on the solution of strong nonlinear oscillators by applying a rational elliptic balance method a rational elliptic balance method is introduced to obtain exact and approximate solutions of nonlinear oscillators by using jacobi elliptic functions to illustrate the applicability of the proposed rational elliptic forms in the solution of nonlinear oscillators we first investigate the exact solution of the non homogenous undamped duffing equation then we introduce first and second order rational elliptic form solutions to obtain approximate solutions of two nonlinear oscillators at the end of the paper we compare the numerical integration values of the angular frequencies with approximate solution results based on the proposed rational elliptic balance method

sparse additive models summary we present a new class of methods for high dimensional non parametric regression and classification called sparse additive models our methods combine ideas from sparse linear modelling and additive non parametric regression we derive an algorithm for fitting the models that is practical and effective even when the number of covariates is larger than the sample size sparse additive models are essentially a functional version of the grouped lasso of yuan and lin they are also closely related to the cosso model of lin and zhang but decouple smoothing and sparsity enabling the use of arbitrary non parametric smoothers we give an analysis of the theoretical properties of sparse additive models and present empirical results on synthetic and real data showing that they can be effective in fitting sparse non parametric models in high dimensional data

a lasso for hierarchical interactions we add a set of convex constraints to the lasso to produce sparse interaction models that honor the hierarchy restriction that an interaction only be included in a model if one or both variables are marginally important we give a precise characterization of the effect of this hierarchy constraint prove that hierarchy holds with probability one and derive an unbiased estimate for the degrees of freedom of our estimator a bound on this estimate reveals the amount of fitting saved by the hierarchy constraint we distinguish between parameter sparsity the number of nonzero coefficients and practical sparsity the number of raw variables one must measure to make a new prediction hierarchy focuses on the latter which is more closely tied to important data collection concerns such as cost time and effort we develop an algorithm available in the r package hiernet and perform an empirical study of our method

a difference scheme for a nonlinear partial integrodifferential equation a difference method for the numerical integration of a nonlinear partial integrodifferential equation is considered the integral term is treated by means of a convolution quadrature suggested by lubich some results from lubich s discretized fractional calculus play a crucial role in proving consistency the verification of stability and convergence is based on the nonnegative character of the real quadratic form associated with the convolution quadrature a stability result is derived that is applicable to equations and numerical methods far more general than those treated in this paper

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

compact finite difference method for the fractional diffusion equation high order compact finite difference scheme for solving one dimensional fractional diffusion equation is considered in this paper after approximating the second order derivative with respect to space by the compact finite difference we use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the local truncation error and discuss the stability using the fourier method then we prove that the compact finite difference scheme converges with the spatial accuracy of fourth order using matrix analysis numerical results are provided to verify the accuracy and efficiency of the proposed algorithm

periodic solution of the generalized rayleigh equation abstract the periodic solutions of a strongly cubic nonlinear oscillator whose motion is described with the generalized rayleigh equation are studied approximate analytic solving methods are introduced a new method based on homotopy and averaging is developed to determine the limit cycle motion the obtained analytical solutions are compared with those calculated by the elliptic harmonic balance method with generalized fourier series and jacobian elliptic functions three types of cubic nonlinearity are considered the coefficients of the linear and cubic terms are positive the coefficient of the linear term is positive and that of the cubic term is negative and the opposite case comparisons of the analytical solution and numerical solution obtained by using the runge kutta method are illustrated with examples

an improved spectral meshless radial point interpolation for a class of time dependent fractional integral equations 2d fractional evolution equation abstract in this paper a kind of spectral meshless radial point interpolation smrpi technique is applied to the fractional evolution equation in two dimensional for arbitrary fractional order the applied approach is based on erudite combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of smrpi in the current work the thin plate splines tps are used as the basis functions and also the discretization technique employed is patterned after an idea of ch lubich it is proved the scheme is unconditional stable with respect to the time variable in l 2 norm and convergent by the order of convergence o t 1 0 1 the results of numerical experiments are compared with analytical solution to confirm the accuracy and efficiency of the presented scheme three numerical examples show that the smrpi has reliable accuracy in irregular domains

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

an iterative multistep kernel based method for nonlinear volterra integral and integro differential equations of fractional order abstract an iterative multistep kernel based method is proposed for the nonlinear volterra integral equations and nonlinear volterra integro differential equations of fractional order which can produce reliable globally smooth numerical solutions an error estimate of the positive definite kernel interpolation and the convergence and error analysis of the proposed iterative scheme are investigated here we focused on positive definite radial basis kernels and further a new and applicable shape parameter selection strategy is proposed the proposed multi step method set up and solve several small local problems instead of a single large problem which makes it suitable for problems with long time simulations in order to show the efficiency and versatility of the proposed method some numerical experiments are reported the comparison of the numerical results with the analytical solutions and the best reported results in the literature confirm the good accuracy and applicability of the proposed method

analysis of the spectral meshless radial point interpolation for solving fractional reaction subdiffusion equation abstract the present paper is devoted to the development of spectral meshless radial point interpolation smrpi technique for solving fractional reaction subdiffusion equation in one and two dimensional cases the time fractional derivative is described in the riemann liouville sense the applied approach is based on a combination of meshless methods and spectral collocation techniques the point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of smrpi it is proved that the scheme is unconditionally stable with respect to the time variable in h 1 and we show convergence order of the time discrete scheme is o t 0 1 in the current work the thin plate splines tps are used to construct the basis functions the results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme

geometrical non linear steady state forced periodic vibration of plates part i model and convergence studies a model for the steady state geometrically non linear periodic vibration of thin rectangular plates under harmonic external excitation is presented the equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method hfem these equations are transformed into the frequency domain by the harmonic balance method hbm and are solved by a continuation method the convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates

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

approximate analytic solutions for singular non linear oscillators mickens 1981 1984 has considered analytic techniques for obtaining approximate solutions to one dimensional nonlinear oscillatory systems x double dot x lambda f x x dot lambda where lambda is a small positive parameter and f is a nonlinear polynomial function of its arguments however in certain cases there is interest in the analysis of physical systems for which the nonlinear function f x x dot lambda is singular for finite values of x or x dot the present investigation is concerned with the use of existing approximate analytic schemes to obtain solutions to singular nonlinear oscillatory differential equations

trace pursuit a general framework for model free variable selection abstractwe propose trace pursuit for model free variable selection under the sufficient dimension reduction paradigm two distinct algorithms are proposed stepwise trace pursuit and forward trace pursuit stepwise trace pursuit achieves selection consistency with fixed p forward trace pursuit can serve as an initial screening step to speed up the computation in the case of ultrahigh dimensionality the screening consistency property of forward trace pursuit based on sliced inverse regression is established finite sample performances of trace pursuit and other model free variable selection methods are compared through numerical studies supplementary materials for this article are available online

misspecified nonconvex statistical optimization for sparse phase retrieval existing nonconvex statistical optimization theory and methods crucially rely on the correct specification of the underlying true statistical models to address this issue we take a first step towards taming model misspecification by studying the high dimensional sparse phase retrieval problem with misspecified link functions in particular we propose a simple variant of the thresholded wirtinger flow algorithm that given a proper initialization linearly converges to an estimator with optimal statistical accuracy for a broad family of unknown link functions we further provide extensive numerical experiments to support our theoretical findings

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

a new spectral meshless radial point interpolation smrpi method for the two dimensional fredholm integral equations on general domains with error analysis in this article we present a numerical method to solve two dimensional fredholm integral equations of the second kind on general domains the method utilizes meshless and spectral collocation techniques but it is not traditional meshless collocation method the point interpolation method with the help of strictly positive definite radial basis functions is used to construct shape functions as a basis functions in the frame of spectral collocation methods these basis functions shape functions have kronecker delta function property since the proposed method is meshless it does not need any domain element and so it is independent of the geometry of the domain the method reduces the solution of the two dimensional integral equation to the solution of a linear system of algebraic equations convergence analysis with error estimates are given with full discussion furthermore some numerical examples are presented to show the validity and efficiency of smrpi

on the convergence analysis stability and implementation of meshless local radial point interpolation on a class of three dimensional wave equations summary in this article the meshless local radial point interpolation method is applied to analyze three space dimensional wave equations of the form utt ut u uxx uyy uzz f x y z t xt x y z rd d 2 3 t 0 subject to given initial and dirichlet boundary conditions the main difficulty of the great number of methods in full 3 d problems is the large computational costs in meshless local radial point interpolation method it does not require any background integration cells so that all integrations are carried out locally over small quadrature domains of regular shapes such as circles or squares in two dimensions and spheres or cubes in three dimensions the point interpolation method with the help of radial basis functions is proposed to construct shape functions that have kronecker delta function property a weak formulation with the heaviside step function converts the set of governing equations into local integral equations on local subdomains a two step time discretization method is employed to evaluate the time derivatives this suggests crank nicolson technique to be applied on the right hand side of the equation the convergence analysis and stability of the method are fully discussed three illustrative examples are presented and satisfactory agreements are achieved it is shown theoretically that the proposed method is unconditionally stable for the second example whereas it is not for the first and third ones copyright 2015 john wiley sons ltd

the limit cycles of the generalized rayleigh lienard oscillator abstract the limit cycles of the generalized rayleigh and mixed rayleigh lienard oscillator x ax 2bx 3 e z 3 z 2 x 2 z 1 x 4 z 4 dot x 2 dot x 0 for a and b 0 are studied by using the jacobian elliptic functions with the generalized harmonic balance method the transitory motion and consequently the limit cycles and their stability are also studied quantitatively with a generalized approximation of the krylov bogoliubov slowly varying amplitude and phase type giving the radius frequency and energy of the limit cycles whether there are zero one or two limit cycles and their stability depends on the values of the parameters z i i 1 2 3 4 these solutions are interesting because they do not depend on the value of e for the simple cases of only two non zero z i parameters plots of the universal functions are given as well as the limit cycle radii approximate solutions in the general case are found to coincide with certain exact solutions of the mixed type oscillators

bounds on the fourier coefficients for the periodic solutions of non linear oscillator equations the differential equations describing nonlinear oscillations as seen in mechanical vibrations electronic oscillators chemical and biochemical reactions acoustic systems stellar pulsations etc are investigated analytically the boundedness of the fourier coefficients for periodic solutions is demonstrated for two special cases and the extrapolation of the results to higher dimensionsal systems is briefly considered

alternating direction implicit orthogonal spline collocation methods for an evolution equation with a positive type memory term new numerical techniques are presented for the solution of a class of linear partial integro differential equations pides with a positive type memory term in the unit square in these methods orthogonal spline collocation osc is used for the spatial discretization and for the time stepping new alternating direction implicit adi methods based on the backward euler the crank nicolson and the second order bdf methods combined with judiciously chosen quadrature rules are considered the adi osc methods are proved to be of optimal accuracy in time and in the l 2 norm in space numerical results confirm the predicted convergence rates and also exhibit optimal accuracy in the l infty and h 1 norms and superconvergence phenomena

on the determination of the limit cycles using the harmonic balance method several methods have been developed to search for limit cycles or bifurcation curves of nonlinear differential equations recently some non perturbative methods have been proposed however the majority of these methods are based on the so called harmonic balance method in this paper we will see how to use the harmonic balance method to obtain information of the number period and form of the limit cycles of certain nonlinear equations the method also gives a sequence of algebraic approximations of the limit cycles we will also compare our results with other non perturbative methods the results can be generalized to other nonlinear differential equations

a new approach for solving a complex valued differential equation abstract in this paper two analytical approximate solving procedures for a complex valued differential equation are developed one of the methods represents the generalization of the krylov bogolubov method for a strong differential equation with complex function the second method is based on the first integrals of the system the differential equation is transformed introducing the perturbed first integrals and the polar co ordinates the solution is obtained applying the straightforward series expansion the solution for the special case of without impact initial conditions is considered the method is applied on the system with strong cubic non linearity the small gyroscopic function and damping function are introduced the analytical approximate solution is compared with numerical exact one and shows a good agreement

choices in the harmonic balance technique we apply the harmonic balance technique in low orders to three types of oscillator model for three equations of the van der pol type with rational nonlinear terms used to model electronic oscillators we examine the choice between a rationalizing before expanding in harmonics and b obtaining fourier coefficients of nonlinear terms alternative b is shown to be more accurate in first order for the models of scott murata and of walker and connelly which apply to circuits with an inverse tangent nonlinear component for these equations in the case of alternative a we find acceptable dependence of the second order corrections on the bifurcation parameter when the harmonic balance method is used to set up a semi classical quantization treatment of a nonlinear conservative oscillator values of energies are slightly improved by the use of alternative b for conservative oscillators with cubic or fifth power forces we compare the standard method using the acceleration equation with an alternative using the energy equation the first is shown to be more accurate in low order

projections of a general binary model on a logistic regression abstract we consider a general binary model for which conditional probability of success given vector of predictors x equals q 1 t x k t x and a family of possibly misspecified logistic regressions fitted to it in the case when x satisfies linearity condition we show that their algebraic structure is uniquely determined and that the vector corresponding to kullback leibler projection on this family is a linear combination of 1 k this generalizes the known result proved by p ruud for k 1 which says that the true and projected vectors are collinear it also follows that the projected vector has the same direction as the first canonical vector which justifies frequent observations that logistic fit yields well performing classifiers even if misspecification is expected in the special case of additive binary model with multivariate normal predictors and when response function q is a convex combination of univariate responses we show that the variance of t x is not larger than the maximal variance of the projected linear combinations for the corresponding univariate problems in the case of balanced additive logistic model it follows that the contribution of i to is bounded by the corresponding coefficient in the convex representation of response function q

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

properties of finite difference models of non linear conservative oscillators finite difference fd approaches to the numerical solution of the differential equations describing the motion of a nonlinear conservative oscillator are investigated analytically a generalized formulation of the duffing and modified duffing equations is derived and analyzed using several fd techniques and it is concluded that although it is always possible to contstruct fd models of conservative oscillators which are themselves conservative caution is required to avoid numerical solutions which do not accurately reflect the properties of the original equation

generalized fourier series for non linear systems application to the study of limit cycles in second order approximation the approximate solution of non linear differential equations is studied to second order 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 xdot g x f x xdot are presented in the generalized van der pol equation with f x e 1 x2 and g x ax 2bx3 a very good second order approximation is given that depends on the values of a b and e

alternating direction implicit euler method for the two dimensional fractional evolution equation in this paper we present a fast and efficient numerical method to solve a two dimensional fractional evolution equation on a finite domain this numerical method combines the alternating direction implicit adi approach with the second order difference quotient in space the backward euler in time and order one convolution quadrature approximating the integral term by using the discrete energy method we prove that the adi scheme is unconditionally stable and the numerical solution converges to the exact one with order o k h x 2 h y 2 where k is the temporal grid size and h x h y are spatial grid sizes in the x and y directions respectively two numerical examples with known exact solution are also presented and the behavior of the error is analyzed to verify the order of convergence of the adi euler method

