positive solutions of kirchhoff type non local elliptic equation a bifurcation approach positive solutions of a kirchhoff type nonlinear elliptic equation with a non local integral term on a bounded domain in n n 1 are studied by using bifurcation theory the parameter regions of existence non existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem local and global bifurcation diagrams of positive solutions for various parameter regions are obtained

nontrivial fucik spectrum of one non selfadjoint operator abstract we provide complete analytical description of the fucik spectrum of the non selfadjoint operator corresponding to four point boundary value problem the non selfadjointness results in a surprising structure of the fucik spectrum with new interesting patterns

the beginning of the fucik spectrum for the p laplacian has a nontrivial solution u here 1 p 2p u div u p 2 u u max u 0 u uu in 1 1 denoting by 1 2 the first two eigenvalues of 2p on w 1 p 0 0 cf the end of this introduction 7p clearly contains in particular 1 1 2 2 and the two lines 1 r and r 1 this generalized notion of spectrum was introduced for p 2 i e in the linear case in the 1970s by fuc ik 21 and dancer 15 in connection article id jdeq 1999 3645 available online at http www idealibrary com on

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

laplace periodogram for time series analysis a new type of periodogram called the laplace periodogram is derived by replacing least squares with least absolute deviations in the harmonic regression procedure that produces the ordinary periodogram of a time series an asymptotic analysis reveals a connection between the laplace periodogram and the zero crossing spectrum this relationship provides a theoretical justification for use of the laplace periodogram as a nonparametric tool for analyzing the serial dependence of time series data superiority of the laplace periodogram in handling heavy tailed noise and nonlinear distortion is demonstrated by simulations a real data example shows its great effectiveness in analyzing heart rate variability in the presence of ectopic events and artifacts

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

on nontrivial solutions of a dirichlet problem whose jumping nonlinearity crosses a multiple eigenvalue let be a bounded domain in rn n 2 with smooth boundary let p r r be a continuous function such that p 0 0 and limt p t t p exist we discuss the existence of nontrivial solutions of the dirichlet problem u p u in u 0 on our hypotheses relate the position of the point p p in the plane with the extended spectrum of they seem to allow noticeably greater freedom for the numbers p than is usually found in the literature in the sense that the interval with end points p is here permitted to contain in its interior an eigenvalue of arbitrarily multiplicity of it also appears that we significantly weaken some technical conditions on p of another paper which treats the crossing of a simple eigenvalue

central limit theorem for linear processes iences mixing sequences or associated sequences the results are important in analyzing the asymptotical properties of some estimators as well as of linear processes 4 1 introduction let be a centered sequence of random variables k 4 and let a 1 i n be a triangular array of numbers many statistical ni procedures produce estimators of the type n 1 1 s a nn i i i1 to give an example let us consider the nonlinear regression model yx gx x where gx is an unknown function and x is the noise now we fix the design points x and we get ni

on the periodic fucik spectrum and a superlinear sturm liouville equation in the first part of the paper a variational characterisation of the periodic eigenvalues the so called fucik spectrum of a semilinear positive homogeneous sturm liouville equation is given the proof relies on the s 1 invariance of the equation in the second part a nonlinear sturm liouville equation with typically an exponential nonlinearity is considered it is proved that under certain conditions this equation is solvable for arbitrary forcing terms the proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the fucik spectrum

structure of the fucik spectrum and existence of solutions for equations with asymmetric nonlinearities let l dom l l2 l2 be a self adjoint operator being open and bounded in rn we give a description of the fucik spectrum of l away from the essential spectrum let be a point in the discrete spectrum of l provided that some non degeneracy conditions are satisfied we prove that the fucik spectrum consists locally of a finite number of curves crossing at each of these curves can be associated to a critical point of the function h x xs21a6 x x l2 restricted to the unit sphere in ker l i the corresponding critical values determine the slopes of these curves we also give global results describing the fucik spectrum and existence results for semilinear equations by performing degree computations between the fucik curves

on the fucik spectrum and a resonance problem for the p laplacian with a nonlinear boundary condition in this paper we prove that there exists a first curve of the fucik spectrum of the problem pu u p 2u in with a nonlinear boundary condition given by u p 2 u u p 1 u p 1 on the boundary of the domain we also prove that there exists a sequence of curves of the fucik spectrum which exist locally in the neighborhood of suitable eigenvalues of the p laplacian with a nonlinear boundary condition finally we study a resonance problem with respect to the fucik spectrum

a nonlinear method for robust spectral analysis a nonlinear spectral analyzer called the l p norm periodogram is obtained by replacing the least squares criterion with an l p norm criterion in the regression formulation of the ordinary periodogram in this paper we study the statistical properties of the l p norm periodogram for time series with continuous and mixed spectra we derive the asymptotic distribution of the l p norm periodogram and discover an important relationship with the so called fractional autocorrelation spectrum that can be viewed as an alternative to the power spectrum in representing the serial dependence of a random process in the frequency domain in comparison with the ordinary periodogram p 2 we show that by varying the value of p in the interval 1 2 the l p norm periodogram can strike a balance between robustness against heavy tailed noise efficiency under regular conditions and spectral leakage for time series with mixed spectra we also show that the l p norm periodogram can detect serial dependence of uncorrelated non gaussian time series that cannot be detected by the ordinary periodogram

a note on the relation between the fucik spectrum and pareto eigenvalues abstract in this paper we summarize some known results concerning the so called fucik spectrum arising in problems with jumping nonlinearities and the pareto spectrum or the k spectrum connected with linear complementarity problems we show a surprising relation between these two notions and illustrate it on several examples

bounded influence propagation tau estimation a new robust method for arma model estimation a new robust and statistically efficient estimator for arma models called the bounded influence propagation estimator is proposed the estimator incorporates an auxiliary model which prevents the propagation of outliers strong consistency and asymptotic normality of the estimator for arma models that are driven by independently and identically distributed iid innovations with symmetric distributions are established to analyze the infinitesimal effect of outliers on the estimator the influence function is derived and computed explicitly for an ar 1 model with additive outliers to obtain estimates for the ar p model a robust durbin levinson type and a forward backward algorithm are proposed an iterative algorithm to robustly obtain arma p q parameter estimates is also presented the problem of finding a robust initialization is addressed which for orders p q 2 is a nontrivial matter numerical experiments are conducted to compare the finite sample performance of the proposed estimator to existing robust methodologies for different types of outliers both in terms of average and of worst case performance as measured by the maximum bias curve to illustrate the practical applicability of the proposed estimator a real data example of outlier cleaning for r r interval plots derived from electrocardiographic data is considered the proposed estimator is not limited to biomedical applications but is also useful in any real world problem whose observations can be modeled as an arma process disturbed by outliers or impulsive noise

on the fu c ik spectrum for the p laplacian with robin boundary condition abstract the aim of this paper is to study the fuc ik spectrum of the p laplacian with robin boundary condition given by p u a u p 1 b u p 1 in u p 2 u u p 2 u on where 0 if 0 it reduces to the fuc ik spectrum of the negative neumann p laplacian the existence of a first nontrivial curve c of this spectrum is shown and we prove some properties of this curve e g c is lipschitz continuous decreasing and has a certain asymptotic behavior a variational characterization of the second eigenvalue 2 of the robin eigenvalue problem involving the p laplacian is also obtained

approximate solution of the backward problem for kirchhoff s model of parabolic type with discrete random noise abstract in this paper we consider the final boundary value problem for non local kirchhoff s model of parabolic type with discrete random noise we first discuss the instability of solutions then we present the regularized solution by the trigonometric method in non parametric regression associated with the truncated expansion method in addition under a prior assumption on the exact solution the convergence rate is obtained in the last section the effectiveness and suitability of our results are justified via numerical experiments

on the fucik spectrum of the laplacian on a torus abstract we study the fucik spectrum of the laplacian on a two dimensional torus t 2 exploiting the invariance properties of the domain t 2 with respect to translations we obtain a good description of large parts of the spectrum in particular for each eigenvalue of the laplacian we will find an explicit global curve in the fucik spectrum which passes through this eigenvalue these curves are ordered and we will show that their asymptotic limits are positive on the other hand using a topological index based on the mentioned group invariance we will obtain a variational characterization of global curves in the fucik spectrum also these curves emanate from the eigenvalues of the laplacian and we will show that they tend asymptotically to zero thus we infer that the variational and the explicit curves cannot coincide globally and that in fact many curve crossings must occur we will give a bifurcation result which partially explains these phenomena

computing eigenvalues and fucik spectrum of the radially symmetric p laplacian eigenvalue problems for the radially symmetric p laplacian are discussed we present algorithms which compute a given number of eigenvalues and fucik curves together with the corresponding eigenfunctions the second order p laplacian equation is transformed into a first order system by a generalized prufer transformation to the first order system we apply shooting algorithms newton s method and in case of the fucik curves a predictor corrector method our approach requires analytical and numerical treatment of generalized sine functions singular as well as regular problems are treated and a detailed error analysis for the approximation of singular problems by regular ones are given numerical results are presented

