the klein gordon equation of generalized hulthen potential in complex quantum mechanics we have investigated the reality of exact bound states of complex and or pt symmetric non hermitian exponential type generalized hulthen potential the klein gordon equation has been solved by using the nikiforov uvarov method which is based on solving the second order linear differential equations by reduction to a generalized equation of hypergeometric type in many cases of interest negative and positive energy states have been discussed for different types of complex potentials

h controller synthesis by j lossless coprime factorization this paper develops a coprime factorization approach to the synthesis of internally stabilizing controllers for a given system such that the mathcal h infty norm of the closed loop is strictly less than a given bound by the use of coprime factorizations it is shown that the mathcal h infty control problem is fundamentally related to the so called analytic systems considered by helton et al regional conference series in mathematics 68 1987 such problems admit a solution if and only if a certain j lossless factorization exists interpreted in the mathcal h infty control context this means that for a controller of the requisite type to exist the plant must admit a certain j lossless coprime factorization the full mathcal h infty synthesis problem requires that two nested j lossless factorizations exist the results are independent of whether discrete time or continuous time systems are being considered it is then shown that j lossless factorization is equivalent to the existen

a unified recurrence operator method for obtaining normalized explicit wavefunctions for shape invariant potentials we construct a unified recurrence operator method for obtaining explicit expressions for the wavefunctions of shape invariant potentials it is found that the normalized coefficients for the energy eigenfunctions satisfy a universal recurrence relation the procedure is illustrated in detail for four potentials we work out the normalized explicit wavefunctions of hulthen potential for which the normalized explicit wavefunctions have not been previously calculated

minimal nonsquare spectral factors we consider the problem of parametrizing the set of all nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis

streets and pedestrian trajectories in an urban district bejan s constructal principle revisited abstract bejan s 1996 analytical optimum for the average travel time between an arbitrary point in a rectangle and its gravitating vertex along a trajectory consisting of a low resistance side of the rectangle street and a high resistance internal segment perpendicular to this side is generalized in the following way a our street length is shorter than the rectangle side b our internal segment of the trajectory is tilted with respect to the street at an angle which is either constant a middle zone of the district or varies in two other zones where trajectories directly converge to either the vertex or the tip of the street wherefrom a pedestrian embarks a bus transporting him to the vertex c our double integrals in the criterion and the first second derivative tests are more cumbersome for evaluation and require computer algebra for calculation the total energy spent by a moving entity is assumed to be linearly dependent on the length of the walkway or street trajectory segments with two different resistance coefficients this energy is minimized for an individual particle such that the tilt angle is determined and zonation is done then another minimum of the area averaged community energy is searched with the total rectangle area as a constraint and the side ratio of the rectangle as a control variable explicit analytical expressions for double integrals and minima are presented alternative criteria involving the travel time non constant specific bus fare and area averaged tilt angle are discussed

j inner outer factorization j spectral factorization and robust control for nonlinear systems the problem of expressing a given nonlinear state space system as the cascade connection of a lossless system and a stable minimum phase system inner outer factorization is solved for the case of a stable system having state space equations affine in the inputs the solution is given in terms of the stabilizing solution of a certain hamilton jacobi equation the stable minimum phase factor is obtained as the solution of an associated nonlinear spectral factorization problem as an application one can arrive at the solution of the nonlinear h sub spl infin control problem for the disturbance feedforward case

nonsquare spectral factors via factorizations of a unitary function abstract we consider the problem of parametrizing the set of nonsquare minimal spectral factors of a rational matrix function taking positive semidefinite values on the imaginary axis the tool here is the use of factorizations of a certain type of the function w 1w where w resp w is the minimal square spectral factor that has all its poles and zeros in the right resp left halfplane

the shifted 1 n expansion and the energy eigenvalues of the hulthen potential for l not 0 the shifted 1 n expansion has been used to obtain the energy values of the hulthen potential for non zero values of the orbital angular momentum the analytic expression for the energies en l yield fairly accurate results for a wide range of values of n l and the screening parameter lambda the energy values obtained by this method have been compared and found to be in excellent agreement with pade approximation calculations

inner outer factorization for nonlinear noninvertible systems this paper considers inner outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor an application of the main result to control of nonminimum phase nonlinear systems is indicated

a master function approach for describing shape invariance parameters with respect to which quantum number they are shape invariant most of the one dimensional shape invariance potentials fall into two classes for each of which one of the quantum numbers describing the shape invariance is assigned on this basis the shape invariance parameters of every class are separately described in terms of a master function its weight function and an appropriate quantum number

a partial differential inequality for dissipative nonlinear systems abstract in this paper we show that storage functions for dissipative systems can be characterised by a partial differential inequality in the weak or viscosity sense the storage functions need not be differentiable this characterisation is stable with respect to perturbations we formulate a version of the positive real lemma for passive systems using the weak sense framework

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

a j spectral factorization approach to h control it is shown that necessary and sufficient conditions for the existence of sub optimal solutions to the model matching problem associated with h control are that two coupled j spectral factorizations exist the second j spectral factor provides a parameterization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of stabilizing non negative definite solutions to two algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given

streets tree networks and urban growth optimal geometry for quickest access between a finite size volume and one point abstract the geometric form of the tree network is deduced from a single mechanism the discovery that the shape of a heat generating volume can be optimized to minimize the thermal resistance between the volume and a point heat sink is used to solve the kinematics problem of minimizing the time of travel between a volume or area and one point the optimal path is constructed by covering the volume with a sequence of volume sizes building blocks which starts with the smallest size and continues with stepwise larger sizes assemblies optimized in each building block is the overall shape and the angle between constituents the speed of travel may vary from one assembly size to the next however the lowest speed is used to reach the infinity of points located in the smallest volume elements the volume to point path that results is a tree network a single design principle the geometric optimization of volume to point access determines all the features of the tree network

supersymmetry and potentials with bound states at arbitrary energies ii for pt i see ibid vol 19 p 2297 1986 it has been shown previously that a potential v0 chi in one dimension which supports no bound states may be used as a reference potential from which by successive applications of the concept of a supersymmetric partner to a given hamiltonian it is possible to find a potential vn chi which supports any specified number n of bound states at any chosen energies ej j 1 n the reflection coefficient of vn is related to the reflection coefficient of v0 various alternative representations of the potentials constructed by this procedure are presented an illustrative example in which vn is constructed by using a sech2 chi barrier as the reference potential is discussed

exact solution of the schrodinger and klein gordon equations for generalised hulthen potentials for the class of generalised hulthen s wave potentials vgh p r sigma i 1p giai exp gamma ir 1 sigma j 1p aj exp gamma jr p or 1 the exact solution is presented for both the schrodinger and klein gordon equations this simultaneous solubility represents a new non perturbative treatment of the kinematical relativistic corrections and is based on a non standard generalisation of the hypergeometric function

a characterization of minimal square spectral factors two different well known approaches to the spectral factorization problem phi s w s w s are connected together by relating the geometric properties of the solution set of the underlying algebraic riccati equation to the structure of the all pass factor of each minimal solution w s

on cancellation in h optimal controllers abstract this note clarifies the mechanism of pole zero cancellations that appeared in the derivation of the h optimal controller via a solution of the corresponding model matching problem the cancellable factor is explicitly expressed using chain scattering matrix descriptions a simple state space formula of the sub optimal controller is also presented based on the chain scattering forms and we show that there is a mixed sensitivity problem that can be solved by calculating a solution of only one riccati equation in each iteration

minimal square spectral factors abstract recently there has been renewed interest in the problem of spectral factorization and in particular the problem of parametrizing all square minimal spectral factors of a given spectrum for instance let us mention two recent papers in this journal one of which dealing with more computational aspects of this problem clements 1993 the other giving a parametrization under additional constraints ferrante et al 1993 this paper is motivated in a large part by ferrante et al 1993 in fact we shall show that a parametrization similar to the one given there can be achieved without one of the additional constraints imposed in ferrante et al 1993 a secondary aim of this paper is to give an overview of several parametrizations available in the literature

a normal form approach to approximate input output linearization for maximum phase nonlinear siso systems a control synthesis scheme is presented for nonlinear single input single output systems which have completely unstable antistable zero dynamics the approach is similar in spirit to linear approaches for nonminimum phase systems and involves the derivation of an input output linearizing controller for a suitably defined nonlinear minimum phase approximation to the original system the linearizing controller achieves an approximately linear input output response and internal stability

inner outer factorization of nonlinear operators abstract a fundamental result in the theory of hardy spaces of analytic matrix and operator valued functions on the unit disk is inner outer factorization for the case of rational matrix functions explicit formulas for the inner and outer factors have recently been obtained in these formulas the functions are presented in terms of a collection of finite matrices as the transfer function of a linear system the key ingredient in the construction is the solution of a linear matrix equation called a stein equation in this paper we obtain extensions of these results to the setting of input output maps of nonlinear discrete time systems a particaular computable class of examples is presented for illustration

a j spectral factorization approach to control necessary and sufficient conditions for the existence of suboptimal solutions to the standard model matching problem associated with mathcal h infty control j control are derived using j spectral factorization theory the existence of solutions to the model matching problem is shown to be equivalent to the existence of solutions to two coupled j spectral factorization problems with the second factor providing a parametrization of all solutions to the model matching problem the existence of the j spectral factors is then shown to be equivalent to the existence of nonnegative definite stabilizing solutions to two indefinite algebraic riccati equations allowing a state space formula for a linear fractional representation of all controllers to be given a virtue of the approach is that a very general class of problems may be tackled within a conceptually simple framework and no additional auxiliary riccati equations are required

a recurrence formula for obtaining certain matrix elements in the base of eigenfunctions of the hamiltonian for a particular screened potential a recurrence formula for obtaining matrix elements of products of powers of exp nu r and 1 exp nu r is derived the matrix elements are calculated in the base of eigenfunctions of the hamiltonian for the effective potential lambda exp nu r 1 exp nu r mu exp nu r 1 exp nu r 2 where lambda nu and mu are positive constants this potential can be considered as a generalisation of a potential suggested by hylleraas and risberg 1941 and by hulthen 1942 in the limit of the parameter nu tending to zero the recurrence formula is transformed into a recurrence formula is transformed into a recurrence formula given by badawi et al 1973 for matrix elements of powers of r for the hydrogen atom

rational spectral factorization using state space methods abstract we describe a state space based algorithm to find the minimum phase spectral factor of any rational spectral density whether it be proper improper or polynomial

on a state space approach to nonlinear h control we study the standard h optimal control problem using state feedback for smooth nonlinear control systems the main theorem obtained roughly states that the l2 induced norm from disturbances to inputs and outputs can be made smaller than a constant 0 if the corresponding h norm for the system linearized at the equilibrium can be made smaller than by linear state feedback necessary and sufficient conditions for the latter problem are by now well known e g from the state space approach to linear h optimal control our approach to the nonlinear h optimal control problem generalizes the state space approach to the linear h problem by replacing the hamiltonian matrix and corresponding riccati equation as used in the linear context by a hamiltonian vector field together with a hamiltonian jacobi equation corresponding to its stable invariant manifold

solution of the relativistic dirac hulthen problem the one particle three dimensional dirac equation with spherical symmetry is solved for the hulthen potential the s wave relativistic energy spectrum and two component spinor wavefunctions are obtained analytically conforming to the standard feature of the relativistic problem the solution space splits into two distinct subspaces depending on the sign of a fundamental parameter in the problem unique and interesting properties of the energy spectrum are pointed out and illustrated graphically for several values of the physical parameters the square integrable two component wavefunctions are written in terms of the jacobi polynomials the nonrelativistic limit reproduces the well known nonrelativistic energy spectrum and results in schrodinger equation with a generalized three parameter hulthen potential which is the sum of the original hulthen potential and its square

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

non minimal factorization of nonlinear systems the observable case minimal factorizations of nonlinear systems have been studied earlier here non minimal factorizations of nonlinear systems are studied for the case where the product contains no extra non observable states observable factorizations a description for all such factorizations is given in terms of invariant foliations of the system and its inverse the results are applied to the special cases of stable factorizations coprime

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

on the lyapunov equation coinvariant subspaces and some problems related to spectral factorizations a geometric approach to stochastic realization theory and hence to spectral factorization problems has been developed by lindquist and picci 1985 1991 and lindquist et al 1995 most of this work was done abstractly fuhrmann and gombani 1998 adopted an entirely hardy space approach to this set of problems studying the set of rectangular spectral factors of given size for a weakly coercive spectral function the parametrization of spectral factors in terms of factorizations of related inner functions as developed in fuhrmann 1995 had to be generalized this led to a further understanding of the partial order introduced by lindquist and picci in the set of stable spectral factors in the present paper we study the geometry of finite dimensional coinvariant subspaces of a vectorial hardy space h 2 via realization theory emphasizing the role of the lyapunov equation in lifting the hardy space metric to the statespace domain we follow this by deriving state space formulas for rectangular spect

stable invariant lagrangian subspaces factorization of symmetric rational matrix functions and other applications several applications of earlier results by the authors concerning various notions of stability of invariant lagrangian subspaces are studied the applications include stability of symmetric minimal factorizations of real symmetric rational matrix functions stability of factorizations of certain classes of symmetric matrix polynomials and stably well posed matricial boundary value problems with symmetries an interpretation of many stability results in terms of equilibria of lie group actions on subspaces is given as well

parametrization of all minimal square spectral factors abstract two different parametrizations of minimal square spectral factors are given these parametrizations extend recent results by picci and pinzoni since they characterize spectral factors without specification on the zero or pole structure

monte carlo approach to dendritic growth the connection between diffusion limited aggregation and the equations of dendritic growth is critically examined a different type of monte carlo simulation is proposed and used to construct two dimensional dendrite like patterns the wavelength occurring for short times is in good agreement with the linear stability analysis the time dependence of the characteristic wavelengths is also determined

intertwining relations for the matrix calogero like models supersymmetry and shape invariance we investigate the intertwining relations for n particle calogero like models with internal degrees of freedom starting from the well known dunkl polychronakos operators we construct a new type of local without exchange operation differential operator these operators intertwine the matrix hamiltonians corresponding to irreducible representations of the permutation group sn in particular cases this method allows us to construct a new class of exactly solvable dirac like equations and a new class of matrix models with shape invariance we establish a connection with the approach of multi dimensional supersymmetric quantum mechanics

shape invariant potentials with pt symmetry suitable complexification of the well known hyperbolic rosen morse oscillator in one dimension is shown to give the fourth and last exactly solvable model which combines the shape and invariance

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

supersymmetric transformations of real potentials on the line a systematic study of supersymmetric or darboux factorizations on the line is performed all possible pairs of supersymmetric transformations with the same factorization energy are reviewed for such pairs there is no condition on this energy iterations of single transformations and of pairs both allow arbitrary modifications of the bound spectrum different iterative methods lead to compact analytical equations depending on the initial potential and its solutions iterations of single transformations are able to transform an even potential into an even potential particular cases of a method based on iteration of pairs conserve either the reflection coefficient at all energies or the norming constants of bound states these results are compared with previous methods established in other contexts the most general supersymmetric transformation of a given potential is finally described both with and without modifications of the bound spectrum

shape invariance and laddering equations for the associated hypergeometric functions introducing the associated hypergeometric functions in terms of two non negative integers we factorize their corresponding differential equation into a product of first order differential operators by four different ways as shape invariance equations these shape invariances are realized by four different types of raising and lowering operators this procedure gives four different pairs of recursion relations on the associated hypergeometric functions

factorization and feedback stabilization for nonlinear systems we establish the equivalence of internal input out stability for two feedback configurations of a nonlinear time varying plant p for which a related plant g is assumed to have a factorization g r with both r and r 1 incrementally stable this extends a factorization principle for stabilizability previously given only for the linear time invariant case as an application of a special case we recover a version of the youla parametrization of stabilizing compensators for the nonlinear case previously presented in the literature we use degree theory to parametrize a collection of solutions of the h control problem for the case of a 1 gain stable or lossless plant in the case of a plant g having a j inner outer factorization this last result combined with the above mentioned factorization principle leads to results on the h control problem for p

nonsquare spectral factorization for nonlinear control systems this paper considers nonsquare spectral factorization of nonlinear input affine state space systems in continuous time more specifically we obtain a parametrization of nonsquare spectral factors in terms of invariant lagrangian submanifolds and associated solutions of hamilton jacobi inequalities this inequality is a nonlinear analogue of the bounded real lemma and the control algebraic riccati inequality by way of an application we discuss an alternative characterization of minimum and maximum phase spectral factors and introduce the notion of a rigid nonlinear system

scaling and universality in the micro structure of urban space abstract we present a broad phenomenological picture of the distribution of the length of open space linear segments l derived from maps of 36 cities in 14 different countries by scaling the zipf plot of l we obtain two master curves for a sample of cities which are not a function of city size we show that a third class of cities is not easily classifiable into these two universality classes the cumulative distribution of l displays power law tails with two distinct exponents b 2 and r 3 we suggest a link between our data and the possibility of observing and modelling urban geometric structures using levy processes

shape invariant potentials depending on n parameters transformed by translation shape invariant potentials in the sense of gendenshtein 1983 jetp lett 38 356 which depend on more than two parameters are not known to date cooper et al 1987 phys rev d 36 2458 posed the problem of finding a class of shape invariant potentials which depend on n parameters transformed by translation but it was not solved we analyse the problem using some properties of the riccati equation and find the general solution

non linear minimal square spectral factorization recent work has identified a correspondence between minimal square spectral factors of a given spectral density invariant subspaces of an associated hamiltonian matrix solutions of an appropriate algebraic riccati equation and minimal unitary left divisors of a certain unitary function on the imaginary line the purpose of this report is to obtain a natural extension of these ideas to a non linear setting at least for the special case where only minimal square stable spectral factors are considered

supersymmetry factorisation of the schrodinger equation and a hamiltonian hierarchy the author presents a systematic procedure for constructing a hierarchy of non relativistic hamiltonians with the property that the adjacent members of the hierarchy are supersymmetric partners i e they share the same eigenvalue spectrum except for the missing ground state and the eigenvectors are simply related