oblivious polynomial evaluation oblivious polynomial evaluation is a protocol involving two parties a sender whose input is a polynomial p and a receiver whose input is a value alpha at the end of the protocol the receiver learns p alpha and the sender learns nothing we describe efficient constructions for this protocol which are based on new intractability assumptions that are closely related to noisy polynomial reconstruction oblivious polynomial evaluation can be used as a primitive in many applications we describe several such applications including protocols for private comparison of data for mutually authenticated key exchange based on possibly weak passwords and for anonymous coupons

maximum tsallis entropy with generalized gini and gini mean difference indices constraints abstract using the maximum entropy principle with tsallis entropy some distribution families for modeling income distribution are obtained by considering income inequality measures maximum tsallis entropy distributions under the constraint on generalized gini and gini mean difference indices are derived it is shown that the tsallis entropy maximizers with the considered constraints belong to generalized pareto family

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

nonlinear oscillator with discontinuity by generalized harmonic balance method a generalized harmonic balance method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn x this method is a modification of the generalized harmonic balance method in which analytical approximate solutions have rational form this approach gives us not only a truly periodic solution but also the frequency of the motion as a function of the amplitude of oscillation we find that this method works very well for the whole range of amplitude of oscillation in the case of the antisymmetric piecewise constant force oscillator and excellent agreement of the approximate frequencies with the exact one has been demonstrated and discussed for the second order approximation we have shown that the relative error in the analytical approximate frequency is 0 24 we also compared the fourier series expansions of the analytical approximate solution and the exact one comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient

oscillators with nonlinear elastic and damping forces abstract generalization to the previous oscillators is done by introducing the nonlinear elastic and damping forces the mathematical model of the system is a second order differential equation with nonlinear elastic and damping terms whose order is integer and or noninteger cveticanin s solving procedure is extended for solving such a strong nonlinear differential equation the approximate solution obtained is a function of initial amplitude and initial phase a damping coefficient and order of damping interaction with an elastic coefficient and order of elasticity for the generalized oscillators are also determined special attention is paid to obtain the relation between initial amplitude and phase on the one hand and initial displacement and velocity on the other hand correction to the frequency of vibration for the linear oscillators with nonlinear damping and for the pure nonlinear oscillators with linear damping is obtained and analyzed analytical results given in this paper are compared with numerically obtained ones and show a good agreement

analytical approximations of the period of a generalized nonlinear van der pol oscillator in this paper analytical approximations for the period of a generalized nonlinear van der pol equation will be obtained by using various asymptotic methods

decoding interleaved reed solomon codes over noisy channels we consider error correction over the non binary symmetric channel nbsc which is a natural probabilistic extension of the binary symmetric channel bsc we propose a new decoding algorithm for interleaved reed solomon codes that attempts to correct all interleaved codewords simultaneously in particular interleaved encoding gives rise to multi dimensional curves and more specifically to a variation of the polynomial reconstruction problem which we call simultaneous polynomial reconstruction we present and analyze a novel probabilistic algorithm that solves this problem our construction yields a decoding algorithm for interleaved rs codes that allows efficient transmission arbitrarily close to the channel capacity in the nbsc model

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

analyses of oscillators with non polynomial damping terms in this paper the properties of the oscillatory motion of the system with non polynomial damping is investigated the two limits for damping are the dry friction and linear viscous damping the mathematical model of the system is a strong nonlinear differential equation with fraction order velocity terms using the modified version of he s homotopy perturbation method the approximate analytic solution is obtained the generating solution is assumed in the form which corresponds to the system with linear viscous damping two special cases are considered first when the coefficient of the damping force is small and second when the damping force is close to dry friction the obtained analytical solutions are compared with numerical ones they show good agreement

homotopy perturbation method for a conservative x1 3 force nonlinear oscillator the homotopy perturbation method is used to obtain the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x 1 3 we find this method works very well for the whole range of initial amplitudes excellent agreement of the approximate frequencies with the exact ones has been demonstrated and discussed only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 0 60 for small and large values of oscillation amplitude while this relative error is as low as 0 050 for the second iteration comparison of the results obtained using this method with those obtained by different harmonic balance methods reveals that the former is more effective and convenient for these types of nonlinear oscillators

