a finite sample distribution free probabilistic lower bound on mutual information for any memoryless communication channel with a binary valued input and a one dimensional real valued output we introduce a probabilistic lower bound on the mutual information given empirical observations on the channel the bound is built on the dvoretzky kiefer wolfowitz inequality and is distribution free a quadratic time algorithm is described for computing the bound and its corresponding class conditional distribution functions we compare our approach to existing techniques and show the superiority of our bound to a method inspired by fano s inequality where the continuous random variable is discretized

two tier formulation of multichannel scattering theory and hypervirial theorems complex dynamics of a multichannel scattering may be treated by a variational procedure but the conventional variational principles are not readily applicable because of i the intrinsic difficulties of unstable fluctuations in the calculated amplitudes as functions of nonlinear parameters and ii lack of criteria to optimize the solutions a two tier theory is formulated in which the complex dynamical mixing and the asymptotic channel sector are treated separately but as a coupled system such that the instability problem i is resolved naturally in a mathematically consistent way even when most of the weakly coupled open channels are neglected the resulting solution is stable but not necessarily optimal modified forms of hypervirial theorems are introduced to optimize the solutions thus rectifying the shortcoming ii thus the reformulated theory for the scattering states coupled with a properly chosen hypervirial theorem can be applied effectively to many body multichannel scattering systems

a geometric proof of total positivity for spline interpolation abstract the total positivity of the spline collocation matrix is the basis of several important results in univariate spline theory this makes it desirable to provide as simple as possible a proof of this total positivity the proofs available in the literature don t qualify since these all rely on certain determinant identities which are not exactly intuitive the authors give a proof that uses nothing more than cramer s rule hard to avoid since total positivity is a statement about determinants and the geometrically obvious fact that a b spline can always be written as a positive combination of b splines on a finer knot sequence the geometric intuition appealed to here stems from the area of computer aided design in which a spline is constructed and manipulated through its b polygon a broken line whose vertices correspond to the b spline coefficients if a knot is added to provide greater potential flexibility the new b polygon is obtained by interpolation to the old this had led lane and riesenfeld to a proof of the variation diminishing property of the spline collocation matrix and is shown here to provide a proof of the total positivity as well

the summation of the ordinary and renormalized perturbation series for the ground state energy of the quartic sextic and octic anharmonic oscillators using nonlinear sequence transformations the strongly divergent rayleigh schrodinger perturbation expansion for the ground state energy of the quartic sextic and octic anharmonic oscillator and a corresponding renormalized perturbation expansion f vinette and j ci ek j math phys 32 3392 1991 are summed by pade approximants by levin s sequence transformation d levin int j comput math b 3 371 1973 and by a closely related sequence transformation which was suggested recently e j weniger comput phys rep 10 189 1989 it is shown that the renormalized perturbation expansion can be summed much more easily than the original perturbation expansion from which it was derived and that levin s sequence transformation diverges and is not able to sum the perturbation expansions the pade summation of the renormalized perturbation expansions gives relatively good results in the quartic and sextic case in the case of the octic anharmonic oscillator even the renormalized perturbation expansion is not pade summable the best results are clearly obtained by the new sequence transformation which for instance is able to sum the renormalized perturbation expansions for the infinite coupling limits of the quartic sextic and octic anharmonic oscillator and which produces at least in the quartic and sextic case extremely accurate results

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

a complex variable form of the heg technique a previously reported simple method for calculating complex matrix eigenvalues is modified to incorporate the traditional heg approach for the case of even parity potentials two examples of resonance calculations are given our matrix and perturbation results agree with each other but are not in full accord with previously published results for one of the test potentials new results are given for the resonances of the inverted gaussian potential

conditioning and accurate computations with pascal matrices a result on the ill conditioning of pascal matrices is proved however it is shown that bidiagonal factorizations of pascal matrices can be applied to perform computations with high relative accuracy

qr decomposition of almost strictly sign regular matrices in this paper l d u factorization of nonsingular sign regular and almost strictly sign regular matrices is analyzed besides nonsingular almost strictly sign regular matrices are characterized through the q r factorization illustrative numerical examples are included

