parametric symmetries in exactly solvable real and pt symmetric complex potentials in this paper we discuss the parametric symmetries in different exactly solvable systems characterized by real or complex pt symmetric potentials we focus our attention on the conventional potentials such as the generalized poschl teller gpt scarf i and pt symmetric scarf ii which are invariant under certain parametric transformations the resulting set of potentials is shown to yield a completely different behavior of the bound state solutions further the supersymmetric partner potentials acquire different forms under such parametric transformations leading to new sets of exactly solvable real and pt symmetric complex potentials these potentials are also observed to be shape invariant si in nature we subsequently take up a study of the newly discovered rationally extended si potentials corresponding to the above mentioned conventional potentials whose bound state solutions are associated with the exceptional orthogonal polynomials eops we discuss the transformations of the corresponding

strict feedforward control systems linearisability and convergent normal forms this article discusses the feedback equivalence of multi inputs feedforward control systems via smooth resp analytic feedback transformations we first address the state resp feedback linearisation problem and provide easily computable algorithms that yield explicit state resp feedback linearising coordinates for systems in strict feedforward form the application of the algorithms does not require checking the commutativity resp involutivity of the distributions associated with the system and the algorithms fail after few steps if the system is not linearisable in the latter case the algorithms are extended to provide coordinate systems bringing the system into a normal form which is a smooth resp analytic counterpart of kang s formal normal form illustrative examples for both the linearisation and convergent normal form include the vertical take off and landing aircraft the multi vehicle wireless testbed among others

asymptotic tracking of a reference state for systems with a feedforward structure we consider the problem of asymptotic tracking for a system which can be written in a feedforward form the data of a bounded reference state trajectory is assumed our solution relies on a lyapunov construction the time varying state feedbacks obtained are bounded and ensure the global uniform asymptotic stability when the signals are periodic and the local exponential stability of the reference state trajectory of the closed loop system

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

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

regulation mechanisms in spatial stochastic development models the aim of this paper is to analyze different regulation mechanisms in spatial continuous stochastic development models we describe the density behavior for models with global mortality and local establishment rates we prove that the local self regulation via a competition mechanism density dependent mortality may suppress a unbounded growth of the averaged density if the competition kernel is superstable

approximate linearization by state feedback and coordinate change a nonlinear system can always be approximated to first order by linear systems it has been shown by jakubczyk and respondek 4 and hunt and su 3 that certain nonlinear systems are the exact transforms of linear systems under nonlinear state coordinate change and nonlinear state feedback in this paper we give necessary and sufficient conditions for a nonlinear system to be approximated to higher order by the transform of a linear system the use of this technique in the design of nonlinear compensators has been suggested recently by the author 6

feedback classification of multi input nonlinear control systems we study the feedback group action on multi input nonlinear control systems with uncontrollable mode we follow slightly an approach proposed in kang and krener w kang and a j krener siam j control optim 30 1992 pp 1319 1337 which consists of analyzing the system and the feedback group step by step we construct a normal form which generalizes on one hand the results obtained in the single input case and on the other hand those recently obtained by the same author in the controllable case we illustrate our results by studying the caltech multi vehicle wireless testbed mvwt and the prototype of planar vertical takeoff and landing aircraft pvtol we also study the notion of bifurcation of controllability for systems with one nonzero uncontrollable mode we first show that the equilibria for those systems is a p dimensional submanifold p equals number of inputs provided that one term in their normal form is nonzero we show that these systems are linearly controllable hence stabilizable at any nearby equilibrium point of the origin

exceptional orthogonal polynomials and the darboux transformation we adapt the notion of the darboux transformation to the context of polynomial sturm liouville problems as an application we characterize the recently described xm laguerre polynomials in terms of an isospectral darboux transformation we also show that the shape invariance of these new polynomial families is a direct consequence of the permutability property of the darboux crum transformation

exact solutions of the schrodinger equation with non central potential by the nikiforov uvarov method the general solutions of the schrodinger equation for a non central potential are obtained by using the nikiforov uvarov method the schrodinger equation with general non central potential is separated into radial and angular parts and energy eigenvalues and eigenfunctions are derived analytically by making special selections the non central potential is reduced to coulomb and hartmann ring shaped potentials and the obtained results are compared with the solutions of coulomb and hartmann potentials given in the literature

