factorization of some confluent heun s differential equations this paper stands for a logic continuation of the previous work a ronveaux factorization of the henn s differential operator appl math comput 141 2003 177 184 we propose a factorization of the confluent double confluent and biconfluent heun s differential equations and investigate their solutions c 2006 elsevier inc all rights reserved

statistical geometry in quantum mechanics a statistical model m is a family of probability distributions characterised by a set of continuous parameters known as the parameter space this possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square integrable functions more precisely by consideration of the square root density function we can regard m as a submanifold of the unit sphere s in a real hilbert space h therefore h embodies the state space of the probability distributions and the geometry of the given statistical model can be described in terms of the embedding of m in s the geometry in question is characterised by a natural riemannian metric the fisher rao metric thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting in particular we focus attention on the variance lower bounds for statistical estimation and establish generalisations of the classical cramer rao and bhattacharyya inequalities described in terms of the geometry of the underlying real hilbert space as a comprehensive illustration of the utility of the geometric framework the statistical model m is then specialised to the case of a submanifold of the state space of a quantum mechanical system this is pursued by introducing a compatible complex structure on the underlying real hilbert space which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real hilbert space geometry the application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the heisenberg uncertainty relations for canonically conjugate observables

bayesian inference in quantum systems abstract bayes criteria are explicitly applied to statistical decision problems in simple quantum mechanical systems the minimum bayes cost is calculated for systems including polarised spins and relativistic spin 1 2 particles the results suggest that in decisions for a given ensemble of particles on the average the bayes cost almost always decreases if one treats the ensemble as a single composite system instead of performing separate measurements on the individual particles the optimal average sample size in observing spin 1 2 particles is also derived these results are potentially relevant to experimental design

on the equivalence of non equivalent algebraic realizations symmetries have been used with great success to determine solutions of differential equations with which they are associated in addition to reducing the order of the equation one can also use the lie algebra of the symmetries to transform the equation into canonical form the canon in this case is determined by the standard realization of the lie algebra following a comment in olver p j 1995 equivalence invariants and symmetry cambridge cambridge university press we conjecture that while lie algebras may have non equivalent realizations in the usual point transformation sense all realizations of the same lie algebra are equivalent when considered on the appropriate ordered jet space we show how this result can have useful implications for ordinary differential equations including linearization for equations thought to be inherently nonlinear

about derivatives of heun s functions from polynomial transformations of hypergeometric equations we discuss how the derivative of solutions of some heun s differential equations can be given in some particular cases from the solution of another heun s equation this work uses some algebraic links between heun and hypergeometric equations c 2008 elsevier inc all rights reserved

sextic anharmonic oscillators and orthogonal polynomials under certain constraints on the parameters a b and c it is known that schrodinger s equation d2 dx2 ax6 bx4 cx2 e a 0 with the sextic anharmonic oscillator potential is exactly solvable in this paper we show that the exact wavefunction is the generating function for a set of orthogonal polynomials p t n x in the energy variable e some of the properties of these polynomials are discussed in detail and our analysis reveals scaling and factorization properties that are central to quasi exact solvability we also prove that this set of orthogonal polynomials can be reduced by means of a simple scaling transformation to a remarkable class of orthogonal polynomials pn e p 0 n e recently discovered by bender and dunne

first integrals and parametric solutions of third order odes admitting mathfrak sl 2 mathbb r a complete set of first integrals for any third order ordinary differential equation admitting a lie symmetry algebra isomorphic to is explicitly computed these first integrals are derived from two linearly independent solutions of a linear second order ode without additional integration the general solution in parametric form can be obtained by using the computed first integrals the study includes a parallel analysis of the four inequivalent realizations of and it is applied to several particular examples these include the generalized chazy equation as well as an example of an equation which admits the most complicated of the four inequivalent realizations

analytical results of zero gap states in periodic potentials we develop a method to construct various classes of one dimensional periodic potentials with two intersecting energy bands analytical exact results for the zero gap states are presented in an explicit form under certain parameter conditions the position of the energies of these zero gap states in the energy bands is identified numerically

confluent heun functions and separation of variables in spheroidal coordinates the helmholtz equation and schrodinger s equation for h2 are separable in respectively oblate and prolate spheroidal coordinates they share the same form of the angular equation in both cases the radial and angular equations have solutions in terms of confluent heun functions we show that the zeros of the wronskian of a pair of solutions to the angular equation give the allowed values of the separation of variables parameter since the heun functions and their derivatives are implemented in maple this provides a new method of calculating the physical values of the separation of variables parameter without programming we also derive the asymptotic forms of the radial solutions of the helmholtz equation and obtain integral relations between the radial and angular solutions

the d dimensional softcore coulomb potential and the generalized confluent heun equation an analysis of the generalized confluent heun equation 2r2 1r y 2r2 1r 0 y e1r e0 y 0 in d dimensional space where i i ei are real parameters is presented with the aid of these general results the quasi exact solvability of the schrodinger eigenproblem generated by the softcore coulomb potential v r e2z r b b 0 is explicitly resolved necessary and sufficient conditions for polynomial solvability are given a three term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly we prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials properties such as the recurrence relations christoffel darboux formulas and moments of the weight function are discussed we also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials an analysis of the generalized confluent heun equation 2r2 1r y 2r2 1r 0 y e1r e0 y 0 in d dimensional space where i i ei are real parameters is presented with the aid of these general results the quasi exact solvability of the schrodinger eigenproblem generated by the softcore coulomb potential v r e2z r b b 0 is explicitly resolved necessary and sufficient conditions for polynomial solvability are given a three term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly we prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials properties such as the recurrence relations christoffel darboux formulas and moments of the weight function are discussed we also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials

new classes of analytic solutions of the two level problem a simple technique to find analytic solutions of the time dependent schrodinger equations is introduced a number of three parametric classes of the two level problem integrable in terms of the gauss hypergeometric function including all presently known analytically solvable models are derived

well posedness of h optimal control problems this paper considers the effect of perturbations of a nominal single input single output linear plant or a band of uncertainty on the solution to certain frequency domain optimal control problems weighted h infty sensitivity minimization mixed and robust sensitivity minimization will be considered along with problems of more general type a brief discussion of h p optimal control problems will also be given typical examples will be presented where optimal sensitivity does not depend continuously on the plant ill posedness and conditions for well posedness will be given it is demonstrated that similar discontinuities can occur for more general problems also suggested are ways of defining an optimization problem so that continuous dependence of the infimum is ensured

incomplete beta function expansions of the solutions to the confluent heun equation several expansions of the solutions to the confluent heun equation in terms of incomplete beta functions are constructed a new type of expansion involving certain combinations of the incomplete beta functions as expansion functions is introduced the necessary and sufficient conditions when the derived expansions are terminated thus generating closed form solutions are discussed it is shown that termination of a beta function series solution always leads to a solution that is necessarily an elementary function

robust controller synthesis for uncertain time varying systems in this paper we develop and present new convex synthesis conditions for robust performance in linear time varying systems subject to time varying perturbations in particular these results are exact for sensitivity minimization in the presence of multiplicative perturbations the methods apply to a number of additional robust performance problems and are always both necessary and sufficient when the system plant is periodic otherwise the conditions provided are always sufficient and a controller can be directly constructed when this condition holds

new solutions of heun s general equation we show that in four particular cases the derivative of the solution of heun s general equation can be expressed in terms of a solution to another heun s equation starting from this property we use the gauss hypergeometric functions to construct series solutions to heun s equation for the mentioned cases each of the hypergeometric functions involved has correct singular behaviour at only one of the singular points of the equation the sum however has correct behaviour

