direct sums of self small mixed groups abstract the class g of self small mixed abelian groups of finite rank has recently been the focus of a number of investigations it is known that up to quasi isomorphism g is dual to the category of locally free torsion free abelian groups of finite rank we utilize the reduced mixed groups in g as building blocks of a more extensive class g the smallest class containing the reduced groups in g that is closed under taking infinite direct sums and direct summands our central result is the determination of a complete set of isomorphism invariants for the groups in g we supplement this broad classification theorem with an investigation of the fine structure of completely decomposable groups in g

adapting to unknown sparsity by controlling the false discovery rate we attempt to recover an n dimensional vector observed in white noise where n is large and the vector is known to be sparse but the degree of sparsity is unknown we consider three different ways of defining sparsity of a vector using the fraction of nonzero terms imposing power law decay bounds on the ordered entries and controlling the lp norm for p small we obtain a procedure which is asymptotically minimax for l r loss simultaneously throughout a range of such sparsity classes the optimal procedure is a data adaptive thresholding scheme driven by control of the false discovery rate fdr fdr control is a relatively recent innovation in simultaneous testing ensuring that at most a certain fraction of the rejected null hypotheses will correspond to false rejections in our treatment the fdr control parameter qn also plays a determining role in asymptotic minimaxity if q lim qn 0 1 2 and also qn log n we get sharp asymptotic minimaxity simultaneously over a wide range of sparse parameter spaces and loss functions on the other hand q lim qn 1 2 1 forces the risk to exceed the minimax risk by a factor growing with q to our knowledge this relation between ideas in simultaneous inference and asymptotic decision theory is new our work provides a new perspective on a class of model selection rules which has been introduced recently by several authors these new rules impose complexity penalization of the form 2 log potential model size actual model size we exhibit a close connection with fdr controlling procedures under stringent control of the false discovery rate

adaptive sparse polynomial chaos expansion based on least angle regression polynomial chaos pc expansions are used in stochastic finite element analysis to represent the random model response by a set of coefficients in a suitable so called polynomial chaos basis the number of terms to be computed grows dramatically with the size of the input random vector which makes the computational cost of classical solution schemes may it be intrusive i e of galerkin type or non intrusive unaffordable when the deterministic finite element model is expensive to evaluate to address such problems the paper describes a non intrusive method that builds a sparse pc expansion first an original strategy for truncating the pc expansions based on hyperbolic index sets is proposed then an adaptive algorithm based on least angle regression lar is devised for automatically detecting the significant coefficients of the pc expansion beside the sparsity of the basis the experimental design used at each step of the algorithm is systematically complemented in order to avoid the overfitting phenomenon the accuracy of the pc metamodel is checked using an estimate inspired by statistical learning theory namely the corrected leave one out error as a consequence a rather small number of pc terms are eventually retained sparse representation which may be obtained at a reduced computational cost compared to the classical full pc approximation the convergence of the algorithm is shown on an analytical function then the method is illustrated on three stochastic finite element problems the first model features 10 input random variables whereas the two others involve an input random field which is discretized into 38 and 30 500 random variables respectively

existence of exotic waves for the nonlinear dispersive mkdv equation the nonlinear dispersive mkdv equation u t u 3 x u 2 x x x 0 is proved to be painleve integrable detailed classification of its travelling waves under certain parameter conditions are obtained by the improved qualitative analysis method abundant type of solutions are shown to exist including exotic traveling waves peaked waves compacted waves looped and cusped waves very particular composite waves having two singular points

combined breathing kink modes in the fpu lattice abstract combined breather kink modes have been observed in several systems here we show that in the fpu lattice small amplitude breathing kink modes are three soliton solutions of an associated mkdv equation since this is integrable the modes can be constructed using the backlund transform as well as finding explicit solutions we consider the stability of the combined mode using variational techniques and illustrate stability for some parameter values and instability for others

