the structure of a nonlinear elliptic operator consider the nonlinear dirichlet problem 1 u u u 3 g for u r u 0 and r n connected and bounded and let i be the ith eigenvalue of u on with u 0 i 1 2 define a map a h h by a u u u u 3 for either the sobolev space w 0 1 2 h h if n 4 or the holder spaces c 0 2 h and c 0 h if is c 2 and define a h r h r by a u a u let g r 2 e r 2 e be the global cusp map given by g s t v s 3 ts t v and let f r e r e be the global fold map given by f t v t 2 v where e is any frechet space

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

the complex cepstrum of higher order cumulants and nonminimum phase system identification a computationally efficient identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase system the method is based on the idea of computing the complex cepstrum of higher order cumulants of the system output in particular the differential cepstrum parameters of the nonminimum phase impulse response are estimated directly from higher order cumulants by least squares solution or two dimensional fft operations the method reconstructs the minimum phase and maximum phase impulse response components separately it is flexible enough to be applied on autoregressive ar moving average ma or arma system without a priori knowledge of the type of the system benchmark simulation examples demonstrate the effectiveness of the method even with short length data records

parameter estimation of exponentially damped sinusoids using higher order statistics a new approach for the estimation of the parameters of exponentially damped sinusoids is introduced based on third or fourth order statistics of the observation signal the method may be seen as an extension of the minimum norm principal eigenvectors method to higher order statistics domains the strong points and limitations of the method are discussed as well as sufficient conditions for the existence of the solution the utilization of the method in the case of finite length signals in the presence of additive gaussian noise white or colored is addressed monte carlo simulations demonstrate the effectiveness of the new method when the additive noise is colored gaussian with unknown autocorrelation sequence for different signal to noise ratios and a single data record the case of an ensemble of data records is studied when the exponentially damped sinusoids are assumed to have random phase

mathematical structures derived from the q multinomial coefficient in tsallis statistics abstract we present q stirling s formula q multinomial coefficient one to one correspondence between q multinominal coefficient and tsallis entropy q pascal s triangle and a conjecture on the q central limit theorem in tsallis statistics for the generalization of the well known fundamental formulas to systems exhibiting power law behaviors the main approach is based on the q product uniquely determined by tsallis entropy which has already been successfully applied to our recent proof of the law of error in tsallis statistics

estimation of close sinusoids in colored noise and model discrimination this paper considers the estimation of the two close dominant frequencies in the signal when it is known a priori that the observation is a sum of two close sinusoids and an additive colored noise whose spectral density is unknown earlier attempts have assumed the additive noise to be independent next we develop decision rules for checking whether the observed signal has only one sinusoid or two close sinusoids all of the earlier studies assumed that there are two close sinusoids in the signal another model discrimination problem considered is the determination of the causal structure of the observed periodicity a rule is given to test whether the observation comes from a sinusoid plus additive noise possible colored or the observation comes from a stationary autoregressive model we also present a numerical study showing the efficacy of the robust estimation procedure for estimating the two frequencies and the decision rules for checking the number of dominant frequencies and the causal mechanism finally we compare the proposed method to well known methods used for two close sinusoids in white noise

exact maximum likelihood parameter estimation of superimposed exponential signals in noise a unified framework for the exact maximum likelihood estimation of the parameters of superimposed exponential signals in noise encompassing both the time series and the array problems is presented an exact expression for the ml criterion is derived in terms of the linear prediction polynomial of the signal and an iterative algorithm for the maximization of this criterion is presented the algorithm is equally applicable in the case of signal coherence in the array problem simulation shows the estimator to be capable of providing more accurate frequency estimates than currently existing techniques the algorithm is similar to those independently derived by kumaresan et al in addition to its practical value the present formulation is used to interpret previous methods such as prony s pisarenko s and modifications thereof

comment on critique of multinomial coefficient method for evaluating tsallis and renyi entropies by a s parvan abstract parvan a s parvan critique of multinomial coefficients method for evaluating tsallis and renyi entropies physica a 389 2010 5645 has recently presented some calculations in order to demonstrate the incorrectness of the results obtained from the generalized multinomial coefficients gmc presented in oikonomou 2007 1 according to parvan the aforementioned approach of studying maximum entropy probability distributions is erroneous in this comment i demonstrate that parvan s arguments do not hold true and that the obtained results from gmc do not present either mathematical or physical discrepancies

nonlinear galerkin methods the finite elements case with the increase in the computing power and the advent of supercomputers the approximation of evolution equations on large intervals of time is emerging as a new type of numerical problem in this article we consider the approximation of evolution equations on large intervals of time when the space discretization is accomplished by finite elements the algorithm that we propose called the nonlinear galerkin method stems from the theory of dynamical systems and amounts to some approximation of the attractor in the discrete finite elements space essential here is the utilization of incremental unknown which is accomplished in finite elements by using hierarchical bases beside a detailed description of the algorithm the article includes some technical results on finite elements spaces and a full study of the stability and convergence of the method

a finite element method for computing the bifurcation function for semilinear elliptic bvps abstract the method of alternative problems can be used to show that a semilinear elliptic boundary value problem lu g x u 0 with gu x u bounded below is equivalent to a finite dimensional problem f c 0 r d c r d in the sense that their solution sets which are not necessarily singletons are in a one to one correspondence this correspondence is based on a map from low frequency to high frequency fourier components of solutions a numerical method is presented for approximating and hence also solutions of the bvp the method uses finite element approximations and avoids the use of eigenfunction expansions existence uniqueness and error estimates for the approximations of and solutions u are derived

on the zeros of the linear prediction error filter for deterministic signals the zeros of a linear prediction error filter polynomial for a class of deterministic signals that are a sum of samples of m exponentially damped undamped sinusoids is studied it is assumed that n samples are available for processing and that they are uncorrupted by noise it is shown that the exponent parameters of the m signals can be determined from m zeros called signal zeros of an lth degree prediction error filter polynomial l m if l lies in between m and n m n m 2 in a special case the rest of the l m zeros of the filter polynomial switch are called extraneous zeros are shown to be approximately uniformly distributed with in the unit circle regardless of the type of exponentials in the signal if the prediction filter coefficients are chosen to have minimum euclidean length the results obtained provide insight into the estimation problem with noisy data

estimating the parameters of exponentially damped sinusoids and pole zero modeling in noise we have presented techniques 1 6 based on linear prediction lp and singular value decomposition svd for accurate estimation of closely spaced frequencies of sinusoidal signals in noise in this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise the estimation procedure presented here makes use of backward prediction in addition to svd first the method is applied to data consisting of one and two exponentially damped sinusoids the choice of one and two signal components facilitates the comparison of estimation error in pole damping factors and pole frequencies to the appropriate cramer rao cr bounds and to traditional methods of linear prediction second our method is applied to an example due to steiglitz 8 in which the data consists of noisy values of the impulse response samples composed of many exponentially damped sinusoids of a linear system having both poles and zeros the poles of the system are accurately determined by our method and the zeros are obtained subsequently using shanks method

an algorithm for pole zero modeling and spectral analysis an explicit connection between fitting exponential models and pole zero models to observed data is made the fitting problem is formulated as a constrained nonlinear minimization problem this problem is then solved using a simplified iterative algorithm the algorithm is applied to simulated data and the performance of the algorithm is compared to previous results

critique of multinomial coefficients method for evaluating tsallis and renyi entropies abstract oikonomou th oikonomou physica a 386 2007 119 has published a calculation which purports to show that the tsallis and renyi entropies can be obtained from the generalized multinomial coefficients in this paper we prove that the method of generalized multinomial coefficients failed to determine the tsallis entropy at equilibrium moreover it is shown that oikonomou s analysis contains mistakes which led to misleading statements related to the jaynes principle of maximum entropy the tsallis and the renyi 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

parameter estimation of exponentially damped sinusoids using a higher order correlation based approach a very common problem in signal processing is parameter estimation of exponentially damped sinusoids from a finite subset of noisy observations when the signal is contaminated with colored noise of unknown power spectral density a cumulant based approach provides an appropriate solution to this problem we propose a new class of estimator namely a covariance type estimator which reduces the deterministic errors associated with imperfect estimation of higher order correlations from finite data length this estimator allows a higher order correlation sequence to be modeled as a damped exponential model in certain slices of the moments plane this result shows a useful link with well known linear prediction based methods such as the minimum norm principal eigenvector method of kumaresan and tufts 1982 which can be subsequently applied to extracting frequencies and damping coefficients from the 1 d correlation sequence this paper discusses the slices allowed in the moments plane the uses and limitations of this estimator using multiple realizations and a single record in a noisy environment monte carlo simulations applied to standard examples are also performed and the results are compared with the kt method and the standard biased estimator based approach the comparison shows the effectiveness of the proposed estimator in terms of bias and mean square error when the signals are contaminated with additive gaussian noise and a single data record with short data length is available

geometric methods for determining system poles from transient response the problem treated is that of identifying the poles of a finite order system by observing its transient decay after cessation of input for a limited time using possibly multiple observation points and experimental repetition various approaches are studied having the common characteristic that a homogeneous matrix equation must be solved several techniques that have been given scant attention in the literature are consolidated by a geometric treatment together with new results including an analytical treatment of the consequences of assuming an excessively high system order derivation of a statistically unbiased estimate for an intermediate parameter in the solution new theorems on error effects a recipe for effective use of the singular value decomposition and a new method for suppression of extraneous poles

