subharmonic and homoclinic bifurcations in a parametrically forced pendulum abstract depending on the parameters of a parametrically forced pendulum system the boundaries of subharmonic and homoclinic bifurcations are calculated on the basis of the melnikov method and of averaging methods it is shown that as a parameter is varied repeated resonances of successively higher periods occur culminating in homoclinic orbits according to the theorem of smale homoclinic bifurcation is the source of the unstable chaotic motions observed for some selected parameter sets the theoretical predictions are tested by numerical calculations very good agreement is found between analytical and numerical results

designing plateaued boolean functions in spectral domain and their classification the design of plateaued functions over gf 2 n also known as 3 valued walsh spectra functions taking the values from the set 0 pm 2 lceil n s 2 rceil has been commonly approached by specifying a suitable algebraic normal form which then induces this particular walsh spectral characterization in this paper we consider the reversed design method which specifies these functions in the spectral domain by specifying a suitable allocation of the nonzero spectral values and their signs we analyze the properties of trivial and nontrivial plateaued functions as affine inequivalent distinct subclasses which are distinguished by their walsh support s f the subset of gf 2 n having the nonzero spectral values in terms of whether it is an affine subspace or not the former class exactly corresponds to partially bent functions and admits linear structures whereas the latter class may contain functions without linear structures a simple sufficient condition on s f which ensures the nonexistence of linear structures is derived and some generic design methods of nontrivial plateaued functions without linear structures are given the extended affine equivalence of plateaued functions is also addressed using the concept of dual of plateaued functions furthermore we solve the problem of specifying disjoint spectra non trivial plateaued functions of maximal cardinality whose concatenation can be used to construct bent functions in a generic manner this approach may lead to new classes of bent functions due to large variety of possibilities to select underlying duals that define these disjoint spectra plateaued functions an additional method of specifying affine inequivalent plateaued functions obtained by applying a nonlinear transform to their input domain is also given

testing and estimating shape constrained nonparametric density and regression in the presence of measurement error in many applications we can expect that or are interested to know if a density function or a regression curve satisfies some specific shape constraints for example when the explanatory variable x represents the value taken by a treatment or dosage the conditional mean of the response y is often anticipated to be a monotone function of x indeed if this regression mean is not monotone in the appropriate direction then the medical or commercial value of the treatment is likely to be significantly curtailed at least for values of x that lie beyond the point at which monotonicity fails in the case of a density common shape constraints include log concavity and unimodality if we can correctly guess the shape of a curve then nonparametric estimators can be improved by taking this information into account addressing such problems requires a method for testing the hypothesis that the curve of interest satisfies a shape constraint and if the conclusion of the test is positive a technique for e

classical laplace estimation for n 3 consistent estimators improved convergence rates and rate adaptive inference abstract we propose a classical laplace estimator alternative for a large class of n 3 consistent estimators including isotonic regression monotone hazard and maximum score estimators the proposed alternative provides a unified method of smoothing easier computation is a byproduct in the maximum score case depending on input parameter choice and smoothness the convergence rate of our estimator varies between n 3 and almost n and its limit distribution varies from chernoff to normal we provide a bias reduction method and an inference procedure which automatically adapts to the correct convergence rate and limit distribution

extension and improvement to the krylov bogoliubov methods using elliptic functions abstract it is shown that the krylov bogoliubov methods that give the approximate oscillatory solution of the equation in terms of jacobi elliptic functions are applicable not only when c 1 0 and c 3 0 but also when c 1 0 and c 3 0 in particular the most precise of these methods the christopher brocklehurst method is discussed in detail its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters

solution of a mixed parity nonlinear oscillator harmonic balance abstract the method of harmonic balance is used to calculate first order approximations to the periodic solutions of a mixed parity nonlinear oscillator first the amplitude in the negative direction is expressed in terms of the amplitude in the positive direction then the two auxiliary equations where the restoring force functions are odd are solved by using a first harmonic term without a constant therefore we obtain the two approximate solutions to the mixed parity nonlinear oscillator one is expressed in terms of the exact amplitude in the negative direction the other in terms of the approximate amplitude these solutions are more accurate than the second approximate solution obtained by the lindstedt poincare method for large amplitudes

on the estimation of an instantaneous transformation for time series summary we modify ramsay s algorithm for estimating monotonic transformations in regression and extend it to autoregression where strict monotonicity is an essential requirement compared with other methods our method can capture some characteristics that are pertinent to the time series and is much easier to implement an order selection method is introduced and developed some real data sets are analysed

unified approach to testing functional hypotheses in semiparametric contexts abstract we suggest a new method with very wide applicability for testing semiparametric hypotheses about functions such as regression means and probability densities the technique is based on characterising hypotheses in terms of functionals which can be estimated root n consistently and constructing test statistics in terms of estimators of the functionals since the tests are semiparametric it is appropriate to assess them on the basis of their ability to detect departures of size n 1 2 from the null hypothesis we show that they do indeed have this property unlike tests constructed in a nonparametric setting their power does not depend critically on choice of a bandwidth and in particular smoothing parameter selection is not an issue that has to be addressed by users of the tests bootstrap methods are suggested for calibrating the tests in a regression setting applications include tests of specification such as partial linear and index models against nonparametric or semiparametric alternatives and tests of monotonicity concavity separability equality of regression functions and base independence of equivalence scales in a density setting they include tests of radial symmetry and stochastic dominance