a bayesian approach to sparse dynamic network identification modeling and identification of high dimensional systems involving signals with many components poses severe challenges to off the shelf techniques for system identification this is particularly so when relatively small data sets as compared to the number signal components have to be used it is often the case that each component of the measured signal can be described in terms of a few other measured variables and these dependences can be encoded in a graphical way via so called dynamic bayesian networks the problem of finding the interconnection structure as well as estimating the dynamic models can be posed as a system identification problem which involves variable selection while this variable selection could be performed via standard selection techniques computational complexity may however be a critical issue being combinatorial in the number of inputs and outputs in this paper we introduce two new nonparametric techniques which borrow ideas from a recently introduced kernel estimator called stable spline as well as from sparsity inducing priors which use 1 type penalties numerical experiments regarding estimation of large scale sparse armax models show that this technique provides a definite advantage over a group lar algorithm and state of the art parametric identification techniques based on prediction error minimization

discrete conservation laws and the convergence of long time simulations of the mkdv equation pseudospectral collocation methods and finite difference methods have been used for approximating an important family of soliton like solutions of the mkdv equation these solutions present a structural instability which make difficult to approximate their evolution in long time intervals with enough accuracy the standard numerical methods do not guarantee the convergence to the proper solution of the initial value problem and often fail by approaching solutions associated to different initial conditions in this frame the numerical schemes that preserve the discrete invariants related to some conservation laws of this equation produce better results than the methods which only take care of a high consistency order pseudospectral spatial discretization appear as the most robust of the numerical methods but finite difference schemes are useful in order to analyze the rule played by the conservation of the invariants in the convergence

piecewise linear solution paths with application to direct weight optimization recently pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression this paper presents a generalization of these algorithms to a wider class of problems it is shown that the approach can be applied to the nonparametric system identification method direct weight optimization dwo and be used to enhance the computational efficiency of this method the most important design parameter in the dwo method is a parameter l controlling the bias variance trade off and the use of parametric optimization with piecewise linear solution paths means that the dwo estimates can be efficiently computed for all values of l simultaneously this allows for designing computationally attractive adaptive bandwidth selection algorithms one such algorithm for dwo is proposed and demonstrated in two examples

how many variables should be entered in a regression equation abstract the optimal number of regressors is determined to minimize mean squared prediction error and is shown to be a small fraction of the number of data points as the number of regressors grows large the sp criterion provides an asymptotically optimal rule for the number of variables to enter

a fast tracking algorithm for generalized lars lasso this letter gives an efficient algorithm for tracking the solution curve of sparse logistic regression with respect to the regularization parameter the algorithm is based on approximating the logistic regression loss by a piecewise quadratic function using rosset and zhu s path tracking algorithm on the approximate problem and then applying a correction to get to the true path application of the algorithm to text classification and sparse kernel logistic regression shows that the algorithm is efficient

sample size determination for logistic regression the problem of sample size estimation is important in medical applications especially in cases of expensive measurements of immune biomarkers this paper describes the problem of logistic regression analysis with the sample size determination algorithms namely the methods of univariate statistics logistics regression cross validation and bayesian inference the authors treating the regression model parameters as a multivariate variable propose to estimate the sample size using the distance between parameter distribution functions on cross validated data sets herewith the authors give a new contribution to data mining and statistical learning supported by applied mathematics

smooth manifolds for semilinear wave equations on r2 on the existence of almost periodic breathers abstract in this paper we prove several nonexistence and existence results for certain real solutions to semilinear wave equations in one space dimension these solutions represent time almost periodic free vibrations which decay toward a constant equilibrium solution as the spatial variable goes to infinity our nonexistence results are of two kinds we first prove that the requirements of almost periodicity in time and of spatial decay at infinity force such solutions to be trivial whenever their spectrum does not interact with the nonlinearity in a certain sense we also prove non existence results without requiring any spectral conditions but this is at the expense of having to impose more stringent conditions on the nonlinearities such as certain convexity properties in the vicinity of the equilibrium solution finally we prove an existence result for time almost periodic free vibrations whose profiles decay exponentially rapidly toward a constant equilibrium solution to accomplish this we convert the wave equation into a dynamical system in which the original spatial variable plays the role of time we then embed that dynamical system into an appropriate banach space of time almost periodic functions so that the one parameter family of stable and unstable manifolds which we construct carry the solutions that we seek the major difficulty to overcome in our construction is a small divisor problem it is related to the fact that the spectrum of the infinitesimal generator for the linearized flow is a pure point spectrum without gaps we discuss several examples and we also stress some important qualitative differences which distinguish the almost periodic case from the purely periodic one