dancer fucik spectrum for fractional schrodinger operators with a steep potential well on rn abstract in this paper we study dancer fucik spectrum of the fractional schrodinger operators which is defined as the set of r 2 such that s u v x u u u in r n has a nontrivial solution u where the potential v has a steep potential well for sufficiently large parameter 0 it is allowed that s v has essential spectrum with finitely many eigenvalues below the infimum of e s s s v many difficulties are caused by general nonlocal operators we develop new techniques to overcome them to construct the first nontrivial curve of dancer fucik point spectrum by minimax methods to show some qualitative properties of the curve and to prove that the corresponding eigenfunctions are foliated schwartz symmetric as applications we obtain the existence of nontrivial solutions for nonlinear schrodinger equations with nonresonant nonlinearity

a simple generalised crossvalidation method of span selection for periodogram smoothing a consistent estimator for the spectral density of a stationary random process can be obtained by smoothing the periodograms across frequency an important component of smoothing is the choice of the span lee 1997 proposed a span selector that was erroneously claimed to be unbiased for the mean squared error the naive use of mean squared error has some important drawbacks in this context because the variance of the periodogram depends on its mean i e the spectrum we propose a new span selector based on the generalised crossvalidation function derived from the gamma deviance this criterion originally developed for use in fitting generalised additive models utilises the approximate full likelihood of periodograms which asymptotically behave like independently distributed chi squared i e gamma random variables the proposed span selector is very simple and easily implemented simulation results suggest that the proposed span selector generally outperforms those obtained under a mean squared error criterion

robust m periodogram a maximum likelihood type m periodogram is developed for observations contaminated by impulse random errors having unknown heavy tailed error distributions the periodogram is defined as being a squared amplitude of a harmonic signal fitting the observations with a nonquadratic residual loss function given by the huber s 1981 minimax robust statistics

rank based estimation for all pass time series models an autoregressive moving average model in which all roots of the autoregressive polynomial are reciprocals of roots of the moving average polynomial and vice versa is called an all pass time series model all pass models are useful for identifying and modeling noncausal and noninvertible autoregressive moving average processes we establish asymptotic normality and consistency for rank based estimators of all pass model parameters the estimators are obtained by minimizing the rank based residual dispersion function given by jaeckel ann math statist 43 1972 1449 1458 these estimators can have the same asymptotic efficiency as maximum likelihood estimators and are robust the behavior of the estimators for finite samples is studied via simulation and rank estimation is used in the deconvolution of a simulated water gun seismogram

multivariable gain phase and sensitivity integral relations and design trade offs we present several extensions of the classical bode integral relations to multivariable systems the main contributions are an extended gain phase integral formula and an extended sensitivity integral relation these results are useful for analyzing fundamental design limitations that arise in linear multivariable feedback systems in particular they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the previously known integral relations applicable to single input single output systems both results show that design limitations in multivariable systems depend strongly upon certain directionality properties

constrained mountain pass algorithm for the numerical solution of semilinear elliptic problems a new numerical algorithm for solving semilinear elliptic problems is presented a variational formulation is used and critical points of a c1 functional subject to a constraint given by a level set of another c1 functional or an intersection of such level sets of finitely many functionals are sought first constrained local minima are looked for then constrained mountain pass points the approach is based on the deformation lemma and the mountain pass theorem in a constrained setting several examples are given showing new numerical solutions in various applications

robust harmonic retrieval via block successive upper bound minimization harmonic retrieval hr is a problem of significance with numerous applications many existing algorithms are explicitly or implicitly developed under gaussian noise assumption which however are not robust against non gaussian noise such as impulsive noise or outliers in this paper by employing the ell p fitting criterion and block successive upper bound minimization bsum technique a variant of the classical relax algorithm named as bsum relax is devised for robust hr it is revealed that the bsum relax successively performs alternating optimization along coordinate directions i e it updates one harmonic by fixing the other k 1 components such that the whole problem is split into k single tone hr problems which are then solved by creating a surrogate function that majorizes the objective function of each subproblem to further refine the frequency component the newton s method that takes linear complexity mathcal o n is derived for updating the frequency estimates we prove that under the single tone case bsum relax converges to a karush kuhn tucker point furthermore the bsum relax is extended to the multidimensional hr case numerical results show that the proposed algorithm outperforms the state of the art methods in heavy tailed noise scenarios

numerical variational methods applied to cylinder buckling we review and compare different computational variational methods applied to a system of fourth order equations that arises as a model of cylinder buckling we describe both the discretization and implementation in particular how to deal with a one dimensional null space we show that we can construct many different solutions from a complex energy surface we examine numerically convergence in the spatial discretization and in the domain size finally we give a physical interpretation of some of the solutions found

