measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

maximal symmetry group of the hamilton jacobi equation relativistic particle in flat spacetime lie s extended group method has been used to obtain the maximal symmetry group of the hamilton jacobi equation for a relativistic particle moving in flat spacetime for a massive particle it is the 21 parameter inhomogeneous pseudo orthogonal group io 4 1 of signature 4 1 for a zero mass particle like a photon or a neutrino this group is an infinite parameter lie group with an infinite parameter invariant subgroup such that the factor group is isomorphic to the inhomogeneous pseudo orthogonal group io 3 1 of signature 3 1

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

the algebraic structure of generalized ermakov systems in three dimensions the characteristic algebra of generalized ermakov systems is sl 2 r the structure of these systems in three dimensions is obtained a subset in the form of an equation of motion with the additional requirement of so 3 symmetry is studied it includes the classical equation of the magnetic monopole the existence of three vectors of poincare type is established consideration is given to weak generalized ermakov systems in which the symmetry breaking occurs in the radial equation

active learning applied to rcs computations with nonuniform sampling using different objective functions an active learning framework is introduced to reduce the number of frequencies and angles one must consider for wideband monostatic scattering computations or measurements this method is used to optimally select those frequencies and angles that would be most informative resulting in nonuniform sampling and often a reduced number of points vis agrave vis uniform sampling in this paper we focus on jointly two dimensional optimal sampling in frequency and incident angle thetas for monostatic scattering the method consists of two basic steps one step involves estimation of model parameters using a least square ls algorithm the next step is to optimally choose the next point frequency and thetas for analysis by the computational model or experiment this new point is selected with the goal of reducing uncertainty in the parametric model quantified via the fisher information matrix iterating these two steps a sequence of numerical computations or measurements are performed each at the most informative point for learning the parameters of the associated simpler parametric model this idea is demonstrated here in the context of reducing the number of points frequencies and orientations at which a computational model must be employed and in order to avoid repeatedly gathering samplings at the edge of the input space an alternative objective function is applied which makes the actively selected points closer to a region of interest

a class of integrable potentials a class of time independent two dimensional integrable potentials all possessing an invariant of the same general form is constructed one of these potentials is superintegrable its invariants realize the symmetry algebra so 3 for negative energies e 2 for zero energy and so 2 1 for positive energies a transformation of coupling constants reveals that in parabolic coordinates this potential is the harmonic oscillator acted on by constant forces this and another potential in the class may be considered as successive extensions of the kepler potential the analytic properties of these integrable systems in the complex time plane are also discussed

symmetry and singularity properties of second order ordinary differential equations of lie s type iii abstract we extend the work of abraham shrauner b abraham shrauner hidden symmetries and linearization of the modified painleve ince equation j math phys 34 1993 4809 4816 on the linearization of the modified painleve ince equation to a wider class of nonlinear second order ordinary differential equations invariant under the symmetries of time translation and self similarity in the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations

hamiltonian structure and integrability of the stationary kuramoto sivashinsky equation in this article the stationary kuramoto sivashinsky ks equation is studied it is shown first that this equation can be transformed into a one dimensional hamiltonian system describing the motion of a particle in a time dependent potential second working with this formalism and coming back then to the initial ks equation a subset of initial conditions is derived for which the corresponding solutions u x will behave like the dominant pole 120 x x0 3 deduced from the painleve analysis and will explode in a finite time finally special attention is given to the well known kuramoto tsuzuki solution and it is shown that provided the independent variable x is initialized in a specified domain this solution explodes also like 120 x x0 3

n dimensional equations with the maximum number of symmetry generators the authors obtain the general class of ordinary differential equations that can be reduced by a point transformation to the free particle equation and given the general form of the lie symmetry generators for these equations they apply these results to obtain the symmetry generators for the n dimensional harmonic oscillator and for a particle in a constant magnetic field