the bloch wave operator generalizations and applications part i the time independent case this is part 1 of a two part review on wave operator theory and methods the basic theory of the time independent wave operator is presented in terms of partitioned matrix theory for the benefit of general readers with a discussion of the links between the matrix and projection operator approaches the matrix approach is shown to lead to simple derivations of the wave operators and effective hamiltonians of lowdin bloch des cloizeaux and kato as well as to some associated variational forms the principal approach used throughout stresses the solution of the nonlinear equation for the reduced wave operator leading to the construction of the effective hamiltonians of bloch and of des cloizeaux several mathematical techniques which are useful in implementing this approach are explained some of them being relatively little known in the area of wave operator calculations the theoretical discussion is accompanied by several specimen numerical calculations which apply the described techniques to a selection of test matrices taken from the previous literature on wave operator methods the main emphasis throughout is on the use of numerical methods which use iterative or perturbation algorithms with simple pade approximant methods being found sufficient to deal with most of the cases of divergence which are encountered the use of damping factors and relaxation parameters is found to be effective in stabilizing calculations which use the energy dependent effective hamiltonian of lowdin in general the computations suggest that the numerical applications of the nonlinear equation for the reduced wave operator are best carried out with the equation split into a pair of equations in which the bloch effective hamiltonian appears as a separate entity the presentation of the theoretical and computational details throughout is accompanied by references to and discussion of many works which have used wave operator methods in physics chemistry and engineering some of the techniques described in this part 1 will be further extended and applied in part 2 of the review which deals with the changes which are required to extend wave operator theory to the case of a time dependent hamiltonian such as that which describes the interaction of a laser pulse with an atom or molecule

comment on the asymptotic iteration method for polynomial potentials it is shown that polynomial potentials can be treated by a method which cuts out most of the complications of the popular aim approach several examples are given to show how expectation values and resonance energies can be calculated using the simplified approach

almost strictly totally negative matrices a real matrix a a i j 1 i j n is said to be almost strictly totally negative if it is almost strictly sign regular with signature e 1 1 1 which is equivalent to the property that all its nontrivial minors are negative in this paper an algorithmic characterization of nonsingular almost strictly totally negative matrices is presented

1 n expansions in nonrelativistic quantum mechanics an extensive number of numerical computations of energy 1 n series using a recursive taylor series method are presented in this paper the series are computed to a high order of approximation and their behavior on increasing the order of approximation is examined

total positivity and neville elimination abstract neville elimination is described in terms of schur complements of matrices and used to improve some well known characterizations of totally positive and strictly totally positive matrices

splitting in a double minimum potential with almost twofold degenerate lower levels an analytic continuation procedure using taylor series is utilised to produce very accurate wavefunctions and eigenvalues for a double minimum potential having almost degenerate eigenvalues comparison is made with results obtained by other means

on nonsingular sign regular matrices abstract an m n matrix a is sign regular if for each k 1 k min m n all k k submatrices of a have determinant with the same nonstrict sign the zero pattern of nonsingular sign regular matrices is analyzed it is proved that the number of zero entries which can appear in a nonsingular sign regular matrix depends on its signature a matrix is totally nonpositive if all its minors are nonpositive a test for recognizing nonsingular totally nonpositive matrices is also provided

part of the d dimensional spiked harmonic oscillator spectra the pseudoperturbative shifted l expansion technique pslet is generalized for states with an arbitrary number of nodal zeros interdimensional degeneracies emerging from the isomorphism between the angular momentum and the dimensionality of the central force schrodinger equation are used to construct part of the d dimensional spiked harmonic oscillator bound state spectra pslet results are found to compare excellently with those from direct numerical integration and generalized variational methods

on an iteration method for eigenvalue problems we discuss a recently proposed asymptotic iteration method for eigenvalue problems we analyse its rate of convergence the use of adjustable parameters to improve it and the relationship with an alternative method based on the same ideas

full rank factorization in quasi ldu form of totally nonpositive rectangular matrices abstract let a a i j r n m be a totally nonpositive matrix with rank a r min n m and a 11 0 in this paper we obtain a characterization in terms of the full rank factorization in quasi ldu form that is a l d u where l r n r is a block lower echelon matrix u r r m is a unit upper echelon totally positive matrix and d r r r is a diagonal matrix with rank l rank u rank d r we use this quasi ldu decomposition to construct the quasi bidiagonal factorization of a moreover some properties about these matrices are studied

the approximation of a totally positive band matrix by a strictly banded totally positive one every nonsingular totally positive m banded matrix is shown to be the product of m totally positive one banded matrices and therefore the limit of strictly m banded totally positive matrices this result is then extended to bi infinite m banded totally positive matrices with linearly independent rows and columns in the process such matrices are shown to possess at least one diagonal whose principal sections are all nonzero as a consequence such matrices are seen to be approximable by strictly m banded totally positive ones

a test for strict sign regularity a characterization of strict sign regular matrices is obtained it is given in an algorithmic form which allows the easy construction of a test to check if a matrix is strictly sign regular the test is based on the neville elimination of several submatrices of a and requires less computational work than those which can be derived from other previously known characterizations of these matrices

summation of strongly divergent perturbation series a new method for summing strongly divergent perturbation series is presented it is based on the change of the power series into a convergent sequence by means of an order dependent mapping obtained from a simple scaling relation the perturbation expansions for a one dimensional integral and for the ground states of the anharmonic oscillator and of the linear confining potential model are accurately summed in the most unfavorable strong coupling limit

