information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

raising and lowering operators factorization and differential difference operators of hypergeometric type starting from the rodrigues formula we present a general construction of raising and lowering operators for orthogonal polynomials of continuous and discrete variables on a uniform lattice in order to have these operators mutually adjoint we introduce orthonormal functions with respect to the scalar product of unit weight using the infeld hull factorization method we generate from the raising and lowering operators the second order self adjoint differential difference operator of hypergeometric type

orthogonal polynomial solutions of linear ordinary differential equations abstract this paper surveys the latest known results concerning the classification of differential equations of the form k 1 n a k x y k x y x having a sequence of polynomial eigenfunctions p n x n 0 that are orthogonal with respect to some real bilinear form since the publishing of the erice report in 1990 several new significant results and generalizations have been discovered

regular linear systems governed by neutral fdes the aim of this paper is to prove that a class of distributed parameter systems governed by neutral fdes provides regular linear systems employing the well established theory of representation transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

fisher based thermodynamics its legendre transform and concavity properties it is shown that the legendre transform structure of thermodynamics can be replicated without any change if one replaces the entropy s by fisher s information measure i also the important thermodynamic property of concavity is shown to be obeyed by i by this use of the fisher information measure we develop a thermodynamics that seems to be able to treat equilibrium and nonequilibrium situations in a manner entirely similar to the conventional one

information measures of hydrogenic systems laguerre polynomials and spherical harmonics fisher s information and shannon s entropy are two complementary information measures of a probability distribution here the probability distributions which characterize the quantum mechanical states of a hydrogenic system are analyzed by means of these two quantities these distributions are described in terms of laguerre polynomials and spherical harmonics whose characteristics are controlled by the three integer quantum numbers of the corresponding states we have found the explicit expression for the fisher information and a lower bound for the shannon entropy with the help of an isoperimetric inequality

fisher info and thermodynamics first law abstract we show starting from first principles that thermodynamics first law can be microscopically obtained for fisher s information measure without need of invoking the adiabatic theorem further it is proved that enforcing the fisher first law requirements in a process in which the probability distribution is infinitesimally varied is equivalent to minimizing fisher s information measure subject to appropriate constraints

information theoretic measures of hyperspherical harmonics the multidimensional spreading of the hyperspherical harmonics can be measured in a different and complementary manner by means of the following information theoretic quantities the fisher information the average density or first order entropic moment and the shannon entropy they give measures of the volume anisotropy of the eigenfunctions of any central potential in the hyperspace contrary to the fisher information which is a local measure because of its gradient functional form the other two quantities have a global character because they are powerlike average density and logarithmic shannon s entropy functionals of the hyperspherical harmonics in this paper we obtain the explicit expression of the first two measures and a lower bound to the shannon entropy in terms of the labeling indices of the hyperspherical harmonics

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

a lagrangean approach to the degree constrained minimum spanning tree problem abstract a known branch and bound algorithm for the degree constrained minimum spanning tree problem is adapted for the use of lagrangean multipliers they improve the bounds from which the edge elimination analysis in the algorithm benefits in particular computational results are given for problems up to 150 nodes both random drawn coordinate problems as well drawn table problems these results are much better than other results reported in literature and they show the lagrangean approach to be more effective when the problems are more difficult to solve

the fisher information based uncertainty relation cramer rao inequality and kinetic energy for the d dimensional central problem the inequality with l being the grand orbital quantum number and its conjugate relation for r2 p 2 are shown to be fulfilled in the d dimensional central problem their use has allowed us to improve the fisher information based uncertainty relation i i const and the cramer rao inequalities r2i d2 p2i d2 in addition the kinetic energy and the radial expectation value r2 are shown to be bounded from below by the fisher information in position and momentum spaces denoted by i and i respectively

