shannon entropy fisher information and uncertainty relations for log periodic oscillators abstract we calculate the time dependent shannon s x and s p entropy and fisher f x and f p information of three log periodic oscillators we obtain a general expression for s x p and f x p in the state n 0 in terms of a c number quantity satisfying a nonlinear differential equation for two out of three oscillators s x p and f x p depend on time but s x s p and f x f p do not the other oscillator behaves as the time independent harmonic oscillator where s x p and f x p are all constants relations among the fisher information and the stam and cramer rao inequalities are also discussed

asymptotic formulae for the quantum renyi entropies of position application to the infinite well general asymptotic formulae are derived by means of the wkb approximation for the continuous and discrete renyi entropies of position of one dimensional quantum systems in energy eigenstates in terms of the corresponding entropies for a microcanonical ensemble of analogous classical systems these results are checked in the simplest particular case of the infinite potential well where the asymptotic formula for continuous entropies holds as an exact identity for the discrete entropies analytical expressions are obtained from which the asymptotic formulae given for the limiting cases of large and small measurement resolution can both be verified

information entropy of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy e n of the classical orthogonal polynomials e email protected p 2 n x logp 2 n x w x dx here an explicit formula for the entropy of general orthogonal polynomials is obtained in terms of the eigenvalues of jacobi matrix associated with the orthogonal polynomials p n x and the weighted moments of p 2 n x m n email protected x kp 2 n x w x dx k 1 2

failure of the method of slowly varying amplitude and phase for non linear singular oscillators it is shown that the method of slowly varying amplitude and phase yields erroneous results in the study of the mathematical properties of nonlinear singular oscillator systems the analytical solution is described in which the phase function is constant and for which a special limiting behavior exists when the wavelength is zero the previous method based on the condition of boundedness cannot be satisfied for nonlinear singular characteristics and the erroneous designation of the expansion parameter is identified

on the solution of strong nonlinear oscillators by applying a rational elliptic balance method a rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using jacobi elliptic functions to illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators we first investigate the exact solution of the non homogenous undamped duffing equation then we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators at the end of the paper we compare the numerical integration values of the angular frequencies with approximate solution results based on the proposed rational elliptic balance method

adomian decomposition method for a reliable treatment of the emden fowler equation in this paper we present a reliable algorithm to determine new exact and new approximate solutions of the generalized emden fowler equation the algorithm rests mainly on adomian decomposition method with an alternative framework designed to overcome the difficulty of the singular point at x 0 the proposed framework reveals quite a number of remarkable features of the scheme used here

a convenient technique for evaluating angular frequency in some nonlinear oscillations in this paper a convenient technique for evaluating angular frequency in some nonlinear oscillations is proposed it is well known that once the restoring force function is given beforehand the period of motion can be determined by an integral the angular frequency has a relation with the period of motion as well as the integral one makes a little modification for the integrand in the integral and a change of variable it is found that if the three divisions are chosen on the integration interval and the trapezoid quadrature rule is used a higher accurate result for the angular frequency can be achieved for the restoring force being an odd function three numerical examples are presented the eardrum type oscillation is studied as well higher accurate results for the angular frequency are obtained in all those examples

extension and improvement to the krylov bogoliubov methods using elliptic functions abstract it is shown that the krylov bogoliubov methods that give the approximate oscillatory solution of the equation in terms of jacobi elliptic functions are applicable not only when c 1 0 and c 3 0 but also when c 1 0 and c 3 0 in particular the most precise of these methods the christopher brocklehurst method is discussed in detail its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters

analysis of the response of a strongly nonlinear damped system using a differential transformation technique the differential transformation technique is applied to solve a nonlinear differential equation that describes the underdamped and the overdamped motion of a system subjected to external excitations numerical results are compared to those obtained by the runge kutta method to illustrate the preciseness and effectiveness of the proposed method

information theoretic measures for a solitonic profile mass schrodinger equation with a squared hyperbolic cosecant potential abstract entropic measures provide analytic tools to help us understand the stability of quantum systems the spreading of the quantum mechanical probability cloud for solitonic profile mass schrodinger equation with a potential v ax v 0 csch 2 ax is studied in position and momentum space by means of global shannon s information entropy information theoretic measures the position information entropy is considered only for x 0 due to the singular point at x 0 the entropy densities s x and s p are demonstrated and the bbm inequality is saturated

solution of a mixed parity nonlinear oscillator harmonic balance abstract the method of harmonic balance is used to calculate first order approximations to the periodic solutions of a mixed parity nonlinear oscillator first the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction then the two auxiliary equations where the restoring force functions are odd are solved by using a first harmonic term without a constant therefore we obtain the two approximate solutions to the mixed parity nonlinear oscillator one is expressed in terms of the exact amplitude in the negative direction the other in terms of the approximate amplitude these solutions are more accurate than the second approximate solution obtained by the lindstedt poincare method for large amplitudes

