an adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures we develop a finite element adaptive strategy with error control for the wave scattering by periodic structures the unbounded computational domain is truncated to a bounded one by an extension of the perfectly matched layer pml technique which attenuates both the outgoing and evanescent waves in the pml region pml parameters such as the thickness of the layer and the medium property are determined through sharp a posteriori error estimates numerical experiments are included to illustrate the competitive behavior of the proposed adaptive method

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

on the use of perfectly matched layers at corners for scattering problems with sign changing coefficients we investigate in a 2d setting the scattering of time harmonic electromagnetic waves by a plasmonic device represented as a non dissipative bounded and penetrable obstacle with a negative permittivity using the t coercivity approach we first prove that the problem is well posed in the classical framework h loc 1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device when the latter has corners for values inside the critical interval unusual strong singularities for the electromagnetic field can appear in that case well posedness is obtained by imposing a radiation condition at the corners to select the outgoing black hole plasmonic wave that is the one which carries energy towards the corners a simple and systematic criterion is given to define what is the outgoing solution finally we propose an original numerical method based on the use of perfectly matched layers at the corners we emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods

on the attainment of the cramer rao bound in lr differentiable families of distributions on presente une preuve rigoureuse qu une atteinte globale de la borne de cramer rao est seulement possible si la famille sous jacente de lois est exponentielle on place la preuve dans le contexte de la differentiabilite l r p v en exigeant une differentiabilite forte dans l r p v de la r ieme racine du rapport de vraisemblance relatif a p v ou p v appartient a une famille de lois de probabilite avec parametre v cr k

periodic solution of the generalized rayleigh equation abstract the periodic solutions of a strongly cubic nonlinear oscillator whose motion is described with the generalized rayleigh equation are studied approximate analytic solving methods are introduced a new method based on homotopy and averaging is developed to determine the limit cycle motion the obtained analytical solutions are compared with those calculated by the elliptic harmonic balance method with generalized fourier series and jacobian elliptic functions three types of cubic nonlinearity are considered the coefficients of the linear and cubic terms are positive the coefficient of the linear term is positive and that of the cubic term is negative and the opposite case comparisons of the analytical solution and numerical solution obtained by using the runge kutta method are illustrated with examples

a simple derivation of the constrained multiple parameter cramer rao bound j d gorman and a o hero 1988 obtained a remarkable extension to the classical multiple parameter cramer rao cr lower bound that accounts for deterministic nonlinear equality constraints on the parameters the virtue of gorman and hero s result is that the constrained cr bound on all of the parameters is obtained by subtracting an easily computed nonnegative definite correction matrix from the unconstrained cr bound matrix the author presents a new simple derivation of the constrained cr bound and a new necessary condition for an estimator to satisfy the constrained cr bound with equality

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

generalized fourier transforms for nonlinear systems a simple generalization is presented of the usual fourier transforms using generalized exponential and fourier series developed previously for nonlinear systems the expressions are given in terms of jacobi elliptic functions in a form as close as possible to the fourier transform there is a pair of transforms for each value of the jacobi parameter m simple applications are presented one of which gives the generalized yukawa potential and another that gives the generalized breit wigner potential both correspond to the generalized klein gordon equation

perfectly matched layers for diffraction gratings in inhomogeneous media stability and error estimates we study the scattering of acoustic or electromagnetic waves at diffraction gratings in inhomogeneous media the refractive index stabilizes to 1 as the distance to the grating increases outgoing solutions are characterized by means of the limiting absorption principle we prove the unique solvability of the problem with perfectly matched layer of finite length further we show that solutions of the latter problem approximate outgoing solutions of the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity contribution of eigenvalues and resonances to the error of approximation is clarified

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

generalized fourier series and limit cycles of generalized van der pol oscillators abstract a very simple generalization using jacobian elliptic functions of the usual fourier series appropriate for non linear systems is used to study the first order approximate solution of the generalized van der pol oscillators x ax 2bx 3 z 3 z 2 x 2 z 1 x 4 dot x 0 a generalized harmonic balance method is used to determine the limit cycles the cases a 0 b 0 and a 0 are considered in some detail for given values of the parameters zi the values of a and b for which limit cycles exist are found as functions of m numerical values for the radius frequency and energy of the limit cycles are given the presence of zero one two three or four limit cycles depends on the value of the parameters of the equation stability bifurcations of fixed points and the limit cycle stability are studied qualitatively

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

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

maximum likelihood reconstruction for emission tomography previous models for emission tomography et do not distinguish the physics of et from that of transmission tomography we give a more accurate general mathematical model for et where an unknown emission density x y z generates and is to be reconstructed from the number of counts n d in each of d detector units d within the model we give an algorithm for determining an estimate of which maximizes the probability p n of observing the actual detector count data n over all possible densities let independent poisson variables n b with unknown means b b 1 b represent the number of unobserved emissions in each of b boxes pixels partitioning an object containing an emitter suppose each emission in box b is detected in detector unit d with probability p b d d 1 d with p b d a one step transition matrix assumed known we observe the total number n n d of emissions in each detector unit d and want to estimate the unknown b b 1 b for each the observed data n has probability or likelihood p n the em algorithm of mathematical statistics starts with an initial estimate 0 and gives the following simple iterative procedure for obtaining a new estimate new from an old estimate old to obtain k k 1 2 new b old b dd 1 n d p b d old b p b d b 1 b

an elliptic perturbation method for certain strongly non linear oscillators abstract an elliptic perturbation method is presented for calculating periodic solutions of strongly non linear oscillators of the formx c1x c3x3 f x x in which the jacobian elliptic functions are employed instead of usual circular functions in the conventional perturbation procedure three types of duffing van der pol equation withf x x 1 x2 xare studied in detail the present method can give a second approximate solution and therefore its accuracy is much higher than that of other methods in which the solutions are only first approximations

periodic solutions of strongly quadratic non linear oscillators by the elliptic lindstedt poincare method abstract the elliptic lindstedt poincare method is used employed to study the periodic solutions of quadratic strongly non linear oscillators of the form x c 1 x c 2 x 2 e f x x in which the jacobian elliptic functions are employed instead of the usual circular functions in the classical lindstedt poincare method the generalized van de pol equation with f x x 0 1 x 2 x 2 is studied in detail comparisons are made with the solutions obtained by using the lindstedt poincare method and runge kutta method to show the efficiency of the present method

exact solution of the quadratic mixed parity helmholtz duffing oscillator abstract in this paper the exact solution of the quadratic mixed parity helmholtz duffing oscillator is derived by using jacobi elliptic functions it is also shown that the exact period of oscillation is given as a function of the complete elliptic integral of the first kind at the end of the paper we examine the stability of the system and determine the regions for which periodic and unbounded motions take place

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

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

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

a hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators abstract a hyperbolic perturbation method is presented for determining the homoclinic solution of certain strongly nonlinear autonomous oscillators of the form x c 1 x c 2 x 2 e f x x in which hyperbolic functions can be employed instead of the usual periodic functions in the perturbation procedure the generalized van der pol oscillator in which f x x 1 x 2 x 2 x is studied to illustrate the accuracy of the present method its predictions are compared with those of runge kutta method

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

lower bounds for parametric estimation with constraints a chapman robbins form of the barankin bound is used to derive a multiparameter cramer rao cr type lower bound on estimator error covariance when the parameter theta in r sup n is constrained to lie in a subset of the parameter space a simple form for the constrained cr bound is obtained when the constraint set theta sub c can be expressed as a smooth functional inequality constraint it is shown that the constrained cr bound is identical to the unconstrained cr bound at the regular points of theta sub c i e where no equality constraints are active on the other hand at those points theta in theta sub c where pure equality constraints are active the full rank fisher information matrix in the unconstrained cr bound must be replaced by a rank reduced fisher information matrix obtained as a projection of the full rank fisher matrix onto the tangent hyperplane of the full rank fisher matrix onto the tangent hyperplane of the constraint set at theta a necessary and sufficient condition involving the forms of the constraint and the likelihood function is given for the bound to be achievable and examples for which the bound is achieved are presented in addition to providing a useful generalization of the cr bound the results permit analysis of the gain in achievable mse performance due to the imposition of particular constraints on the parameter space without the need for a global reparameterization

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

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

the number of limit cycle bifurcation diagrams for the generalized mixed rayleigh lienard oscillator abstract this paper investigates the generalized mixed rayleigh lienard oscillator with highly nonlinear terms not restrict to the number of limit cycles this analysis considers mainly the number of limit cycle bifurcation diagrams of the system first the singularity theory approach is applied to the first order averaged approximation of the system with lower order nonlinear terms to reveal all possible bifurcation diagrams by summarizing the generating rule and structural distinction of different bifurcation diagrams a numerical procedure is then developed calculation suggests that the number of bifurcation diagrams increase very fast as the order of nonlinear terms lastly numerical simulations are adopted to approve the analytical results

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 generalization of the method of harmonic balance a procedure is provided for generalizing the method of harmonic balance to obtain higher order approximations to the periodic solutions of differential equations for two systems one a conservative system and the other nonconservative the procedure is currently being applied to investigate the possible solution behaviors of singular nonlinear oscillators where the usual perturbation methods do not work

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

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

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

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

on the achievability of the cramer rao bound for poisson distribution this correspondence examines the cramer rao cr bound for data obtained in emission tomography the likelihood function involved is the combined probability of independent poisson random variables the expectation of each being a linear function c sub i sup t spl lambda of the parameter vector spl lambda we investigated the achievability of the cr bound in the interior and on the boundary of the domain of the problem for the former we found that the cr bound is achievable if and only if the c sub i vectors are obtained from a basis for r sup n by repeating some vectors multiplied by constant factors a similar result holds for the boundary case the practical implication of the achievability condition is that the cr bound is not attainable for typical emission tomographic systems

generalized exponential circular and hyperbolic functions for nonlinear wave equations wave functions are presented in the form of generalized exponentials that are solutions of some of the most usual linear and nonlinear wave equations the solutions are given in terms of the elliptic functions of jacobi and presented in a form as similar as possible to the usual circular functions some simple theorems are demonstrated to present the solutions as the simplest possible extension of the usual exponentials

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

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

quantum anharmonic symmetrical oscillators using elliptic functions the authors study in the jwkb approximation the energy levels of the symmetric anharmonic oscillators v x ax2 bx4 for different signs and values of a and b comparisons are made with published results for specific cases and with numerical calculations they given an additional example of exact value to add to the very rare catalogue of known examples

periodic solutions of strongly non linear oscillators by the multiple scales method abstract the multiple scales method developed for the systems with small non linearities is extended to the case of strongly non linear self excited systems two types of non linearities are considered quadratic and cubic the solutions are expressed in terms of jacobian elliptic functions higher order approximations of solution as well as modulations of amplitude and phase are derived comparisons to numerical simulations are provided and discussed