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

almost strictly sign regular matrices and neville elimination with two determinant pivoting in 2007 cortes and pena introduced a pivoting strategy for the neville elimination of nonsingular sign regular matrices and called it two determinant pivoting neville elimination has been very useful for obtaining theoretical and practical properties for totally positive negative matrices and other related types of matrices a real matrix is said to be almost strictly sign regular if all its nontrivial minors of the same order have the same strict sign in this paper some nice properties related with the application of neville elimination with two determinant pivoting strategy to almost strictly sign regular matrices are presented

renormalized perturbation series and the semiclassical limit of quantum mechanics the accuracy of the renormalized perturbation series for anharmonic oscillators and other simple quantum mechanical models increases with the quantum number when the adjustable parameter is determined by a simple variational criterion in the semiclassical limit of quantum mechanics such an expansion converges towards the result of the jeffreys wentzel kramers brillouin method to all orders the hypervirial and hellmann feynman theorems facilitate the derivation of the semiclassical series from rayleigh schrodinger perturbation theory

a hill series method for resonances a simple algorithm is given to treat perturbed oscillator bound states and resonances the method is applied to a bound state problem for which the energy is a non analytic function of a perturbation parameter and also to two resonance problems one of which has a spectrum with an unusual dual nature

a factorization theorem for banded matrices in this paper we prove a factorization theorem for strictly m banded totally positive matrices we show that such a matrix is a product of m one banded matrices with positive entries a bi infinite matrix a ai i ca i and ai i n ai i o these matrices arise quite freqiently in applications and recently the have been the subject of several independent investigations 2 3 in this paper we prove a factorization theorem for strictly m handed totally positive matrices we show that such a matrix is a product of m one banded matrices with positive entries let us begin by recalling that a matrix a is totally positive if and only if ull its minors are nonnegative

the hvt technique and the uncertainty relation for central potentials the quantum mechanical hypervirial theorems hvt technique is used to treat the so called uncertainty relation for quite a general class of central potential wells including the reduced poeschl teller and the gaussian one it is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for

nonsingular almost strictly sign regular matrices abstract the class of nonsingular almost strictly totally positive matrices has been characterized m gasca j m pena characterizations and decompositions of almost strictly positive matrices siam j matrix anal appl 28 2006 1 8 in this paper we discuss the class of almost strictly sign regular matrices that includes almost strictly totally positive matrices a characterization is provided for these matrices in terms of their nontrivial minors using consecutive rows and consecutive columns in particular we present a characterization of certain almost strictly sign regular matrices in terms of a very reduced number of boundary almost trivial minors

variational and perturbative schemes for a spiked harmonic oscillator a variational analysis of the spiked harmonic oscillator hamiltonian operator d2 dx2 x2 l l 1 x2 x where is a real positive parameter is reported in this work the formalism makes use of the functional space spanned by the solutions of the schrodinger equation for the linear harmonic oscillator hamiltonian supplemented by a dirichlet boundary condition and a standard procedure for diagonalizing symmetric matrices the eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system valid for positive and relatively large values of the coupling parameter additionally a large coupling perturbative expansion is carried out and the contributions up to fourth order to the ground state energy are explicitly evaluated numerical results are compared for the special case 5 2

solving the schrodinger equation with use of 1 n perturbation theory the large n expansion provides a powerful new tool for solving the schrodinger equation in this paper we present simple recursion formulas which facilitate the calculation we do some numerical calculations which illustrate the speed and accuracy of the technique

hypervirial perturbation theory revisited a wide range of topics in the area of hypervirial perturbation theory hvpt is discussed it is shown that with the use of a few simple procedures hvpt is capable of high accuracy for many problems results from many previous works in the literature are found to be improvable by the careful use of hvpt with appropriate choice of the unperturbed potential and of the origin at which the energy expansion is carried out two multi well problems from the literature are analysed in detail to show the value of a combination of hvpt and finite difference methods