on error saturation nonlinearities in nlms adaptation the effect of a saturation type error nonlinearity in the weight update equation in normalized least mean square nlms adaptation is investigated for system identification for a white gaussian data model nonlinear recursions are derived for the weight mean error and mean square deviation msd that include the effect of an error function erf saturation type nonlinearity on the error sequence driving the algorithm the nonlinear recursion for the msd is solved numerically and shown in excellent agreement with monte carlo simulations supporting the theoretical model assumptions the theory is extended to tracking a markov channel and accurately predicts the tracking behavior as well the saturation behavior of the algorithm is easily studied by varying a single parameter in the error function varying from a linear device to a hard limiter for the white data case the excess mean square error emse is simply related to the msd the tradeoff between the extent of error saturation steady state emse and algorithm convergence rate is studied using these results

performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

the implication problem for functional and inclusion dependencies is undecidable the implication problem for a class of dependencies is the following given a finite set of dependencies determine if they logically imply another dependency we show that the implication problem is undecidable for the class of functional and inclusion dependencies this holds true even if the inclusion dependencies are restricted to be binary it may be noted that the implication problem is known to be decidable for functional and unary inclusion dependencies and also for inclusion dependencies without functional dependencies

stochastic analysis of the lms algorithm with a saturation nonlinearity following the adaptive filter output this paper presents a statistical analysis of the least mean square lms algorithm with a zero memory scaled error function nonlinearity following the adaptive filter output this structure models saturation effects in active noise and active vibration control systems when the acoustic transducers are driven by large amplitude signals the problem is first defined as a nonlinear signal estimation problem and the mean square error mse performance surface is studied analytical expressions are obtained for the optimum weight vector and the minimum achievable mse as functions of the saturation these results are useful for adaptive algorithm design and evaluation the lms algorithm behavior with saturation is analyzed for gaussian inputs and slow adaptation deterministic nonlinear recursions are obtained for the time varying mean weight and mse behavior simplified results are derived for white inputs and small step sizes monte carlo simulations display excellent agreement with the theoretical predictions even for relatively large step sizes the new analytical results accurately predict the effect of saturation on the lms adaptive filter behavior

adaptive filter performance with nonlinearities in the correlation multiplier it is well known that the correlation multiplier used to form the update signal in adaptive transversal filters need not be a true multiplier by using an approximate multiplier such as a sign bit only multiplier implementation can be made less costly in this paper we derive expressions for the convergence rate and misalignment noise of adaptive transversal filters with arbitrary correlation multiplier non linearities the analysis shows any such nonlinearities to always harm performance and quantifies the penalty suffered sign bit only multiplication suffers a significant penalty and power of two multiplication is suggested as a better alternative its performance is almost equal to that of a true multiplier and in a digital implementation at least it is only slightly more difficult to realize than a sign bit only multiplier the theory is developed in a way emphasizing physical understanding and has been verified to be in good agreement with simulation

saturation effects in lms adaptive echo cancellation for binary data the effect of a saturation type error nonlinearity in the weight update equation in least mean square lms adaptive echo cancellation is investigated for an independent binary data model a nonlinear difference equation is derived for the mean norm of the difference between the estimate and the unknown filter to be estimated by the algorithm the difference equation is evaluated numerically it is shown that far end binary data interference is much more deleterious to algorithm transient behavior than far end gaussian data interference the number of additional bits for the same cancellation convergence rates for binary versus gaussian interference of the same power is studied as a function of various system parameters algorithm convergence rates are studied as a function of an arbitrary probability density function pdf for the far end data it is shown that a binary pdf causes the worst degradation and a gaussian shaped pdf causes the least degradation

on error saturation nonlinearities in lms adaptation the effect of a saturation type error nonlinearity in the weight update equation in least mean squares lms adaptation is investigated for a white gaussian data model nonlinear difference equations are derived for the eight first and second moments which include the effect of an error function erf saturation type nonlinearity on the error sequence driving the algorithm a nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures the steady state second moment weight behavior is evaluated exactly for the erf nonlinearity using the above results the tradeoff between the extent of error saturation steady state excess mean square error and rate of algorithm convergence is studied the tradeoff shows that 1 starting with a sign detector the convergence rate is increased by nearly a factor of two for each additional bit and 2 as the number of bits is increased further the additional bit by very little in convergence speed asymptotically approaching the behavior of the linear algorithm

nonlinear quantization effects in the frequency domain complex scalar lms adaptive algorithm nonlinear quantization effects in the frequency domain complex scalar lms adaptive algorithm are analyzed by using conditional expectations the probability density function of the quantizer input conditioned on the weight is derived the density is applied to finding the conditional characteristic function and the mth conditional moment at the quantizer output the first and second conditional moments of the quantizer output are used to derive difference equations that approximate the dynamical behavior of the first and second weight moments these difference equations are solved numerically and compare favorably to simulation results a model of the quantizer as an additive noise source is of no analytical value since the quantizatian noise has negligible effect on the mean square error when the model is valid finally a design approach is proposed for selecting the number of bits in the weight accumulator the moment equations are also used to determine the algorithm mean square error for different quantizer step sizes and the optimum algorithm step size when a fixed amount of input data is available for adaptation

nonlinear quantization effects in the lms and block lms adaptive algorithms a comparison digital implementations of the least mean square lms and block lms blms algorithms are compared with respect to finite word effects the algorithm stalling phenomenon is studied using gaussian data models and conditional expectation arguments it is shown that the blms algorithm requires 1 2 log sub 2 l k fewer bits for the same stalling behavior l block length and k lies between 0 2 and 1 0 depending on the precise definition of algorithm stalling on the other hand the lms algorithm requires log sub 2 l fewer bits than blms for the same level of saturation behavior transient response at algorithm initialization hence the lms algorithm requires 1 2 log sub 2 l k fewer bits than the blms algorithm for the same saturation and stalling effects

efficient nonlinear wiener model identification using a complex valued simplicial canonical piecewise linear filter this paper proposes an efficient adaptive realization of the wiener model for the identification of complex valued nonlinear systems using a two dimensional simplicial canonical piecewise linear filter for the complex valued nonlinear mapping we derive a realization of the wiener model requiring fewer parameters than previous approaches an adaptive implementation of the proposed wiener model is derived and local convergence analysis for the updating algorithm is presented the tradeoff between computational complexity and modeling performance is discussed simulations of a system identification example show that the proposed algorithm can provide similar or better performance than other approaches in terms of computational complexity convergence speed and final mean squared error mse

error estimates with explicit constants for sinc approximation sinc quadrature and sinc indefinite integration error estimates with explicit constants are given for approximations of functions definite integrals and indefinite integrals by means of the sinc approximation although in the literature various error estimates have already been given for these approximations those estimates were basically for examining the rates of convergence and several constants were left unevaluated giving more explicit estimates i e evaluating these constants is of great practical importance since by this means we can reinforce the useful formulas with the concept of verified numerical computations in this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems the function to be approximated is analytic in a suitable region we also improve some formulas themselves to decrease their computational costs numerical examples that confirm the theory are also given

data reusing recurrent neural adaptive filters a class of data reusing learning algorithms for real time recurrent neural networks rnns is analyzed the analysis is undertaken for a general sigmoid nonlinear activation function of a neuron for the real time recurrent learning training algorithm error bounds and convergence conditions for such data reusing algorithms are provided for both contractive and expansive activation functions the analysis is undertaken for various configurations that are generalizations of a linear structure infinite impulse response adaptive filter

stochastic analysis of gradient adaptive identification of nonlinear systems with memory for gaussian data and noisy input and output measurements this paper investigates the statistical behavior of two gradient search adaptive algorithms for identifying an unknown nonlinear system comprised of a discrete time linear system h followed by a zero memory nonlinearity g spl middot the input and output of the unknown system are corrupted by additive independent noises gaussian models are used for all inputs two competing adaptation schemes are analyzed the first is a sequential adaptation scheme where the lms algorithm is first used to estimate the linear portion of the unknown system the lms algorithm is able to identify the linear portion of the unknown system to within a scale factor the weights are then frozen at the end of the first adaptation phase recursions are derived for the mean and fluctuation behavior of the lms algorithm which are in excellent agreement with monte carlo simulations when the nonlinearity is modeled by a scaled error function the second part of the sequential gradient identification scheme is shown to correctly learn the scale factor and the error function scale factor mean recursions for the scale factors show good agreement with monte carlo simulations for slow learning the stationary points of the gradient algorithm closely agree with the stationary points of the theoretical recursions the second adaptive scheme simultaneously learns both the linear and nonlinear portions of the unknown channel the mean recursions for the linear and nonlinear portions show good agreement with monte carlo simulations for slow learning the stationary points of the gradient algorithm also agree with the stationary points of the theoretical recursions

transient and tracking performance analysis of the quantized lms algorithm for time varying system identification this paper investigates the statistical behavior of the finite precision lms adaptive filter in the identification of an unknown time varying stochastic system nonlinear recursions are derived for the mean and mean square behavior of the adaptive weights transient and tracking algorithm performance curves are generated from the recursions and shown to be in excellent agreement with monte carlo simulations our results demonstrate that linear models are inappropriate for analyzing the transient and the steady state algorithm behavior the performance curves indicate that the transient and tracking capabilities cannot be determined from perturbations about the infinite precision case it is shown that the transient phase of the algorithm increases as the digital wordlength or the speed of variation of the unknown system decrease design examples illustrate how the theory can be used to select the algorithm step size and the number of bits in the quantizer

on error saturation nonlinearities for lms adaptation in impulsive noise this correspondence extends the analytic results in n j bershad ldquoon error nonlinearities in lms adaptation rdquo ieee transactions on acoustics speech and signal processing vol assp 36 no 4 pp 440 452 april 1988 to least mean square lms adaptation in impulsive observation noise a scalar recursion for the weight misadjustment is derived for the white input data case monte carlo simulations verify the accuracy of the theoretical model the theoretical recursion is then used to study the effects of the impulse noise on algorithm convergence speed and steady state weight misadjustment for a wide variety of parameter values

analysis of stochastic gradient identification of wiener hammerstein systems for nonlinearities with hermite polynomial expansions this paper investigates the statistical behavior of a sequential adaptive gradient search algorithm for identifying an unknown wiener hammerstein 1958 system whs with gaussian inputs the whs nonlinearity is assumed to be expandable in a series of orthogonal hermite polynomials the sequential procedure uses 1 a gradient search for the unknown coefficients of the hermite polynomials 2 an lms adaptive filter to partially identify the input and output linear filters of the whs and 3 the higher order terms in the hermite expansion to identify each of the linear filters the third step requires the iterative solution of a set of coupled nonlinear equations in the linear filter coefficients an alternative scheme is presented if the two filters are known a priori to be exponentially shaped the mean behavior of the various gradient recursions are analyzed using small step size approximations slow learning and yield very good agreement with monte carlo simulations several examples demonstrate that the scheme provides good estimates of the whs parameters for the cases studied

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods in this paper the theoretical convergence rate of the trapezoidal rule combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable it is well known that the de transformation enables the rule to achieve a much higher rate of convergence than the se transformation and the convergence rate has been analyzed and justified theoretically under a proper assumption here it should be emphasized that the assumption is more severe than the one for the se transformation and there actually exist some examples such that the trapezoidal rule with the se transformation achieves its usual rate whereas the rule with de does not such cases have been observed numerically but no theoretical analysis has been given thus far this paper reveals the theoretical rate of convergence in such cases and it turns out that the de s rate is almost the same as but slightly lower than that of the se by using the analysis technique developed here the theoretical convergence rate of the sinc approximation with the de transformation is also given for a class of functions for which the se transformation is suitable the result is quite similar to above the convergence rate in the de case is slightly lower than in the se case numerical examples which support those two theoretical results are also given

two formulas for numerical indefinite integration we derive two formulas for approximating the indefinite integral over a finite interval the approximation error is 0 c cv uniformly where m is the number of integrand evaluations the integrand is required to be analytic in the interior of the integration interval but may be singular at the endpoints some sample calculations indicate that the actual convergence rate accords with the error bound derived

stochastic analysis of adaptive gradient identification of wiener hammerstein systems for gaussian inputs this correspondence investigates the statistical behavior of two adaptive gradient search algorithms for identifying an unknown wiener hammerstein system whs with gaussian inputs the first scheme attempts to identify the whs with an lms adaptive filter the lms algorithm identifies a scaled version of the convolution of the input and output linear filters of the whs the second scheme attempts to identify the unknown whs with a gradient adaptive whs when the shape of the nonlinearity is known a priori the mean behavior of the gradient recursions are analyzed when the whs nonlinearity is modeled by an error function the mean recursions yield very good agreement with monte carlo simulations for slow learning

a nonlinear analytical model for the quantized lms algorithm the power of two step size case presents a study of the quantization effects in the finite precision lms algorithm with power of two step sizes deterministic nonlinear recursions are presented for the mean and second moment matrix of the weight vector about the wiener weight for white gaussian data models and small algorithm step size spl mu the numerical solutions of these recursions are shown to agree very closely with the monte carlo simulations during all phases of the adaptation process design examples demonstrate the selection of the number of quantizer bits and the adaptation step size spl mu to yield a desired transient behavior and cancellation performance the results obtained indicate that previous models are too conservative in predicting the converged mse for a given number of bits

on rules with existential variables walking the decidability line we consider positive rules in which the conclusion may contain existentially quantified variables which makes reasoning tasks such as conjunctive query answering or entailment undecidable these rules called email protected rules have the same logical form as tuple generating dependencies in databases and as conceptual graph rules the aim of this paper is to provide a clearer picture of the frontier between decidability and non decidability of reasoning with these rules previous known decidable classes were based on forward chaining on the one hand we extend these classes on the other hand we introduce decidable classes based on backward chaining a side result is the definition of a backward mechanism that takes the complex structure of email protected rule conclusions into account we classify all known decidable classes by inclusion then we study the question of whether the union of two decidable classes remains decidable and show that the answer is negative except for one class and a still open case this highlights the interest of studying interactions between rules we give a constructive definition of dependencies between rules and widen the landscape of decidable classes with conditions on rule dependencies and a mixed forward backward chaining mechanism finally we integrate rules with equality and negative constraints to our framework

analysis of stochastic gradient tracking of time varying polynomial wiener systems this paper presents analytical and monte carlo results for a stochastic gradient adaptive scheme that tracks a time varying polynomial wiener 1958 system i e a linear time invariant lti filter with memory followed by a time varying memoryless polynomial nonlinearity the adaptive scheme consists of two phases 1 estimation of the lti memory using the lms algorithm and 2 tracking the time varying polynomial type nonlinearity using a second coupled gradient search for the polynomial coefficients the time varying polynomial nonlinearity causes a time varying scaling for the optimum wiener filter for phase 1 these time variations are removed for phase 2 using a novel coupling scheme to phase 1 the analysis for gaussian data includes recursions for the mean behavior of the lms algorithm for estimating and tracking the optimum wiener filter for phase 1 for several different time varying polynomial nonlinearities and recursions for the mean behavior of the stochastic gradient algorithm for phase 2 the polynomial coefficients are shown to be accurately tracked monte carlo simulations confirm the theoretical predictions and support the underlying statistical assumptions

on weight update saturation nonlinearities in lms adaptation the effect of a saturation type nonlinearity in the weight update equation in least mean square lms adaptation is investigated for a white gaussian data model nonlinear difference equations that include the effect of a 1 e sup ax saturation type nonlinearity on the update term driving the algorithm are derived for the weight first and second moments a nonlinear difference equation for the mean norm is explicitly solved using a differential equation approximation and integration by quadratures the steady state second moment weight behavior is evaluated approximately for the nonlinearity using these results the tradeoff between the extent of weight update saturation steady state excess mean square error and rate of algorithm convergence is studied for the same steady state misadjustment error the tradeoff results are discussed

the double exponential sinc collocation method for singular sturm liouville problems sturm liouville problems are abundant in the numerical treatment of scientific and engineering problems in the present contribution we present an efficient and highly accurate method for computing eigenvalues of singular sturm liouville boundary value problems the proposed method uses the double exponential formula coupled with sinc collocation method this method produces a symmetric positive definite generalized eigenvalue system and has exponential convergence rate numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

numerical indefinite integration by double exponential sinc method we present a numerical method for approximating an indefinite integral by the double exponential sinc method the approximation error of the proposed method with n integrand function evaluations is o exp c 1 n log c 2 n for a reasonably wide class of integrands including those with endpoint singularities the proposed method compares favorably with the existing formulas based on the ordinary sinc method computational results show the accordance of the actual convergence rates with the theoretical estimate

performance analysis of a converged single layer perceptron for nonseparable data models with bias terms rosenblatt s 1985 algorithm is a recursive method used to adjust the weights of a single layer perceptron it is capable of partitioning the input signal space into two regions that are separated by a hyperplane boundary thus when the values of the input signal are linearly separable the algorithm will converge to a stable stationary point that yields zero mean square error the authors examine the stationary points of rosenblatt s algorithm when the data is not linearly separable a system identification model is used to generate the data the model incorporates the effects of bias terms so that the hyperplane boundaries do not necessarily pass through the origin of the signal space an expression is also derived for the probability of an incorrect classification of the output signal when the weights are converged at a stationary point

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

on the lms algorithm with constant and variable leakage factor in a nonlinear environment this paper studies the application of the linear least mean square lms algorithm with leakage to a system having a memoryless saturation type nonlinearity this approach represents an interesting alternative to the nonlinear lms nllms algorithm the major drawback to implement the latter is that the model parameters of the system nonlinearity must be known in contrast the linear lms algorithm with leakage does not require such knowledge it permits to approximate the performance of the nllms by properly selecting a constant leakage factor value to cope with an eventual change of the environment a strategy for a variable leakage is also proposed several numerical simulations are presented with the aim of ratifying the feasibility of the proposed approach

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

function classes for double exponential integration formulas the double exponential de formulas for numerical integration are known to be highly efficient more efficient than the single exponential se formulas in many cases function classes suited to the se formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de formulas this paper identifies function classes suited to the de formulas in a way compatible with the existing theoretical results for the se formulas the de formulas are good for more restricted classes of functions but more efficient for such functions two concrete examples demonstrate the subtlety in the behavior of the de formulas that is revealed by our theoretical analysis

error estimates with explicit constants for the sinc approximation over infinite intervals abstract the sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis even for other functions the sinc approximation works accurately when combined with a proper variable transformation the convergence rate has been analyzed for typical cases including finite semi infinite and infinite intervals recently for verified numerical computations a more explicit computable error bound has been given in the case of a finite interval in this paper such explicit error bounds are derived for other cases

a proof procedure for data dependencies definition d une classe de dependances dependances generant l equalite et les triples et generalisation du processus de poursuite pour traiter ces dependances on montre quelques cas pour lesquels il y a une procedure de decision redondance de l egalite pour decider de l implication des dependances generant les triples et presque redondance pour decider de l implication des dependances generant l egalite

a nonlinear analytical model for the quantized lms algorithm the arbitrary step size case this paper extends conditional moment techniques previously developed for the study of nonlinear versions of the lms algorithm to the study of the effects of quantizers in the finite precision case deterministic nonlinear recursions are derived for the mean and second moment matrix of the weight vector about the wiener weight for white gaussian data models and small algorithm step sizes spl mu these recursions are solved numerically and shown to be in very close agreement with the monte carlo simulations during all phases of the adaptation process a design example is presented that demonstrates how the theory can be used to select the number of quantizer bits and the adaptation step size spl mu to yield a desired transient behavior and cancellation performance

the composite regressor algorithm for iir adaptive systems an adaptive iir algorithm called the composite regressor algorithm cra is developed the algorithm is a generalization of the common equation error a priori output error and a posteriori output error adaptive iir algorithms the cra is analyzed for convergence in a noiseless environment and for bias in a stochastic setting it is determined that by using a parameter called the regressor composition parameter a tradeoff can be obtained between the automatic convergence but large bias results of the equation error algorithm and the difficult convergence condition but small bias results of the output error algorithms in proving results for the cra it is shown that the a posteriori output error algorithm produces estimates with nonzero bias when the adaptive gain is small but bounded away from zero a convergence condition for the a priori output error algorithm is derived for the first time

statistical analysis of the single layer backpropagation algorithm i mean weight behavior the single layer backpropagation algorithm is a gradient descent method that adjusts the connection weights of a single layer perceptron to minimize the mean square error at the output it is similar to the standard least mean square al gorithm except the output of the linear combiner contains a differentiable nonlinearity in this paper we present a statis tical analysis of the mean weight behavior of the single layer backpropagation algorithm for gaussian input signals it is based on a nonlinear system identification model of the desired response which is capable of generating an arbitrary hyper plane decision boundary it is demonstrated that although the weights grow unbounded the algorithm on average quickly learns the correct hyperplane associated with the system iden tification model

double exponential formulas for numerical indefinite integration in this paper we derive a formula for indefinite integration of analytic functions over 1 s where 1 s 1 by means of the double exponential transformation and the sinc method the integrand must be analytic on 1 x 1 but may have a singularity at the end points x 1 the error of the formula behaves approximately as exp c1n logc2n where n is the number of function evaluations of the integrand this error term shows a much faster convergence to zero when n becomes large than that of the known formula by haber also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over 0 s 0 s and over s s several numerical examples indicate high efficiency of the formulas

comparison of the convergence of two algorithms for adaptive fir digital filters the convergence properties of two different algorithms for the updating of the coefficients of an adaptive fir digital filter are investigated and compared with one another these algorithms are the stochastic iteration algorithm and the sign algorithm in this latter algorithm a one bit gradient estimation is used which makes its implementation very simple the convergence is characterized by the residual echo variance after convergence and a parameter that indicates the speed of the convergence it is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained

improved convergence analysis of stochastic gradient adaptive filters using the sign algorithm convergence analysis of stochastic gradient adaptive filters using the sign algorithm is presented in this paper the methods of analysis currently available in literature assume that the input signals to the filter are white this restriction is removed for gaussian signals in our analysis expressions for the second moment of the coefficient vector and the steady state error power are also derived simulation results are presented and the theoretical and empirical curves show a very good match

recent developments of the sinc numerical methods abstract this paper gives a survey of recent developments of the sinc numerical methods a variety of sinc numerical methods have been developed by stenger and his school for a certain class of problems the sinc numerical methods have the convergence rates of o exp n with some 0 where n is the number of nodes or bases used in the methods recently it has turned out that the sinc numerical methods can achieve convergence rates of o exp n log n with some 0 for a smaller but still practically meaningful class of problems and that these convergence rates are best possible the present paper demonstrates these facts for two sinc numerical methods the sinc approximation and the sinc collocation method for two point boundary value problems

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

the price of query rewriting in ontology based data access we give a solution to the succinctness problem for the size of first order rewritings of conjunctive queries in ontology based data access with ontology languages such as owl 2 ql linear datalog and sticky datalog we show that positive existential and nonrecursive datalog rewritings which do not use extra non logical symbols except for intensional predicates in the case of datalog rewritings suffer an exponential blowup in the worst case while first order rewritings can grow superpolynomially unless np p poly we also prove that nonrecursive datalog rewritings are in general exponentially more succinct than positive existential rewritings while first order rewritings can be superpolynomially more succinct than positive existential rewritings on the other hand we construct polynomial size positive existential and nonrecursive datalog rewritings under the assumption that any data instance contains two fixed constants

function classes for successful de sinc approximations the de sinc formulas resulting from a combination of the sinc approximation formula with the double exponential de transformation provide a highly efficient method for function approximation in many cases they are more efficient than the se sinc formulas which are the sinc approximation formulas combined with the single exponential se transformations function classes suited to the se sinc formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de sinc formulas this paper identifies function classes suited to the de sinc formulas in a way compatible with the existing theoretical results for the se sinc formulas furthermore we identify alternative function classes for the de sinc formulas as well as for the se sinc formulas which are more useful in applications in the sense that the conditions imposed on the functions are easier to verify

on the optimum data nonlinearity in lms adaptation the effect of an arbitrary nonlinear operation on the data input to the weight update equation in the lms adaptive algorithm is investigated for a gaussian data model exact difference equations are derived for the weight first and second moments that include the effects of an arbitrary nonlinear operation on the data sequence the difference equations are used to obtain expressions for the transient behavior of the mean square error the mean square error is minimized over the choice of nonlinearity for a fixed transient behavior the best choice of nonlinearity is shown exactly to be of the form x 1 bx2 when the input data are white for an arbitrary data covariance matrix the optimum nonlinearity is shown to be linear when the product of the algorithm step size and input power is much less than unity

statistical analysis of the single layer backpropagation algorithm ii mse and classification performance for pt i see ibid p 583 91 1993 the analysis of pt i is extended to the evaluation of the mean square error and the probability of correct classification it is shown that the mean square error and the corresponding performance surface are such that the single layer perceptron is prevented from correctly classifying with probability one until the weights converge at infinity

optimality of the double exponential formula functional analysis approach in the light of the functional analysis theory we establish the optimality of the double exponential formula the argument consists of the following three ingredients 1 introduction of a number of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity 2 proof of the near optimality of the trapezoidal formula in each space introduced in 1 by showing the near equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures 3 nonexistence theorem for the spaces the characterizing decay rate of which is more rapid than the double exponential

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods part ii indefinite integration in this paper the theoretical convergence rate of the sinc indefinite integration combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable although the de transformation is considered as an enhanced version of the se transformation for sinc related methods the function space for which the de transformation is suitable is smaller than that for se and therefore there exist some examples such that the de transformation is not better than the se transformation even in such cases however some numerical observations in the literature suggest that there is almost no difference in the convergence rates of se and de in fact recently the observations have been theoretically explained for two explicit approximation formulas the sinc quadrature and the sinc approximation the conclusion is that in such cases the de s rate is slightly lower but almost the same as that of the se the contribution of this study is the derivation of the same conclusion for the sinc indefinite integration numerical examples that support the theoretical result are also provided

stochastic gradient identification of polynomial wiener systems analysis and application this paper presents analytical numerical and experimental results for a stochastic gradient adaptive scheme that identifies a polynomial type nonlinear system with memory for noisy output observations the analysis includes the computation of the stationary points the mean square error surface and the stability regions of the algorithm for gaussian data convergence of the mean is studied using l sub 2 and euclidian norms monte carlo simulations confirm the theoretical predictions that show a small sensitivity to the observation noise an application is presented for the identification of a nonlinear time delayed feedback system