entropic integrals of hyperspherical harmonics and spatial entropy of d dimensional central potentials the information entropy of a single particle in a quantum mechanical d dimensional central potential is separated in two parts one depends only on the specific form of the potential radial entropy and the other depends on the angular distribution spatial entropy the latter is given by an entropic like integral of the hyperspherical harmonics which is expressed in terms of the entropy of the gegenbauer polynomials this entropy is expressed in terms of the values of the quadratic logarithmic potential of gegenbauer polynomials cn t at the zeros of these polynomials then this potential for integer is given as a finite expansion of chebyshev polynomials of even order whose coefficients are shown to be wilson polynomials

direct spreading measures of laguerre polynomials the direct spreading measures of the laguerre polynomials l n a x which quantify the distribution of its rakhmanov probability density r n a x 1d n 2x ae x l n a x 2 along the positive real line in various complementary and qualitatively different ways are investigated these measures include the familiar root mean square or standard deviation and the information theoretic lengths of fisher renyi and shannon types the fisher length is explicitly given the renyi length of order q such that email protected n is also found in terms of n a by means of two error free computing approaches one makes use of the lauricella function f a 2 q 1 1q 1q 1 which is based on the srivastava niukkanen linearization relation of laguerre polynomials and another one utilizes the multivariate bell polynomials of combinatorics the shannon length cannot be exactly calculated because of its logarithmic functional form but its asymptotics is provided and sharp bounds are obtained by the use of an information theoretic optimization procedure finally all these spreading measures are mutually compared and computationally analyzed in particular it is found that the apparent quasilinear relation between the shannon length and the standard deviation becomes rigorously linear only asymptotically i e for email protected 1

amplitude frequency relationship obtained using hamiltonian approach for oscillators with sum of non integer order nonlinearities in this paper the hamiltonian approximate analytical approach is extended for solving vibrations of conservative oscillators with the sum of integer and or non integer order strong nonlinearities solution of the nonlinear differential equation is assumed in the form of a trigonometric function with unknown frequency the frequency equation is obtained based on the hypothesis that the derivative of the hamiltonian in amplitude of vibration is zero the accuracy of the approximate solution is treated with two different approaches comparing the analytical value for period of vibration with the period obtained numerically and developing an error estimation method based on the ratio between the averaged residual function and the total constant energy of system the procedures given in the paper are applied for two types of examples an oscillator with a strong nonlinear term and an oscillator where the nonlinearity is of polynomial type

comments on the method of harmonic balance the advantages and limitations of the harmonic balance or describing function approximation scheme for solving nonlinear ordinary differential equations of oscillatory motion are discussed advantages include appicability to equations of any order and with large degrees of nonlinearity ease of determining limit cycle behavior and its stability and overall speed and efficiency the limitation rules are essentially those described by mickens 1983 it is pointed out that perturbation procedures provide better results when the degree of nonlinearity is small

geometrical non linear steady state forced periodic vibration of plates part i model and convergence studies a model for the steady state geometrically non linear periodic vibration of thin rectangular plates under harmonic external excitation is presented the equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method hfem these equations are transformed into the frequency domain by the harmonic balance method hbm and are solved by a continuation method the convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates

amplitude decay of damped non linear oscillators studied with jacobian elliptic functions approximate bounded solutions of the equation x bx c 1 x c 3 x 3 0 with b 0 c 1 0 and c 3 0 are developed in terms of the jacobian elliptic functions cn cd and dn the solutions are found by following the method developed by christopher in 1973 for the case with c 1 0 and c 3 0 formulas for the amplitude decay are given in two different approximations the solutions are compared with runge kutta numerical integration results and shown to be accurate for a wide range of b c 1 c 3 and initial conditions

higher order approximate solutions for nonlinear vibration of a constant tension string a new harmonic balance approach for solving the large amplitude nonlinear vibration of a constant tension string is introduced the coupling of newton s method with harmonic balancing takes the advantage of reducing the deficiency and complexity of the classical harmonic balance method in dealing with the nonlinear systems the solutions are directly induced from a set of linear algebraic equations instead of a set of complicated coupled nonlinear algebraic equations illustrative examples are selected and compared to some published data to verify the accuracy of the higher order solutions

a quantum uncertainty relation based on fisher s information we explore quantum uncertainty relations involving the fisher information functionals ix and ip evaluated respectively on a wavefunction x defined on a d dimensional configuration space and the concomitant wavefunction on the conjugate momentum space we prove that the associated fisher functionals obey the uncertainty relation ixip 4d2 when either x or is real on the other hand there is no lower bound to the above product for arbitrary complex wavefunctions we give explicit examples of complex wavefunctions not obeying the above bound in particular we provide a parametrized wavefunction for which the product ixip can be made arbitrarily small

pure odd order oscillators with constant excitation abstract in this paper the excited vibrations of a truly nonlinear oscillator are analyzed the excitation is assumed to be constant and the nonlinearity is pure without a linear term the mathematical model is a second order nonhomogeneous differential equation with strong nonlinear term using the first integral the exact value of period of vibration i e angular frequency of oscillator described with a pure nonlinear differential equation with constant excitation is analytically obtained the closed form solution has the form of gamma function the period of vibration depends on the value of excitation and of the order and coefficient of the nonlinear term for the case of pure odd order oscillators the approximate solution of differential equation is obtained in the form of trigonometric function the solution is based on the exact value of period of vibration for the case when additional small perturbation of the pure oscillator acts the so called cveticanin s averaging method for a truly nonlinear oscillator is applied two special cases are considered one when the additional term is a function of distance and the second when damping acts to prove the correctness of the method the obtained results are compared with those for the linear oscillator example of pure cubic oscillator with constant excitation and linear damping is widely discussed comparing the analytically obtained results with exact numerical ones it is concluded that they are in a good agreement the investigations reported in the paper are of special interest for those who are dealing with the problem of vibration reduction in the oscillator with constant excitation and pure nonlinear restoring force the examples of which can be found in various scientific and engineering systems for example such mechanical systems are seats in vehicles supports for machines cutting machines with periodical motion of the cutting tools presses etc the examples can be find in electronics electromechanical devices like micro actuators and micro oscillators in music instruments hammers in piano in human voice producing folds voice cords etc

simple analytical results for harmonically trapped quantum gases by making use of the generating function for the associated laguerre polynomials we derive new analytical results for harmonically trapped quantum gases in arbitrary dimensions these exact expressions prove to be very numerically efficient and possess a much simpler mathematical form than those previously reported in the literature as a direct consequence of our work new summation relations for laguerre polynomials are also established

a comparison of structural csp decomposition methods abstract we compare tractable classes of constraint satisfaction problems csps we first give a uniform presentation of the major structural csp decomposition methods we then introduce a new class of tractable csps based on the concept of hypertree decomposition recently developed in database theory and analyze the cost of solving csps having bounded hypertree width we provide a framework for comparing parametric decomposition based methods according to tractability criteria and compare the most relevant methods we show that the method of hypertree decomposition dominates the others in the case of general csps i e csps of unbounded arity we also make comparisons for the restricted case of binary csps finally we consider the application of decomposition methods to the dual graph of a hypergraph in fact this technique is often used to exploit binary decomposition methods for nonbinary csps however even in this case the hypertree decomposition method turns out to be the most general method

combining the methods of harmonic balance and kryloff bogoliuboff in a recent work mickens has described a set of conditions under which the harmonic balance method hbm is implementable the aim here is to broaden the scope of hbm analysis by drawing attention to the possible combination of such methods with those of kryloff bogoliuboff surprisingly these issues are not widely appreciated and as such provide an incentive for this note

elliptic balance solution to two degree of freedom undamped homogeneous systems having cubic nonlinearities abstract the application of the elliptic balance method to the solution of undamped two degree of freedom homogeneous nonlinear systems is described this method uses jacobian elliptic functions in the balance and is based on the concept of averaging with respect to complete elliptic integrals of the first kind to assess the accuracy of the approximate solution thus obtained we consider the motion of a linear vibration absorber attached to a rigid body that is supported symmetrically by incompressible homogeneous and isotropic hyperelastic shear blocks it is shown that the amplitude time response of the model system is well predicted by the elliptic balance method solution even for relatively large parameter values

quantum information entropies and orthogonal polynomials this is a survey of the present knowledge on the analytical determination of the shannon information entropies for simple quantum systems single particle systems in central potentials emphasis is made on d dimensional harmonic oscillator and coulombian potentials in both position and momentum spaces first of all these quantities are explicitly shown to be controlled by the entropic integrals of some classical orthogonal polynomials hermite laguerre and gegenbauer then the connection of these integrals with more common mathematical objects such as the logarithmic potential energy and l p norms of orthogonal polynomials is brie y described third its asymptotic behaviour is discussed for both general and varying weights the explicit computation of these integrals is carried out for the chebyshev and gegenbauer polynomials which have a bounded orthogonality interval as well as for hermite polynomials to illustrate the di culties encountered when the interval is unbounded these results have allowed us to nd the position and momentum entropies of the ground and excited states of the physical systems mentioned above c 2001 elsevier science b v all rights reserved

entropy and information of a harmonic oscillator in a time varying electric field in 2d and 3d noncommutative spaces we analyze the noncommutativity effects on the fisher information fr p and shannon entropies sr p of a harmonic oscillator immersed in a time varying electric field in two and three dimensions we find the exact solutions of the respective time dependent schrodinger equation and use them to calculate the fisher information and the shannon entropy for the simplest case corresponding to the lowest lying state of each system while there is no problem in defining the shannon entropy for noncommutating spaces the definition of the fisher information have to be modified to satisfy the cramer rao inequalities for both systems we observe how the fisher information and shannon entropy in position and momentum change due to the noncommutativity of the space we verify that the bialynicki birula mycielski bbm entropic uncertainty relation still holds in the systems considered