the fucik spectrum of schrodinger operator and the existence of four solutions of schrodinger equations with jumping nonlinearities abstract this paper contains the existence of four solutions of schrodinger equations with jumping nonlinearities the proof procedure is supported by a lot of new results initially a consequence is rendered as a minimax principle on h 1 r n which allows us to achieve the feasibility verification of the ps condition furthermore the constructions of minimal and maximal curves of fucik spectrum in q l see the introduction for the definition of q l warrant an intensive investigation that we encounter some thorny problems is largely due to the absence of compact embedding and the appearance of essential spectrum based on a nontrivial argument we can compute critical groups of homogeneous functional at zero if a b is free of fucik spectrum and a b q l this together with convexity and concavity offers a detailed description of the two curves by a series of sophisticated tricks additionally we present a new version of morse theory in view of the fact that classical version doesn t work directly for weak smooth functional on h 1 r n finally we prove a weak maximum principle for r n which serves as a tool to get a critical point in positive and negative cone respectively and also compute critical groups of critical points of mountain pass type with the help of above preparations we attain the ultimate aim by morse inequalities and various exact homology sequences

gain phase relationships for discrete time systems the classical gain phase relations obtained by bode for minimum phase continuous time systems can be used to establish bounds on the performance of feedback systems it is shown that completely analogous relations can be obtained for discrete time systems and can be used for the same purpose in particular the limitations imposed by plant zeros and poles which lie outside the unit circle are investigated for both genuine discrete time systems namely those defined by difference equations and for sampled data systems one of the results shows that sampled data systems may practically be unstabilizable if their unstable poles lie outside a particular circular region the intended application of these results is to allow preliminary screening of proposed specifications for feasibility in a cad system before embarking on a detailed design

an overview of asymptotic properties of lp regression under general classes of error distributions we survey the asymptotic properties of regression lp estimators under general classes of error distributions it is found that the asymptotic distributions of lp estimators depend crucially on p and the shape of the error distribution near the origin a number of important features arise as a result among which are a use of a small p may yield accelerated convergence rates for lp estimators under certain classes of error distributions b for p 1 lp regression should under some circumstances be undertaken by locally maximizing rather than minimizing the sum of the pth powers of the absolute deviations and c consistent estimation of the sampling distributions of the lp estimators can be achieved by the m out of n bootstrap in general numerical examples are provided to illustrate our theoretical findings and a computational algorithm is suggested for local maximization as may sometimes be required by the lp procedure

on the fucik spectrum in this paper we study the structure of the fucik spectrum of the set of points b a in r 2 for which the equation

bifurcation surfaces stemming from the fucik spectrum abstract let r n be a bounded domain with smooth boundary and g r r be a nonlinear function we prove existence of two dimensional bifurcation surfaces for the elliptic boundary value problem u a u b u g x u in u 0 where u min 0 u u max 0 u and a b r 2 is a pair of parameters we show that these two dimensional bifurcation surfaces stem from the fucik spectrum of the main difficulty in doing that comes from non smoothness of the operators u u in order to overcome this difficulty a variant implicit function theorem and an abstract two dimensional bifurcation theorem are proved these two theorems do not require smoothness of operators and the abstract two dimensional bifurcation theorem can be regarded as an extension of the well known crandall rabinowitz bifurcation theorem and therefore are of interest for their own sake

analytical and numerical results for the fucik spectrum of the laplacian nontrivial solutions of u u vu 0 in rn with zero dirichlet condition on are studied the collection of the pairs v is the so called fucik spectrum f a new variational formulation for parts of f is presented and analyzed based on this formulation a minimization algorithm for the computation of a part of f is developed alternatively an approach using an implicit function argument is discussed analytically and via newton s method also numerically by combining the variational minimization method with newton s method a new bifurcation result is first observed numerically and then proved rigorously by replacing the variational minimization method by the mountain pass algorithm higher curves in f are found numerically several numerical results are discussed including a further example of the previously recorded phenomenon of crossing of fucik curves originating from different eigenvalues

nonparametric spectral density estimation using penalized whittle likelihood abstract the penalized likelihood approach is not well developed in time series analysis even though it has been applied successfully in a number of nonparametric function estimation problems chow and grenander proposed a penalized likelihood type approach to the nonparametric estimation of the spectral density of gaussian processes in this article this estimator is extended to more general stationary processes its practical implementation is developed in some detail and some asymptotic rates of convergence are established its performance is also compared to more widely used alternatives in the field a computational algorithm involving an iterative least squares initialized by the log periodogram is first developed then motivated by an asymptotic linearization an estimator of the integrated squared error between the estimated and true log spectral densities is proposed from this a data dependent procedure for selection of the amount of smoothing is constructed the methodology is illustrated