wobbling kinks in ϕ 4 theory we present a uniform asymptotic expansion of the wobbling kink to any order in the amplitude of the wobbling mode the long range behavior of the radiation is described by matching the asymptotic expansions in the far field and near the core of the kink the complex amplitude of the wobbling mode is shown to obey a simple ordinary differential equation with nonlinear damping we confirm the t ensuremath 1 2 decay law for the amplitude which was previously obtained on the basis of energy considerations

kink soliton characterizing traffic congestion we study traffic congestion by analyzing a one dimensional traffic flow model developing an asymptotic method to investigate the long time behavior near a critical point we derive the modified korteweg de vries mkdv equation as the lowest order model there is an infinite number of kink solitons to the mkdv equation while it has been found by numerical simulations that the kink pattern arising in traffic congestion is uniquely determined irrespective of initial conditions in order to resolve this selection problem we consider higher order corrections to the mkdv equation and find that there is a kink soliton that can deform continuously with the perturbation represented by the addition of these corrections with numerical confirmation we show that this continuously deformable kink soliton characterizes traffic congestion we also discuss the relationship between traffic congestion and the slugging phenomenon in granular flow

the finite quasi baer property this paper discusses extensions of the baer splitting property to the class of self small abelian groups of finite torsion free rank this class is extensive containing the classes of torsion free abelian groups of finite rank and mixed quotient divisible groups the latter in particular includes the class g several applications of the main results are given

on the quasi splitting of exact sequences perhaps the most significant difference between the theories of torsion and torsion free abelian groups is the large number of splitting results in the former and the lack of such in the latter for instance szele s theorem f vol 1 proposition 27 11 has no obvious equivalent in the theory of torsion free abelian groups although it is one of the widely used results in the discussion of torsion and mixed groups let b g qji z p 7 be a subgroup of an abelian group g then b is a direct summand of g if and only if b np g 0 the author was able to extend szele s theorem to valuated p groups in a2 satz 6 81 using methods similar to those which proved useful in the discussion of baer s lemma in al a3 a4 it is the goal of this paper to prove a result which can be considered the equivalent of szele s theorem in the quasi category of torsion free abelian groups before we can describe the result it is necessary to introduce some notation we consider an abelian group a the class of a projective groups of finite a rank is obtained by closing a with respect to direct summands and finite direct sums if g is an abelian group then the a radical of g which is denoted by r g is r g n ker 41 e hom g a finally consider modules b and c over a ring r a homomorphism 1 b c quasi splits if there are a non zero integer n and a map 3 hom c b with cr n id where id denotes the identity map on b if the additive group of b is torsion free then a quasi splitting homomorphism is one to one jl 52 re we say that an abelian group a has the radical splitting property if every exact sequence 0 pa g of torsion free abelian groups in which p is a quasi summand of an a projective group of finite a rank and u p n r g 0 quasi splits szele s original result guarantees the splitting of the sequence in the case in which a z z p 7 for some n w moreover if a z z p z then r g p g theorem 3 1 of this paper establishes that every strongly indecomposable group of finite rank has the

