supersymmetry and algebraic darboux transformations we describe a class of algebraically solvable susy models by considering the deformation of invariant polynomial flags by means of the darboux transformation the algebraic deformations corresponding to the addition of a bound state to a shape invariant potential are particularly interesting the polynomial flags in question are indexed by a deformation parameter m 1 2 and lead to new algebraically solvable models we illustrate these ideas by considering deformations of the hyperbolic p schl teller potential

infinitely many shape invariant potentials and cubic identities of the laguerre and jacobi polynomials we provide analytic proofs for the shape invariance of the recently discovered odake and sasaki phys lett b 679 414 2009 two families of infinitely many exactly solvable one dimensional quantum mechanical potentials these potentials are obtained by deforming the well known radial oscillator potential or the darboux poschl teller potential by a degree l l 1 2 eigenpolynomial the shape invariance conditions are attributed to new polynomial identities of degree 3l involving cubic products of the laguerre or jacobi polynomials these identities are proved elementarily by combining simple identities

decomposition of matrices into 2 involutions abstract we consider the group slnf of all n by n matrices with determinant 1 over a field let res a denote rank a i for any a in slnf a matrix a is called a k involution if a2 i and res a k we prove that every matrix a in slnf is the product of at most res a 1 2 2 2 involutions for n 3 if char f 2 and for n 4 if char f 2

a new first order interior feasible direction method for structural optimization a new interior feasible direction method for optimization problems with linear objective functions but non linear constraints is proposed the method is particularly suitable for weight minimization problems in structural design where the design variables are the cross sectional areas of members and arbitrary stress and displacement constraints are applied four truss structures are analysed and in each case the minimum weight consistent with or lower than that currently available in the literature is achieved although still expensive in its current version the method is comparable in cost and robustness with other methods such as the trust region multiplier method apart from one tolerance value a reconnaissance parameter needs to be specified the fact that the same value of the parameter was used for all the examples and that the method is relatively insensitive to the choice thereof confirms the robustness of the method

general nth order integrals of motion in the euclidean plane the general form of an integral of motion that is a polynomial of order n in the momenta is presented for a hamiltonian system in two dimensional euclidean space the classical and the quantum cases are treated separately emphasizing both the similarities and the differences between the two the main application will be to study nth order superintegrable systems that allow separation of variables in the hamilton jacobi and schrodinger equations respectively

on the robustness and efficiency of the sam algorithm for structural optimization a parameter study involving a feasibility factor adaptive tolerance refinement and active set and move limits is conducted on the interior feasible direction method with successive approximations sam the study involves the sizing design of five structural problems of which one is a large power transmission tower to be designed in adherence with a standard design code it is concluded that a move limit is dispensable for this application while adaptive tolerance refinement makes the algorithm more efficient optimal ranges for the feasibility factor and active set limits are suggested this study reaffirms and quantifies the exceptional stability and robustness of the sam algorithm

extending romanovski polynomials in quantum mechanics some extensions of the third class romanovski polynomials also called romanovski pseudo jacobi polynomials which appear in bound state wavefunctions of rationally extended scarf ii and rosen morse i potentials are considered for the former potentials the generalized polynomials satisfy a finite orthogonality relation while for the latter an infinite set of relations among polynomials with degree dependent parameters is obtained both types of relations are counterparts of those known for conventional polynomials in the absence of any direct information on the zeros of the romanovski polynomials present in denominators the regularity of the constructed potentials is checked by taking advantage of the disconjugacy properties of second order differential equations of schrodinger type it is also shown that on going from scarf i to scarf ii or from rosen morse ii to rosen morse i potentials the variety of rational extensions is narrowed down from types i ii and iii to type iii only

commutators and companion matrices over rings of stable rank 1 abstract we consider the group gl n a of all invertible n by n matrices over a ring a satisfying the first bass stable range condition we prove that every matrix is similar to the product of a lower and upper triangular matrix and that it is also the product of two matrices each similar to a companion matrix we use this to show that when n 3 and a is commutative every matrix in sl n a is the product of two commutators

explicit constraint approximation forms in structural optimization part 1 analyses and projections abstract an analytical study of several alternative forms of constraint approximation relevant to problems in structural design is presented limitations of the existing forms of the constraint approximations are stated and methods are outlined to improve them a new generalized form of constraint approximation is proposed for optimal design problems with and without design variable linking a power form of constraint approximation is derived for bending types of finite elements having nonlinear size stiffness relationships a detailed analysis to determine the relative effectiveness of the various forms is included a rating guideline is proposed to quantify the characteristics of chosen approximations or family of approximations on grounds of accuracy and of conservatism of remaining feasible