harmonic balance approach to periodic solutions of non linear jerk equations the method of harmonic balance is applied to non linear jerk equations which involve the third order time derivative for many types of cubic non linearities the method yields good estimates of the period and displacement amplitude of oscillations for a range of values of initial velocity amplitude when compared with numerical solutions some limitations notably the restriction to zero initial acceleration as well as implications and possible extensions are discussed

analytical study of vibrating systems with strong non linearities by employing saw tooth time transformations abstract a special saw tooth temporal transformation technique is formulated it is simple enough to allow analytical computation of strongly non linear free and forced dynamic responses but at the same time can be applied to the analysis of general classes of non linear problems

option pricing under deformed gaussian distributions abstract in financial literature many have been the attempts to overcome the option pricing drawbacks that affect the black and scholes model starting from the tsallis deformation of the usual exponential function this paper presents in a complete market setup a class of deformed geometric brownian motions flexible enough to reproduce fat tails and to capture the volatility behavior observed in models that consider both stochastic volatility and jumps

iteration method solutions for conservative and limit cycle x1 3 force oscillators abstract an iterative technique is used to calculate a higher order approximation to the periodic solutions of a conservative oscillator for which the elastic force term is proportional to x 1 3 the related van der pol type limit cycle oscillator is also studied

strengths and weaknesses of quantum computing recently a great deal of attention has been focused on quantum computation following a sequence of results bernstein and vazirani in proc 25th annual acm symposium theory comput 1993 pp 11 20 siam j comput 26 1997 pp 1277 1339 simon in proc 35th annual ieee symposium foundations comput sci 1994 pp 116 123 siam j comput 26 1997 pp 1340 1349 shor in proc 35th annual ieee symposium foundations comput sci 1994 pp 124 134 suggesting that quantum computers are more powerful than classical probabilistic computers following shor s result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time it is natural to ask whether all of np can be efficiently solved in quantum polynomial time in this paper we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class np cannot be solved on a quantum turing machine qtm in time o 2 n 2 we also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class np cap conp cannot be solved on a qtm in time o 2 n 3 the former bound is tight since recent work of grover in it proc 28 th annual acm symposium theory comput 1996 shows how to accept the class np relative to any oracle on a quantum computer in time o 2 n 2

improved decoding of reed solomon and algebraic geometry codes given an error correcting code over strings of length n and an arbitrary input string also of length n the list decoding problem is that of finding all codewords within a specified hamming distance from the input string we present an improved list decoding algorithm for decoding reed solomon codes the list decoding problem for reed solomon codes reduces to the following curve fitting problem over a field f given n points x sub i spl middot y sub i sub i 1 sup n x sub i y sub i spl isin f and a degree parameter k and error parameter e find all univariate polynomials p of degree at most k such that y sub i p x sub i for all but at most e values of i spl isin 1 n we give an algorithm that solves this problem for e 1 3 where the result yields the first asymptotic improvement in four decades the algorithm generalizes to solve the list decoding problem for other algebraic codes specifically alternant codes a class of codes including bch codes and algebraic geometry codes in both cases we obtain a list decoding algorithm that corrects up to n spl radic n n d errors where n is the block length and d is the designed distance of the code the improvement for the case of algebraic geometry codes extends the methods of shokrollahi and wasserman see in proc 29th annu acm symp theory of computing p 241 48 1998 and improves upon their bound for every choice of n and d we also present some other consequences of our algorithm including a solution to a weighted curve fitting problem which may be of use in soft decision decoding algorithms for reed solomon codes

cryptographic hardness based on the decoding of reed solomon codes in this paper we investigate the decoding problem of reed solomon rs codes also known as the polynomial reconstruction problem pr from a cryptographic hardness perspective namely we deal with samplable pr instances over parameter choices for which decoding is not known to be feasibly solvable and where part of the solution polynomial is the hidden input we put forth a natural decisional intractability assumption that relates to this decoding problem distinguishing between a single randomly chosen error location and a single randomly chosen nonerror location for a given corrupted rs codeword with random noise we prove that under this assumption pr instances are entirely pseudorandom i e they are indistinguishable from random vectors over the underlying finite field moreover under the same assumption we show that it is hard to extract any partial information related to the hidden input encoded by the corrupted pr instance i e pr instances hide their message polynomial solution in the semantic security sense the above results lay a framework for the exploitation of pr as an intractability assumption for provable security of cryptographic primitives based on this framework we present provably secure cryptographic constructions for 1 a pseudorandom number generator 2 a semantically secure version of the oblivious polynomial evaluation ope protocol and 3 a stateful cipher with a set of interesting properties that include semantic security forward secrecy error correcting decryption and an array of random self reducibility properties with respect to the plaintext choice key choice and partial domain choice

a quantile based study of cumulative residual tsallis entropy measures abstract in the present paper we introduce a quantile based cumulative residual tsallis entropy crte and quantile based crte for order statistics unlike the cumulative residual tsallis entropy measures in the distribution function approach due to sati and gupta 2015 and rajesh and sunoj 2016 respectively the corresponding quantile versions possess some unique properties in many applied works there do not have any tractable distribution functions while the quantile function exists and in such cases the proposed measures become more useful in measuring uncertainty of random variables we obtain some characterizations for distributions based on the quantile versions of crte and derive certain bounds we also study various properties of quantile based crte for order statistics

possible generalization of boltzmann gibbs statistics with the use of a quantity normally scaled in multifractals a generalized form is postulated for entropy namelys q k 1 i 1 w p i q q 1 whereq characterizes the generalization andp i are the probabilities associated withw microscopic configurations w the main properties associated with this entropy are established particularly those corresponding to the microcanonical and canonical ensembles the boltzmann gibbs statistics is recovered as theq 1 limit

determination of periodic solution for a u1 3 force by he s modified lindstedt poincare method abstract this paper applies he s modified lindstedt poincare method to determine the periodic solutions of oscillators in a u 1 3 force the result obtained and comparison with analytical solution provides confirmation for the validity of he s modified lindstedt poincare method

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

tsallis and renyi divergences of generalized jacobi polynomials abstract we consider some discrete probability distribution n x n 1 x n 2 x n n x we show that under suitable conditions the sequence of tsallis divergence d t n x n and the sequence of renyi divergence d r n x n are convergent for any x 1 1

the residue harmonic balance for fractional order van der pol like oscillators when seeking a solution in series form the number of terms needed to satisfy some preset requirements is unknown in the beginning an iterative formulation is proposed so that when an approximation is available the number of effective terms can be doubled in one iteration by solving a set of linear equations this is a new extension of the newton iteration in solving nonlinear algebraic equations to solving nonlinear differential equations by series when fourier series is employed the method is called the residue harmonic balance in this paper the fractional order van der pol oscillator with fractional restoring and damping forces is considered the residue harmonic balance method is used for generating the higher order approximations to the angular frequency and the period solutions of above mentioned fractional oscillator the highly accurate solutions to angular frequency and limit cycle of the fractional order van der pol equations are obtained analytically the results that are obtained reveal that the proposed method is very effective for obtaining asymptotic solutions of autonomous nonlinear oscillation systems containing fractional derivatives the influence of the fractional order on the geometry of the limit cycle is investigated for the first time

