linear dependence of a function set of m variables with vanishing generalized wronskians abstract necessary and sufficient conditions are established for a set of n functions i em e1 which together with their partial derivatives of order at least n 1 are continuous to be linearly dependent one of these conditions is that all generalized wronskians vanish identically

oscillator representations of the 2d conformal algebra and superalgebra for the virasoro algebra and for the neveu schwarz ramond nsr superalgebra oscillator representations are found by localising the miura and the super miura maps from the theory of the korteweg de vries kdv and the super kdv equations respectively

transmutations for darboux transformed operators with applications we solve the following problem let q1 be a continuous complex valued potential of a stationary schr dinger operator defined on a segment a a and q2 be the potential of a darboux transformed schr dinger operator that is where f is a nonvanishing solution of the equation suppose a transmutation operator t1 is known such that for any u c2 a a find an analogous transmutation operator for it is well known that the transmutation operators can be realized in the form of volterra integral operators with continuously differentiable kernels given a kernel k1 of the transmutation operator t1 we find the kernel k2 of t2 in a closed form in terms of k1 as a corollary interesting commutation relations between t1 and t2 are obtained which then are used in order to construct the transmutation operator for the one dimensional dirac system with a scalar potential

lower semicontinuity of multiple integrals and the biting lemma weak lower semicontinuity theorems in the sense of chacon s biting lemma are proved for multiple integrals of the calculus of variations a general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions the biting weak limit of the corresponding integrands is characterised via the young measure and related to the weak limit in the sense of measures finally an example is given which shows that the young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case

the generalized confluent supersymmetry algorithm representations and integral formulas we extend the confluent version of the higher order supersymmetry susy formalism to general linear differential equations of second order closed form representations of transformation functions their wronskians and of the general solutions to susy transformed equations are derived we use these results to construct formulas for resolving multiple integrals of special functions in terms of wronskians

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

arbitrary order jordan chains associated with quantum mechanical hamiltonians representations and integral formulas we consider jordan chains of arbitrary order that are associated with quantum mechanical hamiltonians an explicit formula for the connection between integral and differential representations of the jordan chains is given as a byproduct we obtain a method that allows to resolve certain multiple integrals of special functions

combined state adding and state deleting approaches to type iii multi step rationally extended potentials applications to ladder operators and superintegrability type iii multi step rationally extended harmonic oscillator and radial harmonic oscillator potentials characterized by a set of k integers m1 m2 mk such that m1 m2 mk with mi even resp odd for i odd resp even are considered the state adding and state deleting approaches to these potentials in a supersymmetric quantum mechanical framework are combined to construct new ladder operators the eigenstates of the hamiltonians are shown to separate into mk 1 infinite dimensional unitary irreducible representations of the corresponding polynomial heisenberg algebras these ladder operators are then used to build a higher order integral of motion for seven new infinite families of superintegrable two dimensional systems separable in cartesian coordinates the finite dimensional unitary irreducible representations of the polynomial algebras of such systems are directly determined from the ladder operator action on the constituent one dimensional hamiltonian eigenstates and provide an algebr

korteweg de vries modified korteweg de vries systems and darboux transforms in 1 1 and 2 1 dimensions the jaulent miodek hierarchy m jaulent and i miodek lett math phys 1 243 1976 and its two dimensional analog are considered from the point of view of darboux transforms structures a new form of those transforms as well as old fashioned forms are presented the determinant form of solitons is obtained the combination of different transforms in the 2 1 case allows the buildup of localized solutions in both spatial dimensions including the two dimensional korteweg de vries kdv equation

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

factorization of the classical boussinesq system the scalar equation equivalent to the classical boussinesq or broer kaup system is shown to factorize into a differential operator acting on burgers equation the linearizability of this latter equation provides a very simple explanation for the recently found property of fusion and fission of a class of solitons for this system

on integral and differential representations of jordan chains and the confluent supersymmetry algorithm we construct a relationship between integral and differential representation of second order jordan chains conditions to obtain regular potentials through the confluent supersymmetry algorithm when working with the differential representation are obtained using this relationship furthermore it is used to find normalization constants of wave functions of quantum systems that feature energy dependent potentials additionally this relationship is used to express certain integrals involving functions that are solution of schrodinger equations through derivatives

generalized schrodinger equations with quadratical energy dependence in the potential darboux transformations and application to the heun class we construct darboux transformations of arbitrary order for generalized schrodinger type equations the potentials of which are second degree polynomials in the energy our equations are allowed to contain first derivative terms with arbitrary coefficients such as they occur for example in position dependent mass scenarios our darboux transformations are shown to be applicable to equations from the heun class

mathematics of dispersive water waves a commuting hierarchy of dispersive water wave equations makes a three hamiltonian system which belongs to a general class of nonstandard integrable systems whose theory is developed the modified water wave hierarchy is a bi hamiltonian system its modification bifurcates the water wave hierarchy and the hierarchies of the korteweg de vries and the modified korteweg de vries equations as well as the classical miura map are given new representations through various specializations of nonstandard systems

a spectral study of the second order exceptional x1 jacobi differential expression and a related non classical jacobi differential expression abstract the exceptional x 1 jacobi differential expression is a second order ordinary differential expression with rational coefficients it was discovered by gomez ullate kamran and milson in 2009 in their work they showed that there is a sequence of polynomial eigenfunctions p n n 1 called the exceptional x 1 jacobi polynomials there is no exceptional x 1 jacobi polynomial of degree zero these polynomials form a complete orthogonal set in the weighted hilbert space l 2 1 1 w where w is a positive rational weight function related to the classical jacobi weight among other conditions placed on the parameters and it is required that 0 in this paper we develop the spectral theory of this expression in l 2 1 1 w we also consider the spectral analysis of the extreme non exceptional case namely when 0 in this case the polynomial solutions are the non classical jacobi polynomials p n 2 n 2 we study the corresponding jacobi differential expression in several hilbert spaces including their natural l 2 setting and a certain sobolev space s where the full sequence p n 2 n 0 is studied and a careful spectral analysis of the jacobi expression is carried out

a bochner type characterization theorem for exceptional orthogonal polynomials abstract it was recently conjectured that every system of exceptional orthogonal polynomials is related to a classical orthogonal polynomial system by a sequence of darboux transformations in this paper we prove this conjecture which paves the road to a complete classification of all exceptional orthogonal polynomials in some sense this paper can be regarded as the extension of bochner s result for classical orthogonal polynomials to the exceptional class as a supplementary result we derive a canonical form for exceptional operators based on a bilinear formalism and prove that every exceptional operator has trivial monodromy at all primary poles

derivatives of any order of the confluent hypergeometric function f11 a b z with respect to the parameter a or b the derivatives to any order of the confluent hypergeometric kummer function f f11 a b z with respect to the parameter a or b are investigated and expressed in terms of generalizations of multivariable kampe de feriet functions various properties reduction formulas recurrence relations particular cases and series and integral representations of the defined hypergeometric functions are given finally an application to the two body coulomb problem is presented the derivatives of f with respect to a are used to write the scattering wave function as a power series of the sommerfeld parameter

higher order darboux transformations and wronskian representations for schrodinger equations with quadratically energy dependent potentials we construct higher order darboux transformations for schrodinger equations with quadratically energy dependent potentials similar to the case of conventional darboux also known as supersymmetric transformations both the solutions and the potentials of transformed equations are expressed through wronskians we discuss properties of our darboux transformations and provide an application

conditionally exactly solvable potentials and exceptional orthogonal polynomials it is shown that polynomials associated with solutions of certain conditionally exactly solvable potentials obtained via unbroken as well as broken supersymmetry belong to the category of exceptional orthogonal polynomials some properties of such polynomials e g recurrence relation ladder operators differential equations etc have been obtained

a condition equivalent to linear dependence for functions with vanishing wronskian abstract it is well known that though the vanishing of the wronskian w of a set of functions on an interval i is a necessary condition for it to be linearly dependent it is not a sufficient one since peano in 1989 expressed an interest in finding classes of functions for which w 0 is sufficient for dependence and offered one such example himself others m b cher d r curtiss w hurewicz and g h meisters have provided related results here the author gives a final answer to the question by first generalizing peano s result using the order of a critical point thereby obtaining a dense set of intervals of dependence he then shows that w 0 together with a condition that the intersection of certain subspaces of en is nontrivial is equivalent to the linear dependence of on i the above results are used to establish the dynamical theorem that motion of a particle under the action of a central force field is planar so long as the particle is restricted from the origin the author provides a counterexample for the case in which the particle passes through the origin

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

a hierarchy of coupled korteweg de vries equations and the normalisation conditions of the hilbert riemann problem the authors discuss the normalisation conditions of the hilbert riemann problem which is of importance for the construction of the ladder of soliton solutions for a given spectral problem they provide the prescription for constructing the appropriate normalisation condition all this is applied to a new hierarchy of coupled korteweg de vries equations to find the soliton solution which displays some peculiar phenomena

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

convergence for pseudo monotone semiflows on product ordered topological spaces in this paper we consider a class of pseudo monotone semiflows which only enjoy some weak monotonicity properties and are defined on product ordered topological spaces under certain conditions several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium which improve and extend some corresponding results already known some applications to delay differential equations are presented

linear dependence of quotients of analytic functions of several variables with the least subcollection of generalized wronskians abstract we study linear dependence in the case of quotients of analytic functions in several variables real or complex we identify the least subcollection of generalized wronskians whose identical vanishing is sufficient for linear dependence our proof admits a straight forward algebraic generalization and also constitutes an alternative proof of the previously known result that the identical vanishing of the whole collection of generalized wronskians implies linear dependence motivated by the structure of this proof we introduce a method for calculating the space of linear relations we conclude with some reflections about this method that may be promising from a computational point of view

on a new hierarchy of hamiltonian soliton equations a method is suggested for studying the hamiltonian structure of the nonlinear partial differential equations that can be solved by the use of the spectral transform soliton equations the method is applied to a new hierarchy of n 1 coupled partial differential equations related to a schrodinger like spectral problem it is shown that these soliton equations are integrable hamiltonian equations with commuting flows for n 1 and n 2 a miura like transformation is computed and the corresponding modified equations are explicitly given

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

elementary and binary darboux transformations at rings for a given idempotent p and some element from a differential associative ring we introduce gauge transformation p with the spectral parameter that leaves some linear operators form invariant the explicit form of the is derived for the generalized zakharov shabat problem the maps that factorize darboux transformations are referred to as elementary ones binary transformations that correspond to the iteration of elementary maps with the special choice of solutions of the direct and conjugate problems are introduced and widen the potential reductions set we show an infinitesimal version of iterated transforms the spectral operator and soliton theories applications are outlined the nonabelian n wave interaction equation modification with additional linear terms its zero curvature representation soliton solutions and their stability with respect to infinitesimal deformations are studied

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

the two singular manifold method ii classical boussinesq system for pt i see musette et al ibid vol 24 p 3895 1994 the two singular manifold method a generalization of the singular manifold method of weiss 1983 is applied to the classical boussinesq system also known as the broer kaup system from the point of view of its singularity analysis the important feature of this system is the existence of two principal families with opposite principal parts the usual singular manifold method takes into account only one of these families at a time our generalization takes into account both families and in this way we are able to derive the lax pair and darboux transformation and hence the auto backlund transformation for the classical boussinesq system from its painleve analysis

recursive representation of wronskians in confluent supersymmetric quantum mechanics a recursive form of arbitrary order wronskian associated with transformation functions in the confluent algorithm of supersymmetric quantum mechanics susy is constructed with this recursive form regularity conditions for the generated potentials can be analyzed moreover as byproducts we obtain new representations of solutions to schrodinger equations that underwent a confluent susy transformation

a general approach to lower semicontinuity and lower closure in optimal control theory a self contained approach to lower semicontinuity and lower closure evolves from an extension of relaxed control theory which is based on a central relative weak compactness criterion called tightness and relaxation in all but one variable two lower closure results for outer integral functionals with variable abstract time domain are developed the first of these has a convexity condition for the integrand and generalizes all similar results in the literature the second lower closure result is of a new kind among other things it implies a quite general version of fatou s lemma in several dimensions