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

a quadratic approximation for structural topology optimization in topology optimization it is customary to use reciprocal like approximations which result in monotonically decreasing approximate objective functions in this paper we demonstrate that efficient quadratic approximations for topology optimization can also be derived if the approximate hessian terms are chosen with care to demonstrate this we construct a dual sao algorithm for topology optimization based on a strictly convex diagonal quadratic approximation to the objective function although the approximation is purely quadratic it does contain essential elements of reciprocal like approximations for self adjoint problems our approximation is identical to the quadratic or second order taylor series approximation to the exponential approximation we present both a single point and a two point variant of the new quadratic approximation

products of commutators of dilatations abstract we consider the group sl n f of all n n matrices with determinant 1 over a field let res a denote rank a i for any a in gl n f a matrix a gl n f is called a dilatation if a is similar to a matrix i n 1 a a 1 we prove that if f 4 then every matrix a in sl n f is the product of at most res a 2 1 commutators of dilatations for n 2

a class of exactly solvable potentials related to the jacobi polynomials the author investigates a family of solvable potentials related to the jacobi polynomials this one dimensional potential family depends on three parameters and is restricted to the domain xo so it can be interpreted as the radial part of a central potential in three dimensions with l 0 closed expressions are obtained for the bound state energy spectrum and the wavefunctions the supersymmetric partner of this potential is also determined and it is found not to belong to the same potential family it is shown that this potential family is a special subclass of the general six parameter natanzon potential class and similarities with another subclass the ginocchio potentials are pointed out some aspects of supersymmetric quantum mechanics and shape invariance are also discussed in connection with the potential family under study

exceptional laguerre and jacobi polynomials and the corresponding potentials through darboux crum transformations a simple derivation is presented of the four families of infinitely many shapeinvariant hamiltonians corresponding to the exceptional laguerre and jacobi polynomials the darboux crum transformations are applied to connect the well known shape invariant hamiltonians of the radial oscillator and the darboux p oschl teller potential to the shape invariant potentials of odake sasaki dutta and roy derived the two lowest members of the exceptional laguerre polynomials by this method the method is expanded to its full generality and many other ramifications including the aspects of the generalized bochner problem and the bispectral property of the exceptional orthogonal polynomials are discussed

recurrence relations of the multi indexed orthogonal polynomials in paper ii we presented conjectures of the recurrence relations with constant coefficients for the multi indexed orthogonal polynomials of laguerre jacobi wilson and askey wilson types in this paper we present a proof for the laguerre and jacobi cases their bispectral properties are also discussed which gives a method to obtain the coefficients of the recurrence relations explicitly this paper extends to the laguerre and jacobi cases the bispectral techniques recently introduced by gomez ullate et al j approx theory 204 1 2016 e print arxiv 1506 03651 math ca to derive explicit expressions for the coefficients of the recurrence relations satisfied by exceptional polynomials of hermite type

trust region methods for structural optimization using exact second order sensitivity the performance of multiplier algorithms for structural optimization has been significantly improved by using trust regions the trust regions are constructed using analytical second order sensitivity and within this region the augmented lagrangian is minimized subject to bounds evaluation of first and second derivatives of by the adjoint method does not require derivations of individual implicit constraint functions which makes the method economical eight test problems are considered and a vast improvement over previously used multiplier algorithms has been noted also the algorithm is robust with respect to scaling input parameters and starting designs

structural optimization of guyed trusses here is a strong trend in the design of skeletal structures to wherever practical make increasing use of tensioned cables one of the best examples of the convergence of structural efficiency to a best design can probably be found when studying the evolution of the configurations of power transmission towers from the early days of cantilever truss structures these have now evolved to the much lighter cross rope type in which two separate vertical cable stayed lattice struts support the power conductors through suspension from a cross rope apart from the savings in material significant savings in transportation cost can also be obtained especially since erection often has to occur in remote areas although the cable stayed structures are easy to analyze the determination of a best or least weight design can be complex due to the variety of design variables available these are typically configuration geometry member sizing and prestraining the optimization of a configuration is complex and is best accomplished manually perhaps with the aid of layout optimization methods on the contrary geometric and prestrain optimization and sizing can be treated by direct sensitivity analysis of a chosen topology this paper describes the sizing and prestrain optimization of guyed trusses for the nonlinear analysis the total lagrangian method is used in combination with the pure newton method and adaptive arc length procedures because of the lack of stiffness of a cable structure in its undeformed state soft spring elements are used to stabilize the structure temporarily during the load stepping phase the choice of prestrain instead of prestress variables is justified by both the simplicity of inclusion of prestrain in the sensitivity analysis and by the independence of the prestrain quantity with respect to the sequence of pretensioning of the structure the prestraining of a member can also simply be seen as the lack of fit of a member in a fixed geometry and can in practice be accomplished by specifying member lengths to create a predetermined lack of fit or by specifying turns of a turnbuckle included in the member for the optimization process two optimization methods are used namely the sequential quadratic programming sqp method idesign plba authored by arora1 and the successive approximation method sam algorithm of snyman and stander2 and stander et al 3 using two methods gives optima in which greater confidence can be placed additionally some idea of the degree of difficulty of the newly formulated optimization problems is gained five examples are given to demonstrate the analysis and optimization methodology the optimization of example structures subjected to second order effects show that unexpected results can be obtained a number of papers 4 8 related to the general topic of the sensitivity analysis and optimization of structures with nonlinear geometric

explicit constraint approximation forms in structural optimization part 2 numerical experiences abstract a numerical study of explicit constraint approximation forms as applied to structural design is presented simple forms sla sia and sha and generalized power approximation forms gpa and gha are tested on a set of seven structural problems a total of 13 test cases is considered to study the effects of different problem parameters initial designs linking relationships etc on the design procedure the effect of tuning parameters on the accuracy and conservativeness of constraints for various approximation forms gpa and gha is also investigated results on numerical efficiency are presented as a function of the number of iterations noi required to obtain optimum designs

time dependent potentials associated with exceptional orthogonal polynomials we introduce a method for generating rational extensions of time dependent potentials such that the associated schrodinger equation admits solutions in terms of exceptional orthogonal polynomials our method is applicable to position dependent schrodinger equations as well as to their conventional counterparts for constant mass

performance of some sqp algorithms on structural design problems this paper describes the basic ideas of sequential quadratic programming sqp algorithms for design optimization there are two fundamental differences between the various algorithms i the difinition of the qp subproblem solved at each iteration and ii the descent function used during step size determination the performances of the algorithms can change dramatically depending on how the two steps are executed numerical implementation details of various computational steps are discussed three programs based on sqp algorithms are used to solve 17 structural design problems having 7 to 96 design variables and 10 to 1051 performance constraints besides design variable bounds based on the performance of these programs efficient procedures to execute various steps of the sqp methods are determined it is concluded that the potential constraint strategy where only a subset of the constraints is used to define the qp subproblem is essential for large scale engineering design applications with this strategy the sqp methods are quite robust and have great potential for routine application in engineering design

a class of exactly solvable rationally extended non central potentials in two and three dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions we start from a seven parameter six continuous and one discrete family of non central exactly solvable potentials in three dimensions and construct a wide class of ten parameters six continuous and four discrete family of rationally extended exactly solvable non central real as well as pt symmetric complex potentials the energy eigenvalues and the eigenfunctions of these extended non central potentials are obtained explicitly and it is shown that the wave eigenfunctions of these potentials are either associated with the exceptional orthogonal polynomials or some type of new polynomials which can be further re expressed in terms of the corresponding classical orthogonal polynomials similarly we also construct a wide class of rationally extended exactly solvable non central real as well as complex pt invariant potentials in two dimensions

a unified treatment of exactly solvable and quasi exactly solvable quantum potentials by exploiting the hidden algebraic structure of the schrodinger hamiltonian namely the sl 2 we propose a unified approach of generating both exactly solvable and quasi exactly solvable potentials we obtain in this way two new classes of quasi exactly solvable systems one of which is of periodic type while the other hyperbolic

new successive approximation method for optimum structural design a new procedure using only first order explicit information and consisting of the successive solution of approximate subproblems is proposed the method differs from established sequential quadratic programming methods by not utilizing the lagrangian form and not attempting to store full and explicit hessian information instead each constraint is individually approximated by a quadratic function involving only one coefficient in the quadratic term this coefficient is determined from a two point collocation a novel feature of the method is that each quadratic subproblem is solved by a recently proposed interior feasible direction method an auxiliary problem is formulated to generate a feasible starting point for each subproblem