numerical analysis of semilinear elliptic equations with finite spectral interaction abstract we present an algorithm for solving u f x u g with dirichlet boundary conditions in a bounded domain the nonlinearities are non resonant and have finite spectral interaction no eigenvalue of d is an endpoint of 2 f r which in turn only contains a finite number of eigenvalues the algorithm is based on ideas used by berger and podolak to provide a geometric proof of the ambrosetti prodi theorem and advances work by smiley and chun on the same problem

statistical mechanics based on renyi entropy abstract in this work we show that it is possible to obtain a generalized statistical mechanics thermostatistics based on renyi entropy to be maximized with adequate constraints the equilibrium probability distribution thus obtained has a very interesting property indeed it reminds us the statistical distribution proposed by tsallis known to conveniently describe a variety of phenomena in nonextensive systems moreover some examples are worked out in order to illustrate the main features of the herein introduced formalism

parameter estimation of exponentially damped sinusoids using higher order statistics and matrix pencil it is demonstrated that the higher order statistics based algorithm proposed by c k papadopoulos and c l nikias see ieee trans acoust speech signal process vol 38 no 5 p 814 24 1990 can be improved by using the matrix pencil approach of y hua and t k sarkar see ieee trans acoust speech signal process vol 38 no 8 p 1424 36 1990 the matrix pencil algorithm is computationally more efficient than the papadopoulos nikias algorithm since the computation of the k roots of the k degree polynomial is not needed in the matrix pencil algorithm furthermore it has been shown that the matrix pencil algorithm is less sensitive to noise than the kumaresan tufts method

a numerical study of spontaneous bifurcation abstract in the study of nonlinear boundary value problems it has been observed that bifurcations may take place which are not local to the origin in function space in such problems the analysis of the bifurcation and the computation of solutions is complicated by the nonlocal nature of the bifurcation and the multiplicity of solutions in this work a model problem exhibiting this phenomena which has been of recent interest is numerically studied the method of alternative problems is employed to provide a complete numerical description of the bifurcation occurring in the model problem

arma bispectrum approach to nonminimum phase system identification an identification procedure is proposed for a nongaussian white noise driven linear time invariant nonminimum phase fir finite impulse response system the method is based on parametric modeling of the third moments of the output sequence and uses causal and anticausal autoregressive moving average arma models the magnitude and phase response of the system are expressed in terms of the ar parameters of the arma models in particular the ar part of the causal arma model captures the minimum phase component of the system and the ar part of the anticausal arma captures the maximum phase component both sets of parameters are obtained by solving overdetermined linear systems of equations a model order selection criterion based on third order moments is proposed the arma bispectrum approach is compared to more conventional approaches for magnitude and phase reconstruction it is demonstrated that the proposed identification procedure exhibits improved modeling performance the method does not require knowledge of the non gaussian noise distribution

tsallis renyi and nonextensive gaussian entropy derived from the respective multinomial coefficients abstract we explore the generalization of the ordinary multinomial coefficient based on the deformed q multiplication and q division aim of this study is to construct the appropriate multinomial coefficients from which one can obtain the tsallis renyi and nonextensive gaussian entropy respectively we show that for all three above entropies there are two possible ways to define the generalized multinomial coefficient its consequence is discussed

matrix pencil method for estimating parameters of exponentially damped undamped sinusoids in noise a study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented comparison of this method to a polynomial method svd prony method shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes it is found through perturbation analysis and simulation that for signals with unknown damping factors the pencil method is less sensitive to noise than the polynomial method an expression of the cramer rao bound for the exponential signals is presented

multiplicity results for a semilinear boundary value problem with the nonlinearity crossing higher eigenvalues on etudie le probleme aux valeurs limites au g u h x u 0 ou a est un operateur differentiel lineaire auto adjoint uniformement elliptique d ordre 2

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

low dimensional qualitative and numerical approximation of bifurcation in a semilinear elliptic problem abstract a nonlinear boundary value problem exhibiting a nonlocal bifurcation in its solution set is considered it is shown that for some values of the problem parameters there is only the zero solution while for other values of these parameters two nontrivial solutions also appear these nontrivial solutions do not lie in small neighborhoods of the zero function so that the bifurcation analysis is not local to the origin the method of reduction to alternative finite dimensional problems is applied to give a numerical description of the bifurcation and in particular to describe a surface of bifurcating solutions in addition the solution set of the 1 dimensional galerkin approximation is also shown to have the same qualitative nature as the solution set of the full problem

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded