimension of scattered surfaces the accuracy and performance of the presented methods are validated by comparing them with those of the brute force method

parameterized computational complexity of dodgson and young elections abstract we show that the two np complete problems of dodgson score and young score have differing computational complexities when the winner is close to being a condorcet winner on the one hand we present an efficient fixed parameter algorithm for determining a condorcet winner in dodgson elections by a minimum number of switches in the votes on the other hand we prove that the corresponding problem for young elections where one has to delete votes instead of performing switches is w 2 complete in addition we study dodgson elections that allow ties between the candidates and give fixed parameter tractability as well as w 2 completeness results depending on the cost model for switching ties

efficient quadrature of highly oscillatory integrals using derivatives in this paper we explore quadrature methods for highly oscillatory integrals generalizing the method of stationary phase we expand such integrals into asymptotic series in inverse powers of the frequency the outcome is two families of methods one based on a truncation of the asymptotic series and the other extending an approach implicit in the work of filon filon 1928 proc r soc edinb 49 38 47 1 both kinds of methods approximate the integral as a linear combination of function values and derivatives with coefficients that may depend on frequency we determine asymptotic properties of these methods proving perhaps counterintuitively that their performance drastically improves as frequency grows the paper is accompanied by numerical results that demonstrate the potential of this set of ideas 1 ref 4

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

solvable discrete quantum mechanics q orthogonal polynomials with q 1 and quantum dilogarithm several kinds of q orthogonal polynomials with q 1 are constructed as the main parts of the eigenfunctions of new solvable discrete quantum mechanical systems their orthogonality weight functions consist of quantum dilogarithm functions which are a natural extension of the euler gamma functions and the q gamma functions q shifted factorials the dimensions of the orthogonal spaces are finite these q orthogonal polynomials are expressed in terms of the askey wilson polynomials and their certain limit forms

perturbations around the zeros of classical orthogonal polynomials starting from degree n solutions of a time dependent schrodinger like equation for classical orthogonal polynomials a linear matrix equation describing perturbations around the n zeros of the polynomial is derived the matrix has remarkable diophantine properties its eigenvalues are independent of the zeros the corresponding eigenvectors provide the representations of the lower degree 0 1 n 1 polynomials in terms of the zeros of the degree n polynomial the results are valid universally for all the classical orthogonal polynomials including the askey scheme of hypergeometric orthogonal polynomials and its q analogues

recurrence relations of the multi indexed orthogonal polynomials vi meixner pollaczek and continuous hahn types in previous papers we discussed the recurrence relations of the multi indexed orthogonal polynomials of the laguerre jacobi wilson askey wilson racah and q racah types in this paper we explore those of the meixner pollaczek and continuous hahn types for the m indexed meixner pollaczek and continuous hahn polynomials we present 3 2m term recurrence relations with variable dependent coefficients and 1 2l term l m 1 recurrence relations with constant coefficients based on the latter the generalized closure relations and the creation annihilation operators of the quantum mechanical systems described by the multi indexed meixner pollaczek and continuous hahn polynomials are obtained

unified theory of exactly and quasiexactly solvable discrete quantum mechanics i formalism we present a simple recipe to construct exactly and quasiexactly solvable hamiltonians in one dimensional discrete quantum mechanics in which the schrodinger equation is a difference equation it reproduces all the known ones whose eigenfunctions consist of the askey scheme of hypergeometric orthogonal polynomials of a continuous or a discrete variable the recipe also predicts several new ones an essential role is played by the sinusoidal coordinate which generates the closure relation and the askey wilson algebra together with the hamiltonian the relationship between the closure relation and the askey wilson algebra is clarified

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

hypercomplex fock states for discrete electromagnetic schrodinger operators a bayesian probability perspective abstract we present and study a new class of fock states underlying to discrete electromagnetic schrodinger operators from a multivector calculus perspective this naturally lead to hypercomplex versions of poisson charlier polynomials meixner polynomials among other ones the foundations of this work are based on the exploitation of the quantum probability formulation a la dirac to the setting of bayesian probabilities on which the fock states arise as discrete quasi probability distributions carrying a set of independent and identically distributed i i d random variables by employing mellin barnes integrals in the complex plane we obtain counterparts for the well known multidimensional poisson and hypergeometric distributions as well as quasi probability distributions that may take negative or complex values on the lattice h z n

fixed parameter tractability and completeness ii on completeness for w 1 abstract for many fixed parameter problems that are trivially solvable in polynomial time such as k dominating set essentially no better algorithm is presently known than the one which tries all possible solutions other problems such as feedback vertex set exhibit fixed parameter tractability for each fixed k the problem is solvable in time bounded by a polynomial of degree c where c is a constant independent of k in a previous paper the w hierarchy of parameterized problems was defined and complete problems were identified for the classes w t for t 2 our main result shows that independent set is complete for w 1

calculation of physical optics integrals over nurbs surface using a delaminating quadrature method physical optics po is a widely used radar cross section rcs estimation approach for electrically large scaled objects and it is finally reduced to the calculation of highly oscillatory integrals in this paper the po integrals over non uniform rational b spline nurbs surfaces is proposed to be calculated with a delaminating quadrature method the method has the merits of being very stable accurate and fast and its computation cost does not increase with frequency moreover in order to keep these good properties for po integrals involving critical points stationary phase point resonant point a scheme of integral subdividing is proposed and new approaches for locating the critical points are also presented numerical examples of flat surface cylindrical surface and spherical surface well show the advantages of the new method

parameterized complexity of candidate control in elections and related digraph problems there are different ways for an external agent to influence the outcome of an election we concentrate on control by adding or deleting candidates our main focus is to investigate the parameterized complexity of various control problems for different voting systems to this end we introduce natural digraph problems that may be of independent interest they help in determining the parameterized complexity of control for different voting systems including llull copeland and plurality voting devising several parameterized reductions we provide an overview of the parameterized complexity of the digraph and control problems with respect to natural parameters such as adding deleting only a bounded number of candidates or having only few voters

quadrature methods for multivariate highly oscillatory integrals using derivatives while there exist effective methods for univariate highly oscillatory quadrature this is not the case in a multivariate setting in this paper we embark on a project extending univariate theory to more variables inter alia we demonstrate that in the absence of critical points and subject to a nonresonance condition an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices a direct counterpart of the univariate case this provides a convenient avenue towards the generalization of asymptotic and filon type methods as formerly introduced by the authors in a single dimension to simplices and more generally to polytopes the nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth in effect its violation is equivalent to the presence of stationary points in a single dimension we further explore this issue and propose a technique that often can be used in this situation yet much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals

reply to comment on the darboux transformation and algebraic deformations of shape invariant potentials we reply to the comment on our recent paper made by drs sinha and roy we agree that the backward darboux transformation method used in our paper is equivalent to the approach based on ces potentials but we stress that the emphasis and the results of our paper are different