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

on some classes of mkdv periodic solutions we obtain exact periodic solutions of the positive and negative modified kortweg de vries mkdv equations we examine the dynamical stability of these solitary wave lattices through direct numerical simulations while the positive mkdv breather lattice solutions are found to be unstable the two soliton lattice solution of the same equation is found to be stable similarly a negative mkdv lattice solution is found to be stable we also touch upon the implications of these results for the kdv equation

breather lattice and its stabilization for the modified korteweg de vries equation we obtain an exact solution for the breather lattice solution of the modified korteweg de vries equation numerical simulation of the breather lattice demonstrates its instability due to the breather breather interaction however such multibreather structures can be stabilized through the concurrent application of ac driving and viscous damping terms

wobbling kinks in φ4 and sine gordon theory when the 4 model admits a kink solution it also admits a wobbling kink which satisfies the boundary conditions of a kink but possesses an internal degree of freedom in this paper we develop a formal perturbation series for the wobbling kink in 4 theory and give the first two terms in the series explicitly then we prove that the formal series actually is asymptotic for a rather long time o k ln 1 e for a certain k finally we construct an exact 3 soliton solution of the sine gordon equation that also has the properties of a wobbling kink for the sine gordon equation the wobbling kink seems to be mildly unstable

a systematical way to find breather lattice solutions to the positive mkdv equation in this paper dependent and independent variable transformations are introduced to solve the positive mkdv equation systematically by using knowledge of elliptic equation and jacobian elliptic functions it is shown that different kinds of solutions can be obtained to the positive mkdv equation including many kinds of breather lattice solutions

the modified kdv equation with variable coefficients exact uni bi variable travelling wave like solutions in this paper the modified korteweg de vries mkdv equation with variable coefficients vc mkdv equation is investigated via two kinds of approaches and symbolic computation on the one hand we firstly reduce the vc mkdv equation to a second order nonlinear nonhomogeneous ode using travelling wave like similarity transformation and then we obtain its many types of exact fractional solutions with one travelling wave like variable by applying some fractional transformations to the obtained nonlinear ode on the other hand we reduce the vc mkdv equation to two nonlinear pdes with variable coefficients using the anti tangent and anti hypertangent function transformations respectively and then we given its many types of exact solutions with two different travelling wave like variables by studying the obtained nonlinear pde with variable coefficients c 2008 elsevier inc all rights reserved

jamming transitions and the modified korteweg de vries equation in a two lane traffic flow abstract the two lattice models are presented to simulate the traffic flow on a two lane highway they are the lattice versions of the hydrodynamic model of traffic the one model a is described by the differential difference equation where time is a continuous variable and space is a discrete variable and the other model b is the difference equation in which both time and space variables are discrete the jamming transitions among the freely moving phase the coexisting phase and the uniform congested phase are studied by using the nonlinear analysis and the computer simulation the modified korteweg de vries mkdv equations are derived from the lattice models near the critical point the traffic jam is described by a kink antikink solution obtained from the mkdv equation it is found that the critical point the coexisting curve and the neutral stability line decrease with increasing the rate of lane changing also the computer simulation is performed for the model b it is shown that the coexisting curves obtained from the mkdv equation are consistent with the simulation result

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

piecewise linear regularized solution paths we consider the generic regularized optimization problem argmin l y x j efron hastie johnstone and tibshirani ann statist 32 2004 407 499 have shown that for the lasso that is if l is squared error loss and j 1 is the l 1 norm of the optimal coefficient path is piecewise linear that is is piecewise constant we derive a general characterization of the properties of loss l penalty j pairs which give piecewise linear coefficient paths such pairs allow for efficient generation of the full regularized coefficient paths we investigate the nature of efficient path following algorithms which arise we use our results to suggest robust versions of the lasso for regression and classification and to develop new efficient algorithms for existing problems in the literature including mammen and van de geer s locally adaptive regression splines