coupling three field formulation and meshless mixed galerkin methods using radial basis functions in this work we solve the elliptic partial differential equation by coupling the meshless mixed galerkin approximation using radial basis function with the three field domain decomposition method the formulation has been adopted to increase the efficiency of the numerical technique by decreasing the error and dealing with the ill conditioning of the linear system caused by the radial basis function convergence analysis of the coupled technique is treated and numerical results of some solved examples are given at the end of this paper

new modified multi level residue harmonic balance method for solving nonlinearly vibrating double beam problem abstract in this study a new modified multi level residue harmonic balance method is presented and adopted to investigate the forced nonlinear vibrations of axially loaded double beams although numerous nonlinear beam or linear double beam problems have been tackled and solved there have been few studies of this nonlinear double beam problem the geometric nonlinear formulations for a double beam model are developed the main advantage of the proposed method is that a set of decoupled nonlinear algebraic equations is generated at each solution level this heavily reduces the computational effort compared with solving the coupled nonlinear algebraic equations generated in the classical harmonic balance method the proposed method can generate the higher level nonlinear solutions that are neglected by the previous modified harmonic balance method the results from the proposed method agree reasonably well with those from the classical harmonic balance method the effects of damping axial force and excitation magnitude on the nonlinear vibrational behaviour are examined

robust variable and interaction selection for logistic regression and general index models under the logistic regression framework we propose a forward backward method soda for variable selection with both main and quadratic interaction terms in the forward stage soda adds in predic

an analytical technique for solving a class of strongly nonlinear conservative systems abstract recently an analytical technique has been developed to determine approximate solutions of strongly nonlinear differential equations containing higher order harmonic terms usually a set of nonlinear algebraic equations is solved in this method however analytical solutions of these algebraic equations are not always possible especially in the case of a large oscillation previously such algebraic equations for the duffing equation were solved in powers of a small parameter but the solutions measure desired results when the amplitude is an order of 1 in this article different parameters of the same nonlinear problems are found for which the power series produces desired results even for the large oscillation moreover two or three terms of this power series solution measure a good result when the amplitude is an order of 1 besides these a suitable truncation formula is found in which the solution measures better results than existing solutions the method is mainly illustrated by the duffing oscillator but it is also useful for many other nonlinear problems

a highly accurate trivariate spectral collocation method of solution for two dimensional nonlinear initial boundary value problems abstract in this paper we propose a new numerical method namely the trivariate spectral collocation method for solving two dimensional nonlinear partial differential equations pdes arising from unsteady processes the problems considered are nonlinear pdes defined on regular geometries in the current solution approach the quasi linearization method is used to simplify the nonlinear pdes the solutions of the linearized pdes are assumed to be trivariate lagrange interpolating polynomials constructed using chebyshev gauss lobatto cgl points a purely spectral collocation based discretization is employed on the two space variables and the time variable to yield a system of linear algebraic equations that are solved by iteration the numerical scheme is tested on four typical examples of nonlinear pdes reported in the literature as a single equation or system of equations numerical results confirm that the proposed solution approach is highly accurate and computationally efficient when applied to solve two dimensional initial boundary value problems defined on small time intervals and hence it is a reliable alternative numerical method for solving this class of problems the new error bound theorems and proofs on trivariate polynomial interpolation that we present support findings from the numerical simulations

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

estimation and inference in generalized additive coefficient models for nonlinear interactions with high dimensional covariates in the low dimensional case the generalized additive coefficient model gacm proposed by xue and yang statist sinica16 2006 1423 1446 has been demonstrated to be a powerful tool for studying nonlinear interaction effects of variables in this paper we propose estimation and inference procedures for the gacm when the dimension of the variables is high specifically we propose a groupwise penalization based procedure to distinguish significant covariates for the large p small n setting the procedure is shown to be consistent for model structure identification further we construct simultaneous confidence bands for the coefficient functions in the selected model based on a refined two step spline estimator we also discuss how to choose the tuning parameters to estimate the standard deviation of the functional estimator we adopt the smoothed bootstrap method we conduct simulation experiments to evaluate the numerical performance of the proposed methods and analyze an obesity data set from a genome wide association study as an illustration

an approximate method for one dimensional structures with strong nonlinear and nonhomogenous boundary conditions abstract there is a lack of analytic methods for vibration analysis of structures with strong nonlinear and non homogenous boundary conditions this paper proposes such a method a distinct feature of the method is to introduce the generalized coordinates for nonlinear and nonhomogeneous boundary conditions the total solution of the structure consists of a sum of a spatial particular solution for the boundary conditions and a field solution satisfying linear and homogeneous boundary conditions the field solution can be obtained in terms of the modal expansion the modal coefficients together with the generalized coordinates for the boundary conditions satisfy a set of nonlinear ordinary differential equations which are solved with the harmonic balance method two examples are presented to study the convergence and accuracy of the solution the first example is the longitudinal vibration of a bar subject to a base excitation on the left and restrained by a damper on the right the wave method and the differential quadrature method dqm are compared with the proposed method and are found in good agreement the second example is a beam with nonlinear torsional boundary absorbers the method shows a good accuracy to deal with the complex boundary conditions as compared with the differential quadrature element method dqem the proposed method is also compared with the multi scale method msm and shows an advantage to deal with strong nonlinearities the proposed method is highly promising for vibration analysis of complex structures with nonlinear and nonhomogeneous boundary conditions

model selection for high dimensional quadratic regression via regularization abstractquadratic regression qr models naturally extend linear models by considering interaction effects between the covariates to conduct model selection in qr it is important to maintain the hierarchical model structure between main effects and interaction effects existing regularization methods generally achieve this goal by solving complex optimization problems which usually demands high computational cost and hence are not feasible for high dimensional data this article focuses on scalable regularization methods for model selection in high dimensional qr we first consider two stage regularization methods and establish theoretical properties of the two stage lasso then a new regularization method called regularization algorithm under marginality principle ramp is proposed to compute a hierarchy preserving regularization solution path efficiently both methods are further extended to solve generalized qr models numerical results are also shown to demonstrate performance of the methods s

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

application of the elliptic balance method to a nonlinear singular oscillator abstract new analytical approximate solutions based on jacobi elliptic functions are formulated to compute the angular frequency value of a nonlinear singular oscillator it is shown that when our derived analytical angular frequency solutions are compared to the exact angular frequency value the third order elliptic balance solution provides the best estimate to the exact value since the percentage error value is about 0 0059 which is 216 times lower than the best estimate percentage error value reported in the literature

finite element approximation of a parabolic integro differential equation with a weakly singular kernel we give error estimates for the numerical solution by means of the galerkin finite element method of an integro differential equation of parabolic type with a memory term containing a weakly singular kernel optimal order estimates are shown for spatially semidiscrete and completely discrete methods special attention is paid to the regularity of the exact solution

nonsmooth data error estimates for approximations of an evolution equation with a positive type memory term we study the numerical approximation of an integro differential equation which is intermediate between the heat and wave equations the proposed discretization uses convolution quadrature based on the first and second order backward difference methods in time and piecewise linear finite elements in space optimal order error bounds in terms of the initial data and the inhomogeneity are shown for positive times without assumptions of spatial regularity of the data

meshless spectral method for solution of time fractional coupled kdv equations abstract in this article an efficient and accurate meshless spectral interpolation method is formulated for the numerical solution of time fractional coupled kdv equations that govern shallow water waves meshless shape functions constructed via radial basis functions rbfs and point interpolation are used for discretization of the spatial operator approximation of fractional temporal derivative is obtained via finite differences of order o 2 and a quadrature formula the formulated method is applied to various test problems available in the literature for its validation approximation quality and efficiency of the method is measured via discrete error norms e2 e and erms convergence analysis of the proposed method in space and time is numerically determined by varying nodal points m and time step size respectively stability of the proposed method is discussed and affirmed computationally which is an important ingredient of the current study

a generalized levene s scale test for variance heterogeneity in the presence of sample correlation and group uncertainty we generalize levene s test for variance scale heterogeneity between k groups for more complex data when there are sample correlation and group membership uncertainty following a two stage regression framework we show that least absolute deviation regression must be used in the stage 1 analysis to ensure a correct asymptotic k 12 k 1 distribution of the generalized scale gs test statistic we then show that the proposed gs test is independent of the generalized location test under the joint null hypothesis of no mean and no variance heterogeneity consequently we generalize the recently proposed joint location scale gjls test valuable in settings where there is an interaction effect but one interacting variable is not available we evaluate the proposed method via an extensive simulation study and two genetic association application studies

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed

a high order and unconditionally stable scheme for the modified anomalous fractional sub diffusion equation with a nonlinear source term the aim of this paper is to study the high order difference scheme for the solution of modified anomalous fractional sub diffusion equation the time fractional derivative is described in the riemann liouville sense in the proposed scheme we discretize the space derivative with a fourth order compact scheme and use the grunwald letnikov discretization of the riemann liouville derivative to obtain a fully discrete implicit scheme we analyze the solvability stability and convergence of the proposed scheme using the fourier method the convergence order of method is o t h 4 numerical examples demonstrate the theoretical results and high accuracy of the proposed scheme

interaction pursuit in high dimensional multi response regression via distance correlation feature interactions can contribute to a large proportion of variation in many prediction models in the era of big data the coexistence of high dimensionality in both responses and covariates poses unprecedented challenges in identifying important interactions in this paper we suggest a two stage interaction identification method called the interaction pursuit via distance correlation ipdc in the setting of high dimensional multi response interaction models that exploits feature screening applied to transformed variables with distance correlation followed by feature selection such a procedure is computationally efficient generally applicable beyond the heredity assumption and effective even when the number of responses diverges with the sample size under mild regularity conditions we show that this method enjoys nice theoretical properties including the sure screening property support union recovery and oracle inequalities in prediction and estimation for both interactions and main effects the advantages of our method are supported by several simulation studies and real data analysis

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

the linearization method based on the equivalence of dissipated energies for nonlinearly damped structural systems even though the nonlinear damping characteristics inherently pertain to most structural systems many useful dynamic models are still linear as a consequence a method that is able to equivalently linearize eql the nonlinear damping so that it can be directly applied in these existing linear models is essential in the present paper a novel eql method for nonlinearly damped and single degree of freedom systems is developed the method is theoretically derived by modulating the steady state responses of the original nonlinear system during the linearization the new eql method requires the dissipated energy of the target linear system to equal that of the original nonlinear one in effect this criterion is equivalent to forcing both the phase angles and amplitudes of the two systems to be equal or at least to within a small allowable error furthermore the present paper proves that it is possible for the dissipative energy to be expressed in terms of the fourier coefficients of the modulated signal thus the equivalent viscous damping ratio can be computed from these fourier coefficients this new eql method is numerically tested using examples of bi linear damping models and subjected to experimental measurements both the simulation results and the experimental data verify the validity of the method they also prove that the current method gives the equivalent viscous damping with good accuracy in addition a quality index that signifies how well the eql system reaches is appropriately added

correlation pursuit forward stepwise variable selection for index models summary a stepwise procedure correlation pursuit cop is developed for variable selection under the sufficient dimension reduction framework in which the response variable y is influenced by the predictors through an unknown function of a few linear combinations of them unlike linear stepwise regression cop does not impose a special form of relationship such as linear between the response variable and the predictor variables the cop procedure selects variables that attain the maximum correlation between the transformed response and the linear combination of the variables various asymptotic properties of the cop procedure are established and in particular its variable selection performance under a diverging number of predictors and sample size is investigated the excellent empirical performance of the cop procedure in comparison with existing methods is demonstrated by both extensive simulation studies and a real example in functional genomics