energy levels of a two dimensional anharmonic oscillator characteristic function approach the eigenlevels of a two dimensional quartic anharmonic oscillator are identified as the zeros of a characteristic function derived from the resolvent numerical values are obtained by means of pade approximants

on the density of lyndon roots in factors this work takes another look at the number of runs that a string may contain and provides an alternative proof for the bound we also propose another stronger conjecture that states the following for a fixed order on the alphabet within every factor of a word there are at most as many occurrences of lyndon roots corresponding to runs in the word as the length of the factor only first occurrences of roots in each run are considered

a variational iterative technique applied to quantum mechanical problems a technique for solving nonlinear operator equations iteratively is applied to quantum mechanical calculations it is shown that alternative methods of applying the technique lead to another way of approaching variational calculations and generalisations of perturbation theory respectively

istmo an interval reference point based method for stochastic multiobjective programming problems in this paper we present an interactive algorithm istmo for stochastic multiobjective problems with continuous random variables this method combines the concept of probability efficiency for stochastic problems with the reference point philosophy for deterministic multiobjective problems the decision maker expresses her his references by dividing the variation range of each objective into intervals and by setting the desired probability for each objective to achieve values belonging to each interval these intervals may also be redefined during the process this interactive procedure helps the decision maker to understand the stochastic nature of the problem to discover the risk level s he is willing to assume for each objective and to learn about the trade offs among the objectives

energy levels of two dimensional anharmonic oscillators with sextic and octic perturbations two renormalized versions of inner product and hypervirial techniques are presented together with the hill determinant approach for performing accurate calculations of the energy levels of two dimensional oscillators characterized by the potentials vn 6 x y x2 y2 axxx6 ayyy6 3axyx4 y2 3ayxx2y4 and vn 8 x y x2 y2 axxx8 ayyy8 6b xyx4y4 4axyx6y2 4ayxx2y6 the methods have been used effectively to calculate the energy eigenvalues for many eigenstates over a wide range values of perturbation parameter and various values of coefficients aij the results produced by various techniques are compared with each other

the perturbed three dimensional oscillator a renormalized version of inner product and hypervirial theories is used to calculate the energy eigenvalues for four states of a perturbed three dimensional oscillator and to obtain the rayleigh schrodinger energy perturbation series

the number of runs in a string a run in a string is a nonextendable with the same minimal period periodic segment in a string the set of runs corresponds to the structure of internal periodicities in a string periodicities in strings were extensively studied and are important both in theory and practice combinatorics of words pattern matching computational biology let n be the maximal number of runs in a string of length n it has been shown that n o n the proof was very complicated and the constant coefficient in o n has not been given explicitly we demystify the proof of the linear upper bound for n and propose a new approach to the analysis of runs based on the properties of subperiods the periods of periodic parts of the runs we show that n n and there are at most o 67n runs with periods larger than 87 this supports the conjecture that the number of all runs is smaller than n we also give a completely new proof of the linear bound and discover several new interesting periodicity lemmas

solution approaches for the multiobjective stochastic programming abstract we survey in this paper various solution approaches for multiobjective stochastic problems where random variables can be in both objectives and constraints parameters once a problem requires a stochastic formulation a first step consists in transforming the problem into its deterministic formulation we propose to classify and evaluate such transformations with regards to the many proposed concepts of efficiency the paper addresses also some applications of the multiobjective stochastic programming models

the confined system approximation for solving non separable potentials in three dimensions the hilbert space to which the wavefunction of the three dimensional schrodinger equation belongs has been replaced by where is a bounded region the energy spectrum of the usual unbounded system is then determined by showing that the dirichlet and neumann problems in generate upper and lower bounds respectively to the eigenvalues required highly accurate numerical results for the quartic and sextic oscillators are presented for a wide range of the coupling constants