fisher information kinetic energy and uncertainty relation inequalities by interpolating between fisher information and mechanical kinetic energy we introduce a general notion of kinetic energy with respect to a parameter of schrodinger wavefunctions from a statistical inference perspective kinetic energy is the sum of fisher information and an integral of a parametrized analogue of quantum mechanical current density related to phase a family of integral inequalities concerning kinetic energy and moments are established among which the cramer rao inequality and the weyl heisenberg inequality are special cases in particular the integral inequalities involving the negative order moments are relevant to the study of electron systems moreover by specifying the parameter to a scale we obtain a family of inequalities of uncertainty relation type which incorporate the position and momentum observables symmetrically in a single quantity

criterion for polynomial solutions to a class of linear differential equations of second order we consider the differential equations y 0 x y s0 x y where 0 x s0 x are c functions we prove i if the differential equation has a polynomial solution of degree n 0 then n nsn 1 n 1sn 0 where n n 1 sn 1 0 n 1andsn s n 1 s0 k 1 n 1 2 conversely ii if n n 1 0 and n 0 then the differential equation has a polynomial solution of degree at most n we show that the classical differential equations of laguerre hermite legendre jacobi chebyshev first and second kinds gegenbauer and the hypergeometric type etc obey this criterion further we find the polynomial solutions for the generalized hermite laguerre legendre and chebyshev differential equations

fisher information of orthogonal hypergeometric polynomials the probability densities of position and momentum of many quantum systems have the form x pn2 x x where pn x denotes a sequence of hypergeometric type polynomials orthogonal with respect to the weight function x here we derive the explicit expression of the fisher information i dx x 2 x corresponding to this kind of distributions in terms of the coefficients of the second order differential equation satisfied by the polynomials pn x we work out in detail the particular cases of the classical hermite laguerre and jacobi polynomials for which we find the value of fisher information in closed analytical form and study its asymptotic behaviour in the large n limit

on well posedness of singular neutral equations in the state space c abstract in this paper we study necessary und sufficient conditions for well posedness of scalar singular i e with a difference operator which is nonatomic at zero neutral functional differential equations snfdes in the state space c r 0 we also discuss an ill posed snfde in order to illustrate how the addition of a singularity away from zero in our case we have a perturbing atom at r can destroy well posedness

information entropy of conditionally exactly solvable potentials we evaluate shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials the bialynicki birula mycielski inequality has also been tested for a number of states

fisher information of special functions and second order differential equations we investigate a basic question of analytic information theory namely the evaluation of the fisher information and the relative fisher information with respect to a non negative function for the probability distributions obtained by squaring the special functions of mathematical physics which are solutions of second order differential equations we obtain explicit expressions for these information theoretic properties via the expectation values of the coefficients of the differential equation we illustrate our approach for various nonrelativistic d dimensional wavefunctions and some special functions of physicomathematical interest emphasis is made in the nikiforov uvarov hypergeometric type functions which include and generalize the hermite functions and the gauss and kummer hypergeometric functions among others

non linear characterizations for functions of hypergeometric type and their derivatives of any order abstract to find characterizations for special functions is an important question in applied mathematics and mathematical physics up to now most effort has been invested for polynomial sets which are orthogonal on the real line in this paper two non linear characterizations for a class of functions the so called functions of hypergeometric type are found not only for the functions themselves but also for their derivatives of any order members of this class are the bessel functions the hypergeometric functions the classical orthogonal polynomials as well as many other functions of mathematics and physics applications to the subclass of polynomials of hypergeometric type are made with emphasis on the classical orthogonal polynomials for which the coefficients of the above mentioned non linear characterizations are tabulated

asymptotics of information entropies of some toda like potentials the spreading of the quantum probability density for the highly excited states of a single particle system with an exponential type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the boltzmann shannon information entropy this problem boils down to the calculation of the asymptotics of the entropy like integrals of the modified bessel function of the second kind also called the mcdonald function or basset function the dependence of the two physical entropies on the large quantum number n is given in detail it is shown that the semiclassical wkb position space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single particle systems with any power type potential of the form v x x2k x r and k n the momentum space entropy calculated with a method based on the properties of the mcdonald function is rigorously found to have a behavior of the form ln ln n in strong contrast with the