estimating smooth monotone functions many situations call for a smooth strictly monotone function f of arbitrary flexibility the family of functions defined by the differential equation d 2 f w df where w is an unconstrained coefficient function comprises the strictly monotone twice differentiable functions the solution to this equation is f c 0 c 1 d 1 exp d 1 w where c 0 and c 1 are arbitrary constants and d 1 is the partial integration operator a basis for expanding w is suggested that permits explicit integration in the expression of f in fitting data it is also useful to regularize f by penalizing the integral of w 2 since this is a measure of the relative curvature in f applications are discussed to monotone nonparametric regression to the transformation of the dependent variable in non linear regression and to density estimation

comments on the method of harmonic balance the advantages and limitations of the harmonic balance or describing function approximation scheme for solving nonlinear ordinary differential equations of oscillatory motion are discussed advantages include appicability to equations of any order and with large degrees of nonlinearity ease of determining limit cycle behavior and its stability and overall speed and efficiency the limitation rules are essentially those described by mickens 1983 it is pointed out that perturbation procedures provide better results when the degree of nonlinearity is small

a generalized galerkin s method for non linear oscillators by allowing for frequency variation in the virtual displacement co ordinates a generalized galerkin s procedure is developed this leads to a compact form of galerkin s conditions which in turn yield as many algebraic equations as the number of unknown parameters and frequency ies the independence of the equations is not guaranteed but depends on the nature of the oscillators when they are not independent the possible contradiction serves as a measure of the approximation of the solution the two well known non linear oscillators van der pol s and duffing s are used as the basic examples for the application of the method

calibration using constrained smoothing with applications to mass spectrometry data summary linear regressions are commonly used to calibrate the signal measurements in proteomic analysis by mass spectrometry however with or without a monotone e g log transformation data from such functional proteomic experiments are not necessarily linear or even monotone functions of protein or peptide concentration except over a very restricted range a computationally efficient spline procedure improves upon linear regression however mass spectrometry data are not necessarily homoscedastic more often the variation of measured concentrations increases disproportionately near the boundaries of the instruments measurement capability dynamic range that is the upper and lower limits of quantitation these calibration difficulties exist with other applications of mass spectrometry as well as with other broad scale calibrations therefore the method proposed here uses a functional data approach to define the calibration curve and also the limits of quantitation under the two assumptions i that the variance is a bounded convex function of concentration and ii that the calibration curve itself is monotone at least between the limits of quantitation but not necessarily outside these limits within this paradigm the limit of detection where the signal is definitely present but not measurable with any accuracy is also defined an iterative approach draws on existing smoothing methods to account simultaneously for both restrictions and is shown to achieve the global optimal convergence rate under weak conditions this approach can also be implemented when convexity is replaced by other bounded restrictions examples from addona et al 2009 nature biotechnology 27 663 641 both motivate and illustrate the effectiveness of this functional data methodology when compared with the simpler linear regressions and spline techniques

nonparametric instrumental variable estimation under monotonicity the ill posedness of the inverse problem of recovering a regression function in a nonparametric instrumental variable model leads to estimators that may suer from a very slow logarithmic rate of convergence in this paper we show that restricting the problem to models with monotone regression functions and monotone instruments signicantly weakens the ill posedness of the problem in stark contrast to the existing literature the presence of a monotone instrument implies boundedness of our measure of ill posedness when restricted to the space of monotone functions based on this result we derive a novel non asymptotic error bound for the constrained estimator that imposes monotonicity of the regression function for a given sample size the bound is independent of the degree of ill posedness as long as the regression function is not too steep as an implication the bound allows us to show that the constrained estimator converges at a fast polynomial rate independently of the degree of ill posedness in a large but slowly shrinking neighborhood of constant functions our simulation study demonstrates signicant nite sample performance gains from imposing monotonicity even when the regression function is rather far from being a constant we apply the constrained estimator to the problem of estimating gasoline demand functions from u s data

applications of a generalized galerkin s method to non linear oscillations of two degree of freedom systems abstract a generalized galerkin s method is formulated for multi degree of freedom holonomic systems two autonomous non linear two d o f oscillators are used as the examples for the applications of the method i free oscillations of a non linear mass spring system with two d o f ii two weakly non linearly coupled identical van der pol oscillators the accuracy of the approximate solutions is discussed the effect of different time intervals of integration on the results is also investigated

necessary conditions for the existence of regular p ary bent functions we find some necessary conditions for the existence of regular p ary bent functions from znp to zp where p is a prime in more detail we show that there is no regular p ary bent function f in n variables with w mf larger than n 2 and for a given nonnegative integer k there is no regular p ary bent function f in n variables with w mf n 2 k n 3 2 k respectively for an even n np k an odd n np k respectively where np k is some positive integer which is explicitly determined and the w mf of a p ary function f is some value related to the power of each monomial of f for the proof of our main results we use some properties of regular p ary bent functions such as the macwilliams duality which is proved to hold for regular p ary bent functions in this paper

nonlinear vibration of buckled beams some exact solutions in this paper exact solutions are obtained for the dynamics of buckled beams with different types of end conditions the differential equation for the vibrations of buckled beams with restrictions on axial stretch is nonlinear using the modes of the corresponding linear problem which readily satisfy the boundary conditions the natural frequencies are obtained for the nonlinear problem by using jacobi elliptic functions analytical solutions for the dynamics of buckled beams with various rotational restraints are presented

nonparametric function estimation subject to monotonicity convexity and other shape constraints this paper uses free knot and fixed knot regression splines in a bayesian context to develop methods for the nonparametric estimation of functions subject to shape constraints in models with log concave likelihood functions the shape constraints we consider include monotonicity convexity and functions with a single minimum a computationally efficient mcmc sampling algorithm is developed that converges faster than previous methods for non gaussian models simulation results indicate the monotonically constrained function estimates have good small sample properties relative to i unconstrained function estimates and ii function estimates obtained from other constrained estimation methods when such methods exist also asymptotic results show the methodology provides consistent estimates for a large class of smooth functions two detailed illustrations exemplify the ideas

non parametric estimation of the odds ratios for continuous exposures using generalized additive models with an unknown link function the generalized additive model gam is a powerful and widely used tool that allows researchers to fit non parametrically the effect of continuous predictors on a transformation of the mean response variable such a transformation is given by a so called link function and in gams this link function is assumed to be known nevertheless if an incorrect choice is made for the link the resulting gam is misspecified and the results obtained may be misleading in this paper we propose a modified version of the local scoring algorithm that allows for the non parametric estimation of the link function by using local linear kernel smoothers to better understand the effect that each covariate produces on the outcome results are expressed in terms of the non parametric odds ratio or curves bootstrap techniques were used to correct the bias in the or estimation and to construct point wise confidence intervals a simulation study was carried out to assess the behaviour of the resulting estimates the proposed methodology was illustrated using data from the aids register of galicia nw spain with a view to assessing the effect of the cd4 lymphocyte count on the probability of being aids diagnosed via tuberculosis tb this application shows how the link s flexibility makes it possible to obtain or curve estimates that are less sensitive to the presence of outliers and unusual values that are often present in the extremes of the covariate distributions

generalized fourier series and limit cycles of generalized van der pol oscillators abstract a very simple generalization using jacobian elliptic functions of the usual fourier series appropriate for non linear systems is used to study the first order approximate solution of the generalized van der pol oscillators x ax 2bx 3 z 3 z 2 x 2 z 1 x 4 dot x 0 a generalized harmonic balance method is used to determine the limit cycles the cases a 0 b 0 and a 0 are considered in some detail for given values of the parameters zi the values of a and b for which limit cycles exist are found as functions of m numerical values for the radius frequency and energy of the limit cycles are given the presence of zero one two three or four limit cycles depends on the value of the parameters of the equation stability bifurcations of fixed points and the limit cycle stability are studied qualitatively

discontinuous versus smooth regression given measurements x i y i i 1 n we discuss methods to assess whether an underlying regression function is smooth continuous or differentiable or whether it has discontinuities the variance of the measurements is assumed to be unknown and is estimated simultaneously by regressing squared differences of the data formed with various span sizes on the span size itself we obtain an asymptotic linear model with dependent errors the parameters of this asymptotic linear model include the sum of the squared jump sizes as well as the variance of the measurements both parameters can be consistently estimated with mean squared error rates of convergence of n 2 3 for the sum of squared jump sizes and n 1 for the error variance we derive the asymptotic constants of the mean squared error mse and discuss the dependence of mse on the maximum span size l the test for the existence of jumps is formulated for the null hypothesis that the sum of squared jump sizes is 0 the asymptotic distribution of the test statistic is obtained essentially via a central limit theorem for u statistics we motivate and illustrate the methods with data surrounded by a scientific controversy concerning the question whether the growth of children occurs smoothly or rather in jumps

combining the methods of harmonic balance and kryloff bogoliuboff in a recent work mickens has described a set of conditions under which the harmonic balance method hbm is implementable the aim here is to broaden the scope of hbm analysis by drawing attention to the possible combination of such methods with those of kryloff bogoliuboff surprisingly these issues are not widely appreciated and as such provide an incentive for this note

direct regression models for longitudinal rates of change comparing rates of growth or rates of change across covariate defined subgroups is a primary objective for many longitudinal studies in the special case of a linear trend over time the interaction between a covariate and time will characterize differences in longitudinal rates of change however in the presence of a non linear longitudinal trajectory the standard mean regression approach does not permit parsimonious description or inference regarding differences in rates of change therefore we propose regression methodology for longitudinal data that allows a direct structured comparison of rates across subgroups even in the presence of a non linear trend over time our basic longitudinal rate regression method assumes a proportional difference across covariate groups in the rate of change across time but this assumption can be relaxed rates are compared relative to a generally specified time trend for which we discuss both parametric and non parametric estimating approaches we develop mixed model longitudinal methodology that explicitly characterizes subject to subject variation in rates as well as a marginal estimating equation based method in addition we detail a score test to detect violations of the proportionality assumption and we allow time varying rate effects as a natural generalization simulation results demonstrate potential gains in power for the longitudinal rate regression model relative to a linear mixed effects model in the presence of a non linear trend in time we apply our method to a study of growth among infants born to hiv infected mothers and conclude with a discussion of possible extensions for our methods

a bayesian approach to non parametric monotone function estimation the paper proposes two bayesian approaches to non parametric monotone function estimation the first approach uses a hierarchical bayes framework and a characterization of smooth monotone functions given by ramsay that allows unconstrained estimation the second approach uses a bayesian regression spline model of smith and kohn with a mixture distribution of constrained normal distributions as the prior for the regression coefficients to ensure the monotonicity of the resulting function estimate the small sample properties of the two function estimators across a range of functions are provided via simulation and compared with existing methods asymptotic results are also given that show that bayesian methods provide consistent function estimators for a large class of smooth functions an example is provided involving economic demand functions that illustrates the application of the constrained regression spline estimator in the context of a multiple regression model where two functions are constrained to be monotone

nonparametric kernel regression subject to monotonicity constraints we suggest a method for monotonizing general kernel type estimators for example local linear estimators and nadaraya watson estimators attributes of our approach include the fact that it produces smooth estimates indeed with the same smoothness as the unconstrained estimate the method is applicable to a particularly wide range of estimator types it can be trivially modified to render an estimator strictly monotone and it can be employed after the smoothing step has been implemented therefore an experimenter may use his or her favorite kernel estimator and their favorite bandwidth selector to construct the basic nonparametric smoother and then use our technique to render it monotone in a smooth way implementation involves only an off the shelf programming routine the method is based on maximizing fidelity to the conventional empirical approach subject to monotonicity we adjust the unconstrained estimator by tilting the empirical distribution so as to make the least possible change in the sense of a distance measure subject to imposing the constraint of monotonicity

the reduced monotonic regression method abstract medical researchers often desire to categorize patients into monotonic response groups based on the relationship between continuous variables isotonic regression fits consist of level sets of increasing value for which the estimated response is constant however the number of level sets obtained is often large preventing simple description this article introduces two new nonparametric methods called reduced isotonic regression and reduced monotonic regression the latter being a two sided extension of the former for use when the direction of the trend is unknown using a backward elimination algorithm the new procedures reduce the number of level sets by combining those whose values do not differ greatly for the statistical relations examined here the reduced monotonic method averaged at most 30 of the number of level sets obtained for isotonic regression the method is illustrated with an example that examines the relationship between risk factors for survival among children with leukem

analysis of a beam column system under varying axial forces of elliptic type the exact solution of lame s equation abstract we investigate the dynamical response of a beam column system with hinged ends subjected to an axial pulsating force of elliptic type it is shown that the resulting equation is of the form d 2 y 1 d 2 y 1 d 1 d 2 cn 2 k 2 0 which is the well known lame equation higher transcendental functions bateman manuscript project edited by mcgraw hill new york vol 3 in this paper we obtain the general exact solution of this equation that reveals stable behavior of the beam column system if the assigned initial conditions are of the form y1 0 y10 and y 1 0 0 it is also found that at a certain value of the modulus of the elliptic force the lateral vibrational frequency is independent of the material properties of the beam column system

binary sequences with gold like correlation but larger linear span a new construction of optimal binary sequences identical to the well known family of gold sequences in terms of maximum nontrivial correlation magnitude and family size but having larger linear span is presented the distribution of correlation values is determined for every odd integer spl tau spl ges 3 the construction provides a family that contains 2 sup spl tau 1 cyclically distinct sequences each of period 2 sup spl tau 1 the maximum nontrivial correlation magnitude equals 2 spl tau 1 sup 2 1 with one exception each of the sequences in the family has linear span at least spl tau sup 2 spl tau 2 compared to 2 spl tau for gold sequences the sequences are easily implemented using a quaternary shift register followed by a simple feedforward nonlinearity

periodic solutions of strongly quadratic non linear oscillators by the elliptic lindstedt poincare method abstract the elliptic lindstedt poincare method is used employed to study the periodic solutions of quadratic strongly non linear oscillators of the form x c 1 x c 2 x 2 e f x x in which the jacobian elliptic functions are employed instead of the usual circular functions in the classical lindstedt poincare method the generalized van de pol equation with f x x 0 1 x 2 x 2 is studied in detail comparisons are made with the solutions obtained by using the lindstedt poincare method and runge kutta method to show the efficiency of the present method

monotone smoothing with application to dose response curves and the assessment of synergism a spline based procedure for monotone curve smoothing is proposed and is illustrated by application to dose response curves it is then shown how such smoothing can be applied to assess possible synergism or antagonism of two drugs

jacobi elliptic functions a review of nonlinear oscillatory application problems abstract this review paper is concerned with the applications of jacobi elliptic functions to nonlinear oscillators whose restoring force has a monomial or binomial form that involves cubic and or quadratic nonlinearity first geometric interpretations of three basic jacobi elliptic functions are given and their characteristics are discussed it is shown then how their different forms can be utilized to express exact solutions for the response of certain free conservative oscillators these forms are subsequently used as a starting point for a presentation of different quantitative techniques for obtaining an approximate response for free perturbed nonlinear oscillators an illustrative example is provided further two types of externally forced nonlinear oscillators are reviewed i those that are excited by elliptic type excitations with different exact and approximate solutions ii those that are damped and excited by harmonic excitations but their approximate response is expressed in terms of jacobi elliptic functions characteristics of the steady state response are discussed and certain qualitative differences with respect to the classical duffing oscillator excited harmonically are pointed out parametric oscillations of the oscillators excited by an elliptic type forcing are considered as well and the differences with respect to the stability chart of the classical mathieu equation are emphasized the adjustment of the melnikov method to derive the general condition for the onset of homoclinic bifurcations in a system parametrically excited by an elliptic type forcing is provided and compared with those corresponding to harmonic excitations advantages and disadvantages of the use of jacobi elliptic functions in nonlinear oscillatory application problems are discussed and some suggestions for future work are given

generalized plateaued functions and admissible plateaued functions plateaued functions are very important crypto graphic functions due to their various desirable cryptographic characteristics we point out that plateaued functions are more general than bent functions that is functions with maximum nonlinearity some boolean plateaued functions have large nonlinearity which provides protection against fast correlation attacks when they are used as combiners or filters in stream ciphers and contributes when they are the component functions of the substitution boxes in block ciphers to protection against linear cryptanalysis p ary plateaued functions have attracted recently some attention in the literature and many activities on generalized p ary functions have been carried out this paper increases our knowledge on plateaued functions in the general context of generalized p ary functions we first introduce two new versions of plateaued functions which we shall call generalized plateaued functions and admissible plateaued functions the generalized plateaued functions extend the standard notion of plateaued p ary functions to those whose outputs are in the ring mathbb z p k next we study the generalized plateaued functions and use admissible plateaued functions to characterize the generalized plateaued functions by means of their components finally we provide for the first time two constructions of generalized plateaued functions in particular we generalize a known secondary construction of binary generalized bent functions and derive constructions of binary generalized plateaued functions with different amplitudes

monotone b spline smoothing abstract estimation of growth curves or item response curves often involves monotone data smoothing methods that have been studied in the literature tend to be either less flexible or more difficult to compute when constraints such as monotonicity are incorporated built on the ideas of koenker ng and portnoy and ramsay we propose monotone b spline smoothing based on l 1 optimization this method inherits the desirable properties of spline approximations and the computational efficiency of linear programs the constrained fit is similar to the unconstrained estimate in terms of computational complexity and asymptotic rate of convergence through applications to some real and simulated data we show that the method is useful in a variety of applications the basic ideas utilized in monotone smoothing can be useful in some other constrained function estimation problems

improvement to the averaging method using the jacobian elliptic function the averaging method was improved by using the jacobian elliptic sine sn cosine cn and delta dn functions as generating solutions in order to obtain a highly accurate periodic solution for a strongly nonlinear dynamical system the proposed method can be applied to a relatively general nonlinear system based on the single degree of freedom duffing equation two methods of stability analysis were applied and evaluated for the approximate solutions obtained by the proposed method the approximate solutions obtained by the proposed method for practical examples were compared to the solutions obtained by the shooting method the results confirmed that the proposed method provides a more accurate solution and more accurate stability analysis results than those obtained by the conventional averaging method that uses trigonometric functions as the generating solution

improvement of a krylov bogoliubov method that uses jacobi elliptic functions abstract an improved version of a krylov bogoliubov method that gives the approximate solution of the non linear cubic oscillator x c 1 x c 3 x 3 ef x x dot 0 in terms of jacobi elliptic functions is described compact general expressions are given for the time derivatives of the amplitude and phase similar to those obtained by the usual krylov bogoliubov method which gives the approximate solution in terms of circular functions these expressions are especially simple for quasilinear c3 0 and quasi pure cubic c1 0 oscillators two types of cubic oscillators have been used as examples the linear damped oscillator f x x dot x dot and the van der pol oscillator f x x dot x 2 x dot the approximate solutions of these quasilinear and quasi pure cubic oscillators are simple and accurate the influence of the non linearity on the rate of variation of the amplitude of these two types of cubic oscillators was also studied

generalized monotonic regression using random change points we introduce a procedure for generalized monotonic curve fitting that is based on a bayesian analysis of the isotonic regression model conventional isotonic regression fits monotonically increasing step functions to data in our approach we treat the number and location of the steps as random for each step level we adopt the conjugate prior to the sampling distribution of the data as if the curve was unconstrained we then propose to use markov chain monte carlo simulation to draw samples from the unconstrained model space and retain only those samples for which the monotonic constraint holds the proportion of the samples collected for which the constraint holds can be used to provide a value for the weight of evidence in terms of bayes factors for monotonicity given the data using the samples probability statements can be made about other quantities of interest such as the number of change points in the data and posterior distributions on the location of the change points can be provided the method is illustrated throughout by a reanalysis of the leukaemia data studied by schell and singh

comparing the shapes of regression functions summary does a regression function follow a specified shape do two regression functions have the same shape how can regression functions be grouped based on shape these questions can occur when investigating monotonicity when counting local maxima or when studying variation in families of curves one can address these questions by considering the rank correlation coefficient between two functions this correlation is a generalisation of the rank correlation between two finite sets of numbers and is equal to one if and only if the two functions have the same shape a sample rank correlation based on smoothed estimates of the regression functions consistently estimates the true correlation this sample rank correlation can be used as a measure of similarity between functions in cluster analysis and as a measure of monotonicity or modality

approximations for motion of the oscillators with a non negative real power restoring force abstract oscillators with a non negative real power restoring force are considered in this paper this type of restoring force is related to systems with a quasi zero stiffness characteristic or those in which the restoring force is purely nonlinear in nature examples of these types of restoring force are grounded in real physical and engineering systems periodic motion of such conservative oscillators is described first in a novel way by means of the elliptic function the parameters of which are obtained from the energy conservation law and hamilton s variational principle then the approach is extended to non conservative oscillators by adjusting the elliptic krylov bogoliubov method the methods proposed for the conservative and non conservative systems under consideration have wider applications than the existing one with respect to the power of the restoring force several examples the majority of which are so far unsolved are given to illustrate the methods proposed and to demonstrate their generality which permits unforeseen solutions for motion containing higher harmonics and assuring consistent accuracy regardless of the value of the power of the restoring force the results obtained are compared with numerical results and have excellent accuracy

variable selection in monotone single index models via the adaptive lasso we consider the problem of variable selection for monotone single index models a single index model assumes that the expectation of the outcome is an unknown function of a linear combination of covariates assuming monotonicity of the unknown function is often reasonable and allows for more straightforward inference we present an adaptive lasso penalized least squares approach to estimating the index parameter and the unknown function in these models for continuous outcome monotone function estimates are achieved using the pooled adjacent violators algorithm followed by kernel regression in the iterative estimation process a linear approximation to the unknown function is used therefore reducing the situation to that of linear regression and allowing for the use of standard lasso algorithms such as coordinate descent results of a simulation study indicate that the proposed methods perform well under a variety of circumstances and that an assumption of monotonicity when appropriate noticeably improves performance the proposed methods are applied to data from a randomized clinical trial for the treatment of a critical illness in the intensive care unit

on plateaued functions the focus of this article is on nonlinear characteristics of cryptographic boolean functions first we introduce the notion of plateaued functions that have many cryptographically desirable properties second we establish a sequence of strengthened inequalities on some of the most important nonlinearity criteria including nonlinearity avalanche and correlation immunity and prove that critical cases of the inequalities coincide with characterizations of plateaued functions we then proceed to prove that plateaued functions include as a proper subset all partially bent functions that were introduced earlier by claude carlet 1993 this solves an interesting problem that arises naturally from previously known results on partially bent functions in addition we construct plateaued but not partially bent functions that have many properties useful in cryptography

on estimation for monotone dose response curves abstract estimating monotone dose response curves with independent normal errors has been studied in detail in the literature a simple procedure using a generalization of the studentized maximum modulus technique is proposed for constructing confidence bands for monotone regression functions this procedure is shown to compare well with its predecessors the center of this smooth and efficient confidence band is then used to estimate a smooth monotone regression function its pointwise mean squared errors compare favorably with the maximum likelihood estimate a weighted average of the isotonic regression and the least squares line and the rank transformed regression a numerical example of a binding inhibition assay is included to illustrate the proposed interval and point estimation procedures

nonparametric combinatorial regression for shape constrained modeling we introduce a unified approach for calculating nonparametric shape constrained regression enforcement of the shape constraint often accounts for the impact of a physical phenomenon or a specific property it also improves the model s predicability and facilitates subsequent optimizations the regression models are built by transforming the problem into the combinatorial domain where the shape constraints are imposed by bounding the combinatorial search space we start by addressing isotonicity shape constraint using a dynamic programming algorithm and demonstrate how the problem can be mapped to the graph combinatorics domain next we show how a number of other important shape constraints including unimodality convexity limited level set and limited slope can be addressed using the same framework the flexibility of proposed framework enables solving the shape constrained regression problem with an arbitrary user defined error metric this flexibility is exploited to add robustness against outliers to the model the algorithms are described in detail and their computational complexity is established the performance and effectiveness of the shape constrained regression is evaluated on traces of temperature and humidity measurements from a deployed sensor network where a high degree of accuracy and robustness is demonstrated

a study of a nonlinear vibration isolator with a quasi zero stiffness characteristic a vibration isolator consisting of a vertical linear spring and two nonlinear pre stressed oblique springs is considered in this paper the system has both geometrical and physical nonlinearity firstly a static analysis is carried out the softening parameter leading to quasi zero dynamic stiffness at the equilibrium position is obtained as a function of the initial geometry pre stress and the stiffness of the springs the optimal combination of the system parameters is found that maximises the displacement from the equilibrium position when the prescribed stiffness is equal to that of the vertical spring alone it also satisfies the condition that the dynamic stiffness only changes slightly in the neighbourhood of the static equilibrium position for these values a dynamical analysis of the isolator under asymmetric excitation is performed to quantify the undesirable effects of the nonlinearities it includes considering the possibilities of the appearance of period doubling bifurcation and its development into chaotic motion for this purpose approximate analytical methods and numerical simulations accompanied with qualitative methods including phase plane plots poincare maps and lyapunov exponents are used finally the frequency at which the first period doubling bifurcation appears is found and the effect of damping on this frequency determined

improved lindstedt poincare method for the solution of nonlinear problems abstract we apply the linear delta expansion lde to the lindstedt poincare distorted time method to find improved approximate solutions to nonlinear problems we find that our method works very well for a wide range of parameters in the case of the anharmonic oscillator duffing equation of the non linear pendulum and of more general anharmonic potentials the approximate solutions found with this method converge more rapidly to the exact ones than in the simple lindstedt poincare method

a generalization of the method of harmonic balance a procedure is provided for generalizing the method of harmonic balance to obtain higher order approximations to the periodic solutions of differential equations for two systems one a conservative system and the other nonconservative the procedure is currently being applied to investigate the possible solution behaviors of singular nonlinear oscillators where the usual perturbation methods do not work

monotone nonparametric regression in monotone regression procedures one utilizes only the monotonicity of the regression function in nonparametric regression one utilizes only the assumed smoothness the analytic and asymptotic properties of the estimator are superior in the latter case however monotonicity is not guaranteed we study a hybrid procedure that produces monotone estimators with properties similar to those of nonparametric regression estimators

consistent detection of a monotonic trend superposed on a stationary time series summary consider a time series made up of a signal and a stationary autocorrelated error series a statistic is proposed for examining the hypothesis that the signal term is constant versus the hypothesis that it is monotonic in time the statistic is the ratio of a linear combination of the time series values with coefficients introduced by abelson tukey 1963 to an estimate of the standard error of the linear combination the statistic has asymptotic power 1 for a broad class of monotonic alternatives the procedure is illustrated for the series of river heights at a location on the rio negro in brazil where there is concern that the height is rising due to deforestation of the amazon basin the significance level obtained is 0025

generalized fourier series for the study of limit cycles the approximate solution to first order of non linear differential equations is studied using the method of harmonic balance with generalized fourier series and jacobian elliptic functions as an interesting use of the series very good analytic approximations to the limit cycles of lienard s ordinary differential equation ode x g x f x x are presented specifically it is shown that contrary to an opinion given in a well known textbook on non linear oscillations g x not only modifies the period but influences the topology in the generalized van der pol equation with f x e 1 x2 and g x ax 2bx3 for 0 1 the presence of zero one or three limit cycles is found to depend on the value of ab

a modified elliptic lindstedt poincare method for certain strongly non linear oscillators abstract a modified elliptic lindstedt poincare method is presented for the steady state analysis of strongly non linear oscillators of the form x c 1 x c 3 x 3 ef x x in which a new parameter e is employed such that the value of is always small regardless of the magnitude of the original parameter e therefore the above strongly non linear oscillators with large parameter e is transformed into a small parameter system with respect to

comments on the method of harmonic balance in which jacobi elliptic functions are used abstract a harmonic balance method is presented in which jacobi elliptic functions are used in the trial solution instead of circular functions to obtain approximate periodic solutions of the oscillator x f x dot x 0 conditions for the method to work well are the usual ones of the current method of harmonic balance and that x t must pass through zero the procedure for obtaining a higher order approximation is described and in particular two criteria for chosing the elliptic function parameter m are discussed illustrative examples are presented with f being diverse polynomials of x

oscillator with fraction order restoring force abstract in this paper a new analytical method for solving the differential equations which describe the motion of the oscillator with fraction order elastic force is introduced using the first integral of motion the exact period of vibration in the form of the euler beta function is obtained based on that value the approximate solution of the strong nonlinear fraction order differential equation is formed the suggested procedure is applied for various fraction values but also for the pure quadratic cubic and quintic oscillators the advantage of the suggested method is that is valid for all fraction values 1 the accuracy of the approximate solution is very high as the period of vibration is exactly analytically determined and is independent on the time in the paper the fraction order differential equation with small linear and nonlinear terms is also considered the krylov bogolubov method is extended due to the fact that frequency of vibration depends on the amplitude of vibration the approximate solution is with time variable amplitude phase and frequency the linear damped fraction order oscillator is widely discussed the approximate solution of the differential equation which describes the vibrations of such oscillator is compared with exact numerical one and shows a good agreement

explicit criterions for p ary functions being non bent abstract there has been only limited information on the existence of p ary bent functions recently there has been a result by the authors on finding necessary conditions for the existence of regular p ary bent functions from z p n to z p where p is a prime the general case of p ary bent functions is however an open question for finding necessary conditions for their existence in this paper we complete this open case we state our main result in more detail we find an explicit family of non bent functions we also show that there is no p ary bent function f in n variables with w m f n 2 if n is even w m f n 3 2 if n is odd respectively and for a given nonnegative integer k there is no p ary bent function f in n variables with w m f n 2 k w m f n 3 2 k respectively for an even n n p k for an odd n n p k respectively where n p k is some positive integer which is explicitly determined and w m f is some explicit value related to the power of each monomial of f we point out that if f is not a p ary bent function in n variables and g is any p ary function in n variables such that l g f g g for some ccz transformation l of f then g cannot be bent either this shows that our result produces a larger family of non bent functions

nonparametric analysis of dependently interval censored failure time data failure time studies based on observational cohorts often have to deal with irregular intermittent observation of individuals which produces interval censored failure times when the observation times depend on factors related to a person s failure time the failure times may be dependently interval censored inverse intensity of visit weighting methods have been developed for irregularly observed longitudinal or repeated measures data and recently extended to parametric failure time analysis this article develops nonparametric estimation of failure time distributions using weighted generalized estimating equations and monotone smoothing techniques simulations are conducted for examination of the finite sample performance of proposed estimators this research is motivated in part by the toronto psoriatic arthritis cohort study and the proposed methodology is applied to this study

the monotone smoothing of scatterplots we consider the problem of summarizing a scatterplot with a smooth monotone curve a solution that combines local averaging and isotonic regression is proposed and we demonstrate its use with two examples in the second example the procedure is applied in a regression setting to some data from box and cox 1964 and it is shown how this new procedure generalizes box and cox s well known family of transformations in the same example the bootstrap is applied to obtain a measure of the variability of the procedure

quasi pure cubic oscillators studied using a krylov bogoliubov method abstract a method of krylov bogoliubov type which gives the approximate solution in terms of jacobi elliptic functions is used to study the quasi pure cubic oscillators x c 3 x 3 ef x x 0 explicit approximate solutions with the perturbative terms x x 2 x x quadratic damping and sgn coulomb damping are given and are compared with the numerical or exact solution a simple and accurate expression is derived for the time at which the oscillations with coulomb damping cease

tests for monotonicity of a regression mean with guaranteed level in this paper a nonparametric procedure for testing for monotonicity of a regression mean with guaranteed level is proposed the procedure is based on signs of differences of observations from the response variable the test is calibrated against the most difficult null hypothesis when the regression function is constant and produces an exact test in this context in general the test is conservative the power of the test is good and comparable with that of other nonparametric tests it is shown that the testing procedure has asymptotic power 1 against certain local alternatives the method is also robust against heavy tailed error distributions and even maintains good power when the errors are for example cauchy distributed a simulation study is provided to demonstrate finite sample behaviour of the testing procedure

transition curves in a parametrically excited pendulum with a force of elliptic type this article investigates the equilibria and stability of a pendulum when the support has a prescribed motion defined by an elliptic function stability charts are generated in the parameter plane for different values of the elliptic function modulus numerical integration and floquet theory are used to generate stability charts that are later obtained through harmonic balance analysis it is shown that the size and location of the instability tongues is directly linked to the elliptic function modulus comparisons are also made between the stability charts of mathieu s equation and those of the pendulum when the prescribed motion is defined by an elliptic function

nonparametric estimation of a generalized additive model with an unknown link function mean of a random scalar y conditional on a random vector x e rd let h x e yix x denote the conditional mean function this paper is concerned with estimating h under weak assumptions about its functional form when x is a multidimensional continuous random variable the paper describes a model that nests single index additive and multiplicative models and that can be estimated with a faster rate of convergence in probability than can a fully nonparametric model nonparametric estimation of h is usually unattractive when x is multidimensional the rate of convergence of nonparametric estimators decreases rapidly as the dimension of x increases and nonparametric estimates can be very imprecise with samples of practical size when x is multidimensional the curse of dimensionality this problem can be overcome through the use of dimension reduction methods they reduce the effective dimension of the estimation problem by making assumptions about the functional form of h that are stronger than those of a fully nonparametric estimator but weaker than those of a finite dimensional parametric model one method of dimension reduction is semiparametric single index modeling the best known and most easily estimated semiparametric single index model assumes that 1 1 h x g 3 x where 3 is an unknown d x 1 vector and g is an unknown function the vector 3 can be estimated with an n 1 2 rate of convergence in probability which is the rate that would be obtained if g were known han 1987 hardle and stoker 1989 horowitz and hardle 1996 ichimura 1993 klein and spady 1993 newey and stoker 1993 powell stock and stoker 1989 sherman 1993 stoker 1991 g can be estimated with the rate of convergence n r 2r 1 where r is an integer that depends on the smoothness of g but not on d dimension reduction can also be achieved by using a nonparametric generalized additive model in such a model

explicit criteria for construction of plateaued functions plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics we find explicit criteria for the construction of p ary r plateaued functions with an odd prime p we point out that 0 plateaued functions are bent functions and so plateaued functions generalize the notion of bent functions we first derive an explicit form for the walsh hadamard transform of a p ary r plateaued function we then obtain an upper bound on the degree of p ary r plateaued functions and we classify p ary n 1 plateaued functions in n variables we also obtain explicit criteria for the existence of p ary r plateaued functions accordingly these results lead to improved bounds on the existence of p ary bent functions