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

symmetries of linear systems of second order ordinary differential equations in this paper several problems concerning the lie algebra structure of symmetries and variational symmetries of a general linear system of second order ordinary differential equations are studied in particular a necessary and sufficient condition is obtained in terms of the coefficients of the system for the system s symmetry algebra to be of maximal dimension i e n2 4n 3 and isomorphic to sl n 2 r the well known symmetry algebra of the free particle equation x 0 when this condition is satisfied it is proved that the system is lagrangian and that its variational symmetry algebra is isomorphic to a fixed n2 3n 6 2 dimensional lie algebra whose structure levi mal cev decomposition and realization by means of a matrix algebra is determined for the particular case of isotropic systems which includes as far as is known all the examples treated in the literature explicit formulas for the generators of both the symmetry algebra and the variational symmetry algebra are obtained

discrete models of electromagnetic wave scatterers in a frequency range we investigate the spatial and frequency characteristics of electromagnetic fields scattered by some object frequency functions that allow the expression of all elements of the object scatter matrix are introduced the sampling theorems that are proved for the introduced functions make it possible to characterize the scatterer by a set of discrete parameters the application of the proved theorems provides for the creation of a versatile scatterer discrete model that determines the dependence of scattered fields both on spatial coordinates and frequency a random scatterer is considered and the conditions under which the discrete parameters are statistically independent normalized random variables are stated the article presents the results of a random scatterer modeling

efficient calculation of the electromagnetic dyadic green s function in spherical layered media a numerical methodology is proposed for the evaluation of the electromagnetic fields of an electric or magnetic dipole of arbitrary orientation in spherical stratified media the proposed methodology is based on the numerical solution of the differential equation in the radial coordinate that is satisfied by each coefficient in the spherical harmonics expansion of the governing helmholtz equation more specifically the finite difference solution of this equation may be cast in a pole residue form that allows the analytic evaluation of the spherical harmonics series by means of the watson transformation thus a closed form expression is obtained for the electromagnetic fields in terms of a short series of associated legendre functions p sup m sub spl nu cos spl theta of integer order m and complex degree spl nu the number of terms in the series is strongly dependent on the angle spl theta and decreases very fast when the point of observation moves away from the source the method allows for arbitrary variation in the permittivity and permeability profiles in the radial directions

asymptotic extraction approach for antennas in a multilayered spherical media an efficient algorithm is introduced to enhance the convergence of dyadic green s functions dgf in a layered spherical media where asymptotic expressions have been developed the formulated expressions involve an infinite series of spherical eigenmodes that can be reduced to the simple homogenous media green s function using the addition theorem of spherical hankel functions substantial improvements in the convergence speed have been attained by subtracting the asymptotic series representation from the original dgf the subtracted components are then added to the solution using the homogenous media green s function format

autonomous self similar ordinary differential equations and the painleve connection we demonstrate an intimate connection between nonlinear higher order ordinary differential equations possessing the two symmetries of autonomy and self similarity and the leading order behaviour and resonances determined in the application of the painleve test similar behaviour is seen for systems of first order differential equations several examples illustrate the theory in an integrable case of the abc system the singularity analysis reveals a positive and a negative resonance and the method of leading order behaviour leads naturally to a laurent expansion containing both

estimating correlation dimension from a chaotic time series when does plateau onset occur abstract suppose that a dynamical system has a chaotic attractor a with a correlation dimension d 2 a common technique to probe the system is by measuring a single scalar function of the system state and reconstructing the dynamics in an m dimensional space using the delay coordinate technique the estimated correlation dimension of the reconstructed attractor typically increases with m and reaches a plateau on which the dimension estimate is relatively constant for a range of large enough m values the plateaued dimension value is then assumed to be an estimate of d 2 for the attractor in the original full phase space in this paper we first present rigorous results which state that for a long enough data string with low enough noise the plateau onset occurs at m ceil d 2 where ceil d 2 standing for ceiling of d 2 is the smallest integer greater than or equal to d 2 we then numerical examples illustrating the theoretical prediction in addition we discuss new findings showing how practical factors such as a lack of data and observational noise can produce results that may seem to be inconsistent with the theoretically predicted plateau onset at m ceil d 2

a model based parameter estimation technique for wide band interpolation of periodic moment method impedance matrices with application to genetic algorithm optimization of frequency selective surfaces a model based parameter estimation mbpe technique is introduced in this paper for efficiently interpolating periodic moment method pmm impedance matrices over a wide frequency band in the model only the floquet harmonics that strongly affect the frequency band of interest are employed to approximate the matrix elements while the contributions from all other higher order harmonics are compactly represented by two additional terms the derivation of the model is physics based and the objective is to find the coefficients of the terms in the model by utilizing the values of the impedance matrix elements calculated via pmm at only a few frequency points the number and position of these fitting points can be pre determined from the cutoff frequencies of the floquet harmonics which allows the mbpe interpolation process in this case to be completely automated in other words the number and position of the sampling points are only dictated by the periodicity of the frequency selective surface fss structure and the frequency range of interest unlike many of the other scattering parameter based techniques the shape and the resonances in the response of the fss do not have any impact on the construction of the interpolation model this makes it particularly useful in genetic algorithm ga aided fss design since for a fixed periodicity and frequency range the mbpe interpolation is independent of the scattering response of candidate fss designs several examples of the new pmm mbpe approach are presented including one in which it is used to considerably speed up the ga based design process for a reconfigurable fss

a generalized asymptotic extraction solution for antennas in multilayered spherical media an efficient model is developed to accelerate the convergence of the dyadic green s function s dgf infinite summation when the source and observation points are placed in different layers of a dielectric sphere thereby expediting computational analysis the proposed procedure is based on asymptotic extraction principles in which the quasi static images are extracted from the spectral domain dgf the effectiveness of the approach is demonstrated in a method of moment model where a microstrip antenna as well as a conformal dipole array have been studied

interplay of symmetries null forms darboux polynomials integrating factors and jacobi multipliers in integrable second order differential equations in this work we establish a connection between the extended prellesinger procedure with five other analytical methods which are widely used to identify integrable systems in the contemporary liter

solutions in closed form and as power series to the real lorenz equations using the method of the lie theory of extended groups and for the parameter values b 1 and r 0 we construct explicitly the general exact solution to the real lorenz equations in terms of jacobian elliptic functions in the context of our approach further possible completely integrable cases of the lorenz system are discussed by considering the result of the painleve analysis for 1 b 2 and r 1 9 and negative values of r the latter case r 0 and 0 not following from the painleve test for other positive parameter values and in the form of appropriate power series we find some particular exact solutions which do not possess the painleve property

integrals and symmetries the bernoulli laplace lenz vector the notions of independent constants of motion are clarified emphasising the role of precise definition of the background space using a well defined algorithm that relates symmetries of equations of motion of a classical system newtonian lagrangian or hamiltonian to a conserved object in one to one fashion the symmetries of the coulomb and linear force problems are determined

modified spherical wave functions with anisotropy ratio application to the analysis of scattering by multilayered anisotropic shells we describe a novel and rigorous vector eigenfunction expansion of electric type green s dyadics for radially multi layered uniaxial anisotropic media in terms of the modified spherical vector wave functions which can take into account the effects of anisotropy ratio systematically in each layer the material constitutions e and epsiv macrmu macr are tensors and distribution of sources is arbitrary both the unbounded and scattering dyadic green s functions dgfs for rotationally uniaxial anisotropic media are derived in spherical coordinates r thetas phi the coefficients of scattering dgfs based on the coupling recursive algorithm satisfied by the coefficient matrix are derived and expressed in a compact form with these dgfs obtained the electromagnetic fields in each layer are straightforward once the current source is known a specific model is proposed for the scattering and absorption characteristics of multilayered uniaxial anisotropic spheres and some novel performance regarding anisotropy effects is revealed

an acoustic inverse scattering problem for spheres with radially inhomogeneous compressibility an inverse acoustic scattering problem the main aim of which is to reconstruct the one dimensional variation of the acoustical parameters of a spherical object is investigated the problem is first formulated conventionally through a coupled system of integral equations and then this system is reduced to one dimensional form by using the orthogonality properties of spherical harmonics the inverse problem is solved in an iterative fashion via classical newton algorithm some numerical simulations are carried out to test the feasibility of the method as well as to see the effects of some parameters on the solution it is shown that the method is very effective for the profiles having smooth variations provided that an appropriate initial guess is chosen however some of the classical disadvantages of the newton type algorithms are also observed in numerical experiments which may limit the applicability of the method to a certain extent

symmetries first integrals and the inverse problem of lagrangian mechanics deals with the following question given a symmetry vector field y of a system of second order ordinary differential equations and an associated constant of the motion f is it possible to find a lagrangian l for the system such that y becomes a noether symmetry with respect to l and f its implied noether constant for one degree of freedom systems the answer to this question is affirmative in addition attention is paid to the construction of a suitable constant of the motion f for given symmetry y and vice versa several examples are discussed

jacobi s last multiplier and symmetries for the kepler problem plus a lineal story we calculate the first integrals of the kepler problem by the method of jacobi s last multiplier using the symmetries for the equations of motion also we provide another example which shows that jacobi s last multiplier together with lie symmetries unveils many first integrals neither necessarily algebraic nor rational whereas other published methods may yield just one

