supersymmetric quantum mechanics engineered hierarchies of integrable potentials and related orthogonal polynomials within the context of supersymmetric quantum mechanics and its related hierarchies of integrable quantum hamiltonians and potentials a general programme is outlined and applied to its first two simplest illustrations going beyond the usual restriction of shape invariance for intertwined potentials it is suggested to require a similar relation for hamiltonians in the hierarchy separated by an arbitrary number of levels n by requiring further that these two hamiltonians be in fact identical up to an overall shift in energy a periodic structure is installed in the hierarchy which should allow for its resolution specific classes of orthogonal polynomials characteristic of such periodic hierarchies are thereby generated while the methods of supersymmetric quantum mechanics then lead to generalised rodrigues formulae and recursion relations for such polynomials the approach also offers the practical prospect of quantum modelling through the engineering of quantum potentials from experimental energy spectra in this paper these ideas are presented and solved explicitly for the cases n 1 and n 2 the latter case is related to the generalised laguerre polynomials for which indeed new results are thereby obtained in the context of dressing chains and deformed polynomial heisenberg algebras some partial results for n 3 also exist in the literature which should be relevant to a complete study of the n 3 general periodic hierarchies

adaptive control of a simple nonlinear system without a priori information on the plant parameters the authors present an adaptive control scheme for nonlinear systems of the form x c sup t f x b u where f x is lipschitz c is a constant vector and b is a constant scalar the control scheme achieves asymptotical model matching without a priori knowledge of the sign of the b gain the adaptive scheme is free from singularities in the sense that the estimate of b entering in the denominator of the control law is bounded away from zero the singularity has been overcome through a suitable modification of the parameter estimates which is based on standard least squares covariance matrix properties

on trust region methods for unconstrained minimization without derivatives we consider some algorithms for unconstrained minimization without derivatives that form linear or quadratic models by interpolation to values of the objective function then a new vector of variables is calculated by minimizing the current model within a trust region techniques are described for adjusting the trust region radius and for choosing positions of the interpolation points that maintain not only nonsingularity of the interpolation equations but also the adequacy of the model particular attention is given to quadratic models with diagonal second derivative matrices because numerical experiments show that they are often more efficient than full quadratic models for general objective functions finally some recent research on the updating of full quadratic models is described briefly using fewer interpolation equations than before the resultant freedom is taken up by minimizing the frobenius norm of the change to the second derivative matrix of the model a preliminary version of this method provides some very promising numerical results

cohomological study on variants of the mumford system and integrability of the noumi yamada system the purpose of this paper is twofold the first is to apply the method introduced in the works of nakayashiki and smirnov 11 12 on the mumford system to its variants the other is to establish a relation between the mumford system and the isospectral limit mathcal q g i and mathcal q g ii of the noumi yamada system 15 as a consequence we prove the algebraically completely integrability of the systems mathcal q g i and mathcal q g ii and get explicit descriptions of their solutions

a robust direct adaptive controller this paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics the algorithm is designed based on the reduced order plant which is assumed to be minimum phase and of known order and relative degree but is analyzed with respect to the overall plant which due to the unmodeled dynamics may be nonminimum phase and of unknown order and relative degree it is shown that if the unmodeled dynamics are sufficiently small in the low frequency range then the algorithm guarantees boundedness of all signals in the adaptive loop and small residual tracking errors for any bounded initial conditions in the absence of unmodeled dynamics the residual tracking error is shown to be zero

a globally convergent pole placement indirect adaptive controller a pole displacement indirect adaptive control algorithm is discussed for discrete time linear deterministic plants with arbitrary zeros the global convergence of the resulting closed loop control system is achieved subject to the assumptions that the plant order and a nonzero lower bound on its degree of controllability are known the problem of controllability of the plant model estimate is handled by using both a parameter correction and time varying nonlinear feedback a key property of the algorithm is that the plant estimate reaches a reasonable degree of controllability in a finite time after which the parameter correction and the nonlinear feedback are no longer used

universal characters integrable chains and the painleve equations abstract the universal character is a generalization of the schur polynomial attached to a pair of partitions see adv math 74 1989 57 we prove that the universal character solves the darboux chain the n periodic closing of the chain is equivalent to the painleve equation of type a n 1 1 consequently we obtain an expression of rational solutions of the painleve equations in terms of the universal characters

solutions to the painleve v equation through supersymmetric quantum mechanics in this paper we shall use the algebraic method known as supersymmetric quantum mechanics susy qm to obtain solutions to the painleve v pv equation a second order nonlinear ordinary differential equation for this purpose we will apply first the susy qm treatment to the radial oscillator in addition we will revisit the polynomial heisenberg algebras phas and we will study the general systems ruled by them for first order phas we obtain the radial oscillator while for third order phas the potential will be determined by solutions to the pv equation this connection allows us to introduce a simple technique for generating solutions of the pv equation expressed in terms of confluent hypergeometric functions finally we will classify them into several solution hierarchies

stable adaptive regulation of arbitrary nth order plants this paper presents an algorithm for adaptively stabilizing and asymptotically regulating an arbitrary single input single output linear time invariant plant which is controllable and observable of known order n and has unknown parameters no further assumptions are made no external probing signal is required

classical ladder operators polynomial poisson algebras and classification of superintegrable systems we recall results concerning one dimensional classical and quantum systems with ladder operators we obtain the most general one dimensional classical systems respectively with a third and a fourth order ladder operators satisfying polynomial heisenberg algebras these systems are written in terms of the solutions of quartic and quintic equations they are the classical equivalent of quantum systems involving the fourth and fifth painleve transcendents we use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of lissajous s figures

factorization little or great algorithm the progress of the factorization method since the 1935 work of dirac is briefly reviewed though linked with older mathematical theories the factorization seems an autonomous driving force offering substantial support to the present day darboux and backlund approaches

schlesinger transformations of painleve ii v the explicit form of the schlesinger transformations for the second third fourth and fifth painleve equations is given

darboux transformations and the symmetric fourth painleve equation this paper is concerned with the group symmetries of the fourth painlev equation piv a second order nonlinear ordinary differential equation it is well known that the parameter space of piv admits the action of the extended affine weyl group as shown by noumi and yamada the action of as b cklund transformations of piv provides a derivation of its symmetric form sp4 the dynamical system sp4 is also equivalent to the isomonodromic deformation of an associated three by three matrix linear system lax pair the action of the generators of on this lax pair is derived using the darboux transformation for an associated third order operator

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

global convergence of general derivative free trust region algorithms to first and second order critical points in this paper we prove global convergence for first and second order stationary points of a class of derivative free trust region methods for unconstrained optimization these methods are based on the sequential minimization of quadratic or linear models built from evaluating the objective function at sample sets the derivative free models are required to satisfy taylor type bounds but apart from that the analysis is independent of the sampling techniques a number of new issues are addressed including global convergence when acceptance of iterates is based on simple decrease of the objective function trust region radius maintenance at the criticality step and global convergence for second order critical points

a linear time algorithm for trust region problems we consider the fundamental problem of minimizing a general quadratic function over an ellipsoidal domain also known as the trust region sub problem we give the first provable linear time in the number of non zero entries of the input algorithm for approximately solving this problem specifically our algorithm returns an epsilon approximate solution in runtime of order n sqrt epsilon where n is the number of non zero entries in the input this matches the runtime of nesterov s accelerated gradient descent suitable for the special case in which the quadratic objective is convex and the runtime of the lanczos method which is applicable when the problem is purely quadratic

a pattern search filter method for nonlinear programming without derivatives this paper formulates and analyzes a pattern search method for general constrained optimization based on filter methods for step acceptance roughly a filter method accepts a step that improves either the objective function value or the value of some function that measures the constraint violation the new algorithm does not compute or approximate any derivatives penalty constants or lagrange multipliers a key feature of the new algorithm is that it preserves the division into search and local poll steps which allows the explicit use of inexpensive surrogates or random search heuristics in the search step it is shown here that the algorithm identifies limit points at which optimality conditions depend on local smoothness of the functions and to a greater extent on the choice of a certain set of directions stronger optimality conditions are guaranteed for smoother functions and in the constrained case for a fortunate choice of the directions on which the algorithm depends these directional conditions generalize those given previously for linear constraints but they do not require a feasible starting point in the absence of general constraints the proposed algorithm and its convergence analysis generalize previous work on unconstrained bound constrained and linearly constrained generalized pattern search the algorithm is illustrated on some test examples and on an industrial wing planform engineering design application

geometry of interpolation sets in derivative free optimization we consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated we show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods these bounds involve a constant that reflects the quality of the interpolation set the main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small and at least uniformly bounded this constant is often described through the basis of lagrange polynomials associated with the interpolation set we provide an alternative more intuitive definition for this concept and show how this constant is related to the condition number of a certain matrix this relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded we also derive bounds on the error between the model and the function and between their derivatives directly in terms of this condition number and of this geometry constant

on the robust adaptive stabilization of a class of nominally first order hybrid systems this paper presents a robust adaptive stabilization scheme for a class of nominally first order and controllable hybrid time invariant linear systems involving both continuous and discrete parametrizations and signals the usual assumptions of inverse stability of the plant and knowledge of the high frequency gain are not required the design philosophy relies on the separation of the contributions of continuous and discrete dynamics to the output by generating two associate additive terms to build the control action the controllability of the nominally estimated plant model is maintained by using a hysteresis switching function this allows relaxing the assumption of stability of the plant inverse

integrable boundary conditions for the toda lattice the problem of construction of the boundary conditions for the toda lattice compatible with its higher symmetries is considered it is demonstrated that this problem is reduced to finding the differential constraints consistent with the zs akns hierarchy a method of their construction is offered based on the backlund transformations it is shown that the generalized toda lattices corresponding to the non exceptional lie algebras of finite growth can be obtained by imposing one of the four simplest integrable boundary conditions on both ends of the lattice this fact allows in particular the solution of the reduction problem of the series a toda lattices into the series d lattices deformations of the found boundary conditions are presented which lead to the painleve type equations

spectral curve darboux coordinates and hamiltonian structure of periodic dressing chains a chain of one dimensional schrodinger operators connected by successive darboux transformations is called the darboux chain or dressing chain the periodic dressing chain with period n has a control parameter alpha if alpha not 0 the n periodic dressing chain may be thought of as a generalization of the fourth or fifth depending on the parity of n painleve equations the n periodic dressing chain has two different lax representations due to adler and to noumi and yamada adler s 2 times 2 lax pair can be used to construct a transition matrix around the periodic lattice one can thereby define an associated spectral curve and a set of darboux coordinates called spectral darboux coordinates the equations of motion of the dressing chain can be converted to a hamiltonian system in these darboux coordinates the symplectic structure of this hamiltonian formalism turns out to be consistent with a poisson structure previously studied by veselov shabat noumi and yamada

a study on the fourth q painleve equation a q difference analogue of the fourth painleve equation is proposed its symmetry structure and some particular solutions are investigated

robust pole placement direct adaptive control a pole placement direct adaptive control for not necessarily minimum phase plants is presented which is robust with respect to unmodeled dynamics and bounded disturbances the robustness is achieved by using both a normalized least squares algorithm with dead zone and appropriate nonlinear feedback the resulting closed loop system is shown to be globally stable subject to standard assumptions

brief paper on the stabilizability condition in indirect adaptive control many recent studies on overall stability of adaptive control systems require among other conditions that the identified model be stabilizable at least asymptotically in fact the usual identification methods do not guarantee this property in the passive approach without exciting extra signals or if the model order is subject to over estimation in order to solve this problem it is shown in this paper that certain classical identification algorithms can be easily modified in order to ensure the above stabilizability condition without affecting the other properties

reduction of lattice equations to the painleve equations piv and pv in this paper we construct a new relation between adler bobenko suris equations and painleve equations moreover using this connection we construct the difference differential lax representations of the fourth and fifth painleve equations

on stieltjes relations painleve iv hierarchy and complex monodromy a generalization of the stieltjes relations for the fourth painleve piv transcendents and their higher analogues determined by dressing chains is proposed it is proven that if a rational function from a certain class satisfies these relations it must be a solution of some higher piv equation the approach is based on the interpretation of the stieltjes relations as local trivial monodromy conditions for certain schrodinger equations in the complex domain as a corollary a new class of schrodinger operators with trivial monodromy is constructed in terms of the piv transcendents

a globally convergent adaptive pole placement algorithm without a persistency of excitation requirement this paper presents an indirect adaptive control scheme for deterministic plants which are not necessarily minimum phase global convergence is established for the scheme in the sense that the closed loop poles are asymptotically assigned for the given data sequence and the system input and output remain bounded for all time a key feature of the scheme is that no persistency of excitation condition is required the algorithm uses recursive least squares with variable forgetting factor normalized regression vectors and a matrix gain with constant trace

on an integrable system of q difference equations satisfied by the universal characters its lax formalism and an application to q painleve equations the universal character is a generalization of the schur function attached to a pair of partitions we study an integrable system of q difference equations satisfied by the universal characters which is an extension of the q kp hierarchy and is called the lattice q uc hierarchy we describe the lattice q uc hierarchy as a compatibility condition of its associated linear system lax formalism and explore an application to the q painleve equations via similarity reduction in particular a higher order analogue of the q painleve vi equation is presented

parameter estimation and variable selection for big systems of linear ordinary differential equations a matrix based approach abstractordinary differential equations odes are widely used to model the dynamic behavior of a complex system parameter estimation and variable selection for a big system with linear odes are very challenging due to the need of nonlinear optimization in an ultra high dimensional parameter space in this article we develop a parameter estimation and variable selection method based on the ideas of similarity transformation and separable least squares sls simulation studies demonstrate that the proposed matrix based sls method could be used to estimate the coefficient matrix more accurately and perform variable selection for a linear ode system with thousands of dimensions and millions of parameters much better than the direct least squares method and the vector based two stage method that are currently available we applied this new method to two real datasets a yeast cell cycle gene expression dataset with 30 dimensions and 930 unknown parameters and the standard poor 1500 index stock price dat

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

on some class of reductions for the itoh narita bogoyavlenskii lattice we show a broad class of constraints compatible with the itoh narita bogoyavlenskii lattice hierarchy all these constraints can be written in the form of a discrete conservation law i i 1 i i with an appropriate homogeneous polynomial discrete function i i a

singularity free adaptive pole placement without resorting to persistency of excitation detailed analysis for first order systems abstract this paper presents a novel approach to avoid singularities in adaptive pole placement control algorithms the control scheme is applicable to either minimum or nonminimum phase plants and the closed loop stability is guaranteed in spite of bounded disturbances the algorithm uses a least squares identification algorithm with a particular dead zone that allows the definition of modified estimate that retain the convergence properties of the original estimates explicit modification to the parameters estimates is given that secures well conditioned matrices in the adaptive control law detailed analysis is given for one pole one zero systems

higher order susy linearized nonlinear heisenberg algebras and coherent states using an iterative construction of the first order intertwining technique we find k parametric families of exactly solvable anharmonic oscillators whose spectra consist of a part isospectral to the oscillator plus k additional levels at arbitrary positions below e0 it is seen that the natural ladder operators for these systems give place to polynomial nonlinear algebras and it is shown that these algebras can be linearized the coherent states construction is performed in the nonlinear and linearized cases

nonlinear chains and painleve equations abstract ode systems which define a periodically closed sequence of backlund transformations bt and which are equivalent to the painleve equations p2 p6 are presented transformation properties of painleve equations can be easily derived from the discrete symmetries of these systems

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

adaptive stabilization of nonminimum phase first order continuous time systems presents an adaptive control scheme for first order systems of the form y spl dot a y b sub 0 u spl dot b sub 1 u which may possibly be a nonminimum phase the control scheme achieves asymptotical stabilization without any a priori knowledge on the plant parameters the adaptive scheme is free from singularities in the sense that the plant estimated model is always controllable the singularity has been overcome through a suitable modification of the parameter estimates which is based on standard least squares covariance matrix properties and the use of a hysteresis switching function

susy partners of the truncated oscillator painleve transcendents and backlund transformations in this work the supersymmetric technique is applied to the truncated oscillator to generate hamiltonians ruled by second and third order polynomial heisenberg algebras which are connected to the painleve iv and painleve v equations respectively the aforementioned connection is exploited to produce particular solutions to both nonlinear differential equations and the backlund transformations relating them

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

least frobenius norm updating of quadratic models that satisfy interpolation conditions quadratic models of objective functions are highly useful in many optimization algorithms they are updated regularly to include new information about the objective function such as the difference between two gradient vectors we consider the case however when each model interpolates some function values so an update is required when a new function value replaces an old one we let the number of interpolation conditions m say be such that there is freedom in each new quadratic model that is taken up by minimizing the frobenius norm of the second derivative matrix of the change to the model this variational problem is expressed as the solution of an m n 1 m n 1 system of linear equations where n is the number of variables of the objective function further the inverse of the matrix of the system provides the coefficients of quadratic lagrange functions of the current interpolation problem a method is presented for updating all these coefficients in m n 2 operations which allows the model to be updated too an extension to the method is also described that suppresses the constant terms of the lagrange functions these techniques have a useful stability property that is investigated in some numerical experiments

robust adaptive regulation without persistent excitation a globally convergent adaptive regulator for minimum or nonminimum phase systems subject to bounded disturbances is presented the control strategy is designed for a particular input output representation obtained from the state space representation of the system the leading coefficient of the novel representation is the product of the observability and controllability matrices of the system the controller scheme uses a least squares identification algorithm with a dead zone the dead zone is chosen to obtain convergence properties on the estimates and on the covariance matrix as well this allows the definition of modified estimates which secure well conditioned matrices in the adaptive control law explicit bounds on the plant output are given

an interior affine scaling cubic regularization algorithm for derivative free optimization subject to bound constraints abstract in this paper we introduce an affine scaling cubic regularization algorithm for solving optimization problem without available derivatives subject to bound constraints employing a polynomial interpolation approach to handle the unavailable derivatives of the original objective function we first define an affine scaling cubic model of the approximate objective function which is obtained by the polynomial interpolation approach with an affine scaling method at each iteration a candidate search direction is determined by solving the affine scaling cubic regularization subproblem and the new iteration is strictly feasible by way of an interior backtracking technique the global convergence and local superlinear convergence of the proposed algorithm are established under some mild conditions preliminary numerical results are reported to show the effectiveness of the proposed algorithm

a symmetric system of mixed painleve iii v equations and its integrable origin a mixed symmetric painleve iii v model which describes a hybrid of both equations is defined and obtained by successive self similarity and dirac lagrange multiplier reductions from an integrable 4 boson hierarchy

asymptotic convergence analysis of a new class of proximal point methods finite dimensional local convergence results for self adaptive proximal point methods and nonlinear functions with multiple minimizers are generalized and extended to a hilbert space setting the principle assumption is a local error bound condition which relates the growth in the function to the distance to the set of minimizers a local convergence result is established for almost exact iterates less restrictive acceptance criteria for the proximal iterates are also analyzed these criteria are expressed in terms of a subdifferential of the proximal function and either a subdifferential of the original function or an iteration difference if the proximal regularization parameter mu bf x is sufficiently small and bounded away from zero and f is sufficiently smooth then there is local linear convergence to the set of minimizers for a locally convex function a convergence result similar to that for almost exact iterates is established for a locally convex solution set and smooth functions it is shown that if the proximal regularization parameter has the form mu bf x beta f bf x eta where eta in 0 2 then the convergence is at least superlinear if eta in 0 1 and at least quadratic if eta in 1 2

a dynamical systems approach to the fourth painleve equation we use methods from dynamical systems to study the fourth painleve equation our starting point is the symmetric form of to which the poincare compactification is applied the motion on the sphere at infinity can be completely characterized there are fourteen fixed points which are classified into three different types generic orbits of the full system are curves from one of four asymptotically unstable points to one of four asymptotically stable points with the set of allowed transitions depending on the values of the parameters this allows us to give a qualitative description of a generic real solution of

singularity free multivariable model reference adaptive control in this paper we propose a way to solve the problem of singularities in model reference adaptive control of linear multi input multi output mimo systems using a parameter modification procedure based on the least squares covariance matrix inverse the scheme does not require any explicit prior knowledge about the leading coefficient matrix associated with the control input and secures a uniform lower bound for the determinant of the estimate of this matrix a global convergence analysis is presented

solvable rational extensions of the morse and kepler coulomb potentials we show that it is possible to generate an infinite set of solvable rational extensions from every exceptional first category translationally shape invariant potential this is made by using darboux backlund transformations based on unphysical regular riccati schrodinger functions which are obtained from specific symmetries associated with the considered family of potentials

a derivative free method for solving box constrained underdetermined nonlinear systems of equations abstract a derivative free iterative method for solving bound constrained underdetermined nonlinear systems is presented the procedure consists of a quasi newton method that uses the broyden update formula and a globalized line search that combines the strategy of grippo lampariello and lucidi with the li and fukushima one global convergence results are proved and numerical experiments are presented

a derivative free algorithm for least squares minimization we develop a framework for a class of derivative free algorithms for the least squares minimization problem these algorithms are designed to take advantage of the problem structure by building polynomial interpolation models for each function in the least squares minimization under suitable conditions global convergence of the algorithm is established within a trust region framework promising numerical results indicate the algorithm is both efficient and robust numerical comparisons are made with standard derivative free software packages that do not exploit the special structure of the least squares problem or that use finite differences to approximate the gradients

virtual turning points and bifurcation of stokes curves for higher order ordinary differential equations for a higher order linear ordinary differential operator p its stokes curve bifurcates in general when it hits another turning point of p this phenomenon is most neatly understandable by taking into account stokes curves emanating from virtual turning points together with those from ordinary turning points this understanding of the bifurcation of a stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth painleve equation

integral equations and exact solutions for the fourth painleve equation we consider a special case of the fourth painleve equation given by d 2 d 2 3 5 2 3 1 4 2 v 1 2 1 with v a parameter and seek solutions v satisfying the boundary condition 0 2 equation 1 arises as a symmetry reduction of the derivative nonlinear schrodinger dnls equation which is a completely integrable soliton equation solvable by inverse scattering techniques solutions of equation 1 satisfying 2 are expressed in terms of the solutions of linear integral equations obtained from the inverse scattering formalism for the dnls equation we obtain exact bound state solutions of equation 1 for v n a positive integer using the integral equation representation which decay exponentially as and are the first example of such solutions for the painleve equations additionally using backlund transformations for the fourth painleve equation we derive a nonlinear recurrence relation commonly referred to as a backlund transformation in the context of soliton equations for equation 1 relating v and v 1

benchmarking derivative free optimization algorithms we propose data profiles as a tool for analyzing the performance of derivative free optimization solvers when there are constraints on the computational budget we use performance and data profiles together with a convergence test that measures the decrease in function value to analyze the performance of three solvers on sets of smooth noisy and piecewise smooth problems our results provide estimates for the performance difference between these solvers and show that on these problems the model based solver tested performs better than the two direct search solvers tested

a second order cone based approach for solving the trust region subproblem and its variants we study the trust region subproblem trs of minimizing a nonconvex quadratic function over the unit ball with additional conic constraints despite having a nonconvex objective it is known that the classical trs and a number of its variants are polynomial time solvable in this paper we follow a second order cone soc based approach to derive an exact convex reformulation of the trs under a structural condition on the conic constraint our structural condition is immediately satisfied when there are no additional conic constraints and it generalizes several such conditions studied in the literature as a result our study highlights an explicit connection between the classical nonconvex trs and smooth convex quadratic minimization which allows for the application of cheap iterative methods such as nesterov s accelerated gradient descent to the trs furthermore under slightly stronger conditions we give a low complexity characterization of the convex hull of the epigraph of the nonconvex quadrati

boundedness properties of simple indirect adaptive control systems examining adaptive controllers for a class of first order linear continuous time plants with different estimators and control laws the authors obtain explicit solutions completely describing the nonlinear behavior of the resultant adaptive systems some of these adaptive systems exhibit either finite time escapes or other forms of unbounded behavior due to a loss of stabilizability of the estimated model some fixes for the loss of stabilizability problem are analyzed implications of these results for the general indirect adaptive control case are discussed

on a unified approach to transformations and elementary solutions of painleve equations an algorithmic method is developed for investigating the transformation properties of second order equations of painleve type this method which utilizes the singularity structure of these equations yields explicit transformations which relate solutions of the painleve equations ii vi with different parameters these transformations easily generate rational and other elementary solutions of the equations the relationship between painleve equations and certain new equations quadratic in the second derivative of painleve type is also discussed

third order differential ladder operators and supersymmetric quantum mechanics hierarchies of one dimensional hamiltonians in quantum mechanics admitting third order differential ladder operators are studied each hamiltonian has associated three step darboux pseudo cycles and painleve iv equations as a closure condition the whole hierarchy is generated applying some operations on the cycles these operations are investigated in the frame of supersymmetric quantum mechanics and mainly involve algebraic manipulations a consistent geometric representation for the hierarchy and cycles is built that also helps in understanding the operations three kinds of hierarchies are distinguished and a realization based on the harmonic oscillator hamiltonian is supplied giving an interpretation for the spectral properties of the hamiltonians of each hierarchy

the proximal point algorithm with genuine superlinear convergence for the monotone complementarity problem in this paper we consider a proximal point algorithm ppa for solving monotone nonlinear complementarity problems ncp ppa generates a sequence by solving subproblems that are regularizations of the original problem it is known that ppa has global and superlinear convergence properties under appropriate criteria for approximate solutions of subproblems however it is not always easy to solve subproblems or to check those criteria in this paper we adopt the generalized newton method proposed by de luca facchinei and kanzow to solve subproblems and adopt some ncp functions to check the criteria then we show that the ppa converges globally provided that the solution set of the problem is nonempty moreover without assuming the local uniqueness of the solution we show that the rate of convergence is superlinear in a genuine sense provided that the limit point satisfies the strict complementarity condition

polynomial heisenberg algebras polynomial deformations of the heisenberg algebra are studied in detail some of their natural realizations are given by the higher order susy partners and not only by those of first order as is already known of the harmonic oscillator for even order polynomials here it is shown that the susy partners of the radial oscillator play a similar role when the order of the polynomial is odd moreover it will be proved that the general systems ruled by such kinds of algebras in the quadratic and cubic cases involve painleve transcendents of types iv and v respectively

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

rational solutions of the noumi and yamada system of type a4 1 we completely classify the rational solutions of the noumi and yamada system of type a4 1 which is a generalization of the fourth painleve equation the rational solutions are classified to three by the backlund transformation group

painleve equations from darboux chains i piii pv we show that the painleve equations piii pvi can be derived in a unified way from a periodic sequence of darboux transformations for a schrodinger problem with quadratic eigenvalue dependence the general problem naturally divides into three different branches each described by an infinite chain of equations the painleve equations are obtained by closing the chain periodically at the lowest nontrivial level s the chains provide symmetric forms for the painleve equations from which hirota bilinear forms and lax pairs are derived in this paper part i we analyse in detail the cases piii pv while pvi will be studied in part ii