information entropy of conditionally exactly solvable potentials we evaluate shannon entropy for the position and momentum eigenstates of some conditionally exactly solvable potentials which are isospectral to harmonic oscillator and whose solutions are given in terms of exceptional orthogonal polynomials the bialynicki birula mycielski inequality has also been tested for a number of states

nonlinearizable single input control systems do not admit stationary symmetries the main result of the paper states that almost any analytic single input control system which is truly nonlinear that is not feedback linearizable with controllable linearization at an equilibrium point does not admit any symmetry preserving that point by almost any system we mean that we exclude a small class of odd systems that admit just one nontrivial symmetry conjugated to minus identity the obtained results are based on a recent classification of nonlinear single input systems under formal feedback we also describe symmetries of feedback linearizable systems

analytical solutions of a generalized non central potential in n dimensions we present that n dimensional non relativistic wave equation for the generalized non central potential with arbitrary angular momentum is analytically solvable in the hyperspherical coordinates asymptotic iteration method as a different approach is applied to obtain n dimensional energy eigenvalues and the corresponding eigenfunctions in hyperspherical coordinates the wave function solutions are obtained in terms of hypergeometric functions and jacobi polynomials the bound states of quantum systems under consideration for some special cases such as hartmann and makarov potentials have been discussed in n dimensions

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

quotient submanifolds for static feedback linearization a procedure for finding locally the linearizing output of a single input nonlinear affine system is proposed it relies on successive integrations of one dimensional distributions and projections along these submanifolds the algorithm proceeds recursively reducing the dimension one by one of both the number of coordinates and the number of vector fields until the solution is obtained a variant of the algorithm is also proposed which does not require the computation of the full initial distribution the proof of convergence of this second algorithm shows the importance of a new anti symmetrical product besides providing a new insight into the involutivity condition the algorithm can lead to a simple way of integrating the system of partial differential equations defining the linearizing output

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

extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems in this paper a set of extended quadratic controller normal forms of linearly controllable nonlinear systems is given which is the generalization of the brunovsky form of linear systems a set of invariants under the quadratic changes of coordinates and feedbacks is found it is then proved that any linearly controllable nonlinear system is linearizable to second degree by a dynamic state feedback

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

the superstability of pair potentials of positive type we prove that a pair potential which is continuous l1 and of positive type satisfies a condition of the superstability kind with best possible constants the applications to statistical thermodynamics are mentioned

feedback classification of nonlinear single input control systems with controllable linearization normal forms canonical forms and invariants we study the feedback group action on single input nonlinear control systems we follow an approach of kang and krener based on analyzing step by step the action of homogeneous transformations on the homogeneous part of the same degree of the system we construct a dual normal form and dual invariants with respect to those obtained by kang we also propose a canonical form and a dual canonical form and show that two systems are equivalent via a formal feedback if and only if their canonical forms resp their dual canonical forms coincide we give an explicit construction of transformations bringing the system to its normal dual normal canonical and dual canonical forms we illustrate our results by simple examples on mathbb r 3 and mathbb r 4

free energy density for continuous systems with and without superstability assumptions the thermodynamic limit of the free energy density for a large class of continuous charged systems with stable and sufficiently regular many body forces is studied the main effort is placed on demonstrating rigorously that thermodynamic functions do not depend on the typical boundary conditions among the results the most important and new ones are the following 1 for any superstable or superstable in an extended sense and sufficiently regular many body interactions the thermodynamic limit of the conditioned free energy density exists and is equal to that corresponding to the free boundary condition case 2 a new criterion for the nondependence of the limiting free energy density on the typical boundary condition is formulated and proved this criterion does not require any superstability type of assumptions among several applications of the results we list only two a to the case of two dimensional neutral yukawa gas in the monopole phase which is not even stable according to the standard no

convergence or generic divergence of the birkhoff normal form we prove that the birkhoff normal form of hamiltorlian flows at a nonresonant singular point with given quadratic part is always convergent or generically divergent the same result is proved for the normalization mapping and any formal first integral

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

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