lie symmetries and invariants for the time dependent generalizations of the equation r c1rnl c2rmr 0 in this comment the authors use an invertible point transformation for the study of time dependent equations derived from the equation r c1rn l c2rmr 0 this procedure generalizes some results obtained recently by leach and gorringe 1990

a nonlocal connection between certain linear and nonlinear ordinary differential equations part ii complex nonlinear oscillators in this paper we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions for this purpose we introduce two types of nonlocal transformations which relate specific classes of nonlinear complex ordinary differential equations odes with complex linear odes thereby proving the integrability of the former we also show how to construct the solutions using the two types of nonlocal transformations with several physically interesting cases as examples

the model based parameter estimation of antenna radiation patterns using windowed interpolation and spherical harmonics techniques previously reported for interpolating antenna radiation patterns simultaneously in both the spatial and frequency domains are reexamined here in order to develop new algorithms with improved efficiency and speed while maintaining comparable accuracy to this end interpolation in the frequency domain is constrained to a windowed scheme whereby a series of reduced order pade spl acute rational function fitting models of only three or four sampling frequencies are used to interpolate over a relatively large bandwidth the use of this piece wise technique is shown to reduce the complexity of determining the unknown coefficients without any significant loss in the resulting interpolation accuracy or increase in the required number of sampling frequencies in this paper exact analytical expressions are found for the unknown coefficients that allow the spatial domain interpolation to be performed entirely separate from the frequency domain interpolation the technique is applied to the case of a 0 5 meter dipole modeled from 150 to 950 mhz and for spl theta from 0 spl deg to 90 spl deg using fitting windows with three and four sampling frequencies the results are compared to those obtained using a previously developed generalized simultaneous interpolation procedure in the spatial domain spherical harmonics are introduced as model based fitting functions for antenna radiation patterns specifically interpolation procedures are developed using zonal harmonics and tesseral harmonics to model one and two dimensional far field radiation patterns respectively the case of a z oriented 0 5 meter dipole is considered to demonstrate the efficiency and accuracy obtained by applying these physically based fitting models in the spatial domain zonal harmonics are used to interpolate far field radiation patterns of this antenna for spl theta from 0 spl deg to 180 spl deg for several different frequencies over the range from 150 to 950 mhz this example demonstrates the attractiveness of using spherical harmonics functions to accurately and efficiently interpolate radiation patterns in the spatial domain

ordinary differential equations invariant under translation in the independent variable and rescaling the lagrangian formulation abstract the lagrangian formulation of the class of general second order ordinary differential equations invariant under translation in the independent variable and rescaling is presented the differential equations arising from this analysis are analysed using the painleve test the well known differential equation y y y k y 3 0 is a unique member of this class when k 3 since it is linearisable by a point transformation a wider subset is shown to be linearisable by means of a nonlocal transformation

maximum approximate entropy and r threshold a new approach for regularity changes detection abstract approximate entropy apen has been widely used as an estimator of regularity in many scientific fields it has proved to be a useful tool because of its ability to distinguish different system s dynamics when there is only available short length noisy data incorrect parameter selection embedding dimension m threshold r and data length n and the presence of noise in the signal can undermine the apen discrimination capacity in this work we show that r m a x a p e n m r m a x n a p e n m a x can also be used as a feature to discern between dynamics moreover the combined use of a p e n m a x and r m a x allows a better discrimination capacity to be accomplished even in the presence of noise we conducted our studies using real physiological time series and simulated signals corresponding to both low and high dimensional systems when a p e n m a x is incapable of discerning between different dynamics because of the noise presence our results suggest that r m a x provides additional information that can be useful for classification purposes based on cross validation tests we conclude that for short length noisy signals the joint use of a p e n m a x and r m a x can significantly decrease the misclassification rate of a linear classifier in comparison with their isolated use

sl 3 r as the group of symmetry transformations for all one dimensional linear systems the converse problem of similarity analysis is solved in general for the finite symmetry transformations of any inhomogeneous ordinary linear differential equation of the second order x f2 t x f1 t x f0 t the eight parameter realizations of the symmetry group are obtained in the form f 1p2 f where f stands for transformations of t x that depend exclusively on the fundamental solutions of the equation and where p2 is an arbitrary projective transformation in the plane thus it is shown that the full point symmetry group corresponds to sl 3 r indeed without recourse to the lie algebra also a technique is obtained for calculating the finite point symmetry realization of sl 3 r for any given one dimensional linear system some miscellaneous examples are given

a unification in the theory of linearization of second order nonlinear ordinary differential equations in this letter we introduce a new generalized linearizing transformation glt for second order nonlinear ordinary differential equations snodes the well known invertible point ipt and non point transformations npt can be derived as sub cases of the glt a wider class of nonlinear odes that cannot be linearized through npt and ipt can be linearized by this glt we also illustrate how to construct glts and to identify the form of the linearizable equations and propose a procedure to derive the general solution from this glt for the snodes we demonstrate the theory with two examples which are of contemporary interest

the generalized damped cubic equation integrability and general solution the nonlinear ordinary differential equation of the form u f1u f2u f3u3 o representing a generalized damped cubic equation is fully analysed the author finds the general expression for f2 in terms of f1 and f3 that renders the equation integrable also he shows that if f2 satisfies the above mentioned condition the former nonlinear differential equation can always be reduced through adequate changes of variable to the second painleve transcendent

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve

the simultaneous interpolation of antenna radiation patterns in both the spatial and frequency domains using model based parameter estimation the pade rational function fitting model commonly used for model based parameter estimation mbpe in the frequency domain is enhanced to include spatial dependence in the numerator and denominator coefficients this allows the function to interpolate an antenna radiated electric field pattern in both the frequency and spatial domains simultaneously such that a single set of coefficients can be used to accurately reconstruct an entire radiation pattern at any frequency in the fitting model range a simple procedure is introduced for transforming interpolated electric fields into gain patterns using input impedance versus frequency curves also obtained via mbpe the utility of this method is demonstrated by applying it to a dipole antenna over a frequency range of 150 950 mhz and using a polynomial representation in spl theta for the coefficient spatial dependence it is also used to estimate radiation patterns for a three element yagi array between the frequencies of 470 and 500 mhz using a binomial representation for the spatial variation that includes terms dependent on spl theta as well as spl phi the use of this method for interpolating radiation patterns has at least two significant advantages one being large compression ratios for the amount of data that must be stored to accurately reproduce patterns and the other being a significant decrease in the amount of time required for modeling problems with large computational domains

some remarks on the generalized noether theory of point symmetry transformations of the lagrangian the noether theory of infinitesimal point symmetry transformations is revisited and generalized under the broad scope of the general theory of transformations of lagrangian mechanics the basic generalized point symmetries of a lagrangian function are thus obtained and briefly discussed it is proved that under very general and physically reasonable provisions the point symmetry group of the lagrangian is necessarily a finite lie group and a rather simple technique is then introduced for the explicit calculation of the associated lie algebra next the algebra obeyed by the set of basic noether quantities which also includes the traditional noether constants of motion is examined in this way it is shown that not all the generalized point symmetries of a lagrangian yield an associated conservation law the author presents three simple examples of this not so well known aspect of the noether theory

new classes of conversed quantities associated with non noether symmetries a result of hojman and harleston 1981 has been used to extend the known relationship between conserved quantities and non noether symmetries it is shown that n constants of the motion may be derived from each non noether symmetry of an n dimensional lagrangian system this applies whether the symmetry is continuous or discrete in addition the properties of the cartan form are utilised to to give a new proof of the hojman harleston theorem

escape to infinity under the action of a potential and a constant electromagnetic field escape to infinity is proved for a great variety of potentials including the potential created by an infinite number of sources relativistic escape is studied escape in the presence of a constant electromagnetic field and a potential is also considered

a complete basis set of functionally independent invariants of continuous lie groups using the methods of differential equations the author has obtained the complete set of functionally independent bases of invariants for some continuous lie groups of mathematical physics i the conformal group ii the semi direct product of the weyl group and the one parameter group of psi translation iii the direct product of the poincare group and the one parameter group of psi scaling iv the weyl group v the 14 parameter maximal symmetry group of the schrodinger equation for the coulombic system of atoms and molecules and vi su 3 of elementary particle physics the first four groups are the only groups which together with the poincare group are the maximal symmetry groups of all forms of the non linear klein gordon equation

dynamical noether symmetries the authors use noether s theorem within a restricted class of dynamical transformations involving velocity dependence to obtain first integral constants of the motion not available as noether point transformation constants

on the relation between correlation dimension approximate entropy and sample entropy parameters and a fast algorithm for their calculation abstract we explore the relation between correlation dimension approximate entropy and sample entropy parameters which are commonly used in nonlinear systems analysis using theoretical considerations we identify the points which are shared by all these complexity algorithms and show explicitly that the above parameters are intimately connected and mutually interdependent a new geometrical interpretation of sample entropy and correlation dimension is provided and the consequences for the interpretation of sample entropy its relative consistency and some of the algorithms for parameter selection for this quantity are discussed to get an exact algorithmic relation between the three parameters we construct a very fast algorithm for simultaneous calculations of the above which uses the full time series as the source of templates rather than the usual 10 this algorithm can be used in medical applications of complexity theory as it can calculate all three parameters for a realistic recording of 10 4 points within minutes with the use of an average notebook computer

lie symmetries for the charge monopole problem the authors obtain the lie symmetries for the equation of motion of an electric charge interacting with a magnetic monopole fixed at the origin they also discuss the construction of first integrals for this problem

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

some new aspects of first integrals and symmetries for central force dynamics for the general central force equations of motion in n 1 dimensions a complete set of 2n first integrals is derived in an explicit algorithmic way without the use of dynamical symmetries or noether s theorem the derivation uses the polar formulation of the equations of motion and yields energy angular momentum a generalized laplace runge lenz vector and a temporal quantity involving the time variable explicitly a variant of the general laplace runge lenz vector which generalizes hamilton s eccentricity vector is also obtained the physical meaning of the general laplace runge lenz vector its variant and the temporal quantity is discussed for general central forces their properties are compared for precessing bounded trajectories versus non precessing bounded trajectories as well as unbounded trajectories by considering an inverse square force kepler problem and a cubically perturbed inverse square force newtonian revolving orbit problem

symmetries and integrability of a fourth order euler bernoulli beam equation the complete symmetry group classification of the fourth order euler bernoulli ordinary differential equation where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement is obtained we perform the lie and noether symmetry analysis of this problem in the lie analysis the principal lie algebra which is one dimensional extends in four cases viz the linear exponential general power law and a negative fractional power law it is further shown that two cases arise in the noether classification with respect to the standard lagrangian that is the linear case for which the noether algebra dimension is one less than the lie algebra dimension as well as the negative fractional power law in the latter case the noether algebra is three dimensional and is isomorphic to the lie algebra which is sl 2 r this exceptional case although admitting the nonsolvable algebra sl 2 r remarkably allows for a two parameter family of exact solut

the relationship between the symmetries of and the existence of conserved vectors for the equation r f r l g r r 0 the existence of explicit expressions for conserved vectors for the charge monopole problem and the kepler problem is well known the lie algebras of the point transformations under which the equations of motion are invariant have been found more recently the recent discovery of an explicit expression for a laplace runge lenz like vector for the equation of motion r h l r hh kr 2 r o h h r from which one of the equations for the orbit is easily obtained has prompted the question of what is the lie algebra of the point symmetries of the equation r f r l g r r bold o of which each of the above problems is a member in the case that f r identical to o it is well known that laplace runge lenz like vectors exist the existence of such conserved vectors does not imply a particular algebraic structure of the lie point symmetries of the equation of motion however the existence of such symmetries provides a systematic method for constructing the vectors

on the lie symmetries of the classical kepler problem the standard version of noether s theorem when applied to the classical kepler problem leads to the constants of energy and angular momentum but does not give the hidden symmetry known as the runge lenz vector lie s theory of differential equations is used to obtain all three constants of motion the transformations of solutions under the point transformations to which these constants correspond are studied the results are generalised to n dimensions

a canonical reduction of order for the kepler problem using a new redudion of order technique we obtain four conserved quantities of the darsical planar kepler problem via the symmetries of the equations of motion and an extra vector field by passing to a suitable quotient manifold of the evolution space we deduce that negative energy orbits are closed and periodic without having to solve the differential equation in this paper we look at the classical kepler problem through the application of a new reduction of order technique l we believe that this technique provides a systematic means of producing first integrals from group actions more general than symmetries without relying on the existence of a variational principle moreover it is a powerful tool for investigating global properties such as periodicity the kepler problem is a good vehicle for a demonstration of these features the planar equations of motion are defined by a second order differential equation field on five dimensional evolution space there are three point symmetries which take solutions of the equation of motion into other solutions time translation which leaves the solution curves invariant but alters their starting times rotation about the origin which again leaves the curves invariant but alters the initial angular position and the transformation associated with the runge leuz vector which allows the orientation and eccentricity of the orbits to remain the same but changes the semi latus rectum we wish to show that conserved quantities of the system i e the objects which remain invariant along solution curves can be obtained using alternative techniques to noether s theorem 2 or solving the differential equation 3 41 lie s method 2 provides us with the three point symmetries mentioned above to effect our technique a fourth vector field is required which is a symmetry of the integrable distribution made up of the second order differential equation field and its three symmetries to find this field we modify the usual reduction of order via symmetry method by taking successive quotients of the extended phase space by the three actions for example we project down to a four dimensional manifold by setting the time coordinate to a constant thus identifying all the orbits which only differ by a time translation and we reduce our differential equation field to first order on a two dimensional quotient

a novel technique for the approximation of 3 d antenna radiation patterns in this paper a novel technique for the approximation of three dimensional 3 d antenna radiation patterns is presented the proposed method combines the two principal cuts in order to acquire an adequate estimate of the 3 d antenna radiation solid the absolute error from the theoretical solution is analyzed along with other statistical measures for various types of antenna structures namely omni directional and directive arrays the performance of the method is compared against other existing extrapolation algorithms the proposed technique exhibits low approximation errors and is easily integrated into 3 d radio propagation planning tools such as ray tracing algorithms

the resonance mode theory for exterior problems of electrodynamics and its application to discrete antenna modeling in a frequency range the problem under consideration here is that of building a versatile discrete antenna model in a frequency range the eigenfunction mode expansion or singularity expansion method sem is used in working out the problem a detailed substantiation of sem is provided and an expression determining the dependence of mode amplitudes on exterior source fields is derived a simple example is provided in support of the theory the sem is used in inquiring into the frequency dependence of antenna generalized matrix entries it is intimated that all of the matrix entries contain resonance factors and factors that are hankel transforms of radial coordinate functions a sampling theorem which is a generalized extension of the classic sampling theorem is used for the hankel transform the application of the generalized sampling theorem completes the process of building a versatile discrete antenna model in a frequency range

invariants for dissipative nonlinear systems by using rescaling a rescaling transformation of space and time is introduced in the study of nonlinear dissipative systems that are described by a second order differential equation with a friction term proportional to the velocity t v the transformation is of the form x t where x c t t d dt a2 t this rescaling is used to find each potential for which there exists an exact invariant quadratic in the velocity and to find the invariant the invariants are found explicitly for a power law potential t xm 1 m 1 and an arbitrary coefficient of friction t we show in an example how the rescaling transformation can be chosen to give an asymptotic solution of the equation in cases where the exact invariant does not exist for certain parameters the asymptotic solution is a self similar solution that is an attractor for all initial conditions the technique of applying a rescaling transformation has been useful in other problems and may have additional practical applications

lagrangian formalism for nonlinear second order riccati systems one dimensional integrability and two dimensional superintegrability the existence of a lagrangian description for the second order riccati equation is analyzed and the results are applied to the study of two different nonlinear systems both related with the generalized riccati equation the lagrangians are non natural and the forces are not derivable from a potential the constant value e of a preserved energy function can be used as an appropriate parameter for characterizing the behavior of the solutions of these two systems in the second part the existence of two dimensional versions endowed with superintegrability is proved the explicit expressions of the additional integrals are obtained in both cases finally it is proved that the orbits of the second system that represents a nonlinear oscillator can be considered as nonlinear lissajous figures

exact invariants for a class of time dependent nonlinear hamiltonian systems a method of generalizing a class of invariants for a time dependent linear oscillator is developed for the motion of a mass point in one dimension with a general time dependent nonlinear potential formulas are derived for the allowable time dependent potentials and for the corresponding invariants the method by which these conclusions are reached is interesting theoretically and is explained in detail

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

symmetries of first integrals and their associated differential equations the relationship between the reduction of order through point symmetries and integration is explored with particular emphasis on the loss and gain of point contact for third order symmetries to and from nonlocal symmetries it is seen that reduction of order can even lead to the loss of all point symmetries at the third order level and their replacement at the second order level from nonlocal symmetries it is evident that nonlocal symmetries should be given more attention in applications

characterisation of the algebraic properties of first integrals of scalar ordinary differential equations of maximal symmetry abstract we undertake a study of the first integrals of linear n th order scalar ordinary differential equations with maximal symmetry we establish patterns for the first integrals associated with these equations it is shown that second and third order equations are the pathological cases in the study of higher order differential equations the equivalence of contact symmetries for third order equations to non cartan symmetries of second order equations is highlighted

generalizations of noether s theorem in classical mechanics in this paper a review is presented of various approaches to the generalization of the version of noether s theorem which is presented in most textbooks on classical mechanics its motivation is the controversy still persisting around the possible scope of a noether type theorem allowing for velocity dependent transformations our analysis is centered around the one factor common to all known treatments namely the structure of the related first integral we first discuss the most general framework in which a function of the above mentioned structure constitutes a first integral of a given lagrangian system and show that one cannot really talk about an interrelationship between symmetries and first integrals there we then compare different proposed generalizations of noether s theorem by describing the nature of the restrictions which characterize them when they are situated within the broadest framework we prove a seemingly new equivalence result between the two main approaches that of invaria

properties of second order ordinary differential equations invariant under time translation and self similar transformation the general form of a second order ordinary differential equation invariant under time translation and self similarity is obtained together with its solution in parametric form next two representatives of two different families of such equations are studied in terms of rescaled phase space variables the first example has a simple physical interpretation and its limit cycle properties and period are easily derived similar results are found for the second example but the physical interpretation is less obvious the invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self similar solution sss exists

reflections on the symmetry conservation law duality and the runge lenz vector following recent interest in an earlier work on the kepler problem prince and eliezer 1981 a number of remarks are made concerning the many faceted symmetry conservation law duality in classical mechanics in particular the advantages of associating a number of symmetries with the runge lenz vector are discussed

a spherical wave representation of the dyadic green s function for a wedge a new expansion for the dyadic green s function of a perfectly conducting wedge is derived it is expressed in terms of a discrete series of spherical vector wave functions this expansion converges well except near a spherical surface which passes through the source point the convergence is particularly rapid in the paraxial region this results in the identification of an edge wave which can play an important role in the radiation from sources and scatterers at the edge of the wedge

invariants for time dependent potentials use of self similar techniques abstract we seek a generalization of the derivation by lewis and leach for invariants quadratic in p since the general equations for invariants cubic and quartic in p cannot be solved we specialize to self similar solutions of the equations consequently we can recover only a part of their solutions for quadratic invariants but new pairs of potentials and invariants are obtained where the leading term in the invariants is p3 or p4 more precisely all the potentials are of the form v q t at2 2 q t b q2 t2 where is prescribed and a is arbitrary different functional forms for b lead to different families of solutions where for each family the parameter is allowed to go from to finally a generalization to potentials with three terms is given

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

parametric modeling of radiation patterns and scattering parameters of antennas this paper describes a data driven method to model the radiation patterns over a large angular region and scattering parameters of antennas as a function of the geometry of the antenna the radiation pattern model consists of a linear combination of characteristic basis function patterns cbfps where the expansion coefficients of the cbfps are the functions of geometrical features of the antenna scattering parameters are modeled by means of parameterized state space matrices the obtained models are quick to evaluate and are thus suitable for design activities where multiple simulations are required the proposed method is validated through illustrative examples

a new technique for the derivation of closed form electromagnetic green s functions for unbounded planar layered media a closed form electromagnetic green s function for unbounded planar layered media is derived in terms of a finite sum of hankel functions the derivation is based on the direct inverse hankel transform of a pole residue representation of the spectral domain form of the green s function such a pole residue form is obtained through the solution of the spectral domain form of the governing green s function equation numerically through a finite difference approximation rather than analytically the proposed methodology can handle any number of layers including the general case where the planar media exhibit arbitrary variation in their electrical properties in the vertical direction the numerical implementation of the proposed methodology is straightforward and robust and does not require any preprocessing of the spectrum of the green s function for the extraction of surface wave poles or its quasi static part the number of terms in the derived closed form expression is chosen adaptively with the distance between source and observation point as parameter the development of the closed form green s function is presented for both vertical and horizontal dipoles its accuracy is verified through a series of numerical examples and comparisons with results from other established methods

sl 3 r as the group of symmetry transformations for all one dimensional linear systems iii equivalent lagrangian formalisms the sl 3 r theory of projective transformations of the plane is applied to the lagrangians of all one dimensional newtonian linear systems noether and non noether equivalent lagrangians as well as the associated noether and non noether constants of motion are thus obtained in a completely general and systematic way complete unification is achieved by this group theoretic approach to lagrangians of one dimensional linear systems

jet fields connections and second order differential equations jet fields are defined on fibred manifolds and represented as type 1 1 tensor fields corresponding to cartan ehresmann connections when defined on the first jet bundle they may be used to generalise the second order differential equation fields used in theoretical mechanics and to extend several existing results to field theories