high precision calculation of the eigenvalues for the x2 λx2 1 gx2 potential an analytic continuation procedure using a taylor series is used to produce very accurate wavefunctions and eigenvalues for the one dimensional anharmonic oscillator governed by the potential v x x2 x2 1 gx2

p length and character degrees in p solvable groups abstract let g be a p solvable group where p is a prime we prove that the p length of g is less or equal then the number of distinct irreducible character degrees of g not divisible by p furthermore we prove that the result still holds if we impose some restriction on the field of values of the characters in particular if p 2 we can consider only rational valued characters

boolean functions and finite difference eigenvalues finite difference calculations involving the expectation values of boolean functions are shown to yield detailed information about the properties of the energy levels and eigenfunctions for double well potentials

energy levels of two dimensional anharmonic oscillators with sextic and octic perturbations two renormalized versions of inner product and hypervirial techniques are presented together with the hill determinant approach for performing accurate calculations of the energy levels of two dimensional oscillators characterized by the potentials vn 6 x y x2 y2 axxx6 ayyy6 3axyx4 y2 3ayxx2y4 and vn 8 x y x2 y2 axxx8 ayyy8 6b xyx4y4 4axyx6y2 4ayxx2y6 the methods have been used effectively to calculate the energy eigenvalues for many eigenstates over a wide range values of perturbation parameter and various values of coefficients aij the results produced by various techniques are compared with each other

an accurate method for numerical calculations in quantum mechanics an accurate method for numerical calculations of matrix elements and for solving the eigenvalue problem in quantum mechanics is presented methods for numerical interpolation differentiation and integration provide 15 digit accuracy with double precision arithmetic operations a method for solution of the eigenvalue problem of an ordinary differential equation by using discretization and matrix eigenvalue methods provides 13 to 15 digit accuracy the efficiency of the proposed methods is demonstrated by the applications to bound states for the linear harmonic oscillator anharmonic oscillators the morse potential and the modified poschl teller potential

energy levels for a double well potential in three dimensional system using hill determinant approach abstract energy levels of the schrodinger equation for the double well potential v x y z z 2 1 2 z 2 x 2 y 2 z 2 1 2 a xx x 4 a yy y 4 a zz z 4 2a xy x 2 y 2 2a xz x 2 z 2 2a yz y 2 z 2 in a three dimensional system are calculated using the hill determinant approach for several eigenstates and large values of the perturbation parameters and z2 emphasis is specially placed on the larger values of z2 because the eigenvalues for different states have almost degenerate eigenvalues some of the results calculated by the hill determinant approach are compared with the results produced by other techniques

actions and invariant character degrees iii point subgroup in general we use the same notation as in 6 and 7 wx wxpart of the proof of theorem a depends on the basic properties of thegajendragadkar p special characters 1 and we assume the reader iswxfamiliar with those however we will repeatedly use a deeper fact anirreducible character a of a hall p subgroup

dynamic group approach to the x2 λx2 1 gx2 potential a modified operator method based upon the so 2 1 dynamic group is applied to the one dimensional anharmonic oscillator potential v x x2 x2 1 gx2 a tilting transformation is carried out in order to improve the rate of convergence of the algebraic perturbation series very accurate results are obtained for the energy eigenvalues and for the wave functions especially in the case of small g values

on normal subgroups and simple modules with a given vertex in a p solvable group abstract let n be a normal subgroup of a p solvable group g and let m be a simple fn module where f is an algebraically closed field of characteristic p let q be any p subgroup of g in this paper we count the number of simple fg modules lying over m and having q as a vertex

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

a simple iterative solution of the schrodinger equation in matrix representation form a simple procedure for solving the schrodinger equation is presented it is based upon an iterative solution of the secular equation a large enough convergence rate is obtained by using a basis set of properly scaled functions the effect of the scaling parameter on the convergence rate is studied in order to improve the calculation the method is applied to simple though non trivial quantum mechanical models such as the quartic sextic and octic anharmonic oscillators a double well potential and the linear confining potential highly accurate eigenvalues for all values of the coupling parameter are obtained

character correspondences and subgroups of operator groups abstract let a and g be finite groups with coprime orders and suppose that a acts on g by automorphisms let g a irr a g irr c g a be the glauberman isaacs correspondence let b a and let irr a g we examine the conjecture that g a is an irreducible constituent of the restriction of g b to c g a and show that it is valid if g is supersolvable then we show when the analog of this conjecture for brauer characters and the uno wolf correspondence holds

regime switching for dynamic correlations we propose a new model for the variance between multiple time series the regime switching dynamic correlation we decompose the covariances into correlations and standard deviations and the correlation matrix follows a regime switching model it is constant within a regime but different across regimes the transitions between the regimes are governed by a markov chain this model does not suffer from a curse of dimensionality and it allows analytic computation of multi step ahead conditional expectations of the variance matrix when combined with the armach model taylor modelling financial time series wiley new york and schwert j finance 44 5 1989 1115 for the standard deviations we also present an empirical application which illustrates that our model can have a better fit of the data than the dynamic conditional correlation model proposed by engle j business econ statist 20 3 2002 339

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

a finite difference approach for the calculation of perturbed oscillator energies a simple numerical method for calculating eigenvalues and corresponding eigenvectors of the schrodinger equation for a perturbed oscillator is described the derived results are compared with previously derived numerical data and with available exact values

finite difference calculations of eigenvalues for various potentials the energy levels of the schrodinger equation for the potentials v x sigma 2n 2 lambda 2nx2n v x z2x2 x2n and v x x2 lambda x2n 1 gx2 with 2n 4 6 8 18 20 have been calculated by using a finite difference method for various values of lambda and the quantum number n the obtained results are compared with previously available results

primitive characters of subgroups ofm groups one of the hardest areas in the character theory of solvable groups continues to be the monomial groups a finite group is said to be an m group or monomial if all of its irreducible characters are monomial that is to say induced from linear characters two are still the main problems on m groups are hall subgroups of m groups monomial under certain oddness hypothesis are normal subgroups of m groups monomial in both cases there is evidence that this could be the case the primitive characters of the subgroups in question are the linear characters this is the best result up to date 4 6 recently some idea appears to be taking form in 14 t okuyama proved that if g is an m group and p is a sylow p subgroup of g then n p p is an m group some time later and more generally i m lsaacs shows that if h is a hall subgroup of an m group then n s h h is also an m group 91 since it is well known that normal hall subgroups of m groups are m groups in this case the strong hypothesis of normality makes the problem immediate perhaps it is not only true that hall subgroups of m groups are m groups but something stronger that hall subgroup normalizers also are in this direction we prove theorem a let g be an m group and let h be a hall subgroup o f g i f c e i r r nc h is primitive then c is linear

fully group graded algebras and a theorem of fong abstract it is proved that any algebra fully graded by a finite group over a complete discrete valuation ring with an algebraically closed residue field of characteristic a prime p is morita equivalent to an embedded graded subalgebra which is a crossed product and an explicit way to get a decomposition of unity with a bounded length is shown when the finite group is p solvable a theorem of fong s type for fully graded algebras is obtained

primitive characters of maximal subgroups of solvable groups abstract let g be an m group and suppose that h is a maximal subgroup of odd index and that irr h is primitive it was shown by mark lewis that 1 2 divides g h we show in this situation that either 1 2 g h or 1 1 a number of related results are established for groups that are not necessarily m groups

vertices and normal subgroups in solvable groups abstract for a finite group g and a fixed prime p one can attach to each irreducible brauer character of g a p subgroup q called the vertex of that is unique up to conjugacy in this paper we examine the behavior of the vertices of with respect to normal subgroups when g is assumed to be p solvable for arbitrary finite p solvable groups we develop a correspondence between the set of brauer characters of g with vertex q and the set of brauer characters of a certain subgroup of g with vertex q moreover in the case that g has odd order we extend a result of navarro regarding the behavior with respect to normal subgroups of a correspondence of brauer characters of p degree to a result regarding the behavior with respect to normal subgroups of a correspondence of brauer characters of arbitrary degree

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

asymptotic properties of eigenvalues in variational calculations for double well oscillators simple and accurate formulae for the eigenvalues of the hamiltonian with an anharmonic double well potential are presented special attention is paid to the following three regions of coupling parameters where difficulties arise i in the weak coupling limit the eigenvalues depend on the coupling parameter nonanalytically to obtain accurate explicit expressions in this limit a composite asymptotic expansion is applied ii in the strong coupling limit the derivation of asymptotic expansions is obstructed by the nonuniqueness of solutions to variational equations to overcome this obstacle a restricted variational procedure is suggested iii there exist also critical values of the coupling parameter at which eigenvalues cross zero in the regions around these critical values a dramatic increase of the percentage error occurs a remedy against this critical behaviour is the positive definite shift removing the critical regions the formulae obtained for the eigenvalues provide an accuracy within the errors not exceeding 3 for all values of the coupling parameter from zero to infinity

pseudospectral methods for solving an equation of hypergeometric type with a perturbation almost all regular or singular sturm liouville eigenvalue problems in the schrodinger form email protected x v x j x email protected x email protected email protected email protected r j x l 2 email protected email protected for a wide class of potentials v x may be transformed into the form s x y email protected x y q x email protected email protected a b r by means of intelligent transformations on both dependent and independent variables where s x and t x are polynomials of degrees at most 2 and 1 respectively and l is a parameter the last form is closely related to the equation of the hypergeometric type eht in which q x is identically zero it will be called here the equation of hypergeometric type with a perturbation ehtp the function q x may therefore be regarded as a perturbation it is well known that the eht has polynomial solutions of degree n for specific values of the parameter l i e l email protected n 0 n t 12 n 1 s which form a basis for the hilbert space l 2 a b of square integrable functions pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of ehtp and hence the energies e of the original schrodinger equation specimen computations are performed to support the convergence numerically