small bv solutions of hyperbolic noncooperative differential games the paper is concerned with an n person differential game in one space dimension we state conditions for which the system of hamilton jacobi equations for the value functions is strictly hyperbolic in the positive case we show that the weak solution of a corresponding system of conservation laws determines an n tuple of feedback strategies these yield a nash equilibrium solution to the noncooperative differential game

sufficient conditions for relative minima of broken extremals in optimal control theory we formulate a version of the method of characteristics based on parametrizations of extremals by their terminal values sufficient conditions are given for imbedding a reference trajectory into a local field of broken extremals for a problem with terminal manifold of codimension 1 it is shown that a broken extremal is a relative minimum if i the restrictions of the flow to intervals where the control is continuous have nonsingular partial derivatives with respect to the parameter and ii the switching surfaces are crossed transversally

state local optimality of a bang bang trajectory a hamiltonian approach we give a sufficient condition for a bang bang extremal to be a strong local optimizer for the minimum time problem with fixed endpoints we underline that the conditions imply that the optimum is local with respect to the state and not necessarily to the final time moreover it is given through a finite dimensional minimization problem hence is suited for numerical verification a geometric interpretation through the projection of the hamiltonian flow on the state space is also given

the structure of time optimal trajectories for single input systems in the plane the c nonsingular case for single input c infty systems in the plane in which the control enters linearly we prove if the system is suitably nondegenerate that the time optimal trajectories are finite concatenations of bang bang and singular arcs with local bounds on the number of switchings

the structure of time optimal trajectories for single input systems in the plane the general real analytic case for arbitrary single input real analytic systems in the plane in which the control enters linearly we prove that the time optimal trajectories are finite concatenations of bang bang and singular arcs with local bounds on the number of switchings no nondegeneracy assumption is made other than real analyticity the analysis proceeds by applying our previous results on nondegenerate c infty systems to study the behavior of the trajectories on a neighborhood of every point except for the points in a discrete set of branch points the branch points are then handled by a combination of control theoretic arguments and the use of subanalytic set theory

time optimal synthesis for left invariant control systems on series so 3 consider the control system sigma given by dot x x f ug where x in so 3 u leq 1 and f g in so 3 define two perpendicular left invariant vector fields normalized so that f cos al and g sin al al in 0 pi 4 in this paper we provide an upper bound and a lower bound for n alpha the maximum number of switchings for time optimal trajectories of sigma more precisely we show that n s al leq n al leq n s al 4 where n s al is a suitable integer function of al such that n s al stackrel lower2pt hbox smash sim mbox tiny al to 0 pi 4 alpha the result is obtained by studying the time optimal synthesis of a projected control problem on rr p 2 where the projection is defined by an appropriate hopf fibration finally we study the projected control problem on the unit sphere s 2 it exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations

convergence of generalized aor iterative method for linear systems with strictly diagonally dominant matrices in this paper some improvements on darvishi and hessari on convergence of the generalized aor method for linear systems with diagonally dominant coefficient matrices appl math comput 176 2006 128 133 are presented for bounds of the spectral radius of l w r which is the iterative matrix of the generalized aor gaor method subsequently some new sufficient conditions for convergence of gaor method will be given which improve some results of darvishi and hessari on convergence of the generalized aor method for linear systems with diagonally dominant coefficient matrices appl math comput 176 2006 128 133

improving jacobi and gauss seidel iterations abstract when convergent jacobi or gauss seidel iterations can be applied to solve systems of linear equations a natural question is how convergence rates are affected if the original system is modified by performing some gaussian elimination we prove that if the initial iteration matrix is nonnegative then such elimination improves convergence our results extend those contained in 4

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

clebsch gordan type linearisation relations for the products of laguerre polynomials and hydrogen like functions two series of clebsch gordan type linearisation relations are derived for the most general product of the laguerre polynomials ln1alpha 1 u1x ln2alpha 2 u2x which differ in orders n weights alpha and scaling multipliers u the general form and particular cases of coefficients in the expansion of the polynomial xkln1alpha 1 u1x lnnalpha n unx in terms of the laguerre polynomials are established the applications to hydrogen like functions and morse oscillators are indicated connection with an earlier carlitz expansion the technical links with the hyperspherical harmonics formalism and different approaches to the important koornwinder s positivity theorems are discussed briefly

renyi entropies lq norms and linearization of powers of hypergeometric orthogonal polynomials by means of multivariate special functions the quantification of the spreading of the orthogonal polynomials p n x can be investigated by means of the renyi entropies r q r q being a positive integer number of the associated rakhmanov probability densities r x email protected x p n 2 x where w x is the corresponding weight function the renyi entropies are closely related to the l q norms of the polynomials in this manuscript the l q norms and the associated renyi entropies of the real hypergeometric orthogonal polynomials i e hermite laguerre and jacobi polynomials and the generalized hermite polynomials are expressed in an explicit way in terms of some generalized multivariate special functions of lauricella and srivastava daoust types which are evaluated at some specific values of 2q variables these functions depend on 4q 1 and 6q 2 parameters respectively which are determined by the order q the degree n of the polynomial and the parameters of the orthogonality weight function w x the key idea is based on some extended linearization formulas for these polynomials these results open the way to determine the renyi information entropies of the quantum systems whose wavefunctions are controlled by hypergeometric orthogonal polynomials

expansions in series of varying laguerre polynomials and some applications to molecular potentials the expansion of a large class of functions in series of linearly varying laguerre polynomials i e laguerre polynomials whose parameters are linear functions of the degree is found by means of the hypergeometric functions approach this expansion formula is then used to obtain the brown carlitz generating function which gives a characterization of the exponential function and the connection formula for these polynomials finally these results are employed to connect the bound states of the quantum mechanical potentials of morse and poschl teller which are frequently used to describe molecular systems

viscosity solutions of the bellman equation for exit time optimal control problems with vanishing lagrangians we study the hamilton jacobi bellman equation for undiscounted exit time optimal control problems for fully nonlinear systems and fully nonlinear singular lagrangians using the dynamic programming approach we prove a local uniqueness theorem characterizing the value functions for these problems as the unique viscosity solutions of the corresponding hamilton jacobi bellman equations that satisfy appropriate boundary conditions the novelty of this theorem is in the relaxed hypotheses on the lower bound on the lagrangian and the very general assumptions on the target set as a corollary we show that the value function for the fuller problem is the unique viscosity solution of the corresponding hamilton jacobi bellman equation that vanishes at the origin and satisfies certain growth conditions this implies as special cases first that the value function of this problem is the unique proper viscosity solution of the corresponding hamilton jacobi bellman equation in the class of all functions which are continuous in the plane and null at the origin and second that this value function is the unique viscosity solution of that equation in a class which includes functions which are not bounded below we also apply our results to the degenerate eikonal equation of geometric optics and to the shape from shading equations in image processing our theorem also applies to problems with noncompact targets and unbounded control sets whose lagrangians take negative values

preconditioned gaor methods for solving weighted linear least squares problems in this paper we present the preconditioned generalized accelerated overrelaxation gaor method for solving linear systems based on a class of weighted linear least square problems two kinds of preconditioning are proposed and each one contains three preconditioners we compare the spectral radii of the iteration matrices of the preconditioned and the original methods the comparison results show that the convergence rate of the preconditioned gaor methods is indeed better than the rate of the original method whenever the original method is convergent finally a numerical example is presented in order to confirm these theoretical results

so 2 1 lie algebra the jacobi matrix and the scattering states of the morse oscillator work exploring the connection between so 2 1 lie algebra and the jacobi matrix method for scattering is extended to the morse potential regular and irregular solutions of the three term recursion associated with the jacobi matrix are specified by their asymptotic behaviour and closed form expressions in terms of generalised hypergeometric functions are obtained this example further illustrates the connection between scattering theory and orthogonal polynomials

modified clebsch gordan type expansions for products of discrete hypergeometric polynomials abstract starting from the second order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials p n we find the analytical expressions of the expansion coefficients of any polynomial rm x and of the product rm x qj x in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric polynomials here qj x denotes an arbitrary discrete hypergeometric polynomial of degree j the particular cases in which r m corresponds to the non orthogonal families x m the rising factorials or pochhammer polynomials x m and the falling factorial or stirling polynomials x m are considered in detail the connection problem between discrete hypergeometric polynomials which here corresponds to the product case with m 0 is also studied and its complete solution for all the classical discrete orthogonal hypergeometric cdoh polynomials is given also the inversion problems of cdoh polynomials associated to the three aforementioned nonorthogonal families are solved

sensitivity of spin glass order to temperature changes by a loop expansion around parisi s mean field theory for an ising spin glass it is shown that the overlap of the magnetization patterns belonging to two different temperatures t and t vanishes to any order si t si t 0 while the correlation overlap sisj t sisj t calculated to first loop order and for technical reasons for dimensions d 8 only is found to decay exponentially with a characteristic length approximately t t 1

convolution and hypergroup structures associated with a class of sturm liouville systems product formulas of the type u k u k 0 u k d are obtained for the eigenfunctions of a class of second order regular and regular singular sturm liouville problems on 0 by using the riemann integration method to solve a cauchy problem for an associated hyperbolic differential equation when d is nonnegative which can be guaranteed by a simple restriction on the differential operator of the sturm liouville problem it is possible to define a convolution with respect to which m 0 becomes a banach algebra with the functions u k u 0 as its characters

regular synthesis and sufficiency conditions for optimality we propose a definition of regular synthesis that is more general than those suggested by other authors such as boltyanskii siam j control optim 4 1966 pp 326 361 and brunovsk math slovaca 28 1978 pp 81 100 and an even more general notion of regular presynthesis we give a complete proof of the corresponding sufficiency theorem a slightly weaker version of which had been stated in an earlier article with only a rough outline of the proof we illustrate the strength of our result by showing that the optimal synthesis for the famous fuller problem satisfies our hypotheses we also compare our concept of synthesis with the simpler notion of a family of solutions of the closed loop equation arising from an optimal feedback law and show by means of examples why the latter is inadequate and why the difficulty cannot be resolved by using other concepts of solution such as filippov solutions or the limits of sample and hold solutions recently proposed as feedback solutions by clarke et al ieee trans automat control 42 1997 pp 1394 1407 for equations with a non lipschitz and possibly discontinuous right hand side

convergence of gaor method for doubly diagonally dominant matrices abstract in this paper we obtain bounds for the spectral radius of the matrix l r which is the iterative matrix of the generalized accelerated overrelaxation gaor iterative method moreover we present one convergence theorem of the gaor method finally we present two numerical examples

further results on the bellman equation for optimal control problems with exit times and nonnegative lagrangians abstract in a series of papers we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding bellman equation that satisfies appropriate side conditions the results applied to problems which satisfy a positivity condition on the integral of the lagrangian this positive integral condition assigned a positive cost for remaining outside the target on any interval of positive length in this note we prove a new theorem which characterizes the exit time value function as the unique bounded from below viscosity solution of the bellman equation that vanishes on the target the theorem applies to problems satisfying an asymptotic condition on the trajectories including cases where the positive integral condition is not satisfied our results are based on an extended version of barb lat s lemma we apply the theorem to variants of the fuller problem and other examples where the lagrangian is degenerate

improvements of preconditioned aor iterative method for l matrices in this paper we improve the preconditioned aor method of linear systems considered by evans et al the aor iterative method for new preconditioned linear systems comput appl math 132 2001 461 466 in evans paper the coefficient matrix of linear system have to be an l matrix with a i i 1a i 1 i 0 i 1 n 1 and 0

second order recurrence relation for the linearization coefficients of the classical orthogonal polynomials abstract let pk be any sequence of the classical orthogonal polynomials we give explicitly a second order recurrence relation in k for the coefficients in p i p j k i j i j c i j k p k this result is obtained by a method which is alternative to the one proposed recently by a ronveaux et al applications of the result to some named systems of classical orthogonal polynomials are given

classification of generic singularities for the planar time optimal synthesis this paper is concerned with control systems on the plane with control appearing linearly it is known that under generic conditions the problem of reaching points from the origin in minimum time admits a regular synthesis the minimum time function is piecewise smooth possibly nondifferentiable on a set that is a finite union of embedded submanifolds of dimension 1 or 0 called singularities the purpose of the present paper is to provide a classification of all types of singularities arising under generic conditions

integral formulas associated with products of bessel functions a new partial differential equation approach abstract we investigate the hyperbolic equation with two singular lines 2 2 1 u 2 2 2 1 u 0 for 1 2 under the assumption that a certain asymptotic behavior of u and its partial derivatives is prescribed for tending to the coordinate axes or to infinity intending to establish integral formulas for products of bessel functions which a priori solve such a problem we are looking for an unusual representation of the solution u x y 0

asymptotic approach on conjugate points for minimal time bang bang controls abstract we focus on the minimal time control problem for single input control affine systems x x x u 1 y 1 x in r n with fixed initial and final time conditions x 0 x 0 x t f x 1 and where the scalar control u 1 satisfies the constraint u 1 1 for these systems a concept of conjugate time t c has been defined in e g agrachev et al 2002 23 maurer and osmolovskii 2004 21 and noble and schattler 2002 28 in the bang bang case besides theoretical and practical issues for conjugate time theory are well known in the smooth case see e g agrachev and sachkov 2004 43 and milyutin and osmolovskii 1998 15 and efficient implementation tools are available see bonnard et al 2007 35 the first conjugate time along an extremal is the time at which the extremal loses its local optimality in this work we use the asymptotic approach developed in silva and trelat in press 36 and investigate the convergence properties of conjugate times more precisely for e 0 small and arbitrary vector fields y 1 y m we consider the minimal time problem for the control system x e x x e u 1 e y 1 x e e i 2 m u i e y i x e under the constraint i 1 m u i e 2 1 with the fixed boundary conditions x e 0 x 0 x e t f x 1 of the initial problem under appropriate assumptions the optimal controls of the latter regularized optimal control problem are smooth and the computation of associated conjugate times t c e falls into the standard theory our main result asserts the convergence as e tends to 0 of t c e towards the conjugate time t c of the initial bang bang optimal control problem as well as the convergence of the associated extremals as a byproduct we obtain an efficient algorithmic way to compute conjugate times in the bang bang case

group theoretical foundation of the j matrix theory of scattering group theoretical analysis shows that so 2 1 is an underlying dynamical symmetry for all hamiltonians that are compatible with the jacobi matrix j matrix formalism the class of central potentials with this property is obtained including but not limited to the oscillator coulomb and morse potentials the l2 bases and j matrix elements for these potentials are found so 2 1 invariant transformation of the solutions of the recursion relation for one potential gives those for another potential in the class phase shift and resonance calculations for a single channel potential are carried out in the oscillator basis to illustrate the use of our results

some new preconditioned generalized aor methods for generalized least squares problems in this paper we present some new preconditioned gaor methods for solving generalized least squares problems and their comparison results comparison results show that the convergence rates of the new preconditioned gaor methods are better than those of the preconditioned gaor methods presented by zhou et al 2009 19 and wang et al 2013 18 whenever these methods are convergent lastly numerical experiments are provided to confirm the theoretical results

on parameterized inexact uzawa methods for generalized saddle point problems abstract for the large sparse saddle point problems bai et al recently studied a class of parameterized inexact uzawa piu methods z z bai b n parlett z q wang on generalized successive overrelaxation methods for augmented linear systems numer math 102 2005 1 38 for a special case that the 1 1 block is solved exactly they determined the convergence domain and computed the optimal iteration parameters and the corresponding optimal convergence factor for the induced method in this paper we develop these methods to the large sparse generalized saddle point problems for the obtained parameterized inexact uzawa method we prove its convergence under suitable restrictions on the iteration parameters in particular we determine its quasi optimal iteration parameters and the corresponding quasi optimal convergence factor for the saddle point problems furthermore this piu method is generalized to obtain a framework of accelerated variants of the parameterized inexact uzawa methods for solving both symmetric and nonsymmetric generalized saddle point problems by using the techniques of vector extrapolation matrix relaxation and inexact iteration

numerical methods for generalized least squares problems usually generalized least squares problems are solved by transforming them into regular least squares problems which can then be solved by well known numerical methods however this approach is not very effective in some cases and besides is very expensive for large scale problems in 1979 paige suggested another approach which consists of solving an equivalent equality constrained least squares problem by the orthogonal decomposition the bnp algorithm or the james implicit nullspace iterative methods in this paper we present some new developments of the numerical methods for example 2 cycle sor method and preconditioned conjugate gradient method for generalized least squares problems some numerical comparisons are included as well

the jacobi matrix method in parabolic coordinates expansion of coulomb functions in parabolic sturmians previous analysis of the jacobi matrix method based on the underlying so 2 1 lie algebra is extended to the coulomb hamiltonian in parabolic coordinates the general solution of the generic so 2 1 eigenvalue equation is constructed and special cases which furnish expansions of the coulomb functions k r in a complete set of parabolic sturmians are discussed

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

on generalized successive overrelaxation methods for augmented linear systems for the augmented system of linear equations golub wu and yuan recently studied an sor like method bit 41 2001 71 85 by further accelerating it with another parameter in this paper we present a generalized sor gsor method for the augmented linear system we prove its convergence under suitable restrictions on the iteration parameters and determine its optimal iteration parameters and the corresponding optimal convergence factor theoretical analyses show that the gsor method has faster asymptotic convergence rate than the sor like method also numerical results show that the gsor method is more effective than the sor like method when they are applied to solve the augmented linear system this gsor method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation matrix relaxation and inexact iteration besides we also demonstrate a complete version about the convergence theory of the sor like method

some connection and linearization problems for polynomials in and beyond the askey scheme the connection problem is considered in a hypergeometric function framework for i the two most general families of polynomials belonging to the askey scheme wilson and racah and ii some generalized laguerre and jacobi polynomials falling outside that scheme sister celine cohen and prabhakar jain which are relevant to the study of quantum mechanical systems and include as particular cases the generalizations of the classical families with sobolev type orthogonality in addition using the same method three new linearization like formulae for the gegenbauer polynomials are also derived a linearization formula that generalizes the m n case of dougall s formula the analogue of the m n case of nielsen s inverse linearization formula for hermite polynomials and a connection formula for the squares closed analytical formulae for the corresponding connection and linearization coe cients are given in terms of hypergeometric functions of unit argument which at times can be further simpli ed and expressed as single hypergeometric terms c 2001 published by elsevier science b v

guaranteed cost certification for discrete time linear switched systems with a dwell time this technical note studies the guaranteed cost of a quadratic criterion associated with a linear discrete time switched system for all the set of admissible switching laws the admissible switching laws are here the ones exhibiting a dwell time the approach provided here is to design an upper bound and a parametrized family of lower bounds of the guaranteed cost as close as possible in order to obtain a certification of the guaranteed cost the upper bound is determined via a switched lyapunov function and the lower bounds are obtained via the numerical computation of the cost induced by particular periodic switching laws the features of the proposed approach are illustrated by a numerical example

on the connection and linearization problem for discrete hypergeometric q polynomials abstract in the present paper starting from the second order difference hypergeometric equation on the non uniform lattice x s satisfied by the set of discrete hypergeometric orthogonal q polynomials p n we find analytical expressions of the expansion coefficients of any q polynomial r m x s on x s and of the product r m x s q j x s in series of the set p n these coefficients are given in terms of the polynomial coefficients of the second order difference equations satisfied by the involved discrete hypergeometric q polynomials

product formulas and convolutions for angular and radial spheroidal wave functions product formulas for angular spheroidal wave functions on 0 and for radial spheroidal wave functions on 0 are presented which generalize results for the ultraspherical polynomials and functions as well as for the mathieu functions although these functions cannot be given in closed form the kernels of the product formulas are represented in an explicit and surprisingly simple way in terms of bessel functions so that the exact range of positivity can easily be read off the formulas are used to introduce two families of convolution structures on 0 and 0 many of which provide new hypergroups we proceed from the fact that the spheroidal wave functions are eigenfunctions of sturm liouville equations of confluent heun type and employ a partial differential equation technique based on riemann s integration method

strong optimality for a bang bang trajectory in this paper we give sufficient conditions for a bang bang regular extremal to be a strong local optimum for a control problem in the mayer form strong means that we consider the c0 topology in the state space the controls appear linearly and take values in a polyhedron and the state space and the end point constraints are finite dimensional smooth manifolds in the case of bang bang extremals the kernel of the first variation of the problem is trivial and hence the usual second variation which is defined on the kernel of the first one does not give any information we consider the finite dimensional subproblem generated by perturbing the switching times and we prove that the sufficient second order optimality conditions for this finite dimensional subproblem yield local strong optimality we give an explicit algorithm to check the positivity of the second variation which is based on the properties of the hamiltonian fields

some results on preconditioned gaor methods in this paper we present preconditioned generalized accelerated overrelaxation methods we compare the spectral radii of the iteration matrices of the preconditioned and original methods the comparison results show that the preconditioned gaor methods converge faster than the gaor method whenever the gaor method is convergent finally we present two numerical examples to confirm our theoretical results

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

on automaton recognizability of abnormal extremals for a generic single input planar control system dot x f x u g x x in mathbb r 2 u in 1 1 f 0 0 we analyze the properties of abnormal extremals for the minimum time stabilization to the origin we prove that abnormal extremals are finite concatenations of bang arcs with switchings occurring on the set in which the vector fields f and g are collinear moreover all the generic singularities of one parametric family of extremal trajectories near to abnormal extremals are studied in particular we prove that all possible sequences of these singularities and hence all generic abnormal extremals can be classified by a set of words recognizable by an automaton

generating function for the product of the associated laguerre and hermite polynomials the generating function for the product of the associated laguerre and hermite polynomials is formulated and used to determine the coordinate space wave function of the displaced number states of the harmonic oscillator for the general case of a complex displacement parameter the benefits in this context of an extended form of the associated laguerre polynomials are advertised

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

regular synthesis for time optimal control of single input real analytic systems in the plane for arbitrary single input real analytic systems in the plane in which the control enters linearly we prove the existence of a regular synthesis for the optimal control problem in which it is desired to minimize the integral of a strictly positive real analytic lagrangian that does not depend on the control variable the analysis proceeds by applying our previous results on nondegenerate mathcal c infty systems as well as those on arbitrary real analytic ones to study the local structure of the time optimal trajectories the structure of the optimal trajectories for our problem is derived by reparametrization of time the existence of a synthesis is then proved by using subanalytic set theory

preconditioned aor iterative methods for m matrices linear systems with m matrices often appear in a wide variety of areas in this paper we give general preconditioners for solving the systems with nonsingular m matrix we show that our preconditioners increase the convergence rate of aor iterative methods numerical results for corresponding preconditioned gmres methods are also given

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

one and two dimensional coulomb green s function matrices in parabolic sturmians basis one and two dimensional operators which originate from the asymptotic form of the three body coulomb wave equation in parabolic coordinates are treated within the context of a square integrable basis set the matrix representations of green s functions corresponding to these operators are obtained

time optimal swing up of the planar pendulum this paper presents qualitative results on the global structure of the time optimal trajectories of the planar pendulum on a cart this mechanical system is a benchmark to test nonlinear control methods and various papers addressed the problem of computing time optimal open loop controls relying on the theory of optimal synthesis we provide a discontinuous feedback giving optimal solutions for any initial data the approach is that of geometric control theory

two new preconditioned gaor methods for weighted linear least squares problems abstract in this paper the preconditioned generalized accelerated overrelaxation gaor methods for solving weighted linear least squares problems are considered two new preconditioners are proposed and the convergence rates of the new preconditioned gaor methods are studied comparison results show that the convergence rates of the new preconditioned gaor methods are better than those of the preconditioned gaor methods in the previous literatures whenever these methods are convergent a numerical example is given to confirm our theoretical results

optimal parameter in hermitian and skew hermitian splitting method for certain two by two block matrices the optimal parameter of the hermitian skew hermitian splitting hss iteration method for a real two by two linear system is obtained the result is used to determine the optimal parameters for linear systems associated with certain two by two block matrices and to estimate the optimal parameters of the hss iteration method for linear systems with n by n real coefficient matrices numerical examples are given to illustrate the results

minimal time trajectories for two level quantum systems with two bounded controls in this paper we consider the minimum time population transfer problem for a two level quantum system driven by two external fields with bounded amplitude the controls are modeled as real functions and we do not use the rotating wave approximation after projection on the bloch sphere we treat the time optimal control problem with techniques of optimal synthesis on 2d manifolds based on the pontryagin maximum principle we characterize a restricted set of candidate optimal trajectories properties on this set crucial for complete optimal synthesis are illustrated by numerical simulations furthermore when the two controls have the same bound and this bound is small with respect to the difference of the two energy levels we get a complete optimal synthesis up to a small neighborhood of the antipodal point of the initial condition

general linearization formulae for products of continuous hypergeometric type polynomials the linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials hps of a continuous variable which consists of the expansion of the product of two arbitrary hps in series of an orthogonal hp set here this problem is algebraically solved directly in terms of the coefficients of the second order differential equations satisfied by the involved polynomials general expressions for the expansion coefficients are given in integral form and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis hermite laguerre and jacobi as well as several generalized linearization formulae involving these families the connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions which often reduce to a single function of the same type when possible these functions are evaluated in closed form in some cases sign properties of the coefficients such as positivity or non negativity conditions are derived as a by product from their resulting explicit representations

a generic classification of time optimal planar stabilizing feedbacks consider the problem of stabilization at the origin in minimum time for a planar control system affine with respect to the control for a family of generic vector fields a topological equivalence relation on the corresponding time optimal feedback synthesis was introduced in a previous paper dynamics of continuous discrete and impulsive systems 3 1997 pp 335 371 the set of equivalence classes can be put in a one to one correspondence with a discrete family of graphs this provides a classification of the global structure of generic time optimal stabilizing feedbacks in the plane analogous to the classification of smooth dynamical systems developed by peixoto

verification theorems for hamilton jacobi bellman equations we study an optimal control problem in bolza form and we consider the value function associated to this problem we prove two verification theorems which ensure that if a function w satisfies some suitable weak continuity assumptions and a hamilton jacobi bellman inequality outside a countably h n rectifiable set then it is lower than or equal to the value function these results can be used for optimal synthesis approach

linearization and connection coefficients for hypergeometric type polynomials we consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric type polynomials directly in terms of the corresponding differential equations we illustrate the method by producing explicit formulas for hermite polynomials

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

on convergence of the generalized aor method for linear systems with diagonally dominant coefficient matrices abstract in this paper we obtain bounds for the spectral radius of the matrix l r which is the iterative matrix of generalized accelerated overrelaxation gaor iterative method we consider a linear system hy f where h is an n n diagonally dominant real matrix f is an n real vector of known values and y is an n real vector of unknown variables sufficient conditions for the convergence of gaor method will be presented

resonances in one dimension and fredholm determinants we discuss resonances for schrodinger operators in whole and half line problems one of our goals is to connect the fredholm determinant approach of froese to the fourier transform approach of zworski another is to prove a result on the number of antibound states namely in a half line problem there are an odd number of antibound states between any two bound states

convergence of the generalized aor method a generalized aor gaor method is proposed here to solve the linear system with some special structure the convergence of the gaor method is studied the convergence result obtained here is an extension of the classical convergence result of the regular aor method with weaker condition in comparing with the generalized sor gsor method we show that the gaor method is better than the gsor method under certain conditions in some cases the gaor method is convergent even if the gsor method does not converge finally the gaor method is applied to solve the generalized least squares ls problems three numerical algorithms for the generalized ls problems are given

regularization of chattering phenomena via bounded variation controls in the control theory the term chattering is used to refer to fast oscillations of controls such as an infinite number of switchings over a finite time interval in this paper we focus on three typical instances of chattering the fuller phenomenon referring to situations where an optimal control features an accumulation of switchings in finite time the robbins phenomenon concerning optimal control problems with state constraints where the optimal trajectory touches the boundary of the constraint set an infinite number of times over a finite time interval and the zeno phenomenon for hybrid systems referring to a trajectory that depicts an infinite number of location switchings in finite time from the practical point of view when trying to compute an optimal trajectory for instance by means of a shooting method chattering may be a serious obstacle to convergence in this paper we propose a general regularization procedure by adding an appropriate penalization of the total variation this produces a family of quasi optimal controls whose associated cost converge to the optimal cost of the initial problem as the penalization tends to zero under additional assumptions we also quantify quasi optimality by determining a speed of convergence of the costs