supports of derivations free factorizations and ranks of fixed subgroups in free groups for f a free group of finite rank it is shown that the fixed subgroup of any set b of endomorphisms of f has rank rank f generalizing a recent result of dicks and ventura the proof involves the combinatorics of derivations of groups some related questions are examined

decomposition of generalized frequency response functions for nonlinear systems using symbolic computation a symbolic manipulation procedure which both computes and automatically decomposes the generalized frequency response functions of nonlinear rational polynomial and integrodifferential equation models into a set of closed form nth order transfer functions is presented the symbolic representation exposes the explicit relationship between the model parameters and the nonlinear transfer functions in the frequency domain and leads to important insights into the characterization of nonlinear systems examples for each model type are included to demonstrate the symbolic transfer function approach

duality between smooth min and max entropies in classical and quantum information theory operational quantities such as the amount of randomness that can be extracted from a given source or the amount of space needed to store given data are normally characterized by one of two entropy measures called smooth min entropy and smooth max entropy respectively while both entropies are equal to the von neumann entropy in certain special cases e g asymptotically for many independent repetitions of the given data their values can differ arbitrarily in the general case in this paper a recently discovered duality relation between nonsmooth min and max entropies is extended to the smooth case more precisely it is shown that the smooth min entropy of a system a conditioned on a system b equals the negative of the smooth max entropy of a conditioned on a purifying system c this result immediately implies that certain operational quantities such as the amount of compression and the amount of randomness that can be extracted from given data are related we explain how such relations have applications in cryptographic security proofs

an identification algorithm for polynomial narx models based on simulation error minimization classical prediction error approaches for the identification of non linear polynomial narx narmax models often yield unsatisfactory results for long range prediction or simulation purposes mainly due to incorrect or redundant model structure selection the paper discusses some limitations of the standard approach and suggests two modifications namely a new index based on the simulation error is employed as the regressor selection criterion and a pruning mechanism is introduced in the model selection algorithm the resulting algorithm is shown to be effective in the identification of compact and robust models generally yielding model structures closer to the correct ones computational issues are also discussed finally the identification algorithm is tested on a long range prediction benchmark application

an investigation into the characteristics of non linear frequency response functions part 1 understanding the higher dimensional frequency spaces the characteristics of generalized frequency response functions gfrfs of non linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques it is shown how a systematic analysis can be achieved for a wide class of non linear systems in the frequency domain using the proposed methods the paper is divided into two parts in part 1 the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi dimensional frequency spaces the visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time in part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non linear systems described by the narmax class of models case studies are included in part 2 to illustrate the application of the new methods

non reciprocity in nonlinear elastodynamics abstract reciprocity is a fundamental property of linear time invariant lti acoustic waveguides governed by self adjoint operators with symmetric green s functions the break of reciprocity in lti elastodynamics is only possible through the break of time reversal symmetry on the micro level and this can be achieved by imposing external biases adding nonlinearities or allowing for time varying system properties we present a volterra series based asymptotic analysis for studying spatial non reciprocity in a class of one dimensional 1d time invariant elastic systems with weak stiffness nonlinearities we show that nonlinearity is neither necessary nor sufficient for breaking reciprocity in this class of systems rather it depends on the boundary conditions the symmetries of the governing linear and nonlinear operators and the choice of the spatial points where the non reciprocity criterion is tested extension of the analysis to higher dimensions and time varying systems is straightforward from a mathematical point of view but not in terms of new non reciprocal physical phenomena whereas the connection of non reciprocity and time irreversibility can be studied as well finally we show that suitably defined non reciprocity measures enable optimization and can provide physical understanding of the nonlinear effects in the dynamics enabling one to establish regimes of maximum nonlinearity we highlight the theoretical developments by means of a numerical example

nonlinear system identification using gaussian inputs the paper is concerned with the identification of nonlinear systems represented by volterra expansions and driven by stationary zero mean gaussian inputs with arbitrary spectra that are not necessarily white procedures for the computation of the volterra kernels both in the time as well as in the frequency domain are developed based on cross cumulant information the derived kernels are optimal in the mean squared error sense for noncausal systems order recursive procedures based on minimum mean squared error reduction are derived more general input output representations that result when the volterra kernels are expanded in a given orthogonal base are also considered

gain bounds of higher order nonlinear transfer functions bounds on the gain for high order nonlinear transfer functions of a wide class of nonlinear systems are derived it is shown that the gain bounds can be expressed explicitly in terms of the coefficients of discrete time nonlinear models the bounds can be used to check the convergence of the transfer function series and hence to determine the truncation order in practical applications

efficient computation of higher order frequency response functions for nonlinear systems with and without a constant term the behaviour of a nonlinear system can be profoundly affected by the presence of a constant or dc term in the system governing equation these changes are reflected in the nonlinear frequency response characteristics of the system which provide a powerful insight into the system s dynamics in this article a new and efficient algorithm is presented for computing the higher order volterra frequency response functions from nonlinear time domain models that may contain a constant term a comparison with previous methods is included to demonstrate the significant gains in computational efficiency that are achieved using the new method the algorithm is applicable to systems modelled by nonlinear differential or difference equations and is easily automated several examples are used to illustrate the method and to highlight the importance of dc terms in nonlinear system analysis

unconditional security in quantum cryptography basic techniques to prove the unconditional security of quantum crypto graphy are described they are applied to a quantum key distribution protocol proposed by bennett and brassard 1984 the proof considers a practical variation on the protocol in which the channel is noisy and photos may be lost during the transmission each individual signal sent into the channel must contain a single photon or any two dimensional system in the exact state described in the protocol no restriction is imposed on the detector used at the receiving side of the channel except that whether or not the received system is detected must be independent of the basis used to measure this system

the effects of linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems the associated output frequency response function in the present study a new concept known as the associated output frequency response function aofrf is introduced to facilitate the analysis of the effects of both linear and nonlinear characteristic parameters on the output frequency responses of nonlinear systems based on the aofrf concept the study has shown for the first time that the output frequency responses of a wide class of nonlinear systems that are described by the narx nonlinear auto regressive with exogenous input model can be represented by a polynomial function of both the system linear and nonlinear characteristic parameters of interests to the system analysis moreover an efficient algorithm is derived to determine the structure and coefficients of the aofrf based representation for system output frequency responses finally a case study is provided to demonstrate the effectiveness and advantages of the new aofrf based representation and the implication of the result to the analysis and design of nonlinear systems in the frequency domain

the design of energy transfer filters for energy focus filtering this paper is concerned with extending the energy transfer filters etfs designs to a more general case where the design requirement is to focus signal energy from different input frequency bands into a desired output frequency range etfs are a class of non linear filters recently proposed by the authors to employ non linear effects to transfer signal energy from one frequency band to a different frequency location the new etf designs considered in the present study involve the development of an algorithm for determining output frequencies of non linear systems subject to an input the frequency components of which are located in two separate frequency intervals and a new procedure for the implementation of the filter design the results achieved provide an effective approach to the solution to a more general etf design problem

robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices randomness is a vital resource for modern day information processing especially for cryptography a wide range of applications critically rely on abundant high quality random numbers generated securely here we show how to expand a random seed at an exponential rate without trusting the underlying quantum devices our approach is secure against the most general adversaries and has the following new features cryptographic level of security tolerating a constant level of imprecision in devices requiring only unit size quantum memory for each device component in an honest implementation and allowing a large natural class of constructions for the protocol in conjunction with a recent work by chung et al l2014r it also leads to robust unbounded expansion using just 2 multipart devices when adapted for distributing cryptographic keys our method achieves for the first time exponential expansion combined with cryptographic security and noise tolerance the proof proceeds by showing that the renyi divergence of the outputs of the protocol for a specific bounding operator decreases linearly as the protocol iterates at the heart of the proof are a new uncertainty principle on quantum measurements and a method for simulating trusted measurements with untrusted devices

parameterized convergence bounds for volterra series expansion of narx models evaluation of the convergence bound in the frequency domain for volterra series expansion of nonlinear systems described by narx models is studied this provides new convergence criteria under which the nonlinear system of interest has a convergent volterra series expansion and the new criteria are expressed explicitly in terms of the input magnitude model parameters and frequency variable the new convergence criteria are firstly developed for harmonic inputs and then extended to multi tone and general input cases based on the theoretical analysis a general procedure for calculating the convergence bound is provided the results provide a fundamental basis for nonlinear signal processing using the volterra series theory

computable convergence bounds of series expansions for infinite dimensional linear analytic systems and application this paper deals with the convergence of series expansions of trajectories for semi linear infinite dimensional systems which are analytic in state and affine in input a special case of such expansions corresponds to volterra series which are extensively used for the analysis the simulation and the control of weakly nonlinear finite dimensional systems the main results of this paper give computable bounds for both the convergence radius and the truncation error of the series these results can be used for model simplification and analytic approximation of trajectories with a guaranteed quality they are available for distributed and boundary control systems as an illustration these results are applied to an epidemic population dynamic model in this example it is shown that the truncation of the series at order 2 yields an accurate analytic approximation which can be used for time simulation and control issues the relevance of the method is illustrated by simulations

discrete time subharmonic modelling and analysis traditionally the volterra time and frequency domain analysis tools cannot be applied to severely non linear systems in this paper a new method of building a time domain narx miso model for a class of severely siso non linear systems that exhibit subharmonics is introduced and it is shown how this allows the volterra time and frequency domain analysis to be extended to this class of non linear systems the new approach is based on decomposing the original single input based on a fourier analysis to provide a set of modified input signals which have the same period as the output signal a miso narx model can then be constructed from the decomposed multiple inputs and the single output signal the resulting miso model is shown to meet the basic requirement for the existence of a volterra series representation from which important frequency domain properties can be derived explained and discussed this is done by first introducing the derivation of generalized frequency response functions gfrfs from tim

reconstruction of linear and non linear continuous time system models from input output data using the kernel invariance algorithm abstract a new kernel invariance algorithm kia is introduced to determine both the significant model terms and estimate the unknown parameters in non linear continuous time differential equation models of unknown systems

frequency response functions for nonlinear rational models a recursive algorithm which maps a general class of nonlinear rational model defined as the ratio of two polynomial functions into the frequency domain is derived using the harmonic expansion method the new algorithm provides for the first time a direct analytic map from the time domain rational model parameters to the higher order frequency response functions complex nonlinear time domain behaviours can be analysed and interpreted in the frequency domain and simulated examples are included to illustrate the concepts involved

maximal randomness generation from steering inequality violations using qudits we consider the generation of randomness based upon the observed violation of an einstein podolsky rosen epr steering inequality known as one sided device independent randomness generation we show that in the simplest scenario involving only two parties and two measurements for the uncharacterised party with d outcomes that there exist epr steering inequalities whose maximal violation certifies maximal randomness generation equal to mathrm log d bits we further show that all pure partially entangled full schmidt rank states in all dimensions can achieve maximal violation of these inequalities and thus lead to maximal randomness generation in the one sided device independent setting more generally the amount of randomness that can be generated is given by a semidefinite program which we use to study the behavior for nonmaximal violations of the inequalities

linear time encodable decodable codes with near optimal rate we present an explicit construction of linear time encodable and decodable codes of rate r which can correct a fraction 1 r spl epsiv 2 of errors over an alphabet of constant size depending only on spl epsiv for every 0 0 the error correction performance of these codes is optimal as seen by the singleton bound these are near mds codes such near mds linear time codes were known for the decoding from erasures our construction generalizes this to handle errors as well concatenating these codes with good constant sized binary codes gives a construction of linear time binary codes which meet the zyablov bound and also the more general blokh zyablov bound by resorting to multilevel concatenation our work also yields linear time encodable decodable codes which match forney s error exponent for concatenated codes for communication over the binary symmetric channel the encoding decoding complexity was quadratic in forney s result and forney s bound has remained the best constructive error exponent for almost 40 years now in summary our results match the performance of the previously known explicit constructions of codes that had polynomial time encoding and decoding but in addition have linear time encoding and decoding algorithms

on representability of a finite local ring this article is devoted to the following problem let r be a finite local ring with identity can r be embedded into a matrix ring over a commutative ring the question is reduced to an explicitly described class of test cases

a new method for the design of energy transfer filters this paper is concerned with the development of a new method for the design of energy transfer filters etfs etfs are a new class of non linear filters recently proposed by the authors which employ non linear effects to transfer signal energy from one frequency band to a different frequency location the new method uses the powerful orthogonal least squares ols algorithm to solve the least squares problem associated with the design and compared with previous methods achieves much better filtering performance

non linear output frequency response functions for multi input non linear volterra systems the concept of non linear output frequency response functions nofrfs is extended to the non linear systems that can be described by a multi input volterra series model a new algorithm is also developed to determine the output frequency range of non linear systems from the frequency range of the inputs these results allow the concept of nofrfs to be applied to a wide range of engineering systems the phenomenon of the energy transfer in a two degree of freedom non linear system is studied using the new concepts to demonstrate the significance of the new results

entanglement sampling and applications a natural measure for the amount of quantum information that a physical system e holds about another system a a 1 ldots a n is given by the min entropy rm h min a e in particular the min entropy measures the amount of entanglement between e and a and is the relevant measure when analyzing a wide variety of problems ranging from randomness extraction in quantum cryptography decoupling used in channel coding to physical processes such as thermalization or the thermodynamic work cost or gain of erasing a quantum system as such it is a central question to determine the behavior of the min entropy after some process mathcal m is applied to the system a here we introduce a new generic tool relating the resulting min entropy to the original one and apply it to several settings of interest a simple example of such a process is the one of sampling where a subset s of the systems a 1 ldots a n is selected at random our tool allows us to quantify the entanglement that e has with the selected systems a s i e rm h min a s es as a function of the original rm h min a e we give two applications of this result first it directly provides the first local quantum to classical randomness extractors for use in quantum cryptography as well as decoupling operations acting on only a small fraction a s of the input a moreover it gives lower bounds on the dimension of k out of n fully quantum random access encodings another natural example of such a process is a measurement in e g bb84 bases commonly used in quantum cryptography we establish the first entropic uncertainty relations with quantum side information that are nontrivial whenever e is not maximally entangled with a as a consequence we are able to prove optimality of quantum cryptographic schemes in the noisy storage model this model allows for the secure implementation of two party cryptographic primitives under the assumption that the adversary cannot store quantum information perfectly a special case is the bounded quantum storage model bqsm which assumes that the adversary s quantum memory device is noise free but limited in size ever since the inception of the bqsm it has been a vexing open question to determine whether the security is possible as long as the adversary can only store strictly less than the number of qubits n transmitted during the protocol here we show that security is even possible as long as the adversary s device is not larger than n o log 2 n qubits which finally settles the fundamental limits of the bqsm

analysis of output frequencies of nonlinear systems in this paper an algorithm is derived for the determination of the output frequency ranges of nonlinear systems which extends previous results on the output frequencies of nonlinear systems to a more general situation the new results are significant for the analysis of the output frequency response of a wide class of nonlinear systems

application of higher order spectral analysis to cubically nonlinear system identification in this study digital higher order spectral analysis and frequency domain volterra system models are utilized to yield a practical methodology for the identification of weakly nonlinear time invariant systems up to third order the primary focus is on consideration of random excitation of nonlinear systems and thus the approach makes extensive use of higher order spectral analysis to determine the frequency domain volterra kernels which correspond to linear quadratic and cubic transfer functions although the volterra model is nonlinear in terms of its input it is linear in terms of its unknown transfer functions thus a least squares approach is used to determine the optimal in a least squares sense set of linear quadratic and cubic transfer functions of particular practical note is the fact that the approach of this paper is valid for non gaussian as well as gaussian random excitation it may also be utilized for multitone inputs the complexity of the problem addressed in this paper arises from two principal causes 1 the necessity to work in a 3d frequency space to describe cubically nonlinear systems and 2 the necessity to characterize the non gaussian random excitation by computing higher order spectral moments up to sixth order a detailed description of the approach used to determine the nonlinear transfer functions including considerations necessary for digital implementation is presented

improved structure selection for nonlinear models based on term clustering in this paper the concepts of term clusters and cluster coefficients are defined and used in the context of system identification it is argued that if a certain type of term in a nonlinear model is spurious the respective cluster coefficient is small compared with the coefficients of the other clusters represented in the model once the spurious clusters have been detected the corresponding terms can be deleted from the set of candidate terms the consequences of doing this are i a drastic reduction in the size of the set of candidate terms and consequently a substantial gain in computation time is achieved ii the final estimated model is more likely to reproduce the dynamics of the original system and iii the final model is more robust to overparametrization numerical examples are included to illustrate the new procedure

technical communique magnitude bounds of generalized frequency response functions for nonlinear volterra systems described by narx model in order to reveal the relationship between system time domain model parameters and system frequency response functions new magnitude bounds of frequency response functions for nonlinear volterra systems described by narx model are established the magnitude bound of the nth order generalized frequency response function gfrf can be expressed as a simple n degree polynomial function of the magnitude of the first order gfrf whose coefficients are functions of the model parameters and frequency variables thus the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order gfrf these results demonstrate explicitly the analytical relationship between model parameters and system frequency response functions and provide a significant insight into the magnitude based analysis and synthesis of nonlinear systems in the frequency domain

computation of convergence bounds for volterra series of linear analytic single input systems in this paper the volterra series decomposition of a class of single input time invariant systems analytic in state and affine in input is analyzed input to state convergence results are obtained for several typical norms l 0 t l r as well as exponentially weighted norms from the standard recursive construction of volterra kernels new estimates of the kernel norms are derived the singular inversion theorem is then used to obtain the main result of the paper namely an easily computable bound of the convergence radius guaranteed error bounds for the truncated series are also provided the relevance of the method is illustrated in several examples

generalized frequency response functions and output response synthesis for mimo non linear systems the generation of generalized frequency response functions for multi imput multi output non linear system models is investigated with particular emphasis on the cross kernel transforms a unified framework for the output response of non linear systems in the frequency domain is also introduced

on the convergence of the volterra series representation of the duffing s oscillators subjected to harmonic excitations abstract this paper is concerned with the investigation of the convergence issue of the volterra series representations of the duffing s oscillator subjected to harmonic inputs a simple criterion is proposed to determine the upper limit of the magnitude of the harmonic inputs a comparison between the new proposed criterion and a criterion suggested by tomlinson et al g r tomlinson g manson g m lee a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series journal of sound and vibration 190 1996 751 762 was carried out the results show that the new criterion can provide a more accurate prediction about the convergence of the volterra series representation of the duffing s oscillator

embedding rings in completed graded rings 3 algebras over general k abstract it is shown that any associative algebra r over a commutative ring k such that r n 1 0 can be embedded in a graded k algebra h i such that h i 0 only for 1 i 2 n 1 by the results of the first paper in this series r can therefore be embedded in 2 n 1 1 2 n 1 1 strictly upper triangular matrices over a commutative k algebra for k any integral domain not a field this is in fact a best result in contrast with the case of k a field where one can replace 2 n 1 by n see part 2 of this series more generally if r is a nonunital k algebra with a strictly positive integer valued filtration function then r can be mapped into a completed z graded k algebra s by a homomorphism f such that v a v f a 2 v a 1 where v denotes the given filtration function on r and the grading filtration on s if all values of the filtration function on r are powers of 2 one can even get v f a v a

continuous time non linear system identification in the frequency domain a new algorithm for identifying non linear differential equation models is introduced the algorithm avoids the computation of derivatives by using the generalized frequency response functions to reconstruct the model it is shown that the model can be constructed sequentially by building in first the linear terms then the quadratic terms and so on and provides unbiased estimates in the presence of noise a simulated example and the identification of a model relating the action of waveforces on an offshore structure are included to demonstrate the procedure

explicit codes achieving list decoding capacity error correction with optimal redundancy in this paper we present error correcting codes that achieve the information theoretically best possible tradeoff between the rate and error correction radius specifically for every 0 r 1 and epsiv 0 we present an explicit construction of error correcting codes of rate that can be list decoded in polynomial time up to a fraction 1 r epsiv of worst case errors at least theoretically this meets one of the central challenges in algorithmic coding theory our codes are simple to describe they are folded reed solomon codes which are in fact exactly reed solomon rs codes but viewed as a code over a larger alphabet by careful bundling of codeword symbols given the ubiquity of rs codes this is an appealing feature of our result and in fact our methods directly yield better decoding algorithms for rs codes when errors occur in phased bursts the alphabet size of these folded rs codes is polynomial in the block length we are able to reduce this to a constant depending on epsiv using existing ideas concerning ldquolist recoveryrdquo and expander based codes concatenating the folded rs codes with suitable inner codes we get binary codes that can be efficiently decoded up to twice the radius achieved by the standard gmd decoding

new bound characteristics of narx model in the frequency domain new results about the bound characteristics of both the generalized frequency response functions gfrfs and the output frequency response for the narx non linear autoregressive model with exogenous input model are established it is shown that the magnitudes of the gfrfs and the system output spectrum can all be bounded by a polynomial function of the magnitude bound of the first order gfrf and the coefficients of the polynomial are functions of the narx model parameters these new bound characteristics of the narx model provide an important insight into the relationship between the model parameters and the magnitudes of the system frequency response functions reveal the effect of the model parameters on the stability of the narx model to a certain extent and provide a useful technique for the magnitude based analysis of nonlinear systems in the frequency domain for example evaluation of the truncation error in a volterra series expression of non linear systems and the highest order needed in the

mapping non linear integro differential equations into the frequency domain a recursive algorithm is derived to compute the generalized frequency response functions for a large class of non linear integro differential equations applications to duffing s equation and a modified van der pol model are discussed

the parametric characteristic of frequency response functions for nonlinear systems the characteristic of the frequency response functions of nonlinear systems can be revealed and analyzed by analyzing of the parametric characteristics of these functions to achieve these objectives a new operator is defined and several fundamental and important results about the parametric characteristics of the frequency response functions of nonlinear systems are developed these theoretical results provide a significant and novel insight into the frequency domain characteristics of nonlinear systems and circumvent a large amount of complicated integral and symbolic calculations which have previously been required to perform nonlinear system frequency domain analysis several new results for the analysis and synthesis of nonlinear systems are also developed examples are included to illustrate potential applications of the new results

recursive algorithm for computing the frequency response of a class of non linear difference equation models abstract a recursive algorithm is developed which is both computationally compact and enables the nth order frequency response functions of a large class of non linear difference equation models to be found without restriction on the order n

analysis of nonlinear oscillators using volterra series in the frequency domain the volterra series representation is a direct generalisation of the linear convolution integral and has been widely applied in the analysis and design of nonlinear systems both in the time and the frequency domain the volterra series is associated with the so called weakly nonlinear systems but even within the framework of weak nonlinearity there is a convergence limit for the existence of a valid volterra series representation for a given nonlinear differential equation barrett 1965 1 proposed a time domain criterion to prove that the volterra series converges within a given region for a class of nonlinear systems with cubic stiffness nonlinearity in this paper this time domain criterion is extended to the frequency domain to accommodate the analysis of nonlinear oscillators subject to harmonic excitation a common and severe nonlinear phenomenon called jump a behavior associated with the duffing oscillator and the multi valued properties of the response solution is investigated using the new frequency domain criterion of establishing the upper limits of the nonlinear oscillators to predict the onset point of the jump and the volterra time and frequency domain analysis of this phenomenon are carried out based on graphical and numerical techniques

mapping nonlinear integro differential equations to a generalized describing function form an algebraic relationship is derived which enables the generalized amplitude dependent describing function to be expressed in terms of the parameters of a general nonlinear integro differential equation the algorithm is not restricted to a specific input but treats the whole class of harmonically related phase independent sinusoidal inputs and is illustrated by a number of examples

on quantum renyi entropies a new generalization and some properties the renyi entropies constitute a family of information measures that generalizes the well known shannon entropy inheriting many of its properties they appear in the form of unconditional and conditional entropies relative entropies or mutual information and have found many applications in information theory and beyond various generalizations of renyi entropies to the quantum setting have been proposed most prominently petz s quasi entropies and renner s conditional min max and collision entropy however these quantum extensions are incompatible and thus unsatisfactory we propose a new quantum generalization of the family of renyi entropies that contains the von neumann entropy min entropy collision entropy and the max entropy as special cases thus encompassing most quantum entropies in use today we show several natural properties for this definition including data processing inequalities a duality relation and an entropic uncertainty relation

frequency domain analysis for non linear volterra systems with a general non linear output function for non linear volterra systems which include a non linear state equation and a general non linear output function the system frequency response functions and some related frequency response characteristics are developed and discussed in this study these new results establish the frequency response functions for this general form of non linear systems by extending some existing theory and provide an analytical insight into the relationship between model parameters and frequency response functions and the relationship between model parameters and the magnitude bound of frequency response functions several examples are given to illustrate the new results

external forcing of ecological and epidemiological systems a resonance approach analysing ecological and epidemiological models is complicated by the multiplicity of states that can be excited in the interaction between model nonlinearity and external forcing this multiplicity typically created by a hierarchy of subcritical subharmonics can lead to high amplification of the forcing signal and complex switching under chaos and stochastic externalities our objective is to understand the structure of these possible modes of system oscillation in particular the conditions under which they appear and disappear the analysis is carried out by transforming the problem into a problem of locating the resonances of the system under variable period forcing the hierarchy of resonances identified has a well defined generic structure based on the natural period of oscillation of the model whether stable or unstable when isolated the usefulness and simplicity of this approach is illustrated in a discussion of a range of human diseases

analysis of the effects of time delay in nonlinear systems using generalised frequency response functions abstract a recursive algorithm for computing the generalised frequency response functions gfrfs of nonlinear time delay systems described by nonlinear differential difference equation models is derived using the operator e n and used to analyse the effects of time delay in nonlinear systems the algorithm shows an explicit relationship between the model parameters and the gfrfs such a relationship provides important insight into the behaviour of nonlinear systems the effect of delay on several properties of nonlinear systems such as harmonic generation gain compression expansion and desensitisation is studied by considering the example of a duffing oscillator with retarded damping results of the studies convincingly demonstrate that the system delay has a significant effect on several important properties of nonlinear systems

entanglement detection beyond measuring fidelities one of the most widespread methods to determine if a quantum state is entangled or to quantify its entanglement dimensionality is by measuring its fidelity with respect to a pure state in this letter we find a large class of states whose entanglement cannot be detected in this manner we call them unfaithful we find that unfaithful states are ubiquitous in information theory for small dimensions we check numerically that most bipartite states are both entangled and unfaithful similarly numerical searches in low dimensions show that most pure entangled states remain entangled but become unfaithful when a certain amount of white noise is added we also find that faithfulness can be self activated i e there exist instances of unfaithful states whose tensor powers are faithful to explore how the fidelity approach limits the quantification of entanglement dimensionality we generalize the notion of an unfaithful state to that of a d unfaithful state one that cannot be certified as d dimensionally entangled by measuring its fidelity with respect to a pure state for describing such states we additionally introduce a hierarchy of semidefinite programming relaxations that fully characterizes the set of states of schmidt rank at most d

mapping from parametric characteristics to generalized frequency response functions of non linear systems based on the parametric characteristic of the nth order generalized frequency response function gfrf for non linear systems described by a non linear differential equation nde model a mapping function from the parametric characteristics to the gfrfs is established by which the nth order gfrf can be directly written into a more straightforward and meaningful form in terms of the first order gfrf i e an n degree polynomial function of the first order gfrf the new expression has no recursive relationship between different order gfrfs and demonstrates some new properties of the gfrfs which can explicitly unveil the linear and non linear factors included in the gfrfs and reveal clearly the relationship between the nth order gfrf and its parametric characteristic as well as the relationship between the nth order gfrf and the first order gfrf the new results provide a novel and useful insight into the frequency domain analysis and design of non linear systems based on the gfrfs several examples are given to illustrate the theoretical results

output frequency response function of nonlinear volterra systems an expression for the output frequency response function ofrf which defines the explicit analytical relationship between the output spectrum and the system parameters is derived for nonlinear systems which can be described by a polynomial form differential equation model an effective algorithm is developed to determine the ofrf directly from system simulation or experimental data simulation studies demonstrate the significance of the ofrf concept and verify the effectiveness of the algorithm which evaluates the ofrf numerically these new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain

embedding rings in completed graded rings 4 commutative algebras abstract if k is a commutative ring of finite characteristic and a a commutative associative k algebra with augmentation such that the intersection of the powers of the augmentation ideal is zero it is shown that a can be embedded in a completed graded commutative associative k algebra with degree 0 component just k the corresponding result is false in characteristic 0 though we do not know whether it can fail when k is a field the above finite characteristic result is also true with commutative associative algebra replaced by any variety of associative or nonassociative algebras defined by multilinear identities

estimation of generalized frequency response functions for quadratically and cubically nonlinear systems a new method of estimating the volterra kernels in the frequency domain is introduced based on a non parametric algorithm unlike the traditional non parametric methods using the frequency domain formulations based on discrete fourier transform dft data this new approach uses the time domain measurements directly to estimate the frequency domain response functions

a novel characteristic parameter approach for analysis and design of linear components in nonlinear systems nonlinearity always brings difficulties to the analysis and design of practical systems this study shows that for a wide class of nonlinear systems described by nonlinear differential equations the nonlinear output spectrum can be expressed as a polynomial function with respect to linear model parameters based on the recently developed concepts of parametric characteristic analysis and nonlinear characteristic output spectrum ncos function the coefficients of the new ncos function are independent of the linear parameters of interest and can be directly determined with proposed procedures this reveals an explicit and analytical relationship between the output spectrum of a nonlinear system and its linear components and therefore provides a convenient tool for the analysis and design of nonlinear systems using only linear components or linear feedback control while simultaneously fully considering inherent nonlinear influence this result is a useful complement to the ncos function based analysis and design recently developed in the literature the effectiveness and usefulness of the results are demonstrated through an application to the analysis of an amplifier with a multi tone input for suppressing intermodulation distortion it is shown that the new results reveal a straightforward insight into the parametric optimization of the amplifier for suppression of intermodulation distortion

a volterra series representation for a class of nonlinear infinite dimensional systems with periodic boundary conditions abstract this paper proves the existence of a volterra series representation for the mild solutions of a class of nonlinear infinite dimensional systems more specifically given the evolutionary system operator u t s 0 s t associated with a semilinear evolution equation u t 2 u x 2 f u u 0 u 0 x with periodic boundary conditions it is proved that under suitable conditions the unique mild solution u t u t 0 u 0 t 0 can be expanded by a volterra series a recursive algorithm is given to construct the volterra kernels series terms and a nonlinear heat equation is discussed to illustrate the proposed method

embedding rings in completed graded rings 1 triangular embeddings abstract let k be a commutative ring with 1 r an associative k algebra without 1 and n a positive integer it is shown that the following conditions are equivalent a r can be embedded as a k algebra in the ring of strictly upper triangular n n matrices over an associative k algebra a b r has such an embedding with a further assumed commutative c r can be embedded as a k algebra in a graded associative k algebra c whose only nonzero components are c 1 c nn 1 the proof that the other conditions imply b is essentially a demonstration that a certain kind of k linear category always has a faithful functor into a commutative k algebra subsequent papers in this series determine sufficient conditions for a c above to hold in terms of nilpotence of r under various hypotheses

input output parametric models for non linear systems part ii stochastic non linear systems recursive input output models for non linear multivariate discrete time systems are derived and sufficient conditions for their existence are defined the paper is divided into two parts the first part introduces and defines concepts such as nerode realization multistructural forms and results from differential geometry which are then used to derive a recursive input output model for multivariable deterministic non linear systems the second part introduces several examples compares the derived model with other representations and extends the results to create prediction error or innovation input output models for non linear stochastic systems these latter models are the generalization of the multivariable arm ax models for linear systems and are referred to as narmax or non linear autoregressive moving average models with exogenous inputs

nonlinear influence in the frequency domain alternating series the nonlinear influence on system output spectrum is studied for a class of nonlinear systems which have volterra series expansion it is shown that under certain conditions the system output spectrum can be expressed in an alternating series with respect to some model parameters which define system nonlinearities the magnitude of the system output spectrum can therefore be suppressed by exploiting the properties of alternating series sufficient and necessary conditions in which the output spectrum can be cast into an alternating series are studied these results reveal a novel frequency domain insight into the nonlinear influence on a system and provide a new method for the analysis and design of nonlinear systems in the frequency domain examples are given to illustrate the results

comparisons between harmonic balance and nonlinear output frequency response function in nonlinear system analysis by using the duffing oscillator as a case study this paper shows that the harmonic components in the nonlinear system response to a sinusoidal input calculated using the nonlinear output frequency response functions nofrfs are one of the solutions obtained using the harmonic balance method hbm a comparison of the performances of the two methods shows that the hbm can capture the well known jump phenomenon but is restricted by computational limits for some strongly nonlinear systems and can fail to provide accurate predictions for some harmonic components although the nofrfs cannot capture the jump phenomenon the method has few computational restrictions for the nonlinear damping systems the nofrfs can give better predictions for all the harmonic components in the system response than the hbm even when the damping system is strongly nonlinear

a simple criterion for establishing an upper limit to the harmonic excitation level of the duffing oscillator using the volterra series abstract high order frequency response functions based on the volterra series are employed to represent the input output characteristics of the duffing oscillator subject to sinusoidal excitation from these a series representation of a first order frequency response function of the non linear system is obtained and used to derive a simple criterion which provides an estimate of the magnitude of the input sine wave above which the predicted response begins to diverge the criterion results in a small overestimate of the magnitude of the input force excitation at the undamped natural frequency condition although the derived criterion strictly applies only to the duffing oscillator the procedure can be applied generally and is extended to a two degree of freedom system with a non linear cubic stiffness producing a similar criterion to that of the single degree of freedom system

a bound for the magnitude characteristics of nonlinear output frequency response functions part 1 analysis and computation a bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs it is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included the paper is divided into two parts in part 1 an expression for the output magnitude bound is derived properties of the result are discussed and general procedures for the practical computation of the bound are developed in part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed

a fully quantum asymptotic equipartition property the classical asymptotic equipartition property is the statement that in the limit of a large number of identical repetitions of a random experiment the output sequence is virtually certain to come from the typical set each member of which is almost equally likely in this paper a fully quantum generalization of this property is shown where both the output of the experiment and side information are quantum an explicit bound on the convergence is given which is independent of the dimensionality of the side information this naturally leads to a family of reacutenyi like quantum conditional entropies for which the von neumann entropy emerges as a special case

automatic computation of polyharmonic balance equations for non linear differential systems an automated algorithm is presented which enables the harmonic balance equations for any polynomial type pure or cross product non linear differential system to be written down directly in terms of the coefficients of the governing equation and the complex amplitudes of a general harmonic waveform the system frequency response in the form of a multi input amplitude dependent describing function is therefore readily computed the method is illustrated by means of an example and the results validated against detailed numeric simulation

generalized frequency response function matrix for mimo non linear systems recursive algorithms are derived to compute the generalized frequency response function matrix of multi input multioutput mimo non linear systems as an analytical map from both non linear differential equation models and narx non linear auto regressive models with exogenous inputs models of the system the algorithm is computationally compact and exposes the explicit relationship between the model parameters and the elements of the generalized frequency response function matrix and can thus provide important insights into the behaviour of non linear systems

a bound for the magnitude characteristics of nonlinear output frequency response functions part 2 practical computation of the bound for systems described by the nonlinear autoregressive model with exogenous input in part 1 of this paper the concept of a bound for the output frequency response magnitude characteristics of nonlinear systems was proposed and general calculation and analysis procedures were developed in this part 2 of the paper a new recursive algorithm for the computation of the gain bounds for the generalized frequency response functions of the polynomial nonlinear autoregressive model with exogenous input is proposed and effective procedures for the practical computation of the new bound are developed simulated examples are included to verify the effectiveness of the proposed procedures

embedding rings in completed graded rings 2 algebras over a field abstract it is shown that any z filtered associative algebra r over a field k can be embedded in the completion h of a z graded k algebra hi so that the filtration on r is induced by the grading of h a normal form for the universal such h is found a consequence is that if r is a k algebra such that rn 0 then r can be embedded in a graded k algebra hi of which all homogeneous components except h1 hn 1 are zero and so by the results of part 1 of this series can be embedded in n n strictly upper triangular matrices over a commutative k algebra this last result has also been proved by i v l vov

convergence analysis of volterra series response of nonlinear systems subjected to harmonic excitation volterra series provides a strong platform for non linear analysis and higher order frequency response functions however limited convergence is an inherent difficulty associated with the series and needs to be addressed rigorously prior to its application to a physical system the power series representation of the response of non linear systems subjected to harmonic excitation is investigated in this study the problem of convergence is addressed in terms of the convergence of individual frequency harmonics of the non linear response though the procedure is applicable to general polynomial form non linearity it is illustrated for a duffing oscillator subjected to harmonic excitation a general and structured series expression is obtained for amplitudes of all the response harmonics and convergence is investigated in terms of a non dimensional non linear parameter critical values of this parameter representing the upper limit of excitation level for the convergence are defined for a wide range of excitation frequencies zones of convergence and divergence of the response series are presented graphically for a range of the non dimensional non linear parameter and the number of terms included in the approximation of a response harmonic an algorithm based on ratio test is presented to compute the critical value of the non dimensional non linear parameter results obtained from the suggested algorithm are found to be in close agreement with the exact values the method gives better results compared to previous methods and has wider application in terms of excitation frequency the procedure is also investigated for a two degree of freedom system

output frequency characteristics of nonlinear systems some interesting properties of output frequencies of volterra type nonlinear systems are particularly investigated these results provide a very novel and useful insight into the super harmonic and inter modulation phenomena in output frequency response of nonlinear systems with consideration of the effects incurred by different nonlinear components in the system the new properties theoretically demonstrate several fundamental output frequency characteristics and unveil clearly the mechanism of the interaction or coupling effects between different harmonic behaviors in system output frequency response incurred by different nonlinear components these results have significance in the analysis and design of nonlinear systems and nonlinear filters in order to achieve a specific output spectrum in a desired frequency band by taking advantage of nonlinearities they can provide an important guidance to modeling identification control and signal processing by using the volterra series theory in practice

frequency domain analysis and identification of block oriented nonlinear systems abstract block oriented nonlinear models including wiener models hammerstein models and wiener hammerstein models etc have been extensively applied in practice for system identification signal processing and control in this study analytical frequency response functions including generalized frequency response functions gfrfs and nonlinear output spectrum of block oriented nonlinear systems are developed which can demonstrate clearly the relationship between frequency response functions and model parameters and also the dependence of frequency response functions on the linear part of the model the nonlinear part of these models can be a more general multivariate polynomial function these fundamental results provide a significant insight into the analysis and design of block oriented nonlinear systems effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems compared with some existing frequency domain identification methods the new estimation algorithms do not necessarily require model structure information not need the invertibility of the nonlinearity and not restrict to harmonic inputs simulation examples are given to illustrate these new results

describing functions volterra series and the analysis of non linear systems in the frequency domain a relationship between the higher order frequency response functions of the volterra series model and the well known describing function representation for non linear systems is derived it is shown that a large class of generalized describing functions can be derived from the volterra model and that this approach removes the restriction to specific inputs the concept of a worst case transfer function is introduced and examples are included to illustrate the results

determination of the analytical parametric relationship for output spectrum of volterra systems based on its parametric characteristics the output frequency response function ofrf of volterra systems can be described as a polynomial function of model parameters however the analytical determination of the ofrf is very computationally intensive especially for higher order ofrf to circumvent this problem a numerical method can be adopted provided that a series of simulation or experimental data for this polynomial function are given in this study it is theoretically shown that the analytical parametric relationship of ofrf up to any order can be determined accurately by using a simple least square method and every specific component of the output spectrum can also be determined explicitly based on the ofrf s parametric characteristics practical techniques to obtain a unique and accurate solution for the least square method are discussed this study provides a fundamental result for the determination of the analytical parametric relationship for this kind of system polynomial functions by using numerical methods

approximate observability of infinite dimensional bilinear systems using a volterra series expansion in this paper the approximate observability of a class of infinite dimensional bilinear systems is investigated the observability conditions are discussed based on a volterra series representation of this class of systems a testable observability criterion is derived for the case where the infinitesimal generator is self adjoint or riesz spectral operator the theoretical results are illustrated by examples

orthogonal parameter estimation algorithm for non linear stochastic systems an orthogonal parameter estimation algorithm is derived which allows each parameter in a linear in the parameters non linear difference equation model to be estimated sequentially and quite independently of the other parameters in the model the algorithm can be applied for any persistently exciting input and provides both unbiased estimates in the presence of correlated noise and an indication of which terms to include in the model several simulated examples are included to demonstrate the effectiveness of the algorithm

frequency domain truncation of volterra series for weakly non linear systems subject to harmonic excitation new results for model order reduction for weakly non linear systems in the frequency domain are derived based on a parametric modelling approach

robust feedback synthesis for nonlinear integro differential equation models using generalized describing functions we propose a technique for synthesizing robust feedback systems for plants described by nonlinear integro differential equations nide with considerable coefficient uncertainties the technique is based on the concept of generalized frequency response functions gfrf and exploits recent results that enable ready derivation of these from the nide description of the plant a powerful synthesis procedure that combines the gfrf and quantitative feedback theory qft technique is given the proposed procedure has several advantages over the existing nonlinear qft methods a nonlinear chemical reactor example is solved using the procedure and is found to yield satisfactory results

interpretation of non linear frequency response functions analytical and graphical methods of interpreting generalised frequency response functions for non linear systems are derived it is shown that non linear phenomena can be classified into intra kernel and inter kernel interference and that worst case responses can be computed the results are illustrated using several discrete and continuous time non linear systems

modal analysis of a cracked beam abstract this paper addresses the problem of vibrations of a cracked beam in general the motion of such a beam can be very complex this phenomenon can be attributed to the presence of the non linearity due to the opening and closing of cracks the focus of this paper is the modal analysis of a cantilever beam with a transverse edge crack the non linearity mentioned above has been modelled as a piecewise linear system in an attempt to define effective natural frequencies for this piecewise linear system the idea of a bilinear frequency is utilized the bilinear frequency is obtained by computing the associated frequencies of each of the linear pieces of the piecewise linear system the finite element method is used to obtain the natural frequencies in each linear region in order better to understand the essential non linear dynamics of the cracked beam a piecewise linear two degree of freedom model is studied perturbation methods are used to obtain the non linear normal modes of vibration and the associated period of the motion results of this piecewise linear model problem are shown to justify the definition of the bilinear frequency as the effective natural frequency it is therefore expected that calculating piecewise mode shapes and bilinear frequencies is useful for understanding the dynamics of the infinite degree of freedom cracked beam

strong converse for the classical capacity of entanglement breaking and hadamard channels via a sandwiched renyi relative entropy a strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel along with a corresponding achievability statement for rates below the capacity such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication here we show that such a strong converse theorem holds for the classical capacity of all entanglement breaking channels and all hadamard channels the complementary channels of the former these results follow by bounding the success probability in terms of a sandwiched renyi relative entropy by showing that this quantity is subadditive for all entanglement breaking and hadamard channels and by relating this quantity to the holevo capacity prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results

an investigation into the characteristics of non linear frequency response functions part 2 new analysis methods based on symbolic expansions and graphical techniques in part 1 of this paper the concepts of input and output frequency subdomains were introduced to give insight into the relation between one dimensional and multi dimensional frequency spaces the visualisation of both magnitude and phase responses of third order generalized frequency response functions was also presented in this the second part symbolic expansion techniques are introduced and together with the results achieved in part 1 yield new methods for analysing the properties of generalized frequency response functions case studies are included to illustrate the application of the new methods

min and max relative entropies and a new entanglement monotone two new relative entropy quantities called the min and max relative entropies are introduced and their properties are investigated the well known min and max entropies introduced by renner are obtained from these we define a new entanglement monotone which we refer to as the max relative entropy of entanglement and which is an upper bound to the relative entropy of entanglement we also generalize the min and max relative entropies to obtain smooth min and max relative entropies these act as parent quantities for the smooth renyi entropies eth zurich ph d dissertation 2005 and allow us to define the analogues of the mutual information in the smooth renyi entropy framework further the spectral divergence rates of the information spectrum approach are shown to be obtained from the smooth min and max relative entropies in the asymptotic limit