the distance to a polyhedron abstract we give some exact formulas and some estimates for the distance to a polyhedron in a normed linear space e we show that in the case when e r n endowed with the l norm these estimates are in general better than a recent estimate by cook gerards schrijver and tardos 2

dynamical noether invariants for time dependent nonlinear systems dynamical invariants are derived for time dependent systems with nonlinear equations of motion including nonharmonic damped systems the concept of a dynamical algebra is discussed and its utility for the construction of dynamical invariants for nonharmonic systems is demonstrated finally we show the existence of dynamical invariants for some nonlinear quantum systems

exact order of hoffman s error bounds for elliptic quadratic inequalities derived from vector valued chebyshev approximation in this paper we introduce the exact order of hoffman s error bounds for approximate solutions of elliptic quadratic inequalities elliptic quadratic inequalities are closely related to chebyshev approximation of vector valued functions including complex valued functions the set of chebyshev approximations of a vector valued function defined on a finite set is shown to be hausdorff strongly unique of order exactly 2 s for some nonnegative integer s as a consequence the exact order of hoffman s error bounds for approximate solutions of elliptic quadratic inequalities is exactly 2 s for some nonnegative integer s the integer s called the order of deficiency which is computable quantifies how much the abadie constraint qualification is violated by the elliptic quadratic inequalities

polyhedral sweeping processes with unbounded nonconvex valued perturbation abstract a polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable hilbert space polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes we consider the polyhedral sweeping process as an evolution inclusion with subdifferential operators depending on time the widely used assumption of lipschitz continuity for the multivalued perturbation term is replaced by a weaker notion of h lipschitzness the existence of solutions is proved for this sweeping process

the sharp lipschitz constants for feasible and optimal solutions of a perturbed linear program abstract the purpose of this paper is to derive the sharp lipschitz constants for the feasible solutions and optimal solutions of a linear program with respect to right hand side perturbations the lipschitz constants are given in terms of pseudoinverses of submatrices of the matrices involved and are proven to be sharp

error bounds for nondegenerate monotone linear complementarity problems error bounds and upper lipschitz continuity results are given for monotone linear complementarity problems with a nondegenerate solution the existence of a nondegenerate solution considerably simplifies the error bounds compared with problems for which all solutions are degenerate thus when a point satisfies the linear inequalities of a nondegenerate complementarity problem the residual that bounds the distance from a solution point consists of the complementarity condition alone whereas for degenerate problems this residual cannot bound the distance to a solution without adding the square root of the complementarity condition to it this and other simplified results are a consequence of the polyhedral characterization of the solution set as the intersection of the feasible region z mz q 0 z 0 with a single linear affine inequality constraint

lipschitz continuity of solutions of linear inequalities programs and complementarity problems it is shown that solutions of linear inequalities linear programs and certain linear complementarity problems e g those with p matrices or z matrices but not semidefinite matrices are lipschitz continuous with respect to changes in the right hand side data of the problem solutions of linear programs are not lipschitz continuous with respect to the coefficients of the objective function the lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right hand side perturbation

gauge transformation approach to the exact solution of a generalized harmonic oscillator the exact solution and the invariant operator of a generalized harmonic oscillator are obtained by performing three consecutive gauge transformations on the time dependent schrodinger equation in contrast to previous results they depend only on solutions of a linear differential equation for the associated classical harmonic oscillator non adiabatic and adiabatic berry phases for the system are calculated on the basis of the exact solution

multiparticle susy quantum mechanics and representations of the permutation group the method of multidimensional susy quantum mechanics is applied to the investigation of supersymmetrical n particle systems on a line for the case of separable centre of mass motion new decomposition of the super hamiltonian into block diagonal form with elementary matrix components is constructed matrices of coefficients of these minimal blocks are shown to coincide with matrices of irreducible representations of the permutation group sn which correspond to the young tableaux n m 1m the connections with known generalizations of n particle calogero and sutherland models are established

global regularity theorems for a system of nondifferentiable convex inequalities and linear equalities the best bound is given for the quotient of the distance of an infeasible point and the norm of the residual vector applications to stability theory are then obtained

error bounds for piecewise convex quadratic programs and applications in this paper we establish a local error estimate for feasible solutions of a piecewise convex quadratic program and a global error estimate for feasible solutions of a convex piecewise quadratic program these error estimates provide a unified approach for deriving many old and new error estimates for linear programs linear complementarity problems convex quadratic programs and affine variational inequality problems the approach reveals the fact that each error estimate is a consequence of some reformulation of the original problem as a piecewise convex quadratic program or a convex piecewise quadratic program in a sense even robinson s result on the upper lipschitz continuity of a polyhedral mapping can be considered as a special case of error estimates for approximate solutions of a piecewise convex quadratic program as an application we derive new global error estimates for iterates of the proximal point algorithm for solving a convex piecewise quadratic program

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

error bounds for the linear complementarity problem with a p matrix abstract we introduce a fundamental quantity associated with a p matrix and show how this quantity is useful in deriving error bounds for the linear complementarity problem of the p type we also obtain upper and lower bounds for the quantity introduced

error bounds for monotone linear complementarity problems we give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered when the point satisfies the linear inequalities of the complementarity problem the residual consists of the complementarity condition plus its square root this latter term is essential and without it the error bound cannot hold we also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here

sensitivity theorems in integer linear programming we consider integer linear programming problems with a fixed coefficient matrix and varying objective function and right hand side vector among our results we show that for any optimal solution to a linear program max wx ax b the distance to the nearest optimal solution to the corresponding integer program is at most the dimension of the problem multiplied by the largest subdeterminant of the integral matrixa using this we strengthen several integer programming proximity results of blair and jeroslow graver and wolsey we also show that the chvatal rank of a polyhedron x ax b can be bounded above by a function of the matrixa independent of the vectorb a result which as blair observed is equivalent to blair and jeroslow s theorem that each integer programming value function is a gomory function

error bound and reduced gradient projection algorithms for convex minimization over a polyhedral set consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strongly convex differentiable function the polyhedral set is expressed as the intersection of an affine set with a simpler polyhedral set and a new local error bound for this problem based on projecting the reduced gradient associated with the affine set onto the simpler polyhedral set is studied a class of reduced gradient projection algorithms for solving the case where the simpler polyhedral set is a box is proposed and this bound is used to show that algorithms in this class attain a linear rate of convergence included in this class are the gradient projection algorithm of goldstein and levitin and poljak and an algorithm of bertsekas a new algorithm in this class reminiscent of active set algorithms is also proposed some of the results presented here extend to problems where the objective function is extended real valued and to variational inequality problems

existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation abstract an evolution inclusion is considered in a separable hilbert space the right hand side of the inclusion contains the subdifferential of a time dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed not necessarily bounded values along with this inclusion we consider the inclusion with the perturbation term being convexified we prove the existence of solutions and a density of the solution set of the original inclusion in a closure of the solution set of the inclusion with the convexified perturbation in contrast to the known results of this kind we do not suppose that the convex function has the compactness property and that the values of the perturbation are bounded sets an example of a perturbation with closed unbounded values satisfying the conditions of the main theorems is given

partially solvable quantum many body problems in d dimensional space d 1 2 3 a simple technique employed almost three decades ago to manufacture partially solvable quantum many body problems is revisited a quantum problem is partially solvable if only some of its eigenvalues and eigenfunctions can be exhibited the models thereby generated are characterized by hamiltonians of normal form i e standard kinetic plus momentum independent potential energy in most cases the latter features three body in addition to two body and one body interactions the setting refers to d dimensional space the examples focus on d 1 d 2 and d 2 and include generalizations of and additional results on cases recently discussed in the literature as well as new models

a korovkin s type approximation theorem for periodic functions via the statistical summability of the generalized de la vallee poussin mean the main object of this paper is to prove a korovkin type theorem for the test functions 1 cosx sinx in the space c 2 p r of all continuous email protected functions on the real line r our analysis is based upon the statistical summability involving the idea of the generalized de la vallee poussin mean we also investigate the rate of the de la vallee poussin statistical summability of positive linear operators in the space c 2 p r finally we provide an interesting illustrative example in support of our result

error estimates and lipschitz constants for best approximation in continuous function spaces abstract we use a structural characterization of the metric projection pg f from the continuous function space to its one dimensional subspace g to derive a lower bound of the hausdorff strong unicity constant or weak sharp minimum constant for pg and then show this lower bound can be attained then the exact value of lipschitz constant for pg is computed the process is a quantitative analysis based on the g teaux derivative of pg a representation of local lipschitz constants the equivalence of local and global lipschitz constants for lower semicontinuous mappings and construction of functions

minimum principle sufficiency we characterize the property of obtaining a solution to a convex program by minimizing over the feasible region a linearization of the objective function at any of its solution points minimum principle sufficiency for the case of a monotone linear complementarity problem this mps property is completely equivalent to the existence of a nondegenerate solution to the problem for the case of a convex quadratic program the mps property is equivalent to the span of the hessian of the objective function being contained in the normal cone to the feasible region at any solution point plus the cone generated by the gradient of the objective function at any solution point this in turn is equivalent to the quadratic program having a weak sharp minimum an important application of the mps property is that minimizing on the feasible region a linearization of the objective function at a point in a neighborhood of a solution point gives an exact solution of the convex program this leads to finite termination of convergent algorithms that periodically minimize such a linearization

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

unitary transformation approach to the exact solution for the singular oscillator by performing unitary transformations the exact solution of the time dependent singular oscillator is obtained the invariant operator and the auxiliary equation are rigorously established the non adiabatic berry s phase is calculated

tauberian conditions under which statistical convergence follows from statistical summability c 1 abstract j a fridly and m k khan have recently extended hardy s and landau s tauberian theorems to the case of statistical convergence which was introduced by h fast in 1951 let x k k 0 1 2 be a sequence of real or complex numbers and set n n 1 1 nk 0xk for n 0 1 2 we present necessary and sufficient conditions under which st limxk l follows from st lim n l where l is a finite number if xk is a sequence of real numbers then these are one sided tauberian conditions if xk is a sequence of complex numbers then these are two sided tauberian conditions in particular our conditions are satisfied if xk is statistically slowly decreasing or increasing in the case of real sequences or if xk is statistically slowly oscillating in the case of complex sequences even these special sufficient conditions imply those given by fridy and khan

coupled channel scattering and separation of coupled differential equations by generalized darboux transformations general methods for separation of coupled differential equations are applied in coupled channel scattering theory with appropriate asymptotic conditions it is shown that the problems with thresholds cannot be separated finally it is pointed out that separation of the coupled channel s wave problem without thresholds can be realized by the use of generalized darboux transformations

statistical approximation for new positive linear operators of lagrange type abstract in this paper we prove some approximation results in statistical sense and establish some direct theorems for the positive linear operators constructed by the means of lagrange type polynomials we compute error estimation by using modulus of continuity with the help of matlab and give its algorithm furthermore we show graphically the convergence of our operators to various functions

weak sharp minima in mathematical programming the notion of a sharp or strongly unique minimum is extended to include the possibility of a nonunique solution set these minima will be called weak sharp minima conditions necessary for the solution set of a minimization problem to be a set of weak sharp minima are developed in both the unconstrained and constrained cases these conditions are also shown to be sufficient under the appropriate convexity hypotheses the existence of weak sharp minima is characterized in the cases of linear and quadratic convex programming and for the linear complementarity problem in particular a result of mangasarian and meyer is reproduced that shows that the solution set of a linear program is always a set of weak sharp minima whenever it is nonempty consequences for the convergence theory of algorithms are also examined especially conditions yielding finite termination

sharp lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs the main purpose of this paper to give lipschitz constants for basic optimal solutions or vertices of solution sets and basic feasible solutions or vertices of feasible sets of linear programs with respect to right hand side perturbations the lipschitz constants are given in terms of norms of pseudoinverses of submatrices of the matrices involved and are sharp under very general assumptions there are two mathematical principles involved in deriving the lipschitz constants 1 the local upper lipschitz constant of a hausdorff lower semicontinuous mapping is equal to the lipschitz constant of the mapping and 2 the lipschitz constant of a finite set valued mapping can be inherited by its continuous submappings moreover it is proved that any lipschitz constant for basic feasible solutions can be used as an lipschitz constant for basic optimal solutions feasible solutions and optimal solutions

a condition number for differentiable convex inequalities for a system of differentiable convex inequalities a new bound is given for the absolute error in an infeasible point in terms of the absolute residual by using this bound a condition number is defined for the system of inequalities which gives a bound for the relative error in an infeasible point in terms of the relative residual

on the sensitivity analysis of hoffman constants for systems of linear inequalities relying on a general variational method developed by the authors and lucchetti nonlinear anal to appear the origin of which goes back to ioffe trans amer math soc 251 1979 pp 61 69 we give a formula for the best hoffman constant sigma inf x notin p a b frac ax b infty d x p a b where p a b x ax le b is a nonempty polyhedron in mathbb r n we also sharpen some results of luo and tseng siam j matrix anal appl 15 1994 pp 636 659 by characterizing the continuity set of some hoffman constants and by pointing out their locally lipschitzian character we apply these results to the study of the behavior of the solution set of a linear program inf ax le b u scriptscriptstyle t x with respect to a b u

global error bounds for monotone affine variational inequality problems abstract a global gradient projection type error bound is given for monotone affine variational inequality problems this new bound is based on a previously obtained global error bound for monotone linear complementarity problems and requires no assumptions on the monotone affine variational inequality other than solvability the error bound can be used in conjunction with recent results of tseng and luo to establish global linear convergence of any regular splitting method for solvable symmetric monotone affine variational inequality problems

sharp estimates for hoffman s constant for systems of linear inequalities and equalities we extend the formulae given by aze and corvellec belousov and andronov and ng and zheng for the hoffman constant associated to the system of linear inequalities ax leq b and relate them to that established by li

extension of hoffman s error bound to polynomial systems given any linear system defined by linear inequalities equalities in re n hoffman s error bound says that the distance from any point x in re n to the solution set of the linear system is bounded by a constant independent of x times a certain residual function evaluated at x this paper considers arbitrary possibly nonconvex polynomial systems defined by polynomial inequalities equalities it is shown that the solution sets of such systems are in general holder continuous as the right hand side changes in contrast to the lipschitzian continuity for the linear case also for any convex quadratic inequality system possessing an interior solution the authors show that its solution set behaves in a lipschitzian manner as the right hand side varies this result sharpens the earlier error bound results by mangasarian and robinson for general convex differentiable inequality systems which in addition to the interiority assumption require either an asymptotic constraint qualification con

tauberian theorems for double sequences that are statistically summable c 1 1 abstract let x jk j k 0 1 2 be a double sequence of real or complex numbers and set mn m 1 1 n 1 1 j 0 m k 0 n x jk for m n 0 1 2 we give necessary and sufficient conditions under which st lim mn implies st lim x jk where is a finite number these tauberian conditions are one sided when the xjk are real numbers and they are two sided when the xjk are complex numbers in particular these tauberian conditions are clearly satisfied if xjk is statistically slowly decreasing in the case of real sequences or if xjk is statistically slowly oscillating in the case of complex sequences

exactly solvable three body systems with internal degrees of freedom the approach of multi dimensional susy quantum mechanics is used in an explicit construction of exactly solvable three body and quasi exactly solvable n body matrix problems on a line from intertwining relations with time dependent operators we build exactly solvable non stationary scalar and 2 2 matrix three body models which are time dependent extensions of the calogero model finally we investigate the invariant operators associated with these systems

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

on various new concepts of statistical convergence for sequences of random variables via deferred cesaro mean abstract the paper aims to introduce and investigate various aspects of statistical convergence for sequences of random variables as well as for sequences of real numbers via deferred cesaro summability mean we first established several new theorems presenting the connection among them by using the fundamental limit concepts of sequences of random variables as application of the findings we have proposed different new korovkin type approximation results with algebraic test functions for a sequence of random variables on a banach space and also demonstrated the effectiveness of the findings further several examples have been incorporated to illustrate the findings

new two dimensional quantum models partially solvable by the supersymmetrical approach new solutions for second order intertwining relations in two dimensional susy qm are found via repeated use of the first order supersymmetrical transformations with intermediate constant unitary rotation potentials obtained by this method two dimensional generalized poschl teller potentials appear to be shape invariant the recently proposed method of susy separation of variables is implemented to obtain a part of their spectra including the ground state explicit expressions for energy eigenvalues and corresponding normalizable eigenfunctions are given in an analytic form intertwining relations of higher orders are discussed

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

intertwining relations of non stationary schrodinger operators general first and higher order intertwining relations between non stationary one dimensional schrodinger operators are introduced for the first order case it is shown that the intertwining relations imply some hidden symmetry which in turn results in an r separation of variables the fokker planck and diffusion equations are briefly considered second order intertwining operators are also discussed within a general approach however due to its complicated structure only particular solutions are given in some detail

some tauberian theorems for statistical c 1 summability abstract schoenberg 1959 23 proved that if a sequence x k is bounded and statistically convergent to l then it is statistically summable c 1 to l however the converse is not true in this paper we prove necessary and sufficient tauberian conditions based on generator sequences under which statistical convergence follows from statistical summability c 1

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als

finite termination of the proximal point algorithm this paper concerns the notion of a sharp minimum on a set and its relationship to the proximal point algorithm we give several equivalent definitions of the property and use the notion to prove finite termination of the proximal point algorithm