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

systems of numeration and fractal functions relating to substitutions french abstract let a be a finite alphabet a substitution over a un n n a fixed point for and for each a a a a real number we establish under some assumptions an asymptotic formula concerning the sum s n i n u i n n this result generalizes some previous results from coquet or brillhart erdos and morton moreover relations with self affine functions in a sense which generalizes a definition from kamae are proved the calculi leave over systems of representation of integers and real numbers

determinative vertices of interval family with ωθ stability abstract in this article we pay attention to the stability of interval polynomials with regard to the region an analysis on how to generate the finite set whose stability implies robust stability of the entire family is given those results are intuitive and convenient to operate

on robust hurwitz polynomials in this note kharitonov s theorem on robust hurwitz polynomials is simplified for low order polynomials specifically for n 3 4 and 5 the number of polynomials required to check robust stability is one two and three respectively instead of four furthermore it is shown that for n geq 6 the number of polynomials for robust stability checking is necessarily four thus further simplification is not possible the same simplifications arise in robust schur polynomials by using the bilinear transformation applications of these simplifications to two dimensional polynomials as well as to robustness for single parameters are indicated

guaranteed characterization of stability domains via set inversion a new method is presented for characterizing the set of all values of the parameters of a linear time invariant model that are associated with a stable behavior a formal routh table is used to formulate the problem as one of set inversion which is solved approximately but globally with tools borrowed from interval analysis the method readily extends to the design of controllers stabilizing all models in a given class

robust stability of nested polynomial families we introduce a class of polynomial families for which robust stability analysis is as simple as for polytopic families this class includes the class of polytopic families as a special case stability analysis of polynomial families from this new class does not include the value set construction

a necessary and sufficient condition for schur invariance and generalized stability of polytopes of polynomials polytopes of characteristic polynomials arise when systems experience real parameter variations a necessary and sufficient condition requiring only a finite number of computations is presented for determining whether all the roots of a polytope of polynomials are contained in a desired region this can be any region that is the image of the open left half plane under a real linear fractional transformation since the open unit cycle is such a region this result establishes a finite algorithm for strict schur invariance finite algorithms for generalized stability regions such as maximum bandwidth and maximum setting time are also established

robust stabilization of interval plants using lead or lag compensators abstract a lead or lag compensator robustly stabilizes a family of interval plants if and only if it simultaneously stabilizes the vertex plants

brief robust multivariable pid control for linear parameter varying systems practical control problems often deal with uncertain systems depending on parametric and possibly time varying uncertainties quadratic stability theory and recent advances in convex and nonconvex optimisation algorithms provide useful tools to cope with this kind of problems on the other hand pid is still the controller most commonly used for industrial problems in this paper we consider a certain class of multi input multi output linear parameter varying systems which are encountered in practical control problems and we propose a state space parameterisation which allows to convert pi and pid controller synthesis into an lmi or bmi optimisation problem the proposed technique guarantees uniform exponential stability of the closed loop system a desired rate of convergence and an h norm bound

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

structured singular values and stability analysis of uncertain polynomials part 1 the generalized m abstract one of the primary goals in this two part series is to establish a link between the structured singular value and results in stability analysis of uncertain polynomials another primary goal is to develop an improved method for accurately and efficiently computing the structured singular value for an important class of problems in robust stability analysis to achieve these goal we first introduce a generalized framework of structured singular values and next show how stability problems for uncertain polynomials may be studied in this framework part 1 of this series is devoted entirely to the generalized structured singular values specifically for the case when a certain matrix representing the nominal system is of rank one an analytical expression for the generalized notion is derived in this case which involves solving a convex optimization problem in one real variable and renders the structured singular value readily solvable in particular when the general framework is specialized to that of the standard structured singular value the expression is solved explicitly also for several additional important cases explicit solutions are obtained the framework as well as results will then be used in part 2 of this series to study stability problems for a class of polynomials whose coefficients are affine functions of real or complex uncertainties we demonstrate that the generalized structured singular value is a suitable notion for these problems and its solution unifies a number of results obtained previously via alternative approaches for several problems of interest we further demonstrate that stability conditions based upon the structured singular value and those in the spirit of kharitonov theorem can be derive from one another and hence establish a link between two drastically different type of results

to stabilize an interval plant family it suffices to simultaneously stabilize sixty four polynomials it is shown that stability of three specific polynomial families can be deduced from the stability of a finite number of polynomials these polynomial families are the characteristic polynomials of unity feedback loops with the controller in the forward path and where the plant includes a specific form of parameter uncertainty for the first polynomial family the plant has parameter uncertainty in the even or odd terms of the numerator or denominator polynomial for the second polynomial family the plant has a numerator or denominator which is an interval polynomial for the third polynomial family the plant is interval because of the structure of these results it is shown that they lead to robust stabilization results two examples are included the approach employed here was developed for plants with affine uncertainty it is demonstrated that considerable simplification results if the plants under investigation are interval

stability of a family of polynomials with coefficients bounded in a diamond the robust stability property is examined for family of nth order real polynomials where the coefficients are bounded within a diamond in the n 1 dimensional space it is shown that such a family of polynomials is hurwitz if and only if four specially selected edge polynomials are hurwitz

a simple proof of kharitonov s theorem an alternative proof of kharitonov s theorem on the stability of linear time invariant continuous systems under parameter variations is presented

generalized pi controllability integral controllable systems can be stabilized by all sufficiently low gain integral controllers the concept of integral controllability is extended to two generalized stability regions and to proportional controllability and pi controllability with respect to those regions furthermore closed eigenvalue formulae are derived for the maximal low integral gain and the maximal low proportional gain for generalized stability related eigenvalue formulae are then used in combination to find all low gain pi controllers that place the closed loop eigenvalues in the desired region

extreme point results for robust stabilization of interval plants with first order compensators it has been shown previously that a first order compensator robustly stabilizes an internal plant family if and only if it stabilizes all of the extreme plants these extreme plants are obtained by considering all possible combinations for the extreme values of the numerator and denominator coefficients in this work the authors prove a stronger result namely that it is necessary and sufficient to stabilize only sixteen of the extreme plants these sixteen plants are generated using the kharitonov polynomials associated with the numerator and denominator furthermore when additional information about the compensator is specified sign of the gain and signs and relative magnitudes of the pole and zero then in some cases it is necessary and sufficient to stabilize eight critical plants while in other cases it is necessary and sufficient to stabilize twelve critical plants

on the nyquist envelope of an interval plant family in this note we study the envelope of the nyquist plots generated by an interval plant family and show that this boundary is not always contained in the nyquist plots of the kharitonov plants with this motivation in mind we give a sufficient condition for an envelope point to be contained in the nyquist plot of a kharitonov plant and use it to generate large and critical portions of the nyquist envelope e g we show that the outer nyquist envelope of a stable interval plant is generated by the nyquist plots of the kharitonov plants another by product of this sufficient condition is a framework for developing new extreme point results for interval feedback systems this framework is useful in computing the phase margin and the maximal peaking in the sensitivity and complementary sensitivity functions and in stating a robust version of the circle criterion we also use this framework to easily explain existing extreme point results for the gain margin the h sub spl infin norm and the positive realness of interval plants one conclusion which emerges is this seemingly all important properties of an interval feedback system are deducible from the kharitonov plants

some remarks on the robust schur stability analysis of interval polynomials the schur stability analysis of an interval polynomial family can be quickly performed through a unique suitably defined extreme polynomial the purpose of this article is to provide some improvements with respect to the actually existing methods based on this approach

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

paperfolding morphisms planefilling curves and fractal tiles an interesting class of automatic sequences emerges from iterated paperfolding the sequences generate curves in the plane with an almost periodic structure we generalize the results obtained by davis and knuth on the self avoiding and planefilling properties of these curves giving simple geometric criteria for a complete classification finally we show how the automatic structure of the sequences leads to self similarity of the curves which turns the planefilling curves in a scaling limit into fractal tiles for some of these tiles we give a particularly simple formula for the hausdorff dimension of their boundary

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

technical communique stabilizability considerations and design of rational controllers for a class of time delay systems the problem of stabilization of a linear system characterized by its rational transfer function with an input time delay by a rational controller is considered if the delay is not known at all its interval is infinite the necessary and sufficient condition for the existence of a rational controller is stability of the open loop rational plant moreover the existence of a rational stabilizing controller implies the existence of a constant gain stabilizing controller if the delay is known to lie in a given finite interval two non existence theorems are derived one for the constant gain stabilizing controller and one for any rational stabilizing controller a design method for stabilizing rational controllers based on first order all pass filters cascaded by a constant gain and on violating the conditions for the non existence theorem of the constant gain stabilizer is presented

a dual result to kharitonov s theorem a dual problem to kharitonov s problem involving a diamond instead of a rectangle is considered the results show that a family of polynomials with coefficients varying in the diamond is strictly hurwitz if and only if eight one dimensional exposed edges of the diamond are strictly hurwitz

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

computation of the stability radius of a hurwitz polynomial with diamond like uncertainties this paper considers the stability radius problem of hurwitz polynomials whose coefficients have holder 1 norm bounded uncertainties we show that the solution to this problem demands the computation of the minimum of a piece wise real rational function called the stability radius function it is then shown that the calculations of at the intersection points where changes its representation and at the stationary points where 0 can be reduced to two sets of eigenvalue problems of matrices of the form h 1h where both h and h are frequency independent hurwitz matrices using root locus technique we analyze this function further and prove that in some special cases the minimum of this function can be achieved only at the intersection points extensions of the eigenvalue approach to cover other robust stability problems are also discussed

comments on extreme point results for robust stabilization of interval plants with first order compensators the authors point out a simple alternative proof of the main result theorem 1 of the paper by barmish hollot kraus and tempo 1992 this proof follows immediately on applying the vertex result of hollot and yang 1990 to the generalization of kharitonov s theorem given by chapellat and bhattacharyya 1988 1989 the authors also point out that more general vertex results are known

robust schur stability of a polytope of polynomials the main objective of the authors is to provide a necessary and sufficient condition for a polytope of polynomials to have all its zeros inside the unit circle the criterion obtained serves as a discrete time counterpart for results in s bialas 1985 and f fu and b r barmish 1987 for the continuous case also the results are reduced to operations on n 1 n 1 matrices it is concluded that by the edge result of a c bartlett et al 1987 it suffices to check the exposed edges in order to determine whether a polytope of polynomials has all its zeros in a simply connected region d

on stability of a weighted diamond of real quasi polynomials this paper deals with the stability problem for a weighted diamond of real quasi polynomials we show that under certain conditions on the weights and coefficients in the exponents the stability of the weighted diamond follows from the stability of eight one parameter families edges of quasi polynomials in order to check the diamond for stability it is sufficient to examine only eight one parameter families of quasi polynomials in contrast to the case of a rectangle of quasi polynomials which requires checking the stability of an exponential number of one parameter families of quasi polynomials

robust strong stabilizability of interval plants it suffices to check two vertices it is shown that a proper interval plant is strongly stabilizable if and only if all members are free from unstable pole zero cancellations and two specially selected vertex plants are strongly stabilizable

some discrete time counterparts to extreme point results for robust stability of a diamond of polynomials the main result of the paper is that a family of polynomials with roughly half of all the real coefficients lying in a diamond is robustly schur stable if and only if all the extreme polynomials are schur stable this special polynomial family is called a degenerate diamond of polynomials in addition we obtain the extreme point result for robust schur stability of a so called simplex degenerate diamond polynomial family

closed loop hyperstability of interval plants this note examines the conditions for robust bounded input bounded output closed loop stability of interval plants when there is a nonlinear element in the feedback path specifically the authors present the robust generalized circle criterion and popov s criterion

stability margin evaluation for uncertain linear systems a sufficient stability condition for parameter variation in a perturbed continuous time multivariable linear system represented by a state space model is presented starting with the existence of an algebraic riccati equation a stability bound is derived from the polar decomposition of the nominal system matrix unlike previous work the results are not dependent on the solution of the lyapunov equation and consequently not a function of an arbitrarily selected positive definite matrix in addition the bound would appear to be the tightest possible in that violation of the presented inequality can be shown to lead to instability

hurwitz stability of weighted diamond polynomials abstract in this note we consider real diamond polynomials of degree n whose roots are required to lie in the open left half plane a minimal number of exposed edges whose stability guarantees the stability of the entire polynomial family is selected a condition under which the stability of the vertices of the diamond guarantees the stability of the entire polynomial family is provided some illustrative examples are given

mellin transforms and asymptotics digital sums abstract arithmetic functions related to number representation systems exhibit various periodicity phenomena for instance a well known theorem of delange expresses the total number of ones in the binary representations of the first n integers in terms of a periodic fractal function we show that such periodicity phenomena can be analyzed rather systematically using classical tools from analytic number theory namely the mellin perron formulae this approach yields naturally the fourier series involved in the expansions of a variety of digital sums related to number representation systems

all constant gain stabilizing controllers for an interval delay system with uncertain parameters a computationally tractable design method is presented which explicitly yields the entire set of all constant gain controllers which robustly stabilize a family of systems with uncertainty a desired optimal controller may then be selected from the feasible set the method applies to the class of systems where a delay element e hs with interval finite or infinite uncertainty about h is followed by a plant characterized by a rational transfer function with independent interval coefficients illustrative numerical examples are provided

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

power and exponential sums for generalized coding systems by a measure theoretic approach a measure theoretic approach to study the power and the exponential sums for the usual coding system has been developed since the 1990s in this paper we first introduce a new coding system and then give explicit formulas for the power and the exponential sums for the coding system by the measure theoretic approach an expression for the power sum using the generalized takagi function will also be given

brief condition and algorithm for simultaneous stabilization of linear plants this paper is devoted to the problem of simultaneous stabilization of linear plants it is shown that four given plants p i s i 1 2 3 4 are simultaneously stabilizable if and only if there exist four stable polynomials h i s i 1 2 3 4 which simultaneously satisfy two polynomial equations if we can only find four stable h i s i 1 2 3 4 to satisfy one of the two polynomial equations then two controllers c 1 s and c 2 s having a common denominator or numerator can be obtained such that c 1 s simultaneously stabilizes p 1 s and p 3 s and c 2 s simultaneously stabilizes p 2 s and p 4 s which is called the simultaneous stabilization in groups based on such two controllers a design algorithm for the simultaneous stabilization of three plants is induced

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

robust hurwitz l sup p stability of polynomials with complex coefficients necessary and sufficient conditions for robust stability of a family of polynomials with complex coefficients are provided this work generalizes a frequency domain criteria for robust stability of continuous linear systems reported on by y z tsypkin and b t polyak 1991

invariance of the strict hurwitz property for polynomials with perturbed coefficients given a strictly hurwitz polynomial f lambda lambda n a n 1 lambda n 1 a n 2 lambda n 2 a 1 lambda a 0 it is of interest to know how much the coefficients a i can be perturbed while simultaneously preserving the strict hurwitz property for systems with n leq 4 maximal intervals of the a i are given in a recent paper by guiver and bose 1 in this note a theorem of kharitonov is exploited to obtain a general result for polynomials of any degree

a generalization of kharitonov s four polynomial concept for robust stability problems with linearly dependent coefficient perturbations kharitonov s four polynomial concept is generalized to the case of linearly dependent coefficient perturbations and more general zero location regions to this end a specially constructed scalar function of a scalar variable is instrumental to the robustness analysis the present work is motivated by two fundamental limitations of kharitonov s theorem namely 1 the theorem only applies to polynomials with independent coefficient perturbations and 2 it only applies to zeros in the left hand plane

extreme point results for robust stability of interval plants beyond first order compensators motivated by robust stabilization of interval plants this paper concentrates on a fundamental problem involving a family of polynomials obtained via convex combination of extreme members the objective is to provide conditions under which stability of the extremes implies stability of the entire family although the existing literature contains rather strong extreme point results involving frequency domain analysis there is a paucity of more direct coefficient space criteria this paper deals exclusively with the coefficient space problem to this end we introduce the so called alternating hurwitz minor condition ahmc and show how it can be used to enlarge the class of polynomial families for which extreme point results can be obtained subsequently the ramifications on robust stability of interval plants are fully explained in contrast to existing literature for first order compensators the ahmc makes it possible to deal with classes of higher order compensators in an extreme point context

a survey of extreme point results for robustness of control systems abstract this paper surveys a subset of the body of research which was sparked by kharitonov s theorem the focal point is extreme point results for robust stability and robust performance that is we give conditions under which satisfaction of a performance specification is ascertained for a family of systems by only checking a finite subset of the extreme members of this family the results which are surveyed apply mainly to systems with structured real parametric uncertainty in addition a number of counterexamples are given to illustrate cases for which an extreme point result does not hold for such cases a solution via the so called edge theorem is often possible

frequency domain criteria for l sup p robust stability of continuous linear systems necessary and sufficient criteria in the frequency domain for robust stability are given under the assumption that coefficients of a characteristic polynomial belong to a transformed l sup p ball three cases are considered in detail p infinity interval uncertainty p 2 ellipsoidal uncertainty and p 1 octahedral uncertainty it is shown that frequency domain stability criteria are efficient when one deals with robust stability problems the frequency domain approach considered can be extended to various problems of robust stability

an extreme point result of robust stability of a diamond of polynomials it is shown that a family of polynomials with real coefficients lying in a diamond is hurwitz if and only if eight distinguished extreme polynomials are hurwitz for the case of complex coefficients it is shown by a counterexample that no analogous extreme point result holds

structured singular values and stability analysis of uncertain polynomials part 2 a missing link abstract in part 1 of this paper a generalized notion of structured singular value is introduced and studied specifically in the case when a certain matrix is of rank one in part 2 of this series we demonstrate that the framework developed in part 1 is particularly suitable for a class of stability problems and its unifies several frequency based stability conditions obtained via alternative approaches generally these problems correspond to a class of uncertain polynomials whose coefficients are perturbed in an affine fashion several stability conditions are derived readily from solution of the structured singular value presented in part 1 which either extend or coincide with previously known results in particular for family of interval and diamond polynomials we further show that the stability conditions based on the structured singular value and the corresponding results in the spirit of kharitonov theorem lead to one another thus establishing a link between the two types of drastically different results and accomplishing one of our main goals in this series a related robust stabilization problem is also investigated in this framework

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

on the stability of a weighted diamond of real polynomials recently a subset of the robust stability literature concentrated on the so called diamond of polynomials in this paper we study the weighted diamond qw defined as qw q rn 1 w0 q0 q0 wn qn qn r where the center q the weights w0 w1 3dot wn 0 and the radius r 0 are given for this family we show that an extreme point result holds if and only if a certain interlacing condition on the weights is satisfied

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

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

block bialternate sum and associated stability formulae abstract a new block bialternate sum of a partitioned square matrix with itself is defined and some basic properties established it has similar properties to the bialternate sum but it preserves the block structure of the summand it is used in place of the block kronecker sum and the block lyapunov sum to solve maximal stability problems of stable systems under low gain integral control and of singularly perturbed systems it is also used to find the critical gains for a first order controller with a scalar plant from which all stabilizing first order controllers can be constructed in each stability problem the internal dimensions of the resulting closed formulae are lower than those currently available

nondefinite least squares and its relation to h sub infinity minimum error state estimation the problem of recursive nondefinite least squares state estimation of continuous time stationary processes is solved by applying pontryagin s maximum principle a comparison of the derived solution to the result that is obtained for the h sub infinity minimum error estimation suggests a new interpretation for the h sub infinity optimal estimation mechanism according to this interpretation the estimator tries to optimally estimate the required combination of the states in the l sub 2 norm sense against the worst disturbance signal that stems from a fictitious measurement of this combination

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

robust stability of systems with both parametric and dynamic uncertainties abstract this paper studies the robust stability of a single input single output systems with both parametric and dynamic uncertainties the parametric uncertainties of the system are described by a polytopic perturbation mode and the dynamic uncertainties are characterized by an integral quadratic constraint based on the concepts of biconvex and convex concave functions and their properties the problem of stability checking is discussed and for different uncertainty structures of the system the finite vertex checking results and the edge checking results are presented

on the stability properties of polynomials with perturbed coefficients given a polynomial p c s s n t 1 s n 1 t n 0 which is hurwitz or p d z z n t 1 z n 1 t n 0 which has zeros only within or on the unit circle it is of interest to know how much the coefficients t j can be perturbed while preserving the stability properties in this note a method is presented for obtaining the largest hypersphere centered at t t t 1 t n containing only polynomials which are stable

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

an elementary proof of kharitonov s stability theorem with extensions gives an elementary proof of kharitonov s theorem using simple complex plane geometry without invoking the hermite bieler theorem kharitonov s theorem is a stability result for classes of polynomials defined by letting each coefficient vary independently in an arbitrary interval the result states that the whole class is hurwitz if and only if four special well defined polynomials are hurwitz the paper also gives elementary proofs of two previously known extensions for polynomials of degree less than six the requirement is reduced to fewer than four polynomials and the theorem is generalized to polynomials with complex coefficients