vibrations of a coupled two degree of freedom system abstract in this paper the motion of a two mass system with two degrees of freedom is discussed the masses are connected with three springs the motion of the system is described with a system of two coupled strong non linear differential equations for the case when the non linearity is of a cubic type the analytical solution of the system is obtained it is a combination of a jacobi elliptic function and a trigonometric function an approximate analytical method based on the krylov bogolubov procedure is developed for the system which contains small non linearities two examples are considered the case when all the three stiffnesses are non linear and the case when small damping acts the analytical solutions are compared with numerical ones they show a good agreement

solution of a quadratic nonlinear oscillator by the method of harmonic balance abstract this paper deals with the quadratic nonlinear oscillator qno x x e x 2 0 this oscillator is compared with the nonlinear oscillators x x e x x 0 which are solved by the method of harmonic balance therefore we obtain the first approximate solution to the qno this solution is more accurate than the second approximate solution obtained by the lindstedt poincare method for qnos it has been pointed out that the angular frequency for x 0 should be different from the angular frequency for x 0 if solutions of higher degree of accuracy are required

the limit cycles of the generalized rayleigh lienard oscillator abstract the limit cycles of the generalized rayleigh and mixed rayleigh lienard oscillator x ax 2bx 3 e z 3 z 2 x 2 z 1 x 4 z 4 dot x 2 dot x 0 for a and b 0 are studied by using the jacobian elliptic functions with the generalized harmonic balance method the transitory motion and consequently the limit cycles and their stability are also studied quantitatively with a generalized approximation of the krylov bogoliubov slowly varying amplitude and phase type giving the radius frequency and energy of the limit cycles whether there are zero one or two limit cycles and their stability depends on the values of the parameters z i i 1 2 3 4 these solutions are interesting because they do not depend on the value of e for the simple cases of only two non zero z i parameters plots of the universal functions are given as well as the limit cycle radii approximate solutions in the general case are found to coincide with certain exact solutions of the mixed type oscillators

analytic solution of the system of two coupled differential equations with the fifth order non linearity abstract in this paper an approximate method for solving a system of two coupled second order non linear differential equation with strong and weak non linearity is developed the strong non linearities are of the third and of the fifth order the suggested new method is partially based on the elliptic krylov bogolubov method and on the power series method developed for solving the weak non linear differential equations the one frequency solution is considered for the case of certain relations between the equation parameters p and m and initial conditions as a special case the oscillatory motion of the physical system described with a fourth order strong non linear differential equation is considered a wide analysis of the corresponding equation is done the results of the suggested analytic procedure are compared with numerical ones they show a good agreement

bounds on the fourier coefficients for the periodic solutions of non linear oscillator equations the differential equations describing nonlinear oscillations as seen in mechanical vibrations electronic oscillators chemical and biochemical reactions acoustic systems stellar pulsations etc are investigated analytically the boundedness of the fourier coefficients for periodic solutions is demonstrated for two special cases and the extrapolation of the results to higher dimensionsal systems is briefly considered

entropic integrals of orthogonal hypergeometric polynomials with general supports the boltzmann shannon information entropy of probability measures which involve the continuous hypergeometric type polynomials fpn x g orthogonal with respect to a general weight function x is determined by two integral quantities one with kernel p 2 x x lnp 2 x called as entropy of the polynomial pn x and another one with kernel p 2 x x ln x here an explicit expression for the latter quantity and for a broader family of related integrals is obtained in terms only of the second order dierential equation satised by the involved polynomials for illustration the general formula is applied to evaluate the integrals corresponding to the three classical families of continuous orthogonal polynomials on the real axis of hypergeometric type hermite laguerre and jacobi c 2000 elsevier science b v all rights reserved

model reduction to spectral submanifolds and forced response calculation in high dimensional mechanical systems abstract we show how spectral submanifold ssm theory can be used to extract forced response curves without any numerical simulation in high degree of freedom periodically forced mechanical systems we use multivariate recurrence relations to construct the ssms achieving a major speed up relative to earlier autonomous ssm algorithms the increase in computational efficiency promises to close the current gap between studying lower dimensional academic examples and analyzing larger systems obtained from finite element modeling as we illustrate on two different discretized damped forced beam models using the exact reduction procedure via ssms for obtaining forced response curves we further demonstrate speed gains of several orders in magnitude relative to the available state of the art continuation packages while retaining accuracy

analytical expressions of periodic disequilibrium fluctuations generated by hopf bifurcations in economic dynamics abstract paper constructs approximate analytical expressions of periodic disequilibrium fluctuations business cycles occurring in connection with hopf bifurcations in economic dynamics the kaldorian open economy macrodynamic model with money proposed earlier by the first author is employed as the test model its region of equilibrium stability in parameter space is obtained and a hopf bifurcation curve is identified as a boundary of the region validity of the analytical approximations obtained for the cycles generated by the loss of equilibrium stability on this curve is confirmed by comparison to numerically determined cycles explicit analytical description of such disequilibrium fluctuations is of particular interest in the case of subcritical bifurcations when the cycles are unstable and their numerical determination is difficult as in the case of the test model and may conceivably be useful in testing for subsequent bifurcation of the cycles

entropy and complexity analysis of dirac delta like quantum potentials abstract the dirac delta like quantum mechanical potentials are frequently used to describe and interpret numerous phenomena in many scientific fields including atomic and molecular physics condensed matter and quantum computation the entropy and complexity properties of potentials with one and two dirac delta functions are here analytically calculated and numerically discussed in both position and momentum spaces we have studied the information theoretic lengths of fisher renyi and shannon types as well as the cramer rao fisher shannon and lmc shape complexities of the lowest lying stationary states of one delta and twin delta they allow us to grasp and quantify different facets of the spreading of the charge and momentum of the system far beyond the celebrated standard deviation

choices in the harmonic balance technique we apply the harmonic balance technique in low orders to three types of oscillator model for three equations of the van der pol type with rational nonlinear terms used to model electronic oscillators we examine the choice between a rationalizing before expanding in harmonics and b obtaining fourier coefficients of nonlinear terms alternative b is shown to be more accurate in first order for the models of scott murata and of walker and connelly which apply to circuits with an inverse tangent nonlinear component for these equations in the case of alternative a we find acceptable dependence of the second order corrections on the bifurcation parameter when the harmonic balance method is used to set up a semi classical quantization treatment of a nonlinear conservative oscillator values of energies are slightly improved by the use of alternative b for conservative oscillators with cubic or fifth power forces we compare the standard method using the acceleration equation with an alternative using the energy equation the first is shown to be more accurate in low order

a new approach for solving a complex valued differential equation abstract in this paper two analytical approximate solving procedures for a complex valued differential equation are developed one of the methods represents the generalization of the krylov bogolubov method for a strong differential equation with complex function the second method is based on the first integrals of the system the differential equation is transformed introducing the perturbed first integrals and the polar co ordinates the solution is obtained applying the straightforward series expansion the solution for the special case of without impact initial conditions is considered the method is applied on the system with strong cubic non linearity the small gyroscopic function and damping function are introduced the analytical approximate solution is compared with numerical exact one and shows a good agreement

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

position momentum correlations in the moshinsky atom position momentum correlation in the moshinsky atom is studied by examining the mutual information between wigner functions and position momentum space densities at the one and two particle levels we show that this correlation is independent of the strength of the interparticle potential at the two particle level at the one particle level the correlation decreases as the potential is increased shannon entropies of one and two particle wigner functions are also discussed as measures of the localization in phase space distributions

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

complexity of generalized satisfiability counting problems abstract the class of generalized satisfiability problems introduced in 1978 by schaefer presents a uniform way of studying the complexity of satisfiability problems with special conditions the complexity of each decision and counting problem in this class depends on the involved logical relations in 1979 valiant defined the class p the class of functions definable as the number of accepting computations of a polynomial time nondeterministic turing machine clearly all satisfiability counting problems belong to this class p we prove a dichotomy theorem for generalized satisfiability counting problems that is if all logical relations involved in a generalized satisfiability counting problem are affine then the number of satisfying assignments of this problem can be computed in polynomial time otherwise this function is p complete this gives us a comparison between decision and counting generalized satisfiability problems we can determine exactly the polynomial satisfiability decision problems whose number of solutions can be computed in polynomial time and also the polynomial satisfiability decision problems whose counting counterparts are already p complete moreover taking advantage of a similar dichotomy result proved in 1978 by schaefer for generalized satisfiability decision problems we get as a corollary the implication that the counting counterpart of each np complete generalized satisfiability decision problem is p complete

asymptotics of information entropies of some toda like potentials the spreading of the quantum probability density for the highly excited states of a single particle system with an exponential type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the boltzmann shannon information entropy this problem boils down to the calculation of the asymptotics of the entropy like integrals of the modified bessel function of the second kind also called the mcdonald function or basset function the dependence of the two physical entropies on the large quantum number n is given in detail it is shown that the semiclassical wkb position space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single particle systems with any power type potential of the form v x x2k x r and k n the momentum space entropy calculated with a method based on the properties of the mcdonald function is rigorously found to have a behavior of the form ln ln n in strong contrast with the

improvement to the averaging method using the jacobian elliptic function the averaging method was improved by using the jacobian elliptic sine sn cosine cn and delta dn functions as generating solutions in order to obtain a highly accurate periodic solution for a strongly nonlinear dynamical system the proposed method can be applied to a relatively general nonlinear system based on the single degree of freedom duffing equation two methods of stability analysis were applied and evaluated for the approximate solutions obtained by the proposed method the approximate solutions obtained by the proposed method for practical examples were compared to the solutions obtained by the shooting method the results confirmed that the proposed method provides a more accurate solution and more accurate stability analysis results than those obtained by the conventional averaging method that uses trigonometric functions as the generating solution

entropy of orthogonal polynomials with freud weights and information entropies of the harmonic oscillator potential the information entropy of the harmonic oscillator potential v x 1 2 x2 in both position and momentum spaces can be expressed in terms of the so called entropy of hermite polynomials i e the quantity sn h h2n x log h2n x e x2dx these polynomials are instances of the polynomials orthogonal with respect to the freud weights w x exp x m m 0 here a very precise and general result of the entropy of freud polynomials recently established by aptekarev et al j math phys 35 4423 4428 1994 specialized to the hermite kernel case m 2 leads to an important refined asymptotic expression for the information entropies of very excited states i e for large n in both position and momentum spaces to be denoted by s and s respectively briefly it is shown that for large values of n s 1 2log log 2n e o 1 and s 1 2log log 2n e o 1 so that s s log 2 2n e2 o 1 in agreement with the generalized indetermination relation of byalinicki birula and mycielski commun m

the shannon entropy based uncertainty relation for d dimensional central potentials the uncertainty relation based on the shannon entropies of the probability densities in position and momentum spaces is improved for quantum systems in arbitrary d dimensional spherically symmetric potentials to find this we have used the lp lq norm inequality of de carli and the logarithmic uncertainty relation for the hankel transform of omri applications to some relevant three dimensional central potentials are shown

improvement of a krylov bogoliubov method that uses jacobi elliptic functions abstract an improved version of a krylov bogoliubov method that gives the approximate solution of the non linear cubic oscillator x c 1 x c 3 x 3 ef x x dot 0 in terms of jacobi elliptic functions is described compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual krylov bogoliubov method which gives the approximate solution in terms of circular functions these expressions are especially simple for quasilinear c3 0 and quasi pure cubic c1 0 oscillators two types of cubic oscillators have been used as examples the linear damped oscillator f x x dot x dot and the van der pol oscillator f x x dot x 2 x dot the approximate solutions of these quasilinear and quasi pure cubic oscillators are simple and accurate the influence of the non linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied

quantum expectation values of d dimensional rydberg hydrogenic states by use of laguerre and gegenbauer asymptotics the radial position and momentum p 1 3 expectation values of the d dimensional rydberg hydrogenic states i e states where the electron has a large hyperquantum number n are rigorously determined by means of powerful tools of the modern approximation theory relative to the asymptotics of the varying orthogonal laguerre and gegenbauer polynomials which control the corresponding wavefunctions in position and momentum spaces

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

investigations of information quantifiers for the tavis cummings model abstract in this article a system of two two level atoms interacting with a single mode quantized electromagnetic field in a lossless resonant cavity via a multi photon transition is considered the quantum fisher information negativity classical fisher information and reduced von neumann entropy for the two atoms are investigated we found that the number of photon transitions plays an important role in the dynamics of different information quantifiers in the cases of two symmetric and two asymmetric atoms our results show that there is a close relationship between the different quantifiers also the quantum and classical fisher information can be useful for studying the properties of quantum states which are important in quantum optics and information

analytic approach for the solution of the complex valued strong non linear differential equation of duffing type abstract in this paper an approximate analytic procedure is developed for solving the strong non linear differential equations of the duffing type with complex valued function which describes the dynamical behavior of many real systems the method is based on the elliptic krylov bogolubov procedure where the solutions are the jacobi elliptic functions as an example the self excited vibrations of the rotor with variable shaft rigidity are considered the analytical results of this example are compared with numerical ones and excellent agreement is found between them

averaging method for quasi integrable hamiltonian systems abstract a deterministic averaging method for quasi integrable hamiltonian systems is proposed to predict the approximate response of many degree of freedom autonomous or non autonomous strongly nonlinear systems the form and dimension of the averaged equations depend on the number of degree of freedom and the number of resonant relations of the associated hamiltonian systems in non resonant case the averaged equations for n action variables or n independent integrals of motion are derived in resonant case the averaged equations for n action variables and combinations of angle variables are derived two examples are given to illustrate the application of the proposed method it is shown that the analytical results agree well with those from numerical solution even for systems with very strong nonlinearity

the complexity of homomorphism and constraint satisfaction problems seen from the other side we give a complexity theoretic classification of homomorphism problems for graphs and more generally relational structures obtained by restricting the left hand side structure in a homomorphism for every class c of structures let hom c be the problem of deciding whether a given structure a c has a homomorphism to a given arbitrary structure s we prove that under some complexity theoretic assumption from parameterized complexity theory hom c is in polynomial time if and only if c has bounded tree width modulo homomorphic equivalence translated into the language of constraint satisfaction problems our result yields a characterization of the tractable structural restrictions of constraint satisfaction problems translated into the language of database theory it implies a characterization of the tractable instances of the evaluation problem for conjunctive queries over relational databases

spatial entropy of central potentials and strong asymptotics of orthogonal polynomials the boltzmann shannon information entropy of quantum mechanical systems in central potentials can be expressed in terms of the entropy sn of the classical orthogonal polynomials here an asymptotic formula for the entropy of general orthogonal polynomials on finite intervals is obtained it is shown that this entropy is intimately related to the relative entropy i 0 of the equilibrium measure 0 x and the weight function x of the polynomials to do so the theory of strong asymptotics of orthogonal polynomials on compact sets is used

strong asymptotics of laguerre polynomials and information entropies of two dimensional harmonic oscillator and one dimensional coulomb potentials the information entropies of the two dimensional harmonic oscillator v x y 1 2 x2 y2 and the one dimensional hydrogen atom v x 1 x can be expressed by means of some entropy integrals of laguerre polynomials whose values have not yet been analytically determined here we first study the asymptotical behavior of these integrals in detail by extensive use of strong asymptotics of laguerre polynomials then this result which is also important by itself in a context of both approximation theory and potential theory is employed to analyze the information entropies of the aforementioned quantum mechanical potentials for the very excited states in both position and momentum spaces it is observed in particular that the sum of position and momentum entropies has a logarithmic growth with respect to the main quantum number which characterizes the corresponding physical state finally the rate of convergence of the entropies is numerically examined

an analytical technique for solving a class of strongly nonlinear conservative systems abstract recently an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms usually a set of nonlinear algebraic equations is solved in this method however analytical solutions of these algebraic equations are not always possible especially in the case of a large oscillation previously such algebraic equations for the duffing equation were solved in powers of a small parameter but the solutions measure desired results when the amplitude is an order of 1 in this article different parameters of the same nonlinear problems are found for which the power series produces desired results even for the large oscillation moreover two or three terms of this power series solution measure a good result when the amplitude is an order of 1 besides these a suitable truncation formula is found in which the solution measures better results than existing solutions the method is mainly illustrated by the duffing oscillator but it is also useful for many other nonlinear problems

a generalized iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillators abstract an extended iteration method for calculating the periodic solutions of nonlinear oscillator equations is given the procedure is illustrated by applying it to two truly nonlinear oscillator differential equations

construction of approximate analytical solutions to a new class of non linear oscillator equation the principle of harmonic balance is invoked in the development of an approximate analytic model for a class of nonlinear oscillators typified by a mass attached to a stretched wire by assuming that harmonic balance will hold solutions are devised for a steady state limit cycle and or limit point motion a method of slowly varying amplitudes then allows derivation of approximate solutions by determining the form of the exact solutions and substituting into them the lowest order terms of their respective fourier expansions the latter technique is actually a generalization of the method proposed by kryloff and bogoliuboff 1943

a partial k arboretum of graphs with bounded treewidth the notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies this survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class also some mutual relations between such classes are discussed

on the algebraic structure of combinatorial problems abstract we describe a general algebraic formulation for a wide range of combinatorial problems including satisfiability graph colorability and graph isomorphism in this formulation each problem instance is represented by a pair of relational structures and the solutions to a given instance are homomorphisms between these relational structures the corresponding decision problem consists of deciding whether or not any such homomorphisms exist we then demonstrate that the complexity of solving this decision problem is determined in many cases by simple algebraic properties of the relational structures involved this result is used to identify tractable subproblems of satisfiability and to provide a simple test to establish whether a given set of boolean relations gives rise to one of these tractable subproblems

vibrations in a parametrically excited system abstract this paper deals with vibrations of parametrically excited non linear systems with one degree of freedom the non linearity is cubic and is of the same order as the linear terms the parametric vibrations are excited by a periodical force of jacobi elliptic type the mathematical model of the system is a special type of non linear hill s equation the analytical approximate solution of the equation is obtained applying the elliptic krylov bogolubov method method of variable phase and amplitude developed for strong non linear differential equation of duffing type it enables the regions of unbounded solution to be defined approximately the parameters of a dynamic absorber which transforms the motion to regular are calculated in this paper

generalized fourier series for the study of limit cycles the approximate solution to first order of non linear differential equations is studied using the method of harmonic balance with generalized fourier series and jacobian elliptic functions as an interesting use of the series very good analytic approximations to the limit cycles of lienard s ordinary differential equation ode x g x f x x are presented specifically it is shown that contrary to an opinion given in a well known textbook on non linear oscillations g x not only modifies the period but influences the topology in the generalized van der pol equation with f x e 1 x2 and g x ax 2bx3 for 0 1 the presence of zero one or three limit cycles is found to depend on the value of ab

a modified elliptic lindstedt poincare method for certain strongly non linear oscillators abstract a modified elliptic lindstedt poincare method is presented for the steady state analysis of strongly non linear oscillators of the form x c 1 x c 3 x 3 ef x x in which a new parameter e is employed such that the value of is always small regardless of the magnitude of the original parameter e therefore the above strongly non linear oscillators with large parameter e is transformed into a small parameter system with respect to

forced vibrations of a turbine blade undergoing regularized unilateral contact conditions through the wavelet balance method summary the method of weighted residuals can efficiently enforce time periodic solutions of flexible structures experiencing unilateral contact the harmonic balance method hbm based on fourier expansion of the sought solution is a common formulation although wavelet bases that can sparsely define nonsmooth solutions may be superior this hypothesis is investigated using a full three dimensional blade with unilateral contact conditions on a set of nc discrete contact points located at its tip the unilateral contact conditions are first regularized and a distributional formulation in time is introduced allowing l2 s1 n trial functions to properly approximate in the time domain the solution to the governing equations the mixed wavelet petrov galerkin solutions are found to yield consistent or better results than hbm with higher convergence rates and seemingly more accurate contact force prediction copyright 2014 john wiley sons ltd

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

entropy and complexity analysis of hydrogenic rydberg atoms the internal disorder of hydrogenic rydberg atoms as contained in their position and momentum probability densities is examined by means of the following information theoretic spreading quantities the radial and logarithmic expectation values the shannon entropy and the fisher information as well the complexity measures of cramer rao fisher shannon and lopez ruiz mancini calvet types are investigated in both reciprocal spaces the leading term of these quantities is rigorously calculated by use of the asymptotic properties of the concomitant entropic functionals of the laguerre and gegenbauer orthogonal polynomials which control the wavefunctions of the rydberg states in both position and momentum spaces the associated generalized heisenberg like logarithmic and entropic uncertainty relations are also given finally application to linear l 0 circular l n 1 and quasicircular l n 2 states is explicitly done

comments on the method of harmonic balance in which jacobi elliptic functions are used abstract a harmonic balance method is presented in which jacobi elliptic functions are used in the trial solution instead of circular functions to obtain approximate periodic solutions of the oscillator x f x dot x 0 conditions for the method to work well are the usual ones of the current method of harmonic balance and that x t must pass through zero the procedure for obtaining a higher order approximation is described and in particular two criteria for chosing the elliptic function parameter m are discussed illustrative examples are presented with f being diverse polynomials of x

free vibration of a strong non linear system described with complex functions abstract in this paper two approximate analytical methods for solving strong non linear differential equations with complex functions are developed besides the adopted elliptic krylov bogolubov procedure an alternative analytical method based on the variation of the parameters as well as the elliptic krylov bogolubov is introduced the difference is that the alternative method considers two approximate first order differential equations the suggested procedures are compared and tested for the system with strong cubic non linearity and a small non linearity of van der pol type comparing the approximate analytical solutions with exact numerical solution it is concluded that the difference is negligible

application of the elliptic balance method to a nonlinear singular oscillator abstract new analytical approximate solutions based on jacobi elliptic functions are formulated to compute the angular frequency value of a nonlinear singular oscillator it is shown that when our derived analytical angular frequency solutions are compared to the exact angular frequency value the third order elliptic balance solution provides the best estimate to the exact value since the percentage error value is about 0 0059 which is 216 times lower than the best estimate percentage error value reported in the literature

a classical iteration procedure valid for certain strongly nonlinear oscillators a classical iteration procedure is applied to nonlinear oscillations of conservative single degree of freedom systems with odd nonlinearity with the procedure the analytical approximate frequency and the corresponding periodic solution valid for small as well as large amplitudes of oscillation can be obtained two examples are given to illustrate the accuracy and effectiveness of the method another advantage of this classical iteration approach is that it also works if the linear part of restoring force is zero

the complexity of counting homomorphisms seen from the other side for every class of relational structures c let hom c be the problem of deciding whether a structure a c has a homomorphism to a given arbitrary structure b grohe has proved that under a certain complexity theoretic assumption hom c is solvable in polynomial time if and only if the cores of all structures in c have bounded tree width we prove under a weaker complexity theoretic assumption that the corresponding counting problem hom c is solvable in polynomial time if and only if all structures in c have bounded tree width this answers an open question posed by grohe

oscillator with fraction order restoring force abstract in this paper a new analytical method for solving the differential equations which describe the motion of the oscillator with fraction order elastic force is introduced using the first integral of motion the exact period of vibration in the form of the euler beta function is obtained based on that value the approximate solution of the strong nonlinear fraction order differential equation is formed the suggested procedure is applied for various fraction values but also for the pure quadratic cubic and quintic oscillators the advantage of the suggested method is that is valid for all fraction values 1 the accuracy of the approximate solution is very high as the period of vibration is exactly analytically determined and is independent on the time in the paper the fraction order differential equation with small linear and nonlinear terms is also considered the krylov bogolubov method is extended due to the fact that frequency of vibration depends on the amplitude of vibration the approximate solution is with time variable amplitude phase and frequency the linear damped fraction order oscillator is widely discussed the approximate solution of the differential equation which describes the vibrations of such oscillator is compared with exact numerical one and shows a good agreement

quasi pure cubic oscillators studied using a krylov bogoliubov method abstract a method of krylov bogoliubov type which gives the approximate solution in terms of jacobi elliptic functions is used to study the quasi pure cubic oscillators x c 3 x 3 ef x x 0 explicit approximate solutions with the perturbative terms x x 2 x x quadratic damping and sgn coulomb damping are given and are compared with the numerical or exact solution a simple and accurate expression is derived for the time at which the oscillations with coulomb damping cease