lie point symmetry analysis of a second order differential equation with singularity abstract by using lie symmetry methods we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the ermakov pinney equation in this context nonlinear superposition rule for second order kummer schwarz equation is rediscovered invariance under one dimensional symmetry group is also used to obtain first integrals ermakov lewis invariants our motivation is a type of equations with singular term that arises in many applications in particular in the study of general nls nonlinear schrodinger equations

factorization of the heun s differential operator the differential equation with four regular singularities located at z 0 1 aand called the heun s equation he isy z email protected email protected email protected z email protected bz qz z 1 z a y z 0with email protected email protected email protected email protected and defines the heun s operator h byh y z p 3 z d 2 p 2 z d p 1 z y z with d d dz and p i z polynomials of degree i h can be factorized in the formh l z d m z l z d m z polynomials l l m and m are given explicitly in the cases where this factorization is possible it is shown that the value of the parameters a b and q allowing the factorization coincides with those obtained from the f homotopic transformation y z z r z 1 s z a email protected z forcing email protected z to be solution of a he as y z

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

solution of ordinary differential equations via nonlocal transformations the classical lie analysis provides a useful technique for the solution of ordinary differential equations via point symmetries transformations unfortunately the requirement that an nth order equation possesses at least an n dimensional solvable lie algebra of symmetries is only satisfied by a select number of equations we provide a class of second order ordinary differential equations with less than the required two lie point symmetries that can be solved via nonlocal transformations

new quasi exactly solvable double well potentials a new three parameter family of quasi exactly solvable double well potentials is introduced we show that the solutions of this family of double well potentials are expressed in terms of the heun confluent functions under a certain parameter condition some of the bound state wavefunctions and associated energies can be found exactly in explicit form in particular we develop an analytical method to derive the conditions for the energy eigenvalues of the bound states it is also shown that our analytical results can be applied to construct exact solutions of the nonlinear schrodinger equation with double well potentials and spatially localized nonlinearities

identification and adaptive control towards a complexity based general theory abstract two recent developments will be surveyed here which are pointing the way towards an input output theory of h l 1 adaptive feedback the solution of problems involving 1 feedback performance exact optimization under large plant uncertainty on the one hand the two disc problem of h and 2 optimally fast identification in h on the other taken together these are yielding adaptive algorithms for slowly varying data in h l 1 at a conceptual level these results motivate a general input output theory linking identification adaptation and control learning in such a theory the definition of adaptation is based on system performance under uncertainty and is independent of internal structure presence or absence of variable parameters or even feedback

potentials of the heun class we review different methods of generating potentials such that the one dimensional schrodinger equation odse can be transformed into the hypergeometric equation we compare our results with previous studies and complement the subject with new findings our main result is to derive new classes of potentials such that the odse can be transformed into the heun equation and its confluent cases the generalized heun equation is also considered

commutative algebraic methods for controllability of discrete time polynomial systems abstractin this paper we consider controllability of discrete time polynomial systems first we present a forward accessibility local reachability condition that can be verified in finite time in contrast to conventional conditions second we give a backward accessibility local controllability condition for an invertible system and a condition to verify invertibility finally we derive sufficient conditions to test whether the forward accessible system is reachable and to test the backward accessible system is controllable

the interaction v r ze2 r beta and the confluent heun equation the schrodinger equation with interaction ze2 r beta is shown for general values of the parameters to be reducible to the confluent heun equation

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

symmetry and the chazy equation there are three different actions of the unimodular lie group sl 2 c on a two dimensional space in every case we show how an ordinary differential equation admitting sl 2 as a symmetry group can be reduced in order by three and the solution recovered from that of the reduced equation via a pair of quadratures and the solution to a linear second order equation a particular example is the chazy equation whose general solution can be expressed as a ratio of two solutions to a hypergeometric equation the reduction method leads to an alternative formula in terms of solutions to the lame equation resulting in a surprising transformation between the lame and hypergeometric equations finally we discuss the painleve analysis of the singularities of solutions to the chazy equation

polynomial modular frobenius manifolds the moduli space of frobenius manifolds carries a natural involutive symmetry and a distinguished class so called modular frobenius manifolds lie at the fixed points of this symmetry in this paper a classification of semi simple modular frobenius manifolds which are polynomial in all but one of the variables is begun and completed for three and four dimensional manifolds the resulting examples may also be obtained from higher dimensional manifolds by a process of folding the relationship of these results with orbifold quantum cohomology is also discussed

on the linearization of some second order odes via contact transformations only those second order ordinary differential equations odes that admit an eight dimensional lie algebra of lie point symmetries are linearizable via a point transformation however all second order odes are linearizable via a contact transformation unfortunately the latter result is not useful as one cannot usually find the contact symmetries of second order odes by analysing realizations of three dimensional lie algebras we show how to explicitly construct the linearization of some second order odes via contact transformations

novel relations and new properties of confluent heun s functions and their derivatives of arbitrary order the present paper reveals important properties of the confluent heun s functions we derive a set of novel relations for confluent heun s functions and their derivatives of arbitrary order specific new subclasses of confluent heun s functions are introduced and studied a new alternative derivation of confluent heun s polynomials is presented

algebraic solutions of the lame equation revisited a minor error in the necessary conditions for the algebraic form of the lam e equation to have a finite projective monodromy group and hence for it to have only algebraic solutions is pointed out see f baldassarri on algebraic solutions of lams differential equation j differential equations41 1 1981 44 58 it is shown that if the group is the octahedral group then the degree parameter of the equation may differ by from an integer this possibility was missed the omission affects a recent result on the monodromy of the weierstrass form of the lamequation see r c churchill two generator subgroups of and the hypergeometric riemann and lam e equations j symbolic computa tion 28 4 5 1999 521 545 the weierstrass form which is a differential equation on an elliptic curve may have after all an octahedral projective monodromy group

classification of integrable two component hamiltonian systems of hydrodynamic type in 2 1 dimensions hamiltonian systems of hydrodynamic type occur in a wide range of applications including fluid dynamics the whitham averaging procedure and the theory of frobenius manifolds in 1 1 dimensions the requirement of the integrability of such systems by the generalised hodograph transform implies that integrable hamiltonians depend on a certain number of arbitrary functions of two variables on the contrary in 2 1 dimensions the requirement of the integrability by the method of hydrodynamic reductions which is a natural analogue of the generalised hodograph transform in higher dimensions leads to finite dimensional moduli spaces of integrable hamiltonians in this paper we classify integrable two component hamiltonian systems of hydrodynamic type for all existing classes of differential geometric poisson brackets in 2d establishing a parametrisation of integrable hamiltonians via elliptic hypergeometric functions our approach is based on the godunov type representation of hamiltonian systems and u

on robust performance in h abstract in this paper we consider the robust performance problem in h with scalar perturbations we illustrate how the worst case h performance can be regarded as an h norm of a function of several complex variables based on this characterization we show how the worst case performance can be obtained from the h norm of a single system this system is constructed from the original nominal system with the perturbations replaced by certain all pass functions we show that as the order of the all pass functions goes to infinity this h norm converges to the worst case performance the implications of this characterization for computing the complex structured singular value and robust synthesis are discussed though at present this method does not seem to provide an efficient algorithm for this computation

duality theory for mimo robust disturbance rejection banach space duality theory is used to characterize the solutions of a nonstandard h sup infinity optimization problem which is shown to be allpass in general and unique in the single input single output siso ease the theory leads to a numerical solution of duality and convex optimization which is applied to an example for a limiting case of sharp cutoff filters an explicit solution of the optimal robust disturbance attenuation problem ordap resembling the two arc theorem of complex analysis is derived

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

analytic treatment of the polariton problem for a smooth interface we study the polariton problem for smooth boundaries i e whether or not there exist some localized solutions of maxwell s equations for different types of smooth spatial variation of complex dielectric and or magnetic permittivity tensors for a particular dielectric permittivity profile varying according to the hyperbolic tangent law the singular term is mathematically strongly taken into account in maxwell s equations not in the boundary conditions the problem is then reduced to the canonical form of heun s equation possessing four regular singular points the solution to heun s equation as a power series is constructed and an approximate solution involving a combination of two incomplete betafunctions is derived further the exact eigenvalue solution to the polariton problem as a series in terms of incomplete beta functions or equivalently gauss hypergeometric functions is constructed it is shown that the dispersion relation for the polariton wavenumber does not depend on the interface transition layer width i e it is always exactly the same as the one derived in the limit of abrupt interface we conjecture that the polariton wavenumber eigenvalue depends on either zeros of the dielectric permittivity variance profile or the poles of the logarithmic derivative of the latter

information gains expected from separate and joint measurements of n identical spin 1 2 systems noninformative bayesian analyses an investigator is assumed to have n replicas of a spin 1 2 system but lacks any specific knowledge pertinent to constructing a 2 2 density matrix for it we compute within a noninformative bayesian framework the expected information gains for various measurement schemes first separate and then joint in nature in particular we obtain a result which fully accords with a certain plausibility argument of peres while the parallel result in a recent analysis of massar and popescu does not

applications of fuchsian differential equations to free boundary problems in a couple of recent papers free boundary problems and a class of conformal mappings involving curvilinear quadrilaterals were analysed primarily using transform methods in both the free boundary and conformal mapping problems there is an underlying relation with fuchsian differential equations and an alternative conceptually simpler solution technique for these problems would use solutions of these fuchsian equations thus it was conjectured in those papers that a class of fuchsian differential equations commonly known as heun9s equation has in some special but relatively important cases degenerate solutions these involve hypergeometric functions this complementary solution method is in many cases more convenient than the transform approach the purpose of this paper is to explore the connections with the fuchsian equations directly and use the solutions to solve free boundary problems the specific examples treated here come from a quasi steady approximation to solidification problems and are not without interest in their own right there are few analytical solutions for solidification problems in geometries of practical interest and the solutions found here should be of use in that regard

physical applications of second order linear differential equations that admit polynomial solutions in this paper conditions for the second order linear differential equation to have polynomial solutions are given several applications of these results to schrodinger s equation are discussed conditions under which the confluent biconfluent and general heun equation yields polynomial solutions are explicitly given some new classes of exactly solvable differential equations are also discussed the results of this work are expressed in such a way as to allow direct use without preliminary analysis

heun functions and quasi exactly solvable double well potentials we investigate a type of one dimensional quasi exactly solvable double well potential whose analytical solution can be constructed in terms of the heun functions it is shown that for certain special values of the potential parameters two energy eigenvalues and eigenstates of the lowest part of the energy spectrum can be found exactly in explicit form in addition the wronskian method has been applied to derive the conditions for the energy eigenvalues of the bound states our analytical results may find applications in the tunnelling problem for the double well potentials

the liouville arnold nekhoroshev theorem for non compact invariant manifolds under certain conditions generalized action angle coordinates can be introduced near non compact invariant manifolds of completely and partially integrable hamiltonian systems

lie algebras of differential operators and lie algebraic potentials abstract an explicit characterisation of all second order differential operators on the line which can be written as bilinear combinations of the generators of a finite dimensional lie algebra of first order differential operators is found solving a problem arising in the lie algebraic approach to scattering theory and molecular dynamics one dimensional potentials corresponding to these lie algebras are explicitly classified which include the harmonic oscillator morse one soliton poschl teller mathieu lame confluent hypergeometric and bessel potentials

instability of pole singularities for the chazy equation we prove that the negative resonances of the chazy equation in the sense of painleve analysis can be related directly to its group invariance properties these resonances indicate in this case the instability of pole singularities depending on the value of a parameter in the equation an unstable isolated pole may turn into the familiar natural boundary or split into several isolated singularities in the first case a convergent series representation involving exponentially small corrections can be given this reconciles several earlier approaches to the interpretation of negative resonances on the other hand we also prove that pole singularities with the maximum number of positive resonances are stable the proofs rely on general properties of nonlinear fuchsian equations

eigenvalue singular problem of factorized fourth order self adjoint differential equations this paper deals with the eigenvalue singular problem of the spectral type differential equations of the fourth order self adjoint differential operators1 1 x p 1 x qd 2dx 2 1 x p 2 1 x q 2d 2ydx 2 my 0 y 1 y 1 finite where p 1 q 1 it has been reported in the literature that spectral type differential equations above have general solutions in terms of hypergeometric functions in this work it is showed that the general solution in terms of hypergeometric functions reduces to jacobi orthogonal polynomials as eigenfunctions in the case of eigenvalue singular problem the corresponding eigenvalues are found as application the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform singular structure are reported

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

the integrability of the two state problem in terms of confluent hypergeometric functions the reduction of the two state problem to the confluent hypergeometric equation via complex transformations of independent variables is considered it is shown that all the known analytically integrable cases can be generalized to a single formula a new class of integrable models of the two state problem in terms of confluent hypergeometric functions is presented

solutions for the general confluent and biconfluent heun equations and their connection with abel equations in a recent paper the canonical forms of a new multi parameter class of abel differential equations the so called abel inverse riccati air all of whose members can be mapped into riccati equations were shown to be related to the differential equations for the hypergeometric 2f1 1f1 and 0f1 functions in this paper a connection between the air canonical forms and the general heun ghe confluent che and biconfluent bhe equations is presented this connection fixes the value of one of the heun parameters expresses another one in terms of those remaining and provides closed form solutions in terms of pfq functions for the resulting ghe che and bhe respectively depending on four three and two irreducible parameters this connection also makes evident the relation between the heun parameters such that the solutions admit liouvillian form and suggests a mechanism for relating linear equations with n and n 1 singularities through the canonical forms of a nonlinear equation of one order less

mimo disturbance and plant uncertainty attenuation by feedback this paper investigates the ability of feedback to reduce plant and disturbance uncertainties in the multiple input multiple output mimo case by solving two fundamental problems posed by zames in the late seventies the first problem is termed the mimo extension of the optimal robust disturbance attenuation problem the second is a filtering of plant uncertainty two degree of freedom feedback problem the optimal solutions of these problems are characterized using banach space duality theory and shown to satisfy an allpass condition and an extremal identity moreover the duality description leads to a dual pair of optimizations and the introduction of two nonstandard matrix norms in particular the primal dual optimization reduces naturally to approximate finite dimensional convex optimizations within desired tolerance whereas the computation of the nonstandard matrix norms are shown to be equivalent to specific semi definite programming problems and a numerical solution based on convex programming is provided it is also shown using douglas range inclusion theorem that performance is a monotonic function of uncertainty and some qualitative implications for feedback are derived

convex and finite dimensional conditions for controller synthesis with dynamic integral constraints in this paper the problem of synthesizing controllers when the allowable disturbances and the cost criterion are defined via a finite number of matrix valued dynamic integral constraints is solved this allows one to design optimal control systems for plants subject to fixed input disturbances such as steps sinusoids and periodic disturbances disturbances of fixed and known spectrum disturbances defined as the set of signals in the kernel of a given operator and combinations thereof to penalize particular frequency components of the error variables in the control design process to penalize the maximum amplitude of an error variable and to consider other general cost criteria in the optimization and to solve a large class of robust control synthesis problems necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive finite dimensional linear matrix inequality

euler bernoulli beams from a symmetry standpoint characterization of equivalent equations abstract we completely solve the equivalence problem for euler bernoulli equation using lie symmetry analysis we show that the quotient of the symmetry lie algebra of the bernoulli equation by the infinite dimensional lie algebra spanned by solution symmetries is a representation of one of the following lie algebras 2 a 1 a 1 a 2 3 a 1 or a 3 3 a 1 each quotient symmetry lie algebra determines an equivalence class of euler bernoulli equations save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters lineal mass density and flexural rigidity for each class we provide a simple representative and we explicitly construct transformations that maps a class member to its representative the maximally symmetric class described by the four dimensional quotient symmetry lie algebra a 3 3 a 1 corresponds to euler bernoulli equations homeomorphic to the uniform one constant lineal mass density and flexural rigidity we rigorously derive some non trivial and non uniform euler bernoulli equations reducible to the uniform unit beam our models extend and emphasize the symmetry flavor of gottlieb s iso spectral beams h p w gottlieb isospectral euler bernoulli beam with continuous density and rigidity functions proc r soc lond ser a math phys eng sci 413 1987 235 250

solutions for confluent and double confluent heun equations this paper examines some solutions for confluent and double confluent heun equations in the first place we review two leaver s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation leaver e w j math phys 27 1238 1986 and introduce an additional expansion in series of irregular confluent hypergeometric functions then we find the conditions under which one of these solutions can be written as a linear combination of the others in the second place by means of limiting procedures we generate solutions for the double confluent equation as well as for special limits of both the confluent and double confluent equations finally we present some problems which are ruled by one or the other of these four equations

operator theoretic approach to the optimal two disk problem the nonstandard two disk problem plays a fundamental role in robust feedback optimization here it is shown via banach space duality theory that its solutions satisfy an extremal identity and may be viewed as a dual extremal kernel of a particular l sup 1 optimization problem a novel operator theoretic framework to characterize explicitly its solutions is developed in particular the two disk optimization is shown to be equal to the induced norm of a specific operator defined on a projective tensor product space involving a non hilbert version of a vector valued h sup 2 space moreover this operator is shown to be a combination of multiplication and toeplitz operators under certain conditions existence of maximal vectors is established leading to an explicit formula for the optimal controller an infinite matrix representation with respect to a canonical basis is derived together with an algorithm to compute it the norm of the relevant operator is approximated by special finite dimensional optimizations whose solutions lead to solving semi definite programming problems involving the computation of a matrix projective tensor norm

discrete point symmetries of ordinary differential equations this paper describes a method that enables the user to construct systematically the set of all discrete point symmetries of a given ordinary differential equation ode of order two or greater provided that the ode has at least a one parameter lie group of point symmetries the method is easy to use and is based upon lie s method of constructing continuous symmetries the calculations are simple and computer algebra is not usually required various examples are used to illustrate the method the paper concludes with a proof that every ode whose lie group of point symmetries is isomorphic to the unimodular group has at least four inequivalent real discrete point symmetries

entropies and heun functions associated with positive linear operators we consider a parameterized probability distribution p x p 0 x p 1 x and denote by s x the squared l2 norm of p x the properties of s x are useful in studying the renyi entropy the tsallis entropy and the positive linear operator associated with p x we show that for a family of distributions including the binomial and the negative binomial distributions s x is a heun function reducible to the gauss hypergeometric function 2f1 several properties of s x are derived including integral representations and upper bounds examples and applications are given concerning classical positive linear operators

application of the heun functions to rotor vibratory gyroscope their numerical computation and accuracy estimation abstract for an asymmetric rotor vibratory gyroscope that is oscillating in an elastic suspension means the equation of motion is derived from the eurler lagrange equation and the exact solution is obtained as heun functions hfs a fast and effective method for calculating heun functions by direct calculation of solutions of the heun differential equation hde using standard numerical integration methods is developed three methods of accuracy check are employed in this case the accuracy of the numerical solutions deteriorated in the vicinity of the singularity to overcome this difficulty an optimised method for calculating the hfs is developed which give a uniform accuracy of the calculated values on the interval the optimised method for calculating the numerical hfs by means of the solution of the governing initial value problem gave acceptable accuracy in modelling the behaviour of an asymmetric rotor gyroscope

the 192 solutions of the heun equation a machine generated list of 192 local solutions of the heun equation is given they are analogous to kummer s 24 solutions of the gauss hypergeometric equation since the two equations are canonical fuchsian differential equations on the riemann sphere with four and three singular points respectively tabulation is facilitated by the identification of the automorphism group of the equation with n singular points as the coxeter group d n each of the 192 expressions is labeled by an element of d 4 of the 192 24 are equivalent expressions for the local heun function hl and it is shown that the resulting order 24 group of transformations of hl is isomorphic to the symmetric group s 4 the isomorphism encodes each transformation as a permutation of an abstract four element set not identical to the set of singular points

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

time varying optimal disturbance minimization in presence of plant uncertainty the optimal robust disturbance rejection problem plays an important role in feedback control theory here its time varying version is solved explicitly in terms of duality and operator theory in particular the optimum is shown to satisfy a time varying allpass property moreover optimal performance is given in terms of the norm of a bilinear form the latter depends on a lower triangular projection and a multiplication operator defined on special versions of spaces of compact operators

on reducing the heun equation to the hypergeometric equation abstract the reductions of the heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified heun to hypergeometric reductions are similar to classical hypergeometric identities but the conditions for the existence of a reduction involve features of the heun equation that the hypergeometric equation does not possess namely its cross ratio and accessory parameters the reductions include quadratic and cubic transformations which may be performed only if the singular points of the heun equation form a harmonic or an equianharmonic quadruple respectively and several higher degree transformations this result corrects and extends a theorem in a previous paper which found only the quadratic transformations siam j math anal 10 3 1979 655

exact bound state wavefunctions for potentials varying from the double well to the single well the authors reduce the schrodinger equation on the full line the potential of which varies from the double well to the single well depending on the value of some parameter a to a heun s type equation the solution of which recalls the hypergeometric series the eigenvalue equation which determines the energy of bound states follows from the usual requirement that the wavefunction be zero at infinity transforming this equation into the infinite continued fraction expansion they easily obtain the approximate eigenvalue they compare it with that evaluated exactly and see that the difference is less than 0 5 for any value of the parameter a

factorization of self adjoint ordinary differential equations this paper deals with the factorization of self adjoint differential operators l 2 n email protected ndx email protected b nd ndx n and their spectral type differential equations sufficient conditions of factorization are reported a large class of differential operators and equations that can be factorized is obtained the factorizations of fourth and sixth order operators and equations are explicitly given a particular fourth order spectral type differential equation in which r x 1 x p 1 x q p 1 q 1 is considered its general solution is obtained in terms of hypergeometric functions as application the natural frequencies and mode shapes of mechanical transverse vibrations of a nonuniform structure are found

a perturbative painleve approach to nonlinear differential equations abstract we further improve the painleve test so that negative indices resonances can be treated we demand single valuedness not only for any pole like expansion as in the usual painleve test but also for every solution close to it represented as a perturbation series in a small parameter e order zero is the usual test order one already treated in a preliminary paper reduces to a linear fuchs analysis near a regular singularity and allows the introduction of all missing arbitrary coefficients higher orders lead to the analysis of a linear fuchsian type inhomogeneous equation we obtain an infinite sequence of necessary conditions for the absence of movable logarithmic branch points arising at every integer index whether positive or negative and at every order those arising at negative indices including 1 are new while some conditions may not arise before some high perturbation order we present several illustrative examples we discuss the understanding of negative indices and conclude that they are indistinguishable from positive indices just as in the fuchs theory in particular negative indices give rise to doubly infinite laurent series