perturbative solution for the generalised anharmonic oscillators serious difficulties have been encountered in the past when attempting the straightforward application of the borel method to the summation of perturbation series in quantum mechanics and quantum field theory because of an analytic continuation of the borel transform series which is usually performed by means of conformal mapping techniques the author shows that this continuation may be accomplished by combining the borel method with the first confluent form of the epsilon algorithm of wynn 1960 1968 for the calculation of the ground state energy eigenvalue of the generalised anharmonic oscillators two different approaches are presented both of which lead to accurate results

studies on algebraic methods to solve linear eigenvalue problems generalised anharmonic oscillators a detailed presentation of the recently introduced integration free method with applications to determine the energy levels of the generalised quantum anharmonic oscillators are given numerical calculations are realised for the quartic and the sextic oscillators energy eigenvalues obtained for the ground state as well as for the first few excited states accurate to thirty digits are very impressive and demonstrate the efficiency of the method certain remarks about the selection of the basis functions and a convergence discussion on the presented simple approximation scheme are also included in this paper

satisfactory solution concepts and their relations for stochastic multiobjective programming problems abstract this work deals with the concept of satisfactory solution for stochastic multiobjective programming smp problems based on previous literature we will introduce different concepts of satisfactory solutions for smp problems define a new concept of solution where the decision maker dm sets his her preferences in terms of two aspiration levels for the stochastic objective and two probabilities to reach those levels and establish some relationship between these concepts the results will aim at featuring these concepts and determine the differences between them moreover the paper proposes a new step by step procedure to exchange information between the analyst and dm prior to solving the problem thus the dm will be able to choose the transformation criterion for each stochastic objective and the aspiration level

risk based factorial probabilistic inference for optimization of flood control systems with correlated uncertainties abstract in this paper a risk based factorial probabilistic inference method is proposed to address the stochastic objective function and constraints as well as their interactions in a systematic manner to tackle random uncertainties decision makers risk preferences are taken into account in the decision process statistical significance for each of the linear nonlinear and interaction effects of risk parameters is uncovered through conducting a multi factorial analysis the proposed methodology is applied to a case study of flood control to demonstrate its validity and applicability a number of decision alternatives are obtained under various combinations of risk levels associated with the objective function and chance constraints facilitating an in depth analysis of trade offs between economic outcomes and associated risks dynamic complexities are addressed through a two stage decision process as well as through capacity expansion planning for flood diversion within a multi region multi flood level and multi option context findings from the factorial experiment reveal the multi level interactions between risk parameters and quantify their contributions to the variability of the total system cost the proposed method is compared against the fractile criterion optimization model and the chance constrained programming technique respectively

the rayleigh ritz method still competitive abstract an extended rayleigh ritz method for computing two sided eigenvalue bounds of the one dimensional schrodinger equation is presented the method is based on the b spline approximation over the truncated domain for the low lying eigenvalues and problems with rapidly increasing spectrum the corresponding lower bounds are much more accurate than the known temple bounds numerical results for problems exhibiting different kinds of the eigenfunction behaviour are presented

energy levels of a three dimensional anharmonic oscillator with sextic perturbation renormalized series version of inner product and renormalized series techniques with hill determinant approach are used to calculate the energy eigenvalues for a three dimensional oscillator for several sets of parameters our techniques were modified to treat higher power of perturbation for two eigenstates e0 0 1 and e1 1 0 with odd parity

a characterization of the squares in a fibonacci string a finite fibonacci stringfn is defined as follows f0 b f1 a for every integer n 2 fn fn 1fn 2 for n 1 the length of fn is denoted by the infinite fibonacci stringf is the string which contains every fn n 1 as a prefix apart from their general theoretical importance fibonacci strings are often cited as worst case examples for algorithms which compute all the repetitions or all the abelian squares in a given string in this paper we provide a characterization of all the squares in f hence in every prefix fn this characterization naturally gives rise to a algorithm which specifies all the squares of fn in an appropriate encoding this encoding is made possible by the fact that the squares of fn occur consecutively in runs the number of which is by contrast the known general algorithms for the computation of the repetitions in an arbitrary string require time and produce outputs when applied to a fibonacci string fn

the eigenvalues of the schrodinger equation for spherically symmetric states for various types of potentials in two three and n dimensions by using perturbative and non perturbative methods the energy levels of the schrodinger equation involving various model potentials in two three and n dimensional space are calculated by using perturbative and non perturbative methods the methods have been used effectively to calculate the energy eigenvalues even for high state numbers and large values of the perturbation parameters

the runs conjecture the runs conjecture proposed by kolpakov and kucherov 1999 7 states that the number of occurrences of maximal repetitions runs in a string of length n runs n is at most n we almost solve the conjecture by proving that runs n 1 029n this bound is obtained using a combination of theory and computer verification

converging bounds for the eigenvalues of multiminima potentials in two dimensional space the eigenvalue problem in of schrodinger operators with a polynomial perturbation has been replaced by one corresponding to the system confined in a box with impenetrable walls it is shown that the dirichlet and the von neumann problems in generate upper and lower bounds respectively to the eigenvalues of the unbounded system to illustrate the method rapidly converging two sided bounds for the energy levels of two and three well oscillators are presented using simple trigonometric basis functions

strange an interactive method for multi objective linear programming under uncertainty abstract in the field of investment planning within a time horizon problems typically involve multiple objectives and basic data are uncertain in a large number of cases these decision problems can be written as linear programming problems in which time dependent uncertainties affect the coefficients and the right hand side of constraints given the possibility of defining plausible scenarios on basic data discrete sets of such coefficients are given each with its subjective probability of occurrence the corresponding structure is then characteristic for multi objective stochastic linear programming moslp in the paper an interactive procedure is described to obtain a best compromise for such a moslp problem this algorithm called strange extends the stem method to take the random aspects into account it involves in particular the concepts of stochastic programming with recourse in its interactive steps the efficiency projection techniques are used to provide the decision maker with detailed graphical information on efficient solution families as an illustration of the successive steps a didactic example is solved in some detail and the results of a case study in energy planning are given

pseudospectral methods for solving an equation of hypergeometric type with a perturbation almost all regular or singular sturm liouville eigenvalue problems in the schrodinger form email protected x v x j x email protected x email protected email protected email protected r j x l 2 email protected email protected for a wide class of potentials v x may be transformed into the form s x y email protected x y q x email protected email protected a b r by means of intelligent transformations on both dependent and independent variables where s x and t x are polynomials of degrees at most 2 and 1 respectively and l is a parameter the last form is closely related to the equation of the hypergeometric type eht in which q x is identically zero it will be called here the equation of hypergeometric type with a perturbation ehtp the function q x may therefore be regarded as a perturbation it is well known that the eht has polynomial solutions of degree n for specific values of the parameter l i e l email protected n 0 n t 12 n 1 s which form a basis for the hilbert space l 2 a b of square integrable functions pseudospectral methods based on this natural expansion basis are constructed to approximate the eigenvalues of ehtp and hence the energies e of the original schrodinger equation specimen computations are performed to support the convergence numerically

a characteristic function approach to the discrete spectrum of electrically charged particles the purpose of this article is to obtain a scalar function the zeros of which give the discrete energy values for a system of electrically charged particles the relation between the serial expansion of the characteristic function in powers of the system s eigenvalue and the stieltjes series has been revealed not only for the electrically charged particles but also generally for any positive or negative definite hermitian operator which has only a discrete spectrum the use of pade approximants to express the characteristic function has offered a rapidly convergent scheme to evaluate the system s eigenvalues the first few elements of the pade table for the reciprocal of the characteristic function of certain systems have been given to verify the presented idea numerically the determination of these elements needs the values of certain complicated integrals which we name zeroth order hyperspherical spectral coefficients hsc 0 the first two of these coefficients are investigated and their evaluation is realized analytically