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

nonlinear periodic waves solutions of the nonlinear self dual network equations the new classes of periodic solutions of nonlinear self dual network equations describing the breather and soliton lattices expressed in terms of the jacobi elliptic functions have been obtained the dependences of the frequencies on energy have been found numerical simulations of soliton lattice demonstrate their stability in the ideal lattice and the breather lattice instability in the dissipative lattice however the lifetime of such structures in the dissipative lattice can be extended through the application of ac driving terms

boosting as a regularized path to a maximum margin classifier in this paper we study boosting methods from a new perspective we build on recent work by efron et al to show that boosting approximately and in some cases exactly minimizes its loss criterion with an l1 constraint on the coefficient vector this helps understand the success of boosting with early stopping as regularized fitting of the loss criterion for the two most commonly used criteria exponential and binomial log likelihood we further show that as the constraint is relaxed or equivalently as the boosting iterations proceed the solution converges in the separable case to an l1 optimal separating hyper plane we prove that this l1 optimal separating hyper plane has the property of maximizing the minimal l1 margin of the training data as defined in the boosting literature an interesting fundamental similarity between boosting and kernel support vector machines emerges as both can be described as methods for regularized optimization in high dimensional predictor space using a computational trick to make the calculation practical and converging to margin maximizing solutions while this statement describes svms exactly it applies to boosting only approximately

greedy function approximation a gradient boosting machine function estimation approximation is viewed from the perspective of numerical optimization in function space rather than parameter space a connection is made between stagewise additive expansions and steepest descent minimization a general gradient descent boosting paradigm is developed for additive expansions based on any fitting criterion specific algorithms are presented for least squares least absolute deviation and huber m loss functions for regression and multiclass logistic likelihood for classification special enhancements are derived for the particular case where the individual additive components are regression trees and tools for interpreting such treeboost models are presented gradient boosting of regression trees produces competitive highly robust interpretable procedures for both regression and classification especially appropriate for mining less than clean data connections between this approach and the boosting methods of freund and shapire and friedman hastie and tibshirani are discussed

differential geometric least angle regression a differential geometric approach to sparse generalized linear models sparsity is an essential feature of many contemporary data problems remote sensing various forms of automated screening and other high throughput measurement devices collect a large amount of information typically about few independent statistical subjects or units in certain cases it is reasonable to assume that the underlying process generating the data is itself sparse in the sense that only a few of the measured variables are involved in the process we propose an explicit method of monotonically decreasing sparsity for outcomes that can be modelled by an exponential family in our approach we generalize the equiangular condition in a generalized linear model although the geometry involves the fisher information in a way that is not obvious in the simple regression setting the equiangular condition turns out to be equivalent with an intuitive condition imposed on the rao score test statistics in certain special cases the method can be tweaked to obtain l1 penalized generalized linear model solution paths but the method itself defines sparsity more directly although the computation of the solution paths is not trivial the method compares favourably with other path following algorithms

solitary wave solutions and kink wave solutions for a generalized kdv mkdv equation abstract bifurcation method of dynamical systems is employed to investigate solitary wave solutions and kink wave solutions of the generalized kdv mkdv equation under some parameter conditions their explicit expressions are obtained

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

prediction games and arcing algorithms the theory behind the success of adaptive reweighting and combining algorithms arcing such as adaboost freund schapire 1996a 1997 and others in reducing generalization error has not been well understood by formulating prediction as a game where one player makes a selection from instances in the training set and the other a convex linear combination of predictors from a finite set existing arcing algorithms are shown to be algorithms for finding good game strategies the minimax theorem is an essential ingredient of the convergence proofs an arcing algorithm is described that converges to the optimal strategy a bound on the generalization error for the combined predictors in terms of their maximum error is proven that is sharper than bounds to date schapire freund bartlett and lee 1997 offered an explanation of why adaboost works in terms of its ability to produce generally high margins the empirical comparison of adaboost to the optimal arcing algorithm shows that their explanation is n

decay of bound states in a sine gordon equation with double well potentials we consider a spatially inhomogeneous sine gordon equation with a double well potential describing long josephson junctions with phase shifts we discuss the interactions of symmetric and antisymmetric bound states in the system using a multiple scale expansion we show that the modes decay algebraically in time due to the energy transfer from the discrete to the continuous spectrum in particular exciting the two modes at the same time yields an increased decay rate an external time periodic drive is shown to sustain symmetric state while it damps the antisymmetric one

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed