brief paper polynomial time algorithms for probabilistic solutions of parameter dependent linear matrix inequalities a randomized approach is considered for a feasibility problem on a parameter dependent linear matrix inequality lmi in particular a gradient based and an ellipsoid based randomized algorithms are improved by introduction of a stopping rule the improved algorithms stop after a bounded number of iterations and this bound is of polynomial order in the problem size when the algorithms stop either of the following two events occurs i they find with high confidence a probabilistic solution which satisfies the given lmi for most of the parameter values ii they detect in an approximate sense the non existence of a deterministic solution which satisfies the given lmi for all the parameter values these results are important because the original randomized algorithms have issues to be settled on detection of convergence on the speed of convergence and on the assumption of feasibility the improved algorithms can be adapted for an optimization problem constrained by a parameter dependent lmi a numerical example shows the efficacy of the proposed algorithms

a probabilistic analytic center cutting plane method for feasibility of uncertain lmis many robust control problems can be formulated in abstract form as convex feasibility programs where one seeks a solution x that satisfies a set of inequalities of the form f f x d 0 d d this set typically contains an infinite and uncountable number of inequalities and it has been proved that the related robust feasibility problem is numerically hard to solve in general in this paper we discuss a family of cutting plane methods that solve efficiently a probabilistically relaxed version of the problem specifically under suitable hypotheses we show that an analytic center cutting plane scheme based on a probabilistic oracle returns in a finite and pre specified number of iterations a solution x which is feasible for most of the members of f except possibly for a subset having arbitrarily small probability measure

notes on the scenario design approach the scenario optimization method developed in is a theoretically sound and practically effective technique for solving in a probabilistic setting robust convex optimization problems arising in systems and control design that would otherwise be hard to tackle via standard deterministic techniques in this note we explore some further aspects of the scenario methodology and present two results pertaining to the tightness of the sample complexity bounds we also state a new theorem that enables the user to make a priori probabilistic claims on the scenario solution with one level of probability only

shallow deep and very deep cuts in the analytic center cutting plane method in this paper we show that the cut does not need to go through the query point it can be deep or shallow the primal framework leads to a simple analysis of the potential variation which shows that the inequality needed for convergence of the algorithm is in fact attained at the first iterate of the feasibility step

brief computational complexity of randomized algorithms for solving parameter dependent linear matrix inequalities randomized algorithms are proposed for solving parameter dependent linear matrix inequalities and their computational complexity is analyzed the first proposed algorithm is an adaptation of the algorithms of polyak and tempo syst control lett 43 5 2001 343 and calafiore and polyak ieee trans autom control 46 11 2001 1755 for the present problem it is possible however to show that the expected number of iterations necessary to have a deterministic solution is infinite in order to make this number finite the improved algorithm is proposed the number of iterations necessary to have a probabilistic solution is also considered and is shown to be independent of the parameter dimension a numerical example is provided

a randomized cutting plane method with probabilistic geometric convergence we propose a randomized method for general convex optimization problems namely the minimization of a linear function over a convex body the idea is to generate n random points inside the body choose the best one and cut the part of the body defined by the linear constraint we analyze the convergence properties of the algorithm from a theoretical viewpoint i e under the rather standard assumption that an algorithm for uniform generation of random points in the convex body is available under this assumption the expected rate of convergence for such method is proved to be geometric this analysis is based on new results on the statistical properties of the empirical minimum over a convex body that we obtained in this paper moreover explicit sample size results on convergence are derived in particular we compute the minimum number of random points that should be generated at each step in order to guarantee that in a probabilistic sense the method performs better than the deterministic center of gravity algorithm from a practical viewpoint we show how the method can be implemented using hit and run versions of markov chain monte carlo algorithms and exemplify the performance of this implementable modification via a number of illustrative problems a crucial notion for the hit and run implementation is that of boundary oracle which is available for most optimization problems including linear matrix inequalities and many other kinds of constraints preliminary numerical results for semidefinite programs are presented confirming that the randomized approach might be competitive to modern deterministic convex optimization methods

survey paper research on probabilistic methods for control system design a novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty the main objective of this paper is to provide a broad perspective on this area of research known as probabilistic robust control and to address in a systematic manner recent advances the focal point is on design methods based on the interplay between uncertainty randomization and convex optimization and on the illustration of specific control applications

the quantum relativistic harmonic oscillator spectrum of zeros of its wave functions the distribution of zeros of the wave functions of the quantum relativistic harmonic oscillator is studied an analytic approximate expression for this distribution is derived within the framework of the wkb approximation using potential theory techniques it is proved that this wkb approach gives the right asymptotic distribution which is explicitly calculated

an ellipsoid algorithm for probabilistic robust controller design abstract in this paper a new iterative approach to probabilistic robust controller design is presented which is applicable to any robust controller filter design problem that can be represented as an lmi feasibility problem recently a probabilistic subgradient iteration algorithm was proposed for solving lmis it transforms the initial feasibility problem to an equivalent convex optimization problem which is subsequently solved by means of an iterative algorithm while this algorithm always converges to a feasible solution in a finite number of iterations it requires that the radius of a non empty ball contained into the solution set is known a priori this rather restrictive assumption is released in this paper while retaining the convergence property given an initial ellipsoid that contains the solution set the approach proposed here iteratively generates a sequence of ellipsoids with decreasing volumes all containing the solution set at each iteration a random uncertainty sample is generated with a specified probability density which parameterizes an lmi for this lmi the next minimum volume ellipsoid that contains the solution set is computed an upper bound on the maximum number of possible correction steps that can be performed by the algorithm before finding a feasible solution is derived a method for finding an initial ellipsoid containing the solution set which is necessary for initialization of the optimization is also given the proposed approach is illustrated on a real life diesel actuator benchmark model with real parametric uncertainty for which a h 2 robust state feedback controller is designed

results on the associated jacobi and gegenbauer polynomials abstract recurrence relations for the coefficients of the jacobi polynomial form of the associated jacobi polynomials p n x c are given for certain values of explicit formulae for these coefficients are obtained in particular the classical watson formula for the first associated gegenbauer polynomials is generalized

complexity analysis of an interior cutting plane method for convex feasibility problems we further analyze the convergence and the complexity of a dual column generation algorithm for solving general convex feasibility problems defined by a separation oracle the oracle is called at an approximate analytic center of the set given by the intersection of the linear inequalities which are the previous answers of the oracle we show that the algorithm converges in finite time and is in fact a fully polynomial approximation algorithm provided that the feasible region has a nonempty interior

a cutting plane algorithm for convex programming that uses analytic centers anoracle for a convex sets n accepts as input any pointz in n and ifz s then it returns yes while ifz s then it returns no along with a separating hyperplane we give a new algorithm that finds a feasible point ins in cases where an oracle is available our algorithm uses the analytic center of a polytope as test point and successively modifies the polytope with the separating hyperplanes returned by the oracle the key to establishing convergence is that hyperplanes judged to be unimportant are pruned from the polytope if a ball of radius 2 l is contained ins ands is contained in a cube of side 2l 1 then we can show our algorithm converges after o nl2 iterations and performs a total of o n4l3 tnl2 arithmetic operations wheret is the number of arithmetic operations required for a call to the oracle the bound is independent of the number of hyperplanes generated in the algorithm an important application in which an oracle is available is minimizing a convex function overs

on the properties of special functions on the linear type lattices abstract we present a general theory for studying the difference analogs of special functions of hypergeometric type on the linear type lattices i e the solutions of the second order linear difference equation of hypergeometric type on a special kind of lattices the linear type lattices in particular using the integral representation of the solutions we obtain several difference recurrence relations for such functions finally applications to q classical polynomials are given

on quasi orthogonal polynomials their differential equations discriminants and electrostatics abstract in this paper we develop a general theory of quasi orthogonal polynomials we first derive three term recurrence relation and second order differential equations for quasi orthogonal polynomials we also give an expression for their discriminants in terms of the recursion coefficients of the corresponding orthogonal polynomials in addition we investigate an electrostatic equilibrium problem where the equilibrium position of movable charges is attained at the zeros of the quasi orthogonal polynomials the examples of the freud weight w x e x 4 2 t x 2 and the jacobi weight w x 1 x 1 x are discussed in some detail finally we consider the nonlinear orthogonality preserving transformation and related matrix problem

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

stochastic algorithms for exact and approximate feasibility of robust lmis in this note we discuss fast randomized algorithms for determining an admissible solution for robust linear matrix inequalities lmis of the form f x spl delta spl les 0 where x is the optimization variable and spl delta is the uncertainty which belongs to a given set spl delta the proposed algorithms are based on uncertainty randomization the first algorithm finds a robust solution in a finite number of iterations with probability one if a strong feasibility condition holds in case no robust solution exists the second algorithm computes an approximate solution which minimizes the expected value of a suitably selected feasibility indicator function the theory is illustrated by examples of application to uncertain linear inequalities and quadratic stability of interval matrices

sequential randomized algorithms for robust convex optimization sequential randomized algorithms are considered for robust convex optimization which minimizes a linear objective function subject to a parameter dependent convex constraint employing convex optimization and random sampling of parameter these algorithms enable us to obtain a suboptimal solution within reasonable computational time the suboptimal solution is feasible in a probabilistic sense and the suboptimal value belongs to an interval which contains the optimal value the maximum of the interval is the optimal value of the robust convex optimization plus a specified tolerance on the other hand its minimum is the optimal value of the chance constrained optimization which is a probabilistic relaxation of the robust convex optimization with high probability

results on the associated classical orthogonal polynomials abstract let p k x be any system of the classical orthogonal polynomials and let p k x c be the corresponding associated polynomials of order c c n second order recurrence relation in k is given for the connection coefficient a n 1 k c in p n 1 x c k 0 n 1 a n 1 k c p k x this result is obtained thanks to a new explicit form of the fourth order differential equation satisfied by p n 1 c

a statistical learning theory approach for uncertain linear and bilinear matrix inequalities abstract in this paper we consider the problem of minimizing a linear functional subject to uncertain linear and bilinear matrix inequalities which depend in a possibly nonlinear way on a vector of uncertain parameters motivated by recent results in statistical learning theory we show that probabilistic guaranteed solutions can be obtained by means of randomized algorithms in particular we show that the vapnik chervonenkis dimension vc dimension of the two problems is finite and we compute upper bounds on it in turn these bounds allow us to derive explicitly the sample complexity of these problems using these bounds in the second part of the paper we derive a sequential scheme based on a sequence of optimization and validation steps the algorithm is on the same lines of recent schemes proposed for similar problems but improves both in terms of complexity and generality the effectiveness of this approach is shown using a linear model of a robot manipulator subject to uncertain parameters

uncertain convex programs randomized solutions and confidence levels many engineering problems can be cast as optimization problems subject to convex constraints that are parameterized by an uncertainty or instance parameter two main approaches are generally available to tackle constrained optimization problems in presence of uncertainty robust optimization and chance constrained optimization robust optimization is a deterministic paradigm where one seeks a solution which simultaneously satisfies all possible constraint instances in chance constrained optimization a probability distribution is instead assumed on the uncertain parameters and the constraints are enforced up to a pre specified level of probability unfortunately however both approaches lead to computationally intractable problem formulations

on the fourth order difference equation for the associated meixner polynomials three equivalent forms of the fourth order difference equation obeyed by the associated meixner polynomials with a nonnegative real association parameter are derived from a refinement of a recent result due to letessier et al 1996

a probabilistic ellipsoid algorithm for linear optimization problems with uncertain lmi constraints in this paper a probabilistic algorithm based on the deep cut ellipsoid method is proposed to solve a linear optimization problem subject to an uncertain linear matrix inequality lmi first a deep cut ellipsoid algorithm is introduced to address probabilistic feasibility of the uncertain lmi objective cuts are then defined to search for the optimal solution the final probabilistic ellipsoid algorithm is a combination of feasibility cuts and objective cuts it is shown that in a finite number of iterations the ellipsoid algorithm either returns a suboptimal probabilistically feasible solution with a high confidence level or finds the problem infeasible furthermore the bounds of the suboptimal value are provided with probabilistic guarantees

further polynomial type eigenfunctions the method developed previously by the author for obtaining eigenfunctions in the form polynomial exponential polynomial for a linear second order differential equation in normal form is extended to embrace the general case when the interaction potential or square of the refractive index profile has the form polynomial polynomial various situations arise in the development of the theory some of which can be concluded algebraically but the majority require numerical calculations regarding the rank of quite a general matrix containing many unknown parameters

on the zeros of eigenfunctions of polynomial differential operators it is shown that for polynomial eigenfunctions of an ordinary polynomial differential operator with coefficients depending only on the independent variable it is possible to determine the density of nodes around the mean without solving the corresponding eigenvalue problem this is done by means of the first few moments which can be directly expressed in terms of the above mentioned coefficients also very simple expressions for the asymptotic values i e when the degree of the polynomial becomes very large of these quantities are found for illustration these results are applied to various orthogonal polynomials which satisfy ordinary differential equations of second fourth and or sixth order

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

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

spectral properties of solutions of hypergeometric type differential equations abstract the second order differential equation x y x y y 0 is usually called equation of hypergeometric type provided that are polynomials of degree not higher than two and one respectively and is a constant their solutions are commonly known as hypergeometric type functions htfs in this work a study of the spectrum of zeros of those htfs for which 1 2 1 v r and are independent of is done within the so called semiclassical or wkb approximation specifically the semiclassical or wkb density of zeros of the htfs is obtained analytically in a closed way in terms of the coefficients of the differential equation that they satisfy applications to the gaussian and confluent hypergeometric functions as well as to hermite functions are shown

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

on the elementary schrodinger bound states and their multiplets the problem of the existence of elementary bound states is discussed a trivial observation that every elementary wave function i r is an exact bound state for an appropriate potential v r v i r r is shown to lead to a very transparent form of the quasiexact qe solvability condition v i v j for doublets and multiplets of the s in this sense the particular class of elementary ansatze i r r polynomial r2 exp r polynomial r2 also defines the particular class of qe solvable potentials they have an elementary nonpolynomial rational form possibly also with a strongly singular repulsive core at the origin the properties of these forces are discussed in detail

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

newton sum rules of zeros of semiclassical orthogonal polynomials abstract the distribution of zeros of the semiclassical orthogonal polynomials with weights w x x w c x wc x denoting a classical weight function and x being equal to x c or x2 c2 is investigated via the newton sum rules of zeros i e the sums of the rth powers of zeros recursion relations satisfied by these sum rules for semiclassical legendre laguerre and hermite polynomials are explicitly given extensions to other semiclassical polynomials are indicated

quasi orthogonal polynomials quadrature and interpolation abstract we investigate the quasi orthogonal polynomials by expressing them as characteristic polynomials of symmetric matrices many properties of zeros of quasi orthogonal polynomials are derived from this formulation we derive new formulae for quadrature and interpolation based on the zeros of quasi orthogonal polynomials and prove the mean convergence of lagrange interpolation

fourth order differential equation satisfied by the associated of any order of all classical orthogonal polynomials a study of their distribution of zeros abstract the first associated numerator polynomials of all classical orthogonal polynomials satisfy one fourth order differential equation valid for the four classical families but for the associated of arbitrary order the differential equations are only known separately in this work we introduce a program built in mathematica symbolic language which is able to construct the unique differential equation satisfied by the associated of any order of the classical class then we use this differential equation in order to study the distribution of zeros of these polynomials via their newton sum rules i e the sums of the kth power of zeros which are closely related with the moments of such a distribution

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

associated laguerre and hermite polynomials explicit orthogonality relations are found for the associated laguerre and hermite polynomials one consequence is the construction of the n 1 n pade approximation to a 1 b x a b x where a b x is the second solution to the confluent hypergeometric differential equation that does not grow rapidly at infinity

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

quasi orthogonality and differential equations abstract the following theorem is proved quasi orthogonal polynomials corresponding to the semi classical or laguerre hahn families and the orthogonal polynomials themselves verify differential equations of same order reducible to two or four by the hahn theorem

polynomial type eigenfunctions the differential equation for the harmonic oscillator is generalised to include an interaction potential containing a positive definite quadratic denominator conditions are developed under which certain eigenfunctions take the form of an exponential function multiplied by a polynomial the problem reduces to that of finding the eigenvalues of a certain matrix that is tridiagonal in form properties of the eigenvalues of this matrix are investigated since they are functions of a parameter occurring in the positive definite quadratic form the asymptotic forms of these eigenvalues are developed together with computed results expressed as curves showing the variation of the eigenvalues with respect to this parameter

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

fourth order differential equations for numerator polynomials the author gives explicitly the fourth order differential equation satisfied by the numerator polynomials associated polynomials of the classical orthogonal polynomials the coefficients of the differential equation are at most a quadratic combination of the polynomials sigma and tau and their derivatives defined via the relation sigma rho te this differential equation is therefore valid for the jacobi laguerre hermite and bessel associated polynomials

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

method of centers for minimizing generalized eigenvalues abstract we consider the problem of minimizing the largest generalized eigenvalue of a pair of symmetric matrices each of which depends affinely on the decision variables although this problem may appear specialized it is in fact quite general and includes for example all linear quadratic and linear fractional programs many problems arising in control theory can be cast in this form the problem is nondifferentiable but quasiconvex so methods such as kelley s cutting plane algorithm or the ellipsoid algorithm of shor nemirovsky and yudin are guaranteed to minimize it in this paper we describe relevant background material and a simple interior point method that solves such problems more efficiently the algorithm is a variation on huard s method of centers using a self concordant barrier for matrix inequalities developed by nesterov and nemirovsky nesterov and nemirovsky have also extended their potential reduction methods to handle the same problem since the problem is quasiconvex but not convex devising a nonheuristic stopping criterion i e one that guarantees a given accuracy is more difficult than in the convex case we describe several nonheuristic stopping criteria that are based on the dual of a related convex problem and a new ellipsoidal approximation that is slightly sharper in some cases than a more general result to nesterov and nemirovsky the algorithm is demonstrated on an example determining the quadratic lyapunov function that optimizes a decay rate estimate for a differential inclusion

spectral properties of the biconfluent heun differential equation abstract the spectrum of zeros of the polynomial solutions of the biconfluent heun differential equation is investigated by two different methods first the spectral newton sums i e the sums of the r th powers of the zeros are given in a rigorous and recurrent way second the density of zeros i e the number of zeros per unit interval is calculated in an explicit way within the so called semiclassical or bkw approximation this is done by using a general theorem which applies to a linear second order differential equation under certain conditions applications to the radial schrodinger equations associated to some quantum mechanical systems rotating harmonic oscillator confinement potentials as well as to the bessel differential equation are also shown

a convex approach to robust stability for linear systems with uncertain scalar parameters in this paper robust stability for linear systems with several uncertain complex and or real scalar parameters is studied a countable family of conditions sufficient for robust stability is given in terms of solvability of some simple linear matrix inequalities lmis these conditions are of increasing precision and it is shown conversely that robust stability implies solvability of these lmis from a certain rank and beyond this result constitutes an extension of the characterization by solvability of lyapunov inequality of the asymptotic stability for usual linear systems it is based on the search of parameter dependent quadratic lyapunov functions polynomial of increasing degree in the parameters

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

properties of the polynomials associated with the jacobi polynomials abstract power forms and jacobi polynomial forms are found for the polynomials w associated with jacobi polynomials also some differential difference equations and evalua tions of certain integrals involving wii 1 are given 1 introduction let w be a nonnegative function defined on an interval a b forwhich all momentsm j x w x dx n 0 1 exist and are finite m0 0 let be the monic polynomials orthogonal on a b with respect to w the polynomials 1 1 q x n pax rztt wu at pn t jt n 0 1 j a x are called the polynomials associated with the pn as is well known pn x a x are linearly independent solutions of the recurrence formula i 2 yn i x a y bny x q 0 1 where l3 an xpn p p pn n 0 o o b pn pn pn x pn x n l and f g f f x g x w x dx ja the initial conditions arep i x 0 pa x 1

bivariate second order linear partial differential equations and orthogonal polynomial solutions in this paper we construct the main algebraic and differential properties and the weight functions of orthogonal polynomial solutions of bivariate second order linear partial differential equations which are admissible potentially self adjoint and of hypergeometric type general formulae for all these properties are obtained explicitly in terms of the polynomial coefficients of the partial differential equation using vector matrix notation moreover rodrigues representations for the polynomial eigensolutions and for their partial derivatives of any order are given as illustration these results are applied to a two parameter monic appell polynomials finally the non monic case is briefly discussed 2011 elsevier inc

brief paper stochastic ellipsoid methods for robust control multiple updates and multiple cuts efficient randomized algorithms are developed for solving robust feasibility problems with multiple parameter dependent convex constraints two complementary strategies are presented both of which exploit the multiplicity to achieve fast convergence one is the stochastic ellipsoid method with multiple updates in each iteration of this algorithm an ellipsoid which describes a candidate of the solution set is updated many times via the multiple constraints with one random sample while at most one update is allowed in the original method the other is the stochastic ellipsoid method with multiple cuts here a new update rule is presented to construct a smaller ellipsoid directly via multiple subgradients given by the constraints a quantitative analysis of the volume of the ellipsoid is also provided which guarantees the advantage of the proposed algorithm over the original one the above features lead to a reduction of the total number of random samples necessary for convergence which is extensively demonstrated through numerical examples

the three term recurrence relation and the differentiation formulas for hypergeometric type functions abstract the functions of hypergeometric type are the solutions y y x of the differential equation z y z y y 0 where are polynomials of degrees not higher than 2 and 1 respectively and is a constant here we consider a class of functions of hypergeometric type with the additional condition that 1 2 1 0 being a complex number in general moreover we assume that the coefficients of the polynomials and have no dependence on to this class of functions belong gauss kummer and hermite functions the classical orthogonal polynomials and many other functions encountered in linear and non linear physics we obtain two important structural properties of these functions i the so called three term recurrence relation which correlates three functions of successive orders and ii the differentiation formulas also called ladder or structure relations or even differential recurrence relations which relate the first derivative y z with the functions y z and y 1 z or y 1 z finally these three relationships are applied to the polynomials of hypergeometric type which form a broad subclass of functions y where is a positive integer number and the associated contour is closed for completeness the explicit expressions corresponding to all classical orthogonal polynomials jacobi laguerre hermite and bessel are tabulated

algebraic and spectral properties of some quasiorthogonal polynomials encountered in quantum radiation the nodal structure of the wavefunctions of a large class of quantum mechanical potentials is often governed by the distribution of zeros of real quasiorthogonal polynomials it is known that these polynomials i may be described by an arbitrary linear combination of two orthogonal polynomials pn x and ii have real and simple zeros here the three term recurrence relation the second order differential equation and the distribution of zeros of quasiorthogonal polynomials of the classical class i e when pn x is a jacobi laguerre or hermite polynomial are derived and analyzed specifically the exact values of the newton sum rules and the wkb density of zeros of these polynomials are found

quasi orthogonality and zeros of some 2ϕ2 and 3ϕ2 polynomials abstract we state and prove the q extension of a result due to johnston and jordaan cf johnston and jordaan 2015 and make use of this result the orthogonality of q laguerre little q jacobi q meixner and al salam carlitz i polynomials as well as contiguous relations satisfied by the polynomials to establish the quasi orthogonality of certain 2 2 and 3 2 polynomials the location and interlacing properties of the real zeros of these quasi orthogonal polynomials are studied interlacing properties of the zeros of q laguerre quasi orthogonal polynomials l n z q when 2 1 with those of l n 1 1 z q and l n 1 z q are also considered

randomized methods for design of uncertain systems in this paper we study randomized methods for feedback design of uncertain systems the first contribution is to derive the sample complexity of various constrained control problems in particular we show the key role played by the binomial distribution and related tail inequalities and compute the sample complexity this contribution significantly improves the existing results by reducing the number of required samples in the randomized algorithm these results are then applied to the analysis of worst case performance and design with robust optimization the second contribution of the paper is to introduce a general class of sequential algorithms denoted as sequential probabilistic validation spv in these sequential algorithms at each iteration a candidate solution is probabilistically validated and corrected if necessary to meet the required specifications the results we derive provide the sample complexity which guarantees that the solutions obtained with spv algorithms meet some pre specified probabilistic accuracy and confidence the performance of these algorithms is illustrated and compared with other existing methods using a numerical example dealing with robust system identification

co recursive orthogonal polynomials and fourth order differential equations abstract we give explicitly the 4th order differential equation satisfied by the co recursive polynomials of the classical orthogonal polynomials we emphasize the legendre and bessel cases for which the co recursive differential equation coincide with the equation satisfied by the numerator polynomial differential equation and we propose new conjectures

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

probabilistic robust design with linear quadratic regulators in this paper we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators lqr we consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism the approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters in particular we propose an iterative algorithm for designing a controller which is based upon subgradient iterations at each step of the sequence we first generate a random sample and then we perform a subgradient step for a convex constraint defined by the lqr problem the main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps in addition at a fixed step we compute a lower bound of the probability that a quadratically stabilizing controller is found

on the polynomial solutions of ordinary differential equations of the fourth order the density of zeros of polynomial solutions of ordinary differential equations of the fourth order with coefficients depending only on the independent variable is analyzed the first four moments of such a density are given directly in terms of the coefficients which characterize the differential operator application to the nonclassical orthogonal polynomials corresponding to the names krall legendre krall laguerre and krall jacobi is done global asymptotic properties of the zeros of these polynomials are also obtained

on tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty we present efficiently verifiable sufficient conditions for the validity of specific np hard semi infinite systems of linear matrix inequalities lmis arising from lmis with uncertain data and demonstrate that these conditions are tight up to an absolute constant factor in particular we prove that given an n times n interval matrix cal u rho a mid a ij a ij leq rho c ij one can build a computable lower bound accurate within the factor pi over 2 on the supremum of those rho for which all instances of cal u rho share a common quadratic lyapunov function we then obtain a similar result for the problem of quadratic lyapunov stability synthesis finally we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube

quick construction of recurrence relations for the jacobi coefficients abstract recurrence relations are given for the jacobi series coefficients of functions which satisfy linear differential equations of first to third order with polynomial coefficients

second order difference equations for certain families of discrete polynomials abstract in this paper we will consider two algorithms which allow us to obtain second order linear difference equations for certain families of polynomials the corresponding algorithms can be implemented in any computer algebra system in order to obtain explicit expressions of the coefficients of the difference equations

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

repetitive scenario design repetitive scenario design rsd is a randomized approach to robust design based on iterating two phases a standard scenario design phase that uses n scenarios design samples followed by randomized feasibility phase that uses n o test samples on the scenario solution we give a full and exact probabilistic characterization of the number of iterations required by the rsd approach for returning a solution as a function of n n o and of the desired levels of probabilistic robustness in the solution this novel approach broadens the applicability of the scenario technology since the user is now presented with a clear tradeoff between the number n of design samples and the ensuing expected number of repetitions required by the rsd algorithm the plain one shot scenario design becomes just one of the possibilities sitting at one extreme of the tradeoff curve in which one insists in finding a solution in a single repetition this comes at the cost of possibly high n other possibilities along the tradeoff curve use lower n values but possibly require more than one repetition