center manifolds and contractions on a scale of banach spaces we give a new proof for the existence of a ck center manifold at a nonhyperbolic equilibrium point of a finite dimensional vector field of class ck the problem is reduced to a fixed point problem on a scale of banach spaces these banach spaces consist of mappings with a certain maximal exponential growth at infinity we give conditions under which there is a unique fixed point depending differentiably on the parameters the main difficulty is that the mappings under consideration become only differentiable after composition with appropriate embeddings on the scale of banach spaces

center manifolds for invariant sets abstract we derive a general center manifolds theory for a class of compact invariant sets of flows generated by a smooth vector field in r n by applying the hadamard graph transform technique it is shown that associated to a natural dynamical characteristic of the linearized flow along the invariant set there exists an invariant manifold called a center manifold of the invariant set which contains every locally bounded solution in particular contains the invariant set and is persistent under small perturbations

a center manifold for second order semilinear differential equations on the real line and applications to the existence of wave trains for the gurtin mccamy equation

polynomial growth rates abstract we consider linear equations v a t v with a polynomial asymptotic behavior that can be stable unstable and central we show that this behavior is exhibited by a large class of differential equations by giving necessary and sufficient conditions in terms of generalized polynomial lyapunov exponents for the existence of polynomial behavior in particular any linear equation in block form in a finite dimensional space with three blocks having polynomial lyapunov exponents respectively negative positive and zero has a nonuniform version of polynomial trichotomy which corresponds to the usual notion of trichotomy but now with polynomial growth rates we also obtain sharp bounds for the constants in the notion of polynomial trichotomy in addition we establish the persistence under sufficiently small nonlinear perturbations of the stability of a nonuniform polynomial contraction

reduction of peak factor in orthogonal multicarrier modulation by amplitude limiting and coding a main disadvantage of orthogonal multicarrier modulation is that it exhibits a high peak to average power ratio papr because of this all circuits between the output of the modulator and the input of the demodulator must be linear within a large dynamic range to avoid distortions and spectral spreading it is therefore desirable to limit the papr of the multicarrier signal a method based on amplitude limiting and forward error correcting al fec coding is proposed for reduction of the papr the performance of the al fec is compared to two methods based on block coding specially constructed to reduce the papr the al fec is superior as compared to the existing methods moreover its practical realization is very simple for any number of carriers

lyapunov functions for trichotomies with growth rates abstract we consider linear equations x a t x that may exhibit stable unstable and central behaviors in different directions with respect to arbitrary asymptotic rates e c t determined by a function t for example the usual exponential behavior with t t is included as a very special case and when t log t we obtain a polynomial behavior we emphasize that we also consider the general case of nonuniform exponential behavior which corresponds to the existence of what we call a nonuniform exponential trichotomy this is known to occur in a large class of nonautonomous linear equations our main objective is to give a complete characterization in terms of strict lyapunov functions of the linear equations admitting a nonuniform exponential trichotomy this includes criteria for the existence of a nonuniform exponential trichotomy as well as inverse theorems providing explicit strict lyapunov functions for each given exponential trichotomy in the particular case of quadratic lyapunov functions we show that the existence of strict lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the lyapunov functions as an application of the characterization of nonuniform exponential trichotomies in terms of strict lyapunov functions we establish the robustness of nonuniform exponential trichotomies under sufficiently small linear perturbations we emphasize that in comparison with former works our proof of the robustness is much simpler even when t t

low complexity selected mapping schemes for peak to average power ratio reduction in ofdm systems orthogonal frequency division multiplexing ofdm is an attractive transmission technique for high bit rate communication systems one major drawback of ofdm is the high peak to average power ratio papr of the transmitter s output signal the selected mapping slm approach provides good performance for papr reduction but it requires a bank of inverse fast fourier transforms iffts to generate a set of candidate transmission signals and this requirement usually results in high computational complexity in this paper we propose a kind of low complexity conversions to replace the ifft blocks in the conventional slm method based on the proposed conversions we develop two novel slm schemes with much lower complexity than the conventional one the first method uses only one ifft block to generate the set of candidate signals while the second one uses two ifft blocks computer simulation results show that as compared to the conventional slm scheme the first proposed approach has slightly worse papr reduction performance and the second proposed one reaches almost the same papr reduction performance

inertial manifolds for nonlinear evolutionary equations abstract in this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations in particular for ordinary and partial differential equations these manifolds which are finite dimensional invariant lipschitz manifolds seem to be an appropriate tool for the study of questions related to the long time behavior of solutions of the evolutionary equations the inertial manifolds contain the global attractor they attract exponentially all solutions and they are stable with respect to perturbations furthermore in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

center manifolds for infinite delay we obtain a c1 center manifold theorem for perturbations of delay difference equations in banach spaces with infinite delay our results extend in several directions the existing center manifold theorems besides considering infinite delay equations we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable unstable and central behaviors with respect to arbitrary asymptotic rates ec n for some diverging sequence n this includes as a very special case the usual exponential behavior with n n

strongly convex functions of higher order abstract the notion of strongly n convex functions with modulus c 0 is introduced and investigated relationships between such functions and n convex functions in the sense of popoviciu as well as generalized convex functions in the sense of beckenbach are given characterizations by derivatives are presented some results on strongly jensen n convex functions are also given

support type properties of convex functions of higher order and hadamard type inequalities abstract it is well known that every convex function f i r where i r is an interval admits an affine support at every interior point of i i e for any x 0 int i there exists an affine function a i r such that a x 0 f x 0 and a f on i convex functions of higher order precisely of an odd order have a similar property they are supported by the polynomials of degree no greater than the order of convexity in this paper the attaching method is developed it is applied to obtain the general result theorem 2 from which the mentioned above support theorem and some related properties of convex functions of higher both odd and even order are derived they are applied to obtain some known and new hadamard type inequalities between the quadrature operators and the integral approximated by them it is also shown that the error bounds of quadrature rules follow by inequalities of this kind

performance analysis of deliberately clipped ofdm signals we analyze the performance of the clipped orthogonal frequency division multiplexing ofdm system in terms of peak power reduction capability and degradation of channel capacity the clipping is performed on the baseband ofdm signals with and without oversampling followed by the ideal low pass filter first the effect of the envelope clipping on the peak to average power ratio papr and the instantaneous power of the band limited ofdm signal is studied we then discuss the channel capacity of the oversampled and clipped ofdm signals over the additive white gaussian noise and ideally interleaved rayleigh fading channels the capacity is calculated based on the assumption that the distortion terms caused by the clipping are gaussian it is shown that the snr penalty due to the clipping can be considerably alleviated by using optimal coding and reducing the information data rate the results are justified by the simulation results using near optimal turbo codes

novel low complexity slm schemes for papr reduction in ofdm systems selected mapping slm schemes are commonly employed to reduce the peak to average power ratio papr in orthogonal frequency division multiplexing ofdm systems it has been shown that the computational complexity of the traditional slm scheme can be substantially reduced by adopting conversion vectors obtained by using the inverse fast fourier transform ifft of the phase rotation vectors in place of the conventional ifft operations c l wang and y ouyang low complexity selected mapping schemes for peak to average power ratio reduction in ofdm systems ieee trans signal process vol 53 no 12 pp 4652 4660 dec 2005 to ensure that the elements of these phase rotation vectors have an equal magnitude conversion vectors should have the form of a perfect sequence this paper presents three novel classes of perfect sequence each of which comprises certain base vectors and their cyclically shifted versions three novel low complexity slm schemes are then proposed based upon the unique structures of these perfect sequences it is shown that while the papr reduction performances of the proposed schemes are marginally poorer than that of the traditional slm scheme the three schemes achieve a substantially lower computational complexity

on the classes of higher order jensen convex functions and wright convex functions ii abstract the classes of n wright convex functions and n jensen convex functions are compared with each other it is shown that for any odd natural number n the first one is the proper subclass of the second one to reach this aim new tools connected with measure theory are developed

center manifolds under nonuniform hyperbolicity maximal regularity abstract we establish the existence of invariant center manifolds with maximal c r regularity for a nonautonomous dynamics with discrete time we consider the general case of perturbations of a nonuniform exponential trichotomy our proof uses the fiber contraction principle and allows linear perturbations without any further effort

the multitone channel the maximum bit rate of multitone qam quadrature amplitude modulation over a general linear channel is found first the overall bit rate for an awgn channel with a two level transfer function is maximized using a multitone qam system the power distribution between the tones and the number of bits symbol per tone is optimized for a given symbol error rate extending these results to the general channel it is shown that the optimum power division for multitone signals is similar to the water pouring solution of information theory furthermore multitone qam performance is about 9 db worse than the channel capacity independent of the channel characteristics the multitone results throughout are compared to those of an equivalent single tone linearly equalized system the comparison shows that the multitone system is useful for some channels e g those with deep nulls in the transfer function the maximum bit error rate over a twisted pair channel which is performance dominated by near end crosstalk next is also found

smooth center manifolds for nonuniformly partially hyperbolic trajectories abstract we establish the existence of unique smooth center manifolds for ordinary differential equations v a t v f t v in banach spaces assuming that v a t v admits a nonuniform exponential trichotomy this allows us to show the existence of unique smooth center manifolds for the nonuniformly partially hyperbolic trajectories in addition we prove that the center manifolds are as regular as the vector field our proof of the c k smoothness of the manifolds uses a single fixed point problem in an appropriate complete metric space to the best of our knowledge we establish in this paper the first smooth center manifold theorem in the nonuniform setting

viscous singular shock profiles for a system of conservation laws modeling two phase flow abstract this paper is concerned with singular shocks for a system of conservation laws via the dafermos regularization u t f u x t u x x for a system modeling incompressible two phase fluid flow the existence of viscous profiles is proved using geometric singular perturbation theory the weak convergence and the growth rate of the viscous solution are also derived the weak limit is the sum of a piecewise constant function and a measure supported on a shock line and the maximum value of the viscous solution is of order exp 1

lyapunov sequences for exponential trichotomies abstract for a linear cocycle with discrete time we give a complete characterization of nonuniform exponential trichotomies in terms of strict lyapunov sequences we also obtain inverse theorems by constructing explicitly strict lyapunov sequences for each nonuniform exponential trichotomy these are constructed in terms of lyapunov norms with respect to which the nonuniform behavior of the trichotomies becomes uniform we also obtain a corresponding version of the results for cocyles over measure preserving transformations

resident invader dynamics in infinite dimensional systems abstract motivated by evolutionary biology we study general infinite dimensional dynamical systems involving two species the resident and the invader sufficient conditions for competition exclusion phenomena are given when the two species play similar but distinct strategies those conditions are based on invasibility criteria for instance evolutionarily stable strategies in the framework of adaptive dynamics these types of questions were first proposed and studied by s geritz et al 20 and s geritz 19 for a class of ordinary differential equations we extend and generalize previous work in two directions firstly we consider analytic semiflows in infinite dimensional spaces secondly we devise an argument based on hadamard s graph transform method that does not depend on the monotonicity of the two species system our results are applicable to a wide class of reaction diffusion models as well as models with nonlocal diffusion operators

ck centre unstable manifolds we consider the existence and smoothness of global centre unstable manifolds for finite and infinite dimensional flows or maps we show that every global centre unstable manifold can be expressed as a graph of a c k map provided that the nonlinearities are c k smooth the proofs are based on a lemma by d henry on a necessary and sufficient condition for a lipschitz map to be continuously differentiable

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

center manifolds for smooth invariant manifolds this is the published version also available here http dx doi org 10 1090 s0002 9947 00 02443 0 first published in trans amer math soc in 2000 published by the american mathematical society

polynomial contractions and lyapunov regularity abstract the purpose of this note is twofold to introduce the notion of polynomial contraction for a linear nonautonomous dynamics with discrete time and to show that it persists under sufficiently small linear and nonlinear perturbations the notion of polynomial contraction mimics the notion of exponential contraction but with the exponential decay replaced by a polynomial decay we show that this behavior is exhibited by a large class of dynamics by giving necessary conditions in terms of polynomial lyapunov exponents finally we establish the persistence of the asymptotic stability of a polynomial contraction under sufficiently small linear and nonlinear perturbations we also consider the case of nonuniform polynomial contractions for which the lyapunov stability is not uniform

peak to average power ratio in high order ofdm the problem of peak to average power ratio papr of high order orthogonal frequency division modulation ofdm is considered using results on level crossing of random processes an upper bound on the probability that the papr of an ofdm signal will exceed a given value is derived numerical computations are used to show that this bound is tight for low pass ofdm systems the central limit theorem is used to find an asymptotic expression for the bound when the number of carriers n grows to infinity the central limit theorem is also used to find an asymptotic expression for another bound that is based on the envelope of the ofdm signal and is tighter for bandpass systems it is shown that effectively the papr grows as 2lnn and not linearly with n and by developing a lower bound on the probability that the papr of an ofdm signal will exceed a given value it is shown that asymptotically most ofdm symbols have a papr close to 2lnn some approaches to coping with the papr problem are discussed in light of the obtained results

persistent homoclinic orbits for a perturbed nonlinear schrodinger equation the persistence of homoclinic orbits for certain perturbations of the integrable nonlinear schrodinger equation under even periodic boundary conditions is established more specifically the existence of a symmetric pair of homoclinic orbits is established for the perturbed nls equation through an argument that combines melnikov analysis with a geometric singular perturbation theory for the pde 1996 john wiley sons inc

average power reduction techniques for multiple subcarrier intensity modulated optical signals we describe two classes of simple effective techniques for reducing the average optical power requirement in intensity modulated optical systems using multiple bpsk or qpsk subcarriers the first class of techniques involves block coding between the information bits to be transmitted and the symbol amplitudes modulated onto the subcarriers in order to increase the minimum value of the multiple subcarrier electrical waveform the second class of techniques involves replacing the fixed dc bias by a bias signal that varies on a symbol by symbol basis these two classes of techniques can be applied separately or in tandem the reduction in power requirement increases with the number of subcarriers and with eight subcarriers can be as high as about 3 6 db and 3 2 db with bpsk and qpsk respectively the techniques described are applicable as long as all subcarriers originate from a single transmitter and are symbol synchronized

invariant foliations near normally hyperbolic invariant manifolds for semiflows let m be a compact c1 manifold which is invariant and normally hyperbolic with respect to a c1 semiflow in a banach space then in an neighborhood of m there exist local c1 center stable and center unstable manifolds w cs and w cu respectively here we show that w cs and w cu may each be decomposed into the disjoint union of c1 submanifolds leaves in such a way that the semiflow takes leaves into leaves of the same collection furthermore each leaf intersects m in a single point which determines the asymptotic behavior of all points of that leaf in either forward or backward time

approximation dynamics and the stability of invariant sets abstract we introduce the concept of a weakly normally hyperbolic set for a system of ordinary differential equations this concept includes the notion of a hyperbolic flow as well as that of a normally hyperbolic invariant manifold moreover it has the property that it is closed under finite set products consequently the theory presented here can be used for the study of perturbations of the dynamics of coupled systems of weakly normally hyperbolic sets our main objective is to show that under a small c 1 perturbation a weakly normally hyperbolic set k is preserved by a homeomorphism where the image k y is a compact invariant set with a related hyperbolic structure for the perturbed equation in addition the homeomorphism is close to the identity in c 0 1 and the perturbed dynamics on k y are close to the original dynamics on k

properties of center manifolds the center manifold has a number of puzzling properties associated with the basic questions of existence uniqueness differentiability and analyticity which may cloud its profitable application in e g bifurcation theory this paper aims to deal with some of these subtle properties regarding existence and uniqueness it is shown that the cut off function appear ing in the usual existence proof is responsible for the selection of a single center manifold thereby hiding the inherent nonuniqueness conditions are given for different center manifolds at an equilibrium point to have a nonempty intersection this intersection will include at least the families of stationary and periodic solutions crossing through the equilibrium in the case of nonuniqueness the difference between any two center manifolds will be less than el exp e2x l with el and e2 constants which explains why the formal taylor expansions of different center manifolds are the same while the expansions do not converge the differentiability of a center manifold will in certain cases decrease when moving out of the origin and a simple example shows how the differentiability may be lost center manifolds are usually not analytic however an analytic manifold may exist which contains all periodic solutions of a certain type but which may otherwise not be invariant using this manifold a new and very simple proof of the lyapunov subcenter theorem is given

center manifolds for difference equations smooth parameter dependence abstract for sufficiently small c 1 perturbations of nonautonomous linear difference equations with a nonuniform exponential trichotomy we establish the existence of center manifolds with the optimal c 1 regularity we also consider the case of parameter dependent perturbations and we obtain the c 1 dependence of the center manifolds on the parameter in addition we consider arbitrary growth rates with the usual exponential estimates of the form e c t in the notion of exponential trichotomy replaced by e c t where is now an arbitrary function the proof of the regularity both of the center manifold and of its dependence on the parameter is based on the fiber contraction principle the most technical part of the argument concerns the continuity of the fiber contraction that essentially needs a direct argument