entropy of orthogonal polynomials with freud weights and information entropies of the harmonic oscillator potential the information entropy of the harmonic oscillator potential v x 1 2 x2 in both position and momentum spaces can be expressed in terms of the so called entropy of hermite polynomials i e the quantity sn h h2n x log h2n x e x2dx these polynomials are instances of the polynomials orthogonal with respect to the freud weights w x exp x m m 0 here a very precise and general result of the entropy of freud polynomials recently established by aptekarev et al j math phys 35 4423 4428 1994 specialized to the hermite kernel case m 2 leads to an important refined asymptotic expression for the information entropies of very excited states i e for large n in both position and momentum spaces to be denoted by s and s respectively briefly it is shown that for large values of n s 1 2log log 2n e o 1 and s 1 2log log 2n e o 1 so that s s log 2 2n e2 o 1 in agreement with the generalized indetermination relation of byalinicki birula and mycielski commun m

asymptotics of the information entropy of the airy function the boltzmann shannon information entropy of linear potential wavefunctions is known to be controlled by the information entropy of the airy function ai x here the entropy asymptotics is analysed so that the first two leading terms previously calculated in the wkb approximation as well as the following term already conjectured are derived by using only the specific properties of the airy function

numerical methods for a class of singular integro differential equations based on semigroup approximation in this paper computational methods for singular integro differential equations of neutral type are discussed the study is based on a well posed state space formulation and uses semigroup theoretical framework to analyze the convergence of semi and fully discrete numerical schemes feasibility of the fully discrete scheme is demonstrated by applying it to abel volterra equations of the first kind and to a singular neutral functional differential equation

cramer rao information plane of orthogonal hypergeometric polynomials the classical hypergeometric polynomials pn x n 0 which are orthogonal with respect to a weight function x defined on a real interval are analyzed in the cramer rao information plane that is the plane defined by both fisher information and variance of the probability density n x pn x 2 x the rakhmanov density n x of these polynomials which describes the probability density of the quantum states for various physical prototypes in an exact manner and for numerous physical systems to a very good approximation is discussed in detail

a multiperiod degree constrained minimal spanning tree problem abstract the multiperiod degree constrained minimal spanning tree mdcmst problem consists of scheduling the installation of links in a network so as to connect a set of terminal nodes to a central node with minimal present value of expenditures the network design is subject to degree constraint which limits the number of links incident on each terminal node to a prespecified number due to the number of ports available on it some of the terminal nodes in the network are active at the beginning of the planning horizon while others are activated over time we formulate this problem as an integer programming problem we suggest a lagrangian based heuristic to solve the integer programming formulation of the network topology problem lower bounds found as a byproduct of the solution procedure are used to estimate the quality of the solution given by the heuristic experimental results over a wide range of problem structures show that the lagrangian based heuristic method yields verifiably good solutions to this hard problem

on minimum fisher information distributions with restricted support and fixed variance fisher information is of key importance in estimation theory it also serves in inference problems as well as in the interpretation of many physical processes the mean squared estimation error for the location parameter of a distribution is bounded by the inverse of the fisher information associated with this distribution in this paper we look for minimum fisher information distributions with a restricted support more precisely we study the problem of minimizing the fisher information in the set of distributions with fixed variance defined on a bounded subset s of r or on the positive real line we show that the solutions of the underlying differential equation can be expressed in terms of whittaker functions then in the two considered cases we derive the explicit expressions of the solutions and investigate their behavior we also characterize the behavior of the minimum fisher information as a function of the imposed variance

lie algebras and polynomials in one variable a classification of linear differential and difference equations in one variable having polynomial solutions the generalized bochner problem is presented the idea of the approach is based on consideration of the eigenvalue problem for a polynomial in generators of the algebra sl2 r for differential equations or sl2 r q for finite difference equation given in the finite dimensional projectivized representation

singular functional differential equations of neutral type in banach spaces abstract the well posedness of a large class of singular partial differential equations of neutral type is discussed here the term singularity means that the difference operator of such equations is nonatomic at zero this fact offers many difficulties in applying the usual methods of perturbation theory and laplace transform technique and thus makes the study interesting our approach is new and it is based on functional analysis of semigroup of operators in an essential way and allows us to introduce a new concept of solutions for such equations finally we study the well posedness of a singular reaction diffusion equation of neutral type in weighted lebesgue s spaces

fisher like atomic divergences mathematical grounds and physical applications abstract two different local divergence measures the fisher fd and the jensen fisher jfd ones are compared in this work by applying them to atomic one particle densities in position and momentum spaces they are defined in terms of the absolute and the relative fisher information functionals the analysis here afforded includes not only neutral atoms but also singly charged cations the results are interpreted and justified according to i shell filling patterns ii short and long range behaviors of the atomic densities and iii the value of the atomic ionization potential the strengths of the fd measure as compared to the jfd one are emphasized

uncertainty relation for fisher information of d dimensional single particle systems with central potentials an uncertainty fisher information relation in quantum mechanics is derived for multidimensional single particle systems with central potentials it is based on the concept of fisher information in the two complementary position and momentum spaces which is a gradient functional of the corresponding probability distributions the lower bound of the product of position and momentum fisher informations is shown to depend on the orbital and magnetic quantum numbers of the physical state and the space dimensionality applications to various elementary systems is discussed

on neutral functional differential equations with nonatomic difference operator where the difference operator d is nonatomic at zero here x denotes the function defined by x s x t s y d s d 0 the case of d l being bounded linear functionals c r 0 w r with d atomic at zero has been studied extensively in the literature cf e g s introducing a state which in the continuous case corresponds to dx x instead of x t x as in the retarded case burns herdman and stech proved in 3 that problem 1 1 gives rise to a c semigroup in r x lp a 0 fp 1 dp co provided d is a continuous functional on c r 0 w atomic at zero and l is continuous from w r 0 w into r in 3 it was also shown by an example that atomicity of d at zero is not necessary for 1 1 to generate a c semigroup on r x lp r 0 w

parameter based fisher s information of orthogonal polynomials the fisher information of the classical orthogonal polynomials with respect to a parameter is introduced its interest justified and its explicit expression for the jacobi laguerre gegenbauer and grosjean polynomials found

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

some well posed functional equations which generate semigroups consider the abstract linear functional equation fe dx t f t t 0 x t t t 0 in a banach space b a theorem is proven which contains the following result as a special case let y r b be a lp space or c0 space on r t8 with a suitable weight function and with values in b let d be a closed unbounded causal linear operator in y r b which commutes with translations suppose that d i has a continuous causal inverse for some complex and that d restricted to those functions in y r b which vanish on r 0 has a continuous causal inverse then fe generates a strongly continuous semigroup of translation type on a banach space which is essentially the cross product of the restriction of the domain of d to r and y r b examples with b cn on how the theory applies to a neutral functional differential equation a difference equation a volterra integrodifferential equation with nonintegrable kernel but integrable resolvent and a fractional order functional differential equation are given also an abstract neutral functional differential equation in a hilbert space is studied and applications to an abstract volterra integrodifferential equation in a banach space are indicated

recurrence relations for radial wavefunctions for the nth dimensional oscillators and hydrogenlike atoms we present a general procedure for finding recurrence relations of the radial wavefunctions for nth dimensional isotropic harmonic oscillators and hydrogenlike atoms

neutral functional integro differential equations with weakly singular kernels the well posedness of a general functional differential equation of neutral type is considered where the difference operator does not have an atom at zero the results are applicable to certain classes of singular integro differential equations that occur in aeroelasticity the equations are considered as dynamical systems on product spaces

general recurrence and ladder relations of hypergeometric type functions a method for the explicit construction of general linear sum rules involving hypergeometric type functions and their derivatives of any order is developed this method only requires the knowledge of the coefficients of the differential equation that they satisfy namely the hypergeometric type differential equation special attention is paid to the differential recurrence or ladder relations and to the fundamental three term recurrence formulas most recurrence and ladder relations published in the literature for numerous special functions including the classical orthogonal polynomials are instances of these sum rules moreover an extension of the method to the generalized hypergeometric type functions is also described allowing us to obtain explicit ladder operators for the radial wave functions of multidimensional hydrogen like atoms where the varying parameter is the dimensionality