products of commutators of transvections over local rings abstract let r be a commutative local ring and m the maximal ideal of r we prove that if r m 3 then every matrix in sl n r n 2 is a product of at most n 2 2 commutators of transvections and the length res a 2 2 cannot be reduced for n 2

effective two point function approximation for design optimization anew two pointapproximationofafunctionthatisverystraightforwardtobuildandthatprovidesaccurateand stable approximation is presented earlier developments of the two point approximations had either incomplete matches at two data points or needed the solution of additional equations to get all of the parameters the present two point approximation is an incomplete second order taylor series expansion in terms of intervening variables thehessianmatrixhasonlydiagonalelements anditdependsondesignvariables theexponentofeachintervening design variable and the unknown constant of the second order terms are evaluated by matching the derivatives and the value of the approximation with the previous data point gradients and the value of the original function respectively all of theunknownsareidentie ed in aclosedform both thefunctionandthegradientof thetwo point approximation are equal to those of the original function at two data points several examples are given to show the accuracy and efe ciency of this method

comprehensive analysis of conditionally exactly solvable models we study a quantum mechanical potential introduced previously as a conditionally exactly solvable ces model besides an analysis following its original introduction in terms of the point canonical transformation we also present an alternative supersymmetric construction of it we demonstrate that from the three roots of the implicit cubic equation defining the bound state energy eigenvalues there is always only one that leads to a meaningful physical state finally we demonstrate that the present ces interaction is in fact an exactly solvable natanzon class potential

invariance properties of wronskian type determinants of classical and classical discrete orthogonal polynomials abstract given a finite set f f 1 f k of nonnegative integers written in increasing order of magnitude and a classical discrete family p n n of orthogonal polynomials charlier meixner krawtchouk or hahn we consider the casorati determinant det p f i x j 1 i j 1 k in this paper we prove an invariance property of this kind of casorati determinants when the set f is substituted by the set i f 0 1 2 max f max f f f f our approach uses orthogonal polynomials that are eigenfunctions of higher order difference operators krall discrete polynomials these polynomials are orthogonal with respect to certain christoffel transforms of the classical discrete measures by passing to the limit this invariance property is extended to wronskian type determinants whose entries are hermite laguerre and jacobi polynomials

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

an extended class of orthogonal polynomials defined by a sturm liouville problem abstract we present two infinite sequences of polynomial eigenfunctions of a sturm liouville problem as opposed to the classical orthogonal polynomial systems these sequences start with a polynomial of degree one we denote these polynomials as x 1 jacobi and x 1 laguerre and we prove that they are orthogonal with respect to a positive definite inner product defined over the compact interval 1 1 or the half line 0 respectively and they are a basis of the corresponding l 2 hilbert spaces moreover we prove a converse statement similar to bochner s theorem for the classical orthogonal polynomial systems if a self adjoint second order operator has a complete set of polynomial eigenfunctions p i i 1 then it must be either the x 1 jacobi or the x 1 laguerre sturm liouville problem a rodrigues type formula can be derived for both of the x 1 polynomial sequences

hybrid optimization strategy beyond local optima in aerospace panel designs convergence to local optima rather than the global optimum is a common shortcoming of directed seareh gradient based optimization techniques successive convergence on better local optima is an efficient way of finding the global optimum and is achieved by the proposed hybrid global optimization strategy which combines ideas from the improving hit and run and the tunneling global optimization methods the searching technique of improving hit and run is adapted to find suitable starting points to converge on other better local optima the suitability of candidate starting points is confirmed by the concept of the tunneling method the strategy presented has unique features due to the underlying structural theory being exact in the sense of differential equations being solved for the member or element stiffnesses so that they are transcendental functions of the load factor hence the buckling eigenproblem for the entire structure is a transcendental one instead of the linear one associated with the more usual finite element method formulations constrained minimum mass aerospace panel stability design problems are used to demonstrate the efficiency and robustness of the hybrid global optimization strategy

exceptional orthogonal polynomials exactly solvable potentials and supersymmetry we construct two new exactly solvable potentials giving rise to bound state solutions to the schrodinger equation which can be written in terms of the recently introduced laguerre or jacobi type x1 exceptional orthogonal polynomials these potentials extending either the radial oscillator or the scarf i potential by the addition of some rational terms turn out to be translationally shape invariant as their standard counterparts and isospectral to them