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

a study of nonlinear oscillators with u1 3 force by he s variational iteration method this paper applies he s variational iteration method to determine the periodic solutions of oscillators in a u1 3 force with the procedure the excellent approximate frequencies and the corresponding periodic solutions can easily be obtained

invariants and approximate solutions for certain non linear oscillators by means of the field method certain strongly non linear conservative oscillators are approached with the field method which is combined with the convolution integral method a complete set of their adiabatic invariants are derived on the basis of which approximate solutions for motion can be obtained

strongly nonlinear vibrations of damped oscillators with two nonsmooth limits a family of strongly nonlinear oscillators with a generalized power form elastic force and viscous damping is considered an explicit analytical solution is obtained as a combination of smooth and nonsmooth functions two different nonsmooth functions involved into the solution are associated with two different nonsmooth limits of the oscillator as the exponent becomes either zero or infinity as a result the solution is drastically simplified to give the best match with numerical tests if approaching any of the two limits

limit cycles of non smooth oscillators abstract an artificial parameter linstedt poincare method based on the introduction of a linear stiffness term a change of independent variable and the expansion of both the solution and the frequency of oscillation in power series of the artificial parameter and the theory of generalized functions are used to determine the frequency of limit cycles of non smooth oscillators the method is applied to three non smooth oscillators and it is shown that the first order approximation coincides with those obtained by means of harmonic balance and parameter expansion techniques whereas higher order ones are given in terms of convergent series

a convolution integral method for certain strongly nonlinear oscillators a modification of the convolution integral method for linear oscillators is presented for the analysis of certain strongly nonlinear oscillators the modification provides an iteration scheme two examples are given to illustrate the effectiveness of the proposed method

an artificial parameter decomposition method for nonlinear oscillators applications to oscillators with odd nonlinearities abstract a method for obtaining series solutions of nonlinear second order ordinary differential equations based on the introduction of an artificial parameter is presented and shown to be identical to the well known adomian s decomposition technique the method is formulated in both integral and differential forms for the determination of the limit cycle of oscillators with odd nonlinearities two differential forms and one integral form of the artificial parameter method are presented these versions are based on introducing a linear stiffness term with an unknown frequency and the use of either the original independent variable or a new independent variable that depends linearly on the unknown frequency of the oscillator the three formulations provide identical results and their application to eight oscillators with odd nonlinearities shows that the artificial parameter technique presented in this paper predicts the same frequency of oscillation as the harmonic balance and iterative techniques as well as modified linstedt poincare methods however the method presented here is based on the introduction of an artificial parameter and does not require the presence of small perturbation parameters in the ordinary differential equation it is also shown that two and three level iterative methods yield the same frequency of oscillation as the artificial parameter technique presented in this paper provided that the initial iterate of the former coincides with the leading order solution of the latter and only one iteration of iterative techniques and only the second approximation of the artificial parameter method are determined

determination of zero coupon and spot rates from treasury data by maximum entropy methods abstract an interesting and important inverse problem in finance consists of the determination of spot rates or prices of the zero coupon bonds when the only information available consists of the prices of a few coupon bonds a variety of methods have been proposed to deal with this problem here we present variants of a non parametric method to treat with such problems which neither imposes an analytic form on the rates or bond prices nor imposes a model for the random evolution of the yields the procedure consists of transforming the problem of the determination of the prices of the zero coupon bonds into a linear inverse problem with convex constraints and then applying the method of maximum entropy in the mean this method is flexible enough to provide a possible solution to a mispricing problem

new classes of lorenz curves by maximizing tsallis entropy under mean and gini equality and inequality constraints abstract in this paper by using the entropy maximization principle with tsallis entropy new distribution families for modeling the income distribution are derived also new classes of lorenz curves are obtained by applying the entropy maximization principle with tsallis entropy under mean and gini index equality and inequality constraints

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

forced vibrations of oscillators with a purely nonlinear power form restoring force abstract harmonically excited oscillators with non negative real power geometric nonlinearities and no linear term in the restoring force are considered perturbation approaches are developed for the cases of weak and strong nonlinearity frequency amplitude equations are derived for an arbitrary value of the non negative real power of the restoring force as well as analytical expressions for the steady state response at the frequency of excitation it is shown that the system response is of a softening type for the powers lower than unity and of a hardening type for the powers higher than unity frequency response curves of the antisymmetric constant force oscillator the restoring force of which has a zero power are also discussed comparisons with numerical results are presented for confirmation of the analytical results obtained

on the variational iteration method and other iterative techniques for nonlinear differential equations abstract a variety of iterative methods for the solution of initial and or boundary value problems in ordinary and partial differential equations is presented these iterative procedures provide the solution or an approximation to it as a sequence of iterates for initial value problems it is shown that these iterative procedures can be written in either an integral or differential form the integral form is governed by a volterra integral equation whereas the differential one can be obtained from the volterra representation by simply differentiation it is also shown that integration by parts variation of parameters adjoint operators green s functions and the method of weighted residuals provide the same volterra integral equation and that this equation in turn can be written as that of the variational iteration method it is therefore shown that the variational iteration method is nothing else by the picard lindelof theory for initial value problems in ordinary differential equations and banach s fixed point theory for initial value problems in partial differential equations and the convergence of these iterative procedures is ensured provided that the resulting mapping is lipschitz continuous and contractive it is also shown that some of the iterative methods for initial value problems presented here are special cases of the bellman kalaba quasilinearization technique provided that the nonlinearities are differentiable with respect to the dependent variable and its derivatives but such a condition is not required by the techniques presented in this paper for boundary value problems it is shown that one may use the iterative procedures developed for initial value problems but the resulting iterates may not satisfy the boundary conditions and two new iterative methods governed by fredholm integral equations are proposed it is shown that the resulting iterates satisfy the boundary conditions if the first one does so the iterative integral formulation presented here is applied to ten nonlinear oscillators with odd nonlinearities and it is shown that its results coincide with those of differential two and three level iterative techniques harmonic balance procedures and standard and modified linstedt poincare techniques the method is also applied to two boundary value problems

nonextensive statistical mechanics and economics abstract ergodicity this is to say dynamics whose time averages coincide with ensemble averages naturally leads to boltzmann gibbs bg statistical mechanics hence to standard thermodynamics this formalism has been at the basis of an enormous success in describing among others the particular stationary state corresponding to thermal equilibrium there are however vast classes of complex systems which accommodate quite badly or even not at all within the bg formalism such dynamical systems exhibit in one way or another nonergodic aspects in order to be able to theoretically study at least some of these systems a formalism was proposed 14 years ago which is sometimes referred to as nonextensive statistical mechanics we briefly introduce this formalism its foundations and applications furthermore we provide some bridging to important economical phenomena such as option pricing return and volume distributions observed in the financial markets and the fascinating and ubiquitous concept of risk aversion one may summarize the whole approach by saying that bg statistical mechanics is based on the entropy s bg k i p i ln p i and typically provides exponential laws for describing stationary states and basic time dependent phenomena while nonextensive statistical mechanics is instead based on the entropic form s q k 1 i p i q q 1 with s 1 s bg and typically provides for the same type of description asymptotic power laws

new functional forms of lorenz curves by maximizing tsallis entropy of income share function under the constraint on generalized gini index abstract the lorenz curve is one of the most powerful tools in the analysis of the size distribution of income and wealth in the past decades many authors have proposed different functional forms for estimating lorenz curves using a variety of approaches in this paper new functional forms are derived by maximizing tsallis entropy of income share function subject to a given generalized gini index the obtained lorenz curves are fitted to the income data sets of three asian countries in 1988 and their relative performances with respect to some well known parametric models of lorenz curves are compared using two types of goodness of fit measures

new measure selection for hunt devolder semi markov regime switching interest rate models in this paper we construct the minimal entropy martingale for semi markov regime switching interest rate models using some general entropy measures we prove that for the one period model the minimal entropy martingale for semi markov processes in the case of the tsallis and kaniadakis entropies are the same as in the case of shannon entropy

a pseudorandom generator from any one way function pseudorandom generators are fundamental to many theoretical and applied aspects of computing we show how to construct a pseudorandom generator from any one way function since it is easy to construct a one way function from a pseudorandom generator this result shows that there is a pseudorandom generator if and only if there is a one way function

vibrations of the nonlinear oscillator with quadratic nonlinearity abstract in this paper the vibration of a mass spring oscillator with strong quadratic nonlinearity and one degree of freedom is analyzed the both strong and hard springs are considered the restoring force in the spring which is the function of the quadratic deformation has to satisfy the condition of antisymmetry the mathematical model is an ordinary second order differential equation where the quadratic nonlinear term changes the sign the quantitative and the qualitative analysis of the equation is done the exact analytical solution is obtained it depends on the jacobi elliptic function

solutions of nonlinear oscillators with fractional powers by an iteration procedure a modified iteration procedure is applied to nonlinear oscillations with fractional powers with the procedure the excellent approximate frequencies and the corresponding periodic solutions can be easily obtained

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

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

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

a non simultaneous variational approach for the oscillators with fractional order power nonlinearities abstract this paper is concerned with a class of conservative oscillators the restitution force of which is of a power form which includes positive non integer exponents it is shown how an approximate lagrangian and hamilton s variational principle can be used to obtain a second order approximate solution for their free vibrations due to the fact that in a general case when the restoring force is multi term the period cannot be obtained from the energy conservation law in a closed form the problem is formulated as a one point boundary value problem and a non simultaneous variation is introduced the explicit expressions for the amplitudes and frequency of oscillations are derived in which there are no restrictions on the values of the non integer powers the analytically obtained results are compared with numerical results as well as with some approximate analytical results from the literature

periodic solutions of strongly quadratic non linear oscillators by the elliptic perturbation method abstract the elliptic perturbation method is applied to the study of the periodic solutions of strongly quadratic non linear oscillators of the formx c1x c 2x2 f x x in which the jacobian elliptic functions are employed the generalized van der pol equation withf x x 0 1x 2x2is 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

a fast continuation scheme for accurate tracing of nonlinear oscillator frequency response functions a new algorithm is proposed to combine the split frequency harmonic balance method sf hbm with arc length continuation alc for accurate tracing of the frequency response of oscillators with non expansible nonlinearities alc is incorporated into the sf hbm in a two stage procedure stage i involves finding a reasonably accurate response frequency and solution using a relatively large number of low frequency harmonics this step is achieved using the sf hbm in conjunction with alc stage ii uses the sf hbm to obtain a very accurate solution at the frequency obtained in stage i to guarantee rapid path tracing the frequency axis is appropriately subdivided this gives high chance of success in finding a globally optimum set of harmonic coefficients when approaching a turning point however arc lengths are adaptively reduced to obtain a very accurate solution the combined procedure is tested on three hardening stiffness examples a duffing model an oscillator with non expansible stiffness and single harmonic forcing and an oscillator with non expansible stiffness and multiple harmonic forcing the results show that for non expansible nonlinearities and multiple harmonic forcing the proposed algorithm is capable of tracing out frequency response functions with high accuracy and efficiency

maximum entropy lorenz curves abstract for describing the distribution of income a family of maximum entropy density functions is derived under optional side conditions on the distance of the mean income from the minimum income and the absolute gini index of concentration many of the resulting lorenz curves obtained from the reciprocal of the density function by repeated quadrature turn out to be well known models

the role of constraints within generalized nonextensive statistics the gibbs jaynes path for introducing statistical mechanics is based on the adoption of a specific entropic form sand of physically appropriate constraints for instance for the usual canonical ensemble one adopts i ii ipi 1 and iii ei eigenvalues of the hamiltonian u1 internal energy equilibrium consists in optimizing s1 with regard to pi in the presence of constraints ii and iii within the recently introduced nonextensive statistics i is generalized into sq k 1 ipiq q 1 q 1 reproduces s1 ii is maintained and iii is generalized in a manner which might involve piq in the present effort we analyze the consequences of some special choices for iii and their formal and practical implications for the various physical systems that have been studied in the literature to illustrate some mathematically relevant points we calculate the specific heat respectively associated with a nondegenerate two level system as well as with the classical and quantum harmonic oscillators