weak solutions of a class of quasilinear hyperbolic integro differential equations describing viscoelastic materials global weak solutions of scalar second order quasilinear hyperbolic integro differential equations with singular kernels are constructed perturbations of rest states are shown to propagate with finite speed smoothing effects of the solution operator are exhibited and conditions for the asymptotic stability of rest states are given the equations arise in viscoelasticity

a dynamic regularized radial basis function network for nonlinear nonstationary time series prediction in this paper constructive approximation theorems are given which show that under certain conditions the standard nadaraya watson 1964 regression estimate nwre can be considered a specially regularized form of radial basis function networks rbfns from this and another related result we deduce that regularized rbfns are m s consistent like the nwre for the one step ahead prediction of markovian nonstationary nonlinear autoregressive time series generated by an i i d noise processes additionally choosing the regularization parameter to be asymptotically optimal gives regularized rbfns the advantage of asymptotically realizing minimum m s prediction error two update algorithms one with augmented networks infinite memory and the other with fixed size networks finite memory are then proposed to deal with nonstationarity induced by time varying regression functions for the latter algorithm tests on several phonetically balanced male and female speech samples show an average 2 2 db improvement in the predicted signal noise error ratio over corresponding adaptive linear predictors using the exponentially weighted rls algorithm further rls filtering of the predictions from an ensemble of three such rbfns combined with the usual autoregressive inputs increases the improvement to 4 2 db on average over the linear predictors

lipschitz metric for the periodic camassa holm equation we study stability of conservative solutions of the cauchy problem for the periodic camassa holm equation ut uxxt 3uux 2uxuxx uuxxx 0 with initial data u0 in particular we derive a new lipschitz metric dd with the property that for two solutions u and v of the equation we have dd u t v t e ct dd u0 v0 the relationship between this metric and usual norms in h 1 per and l 1 per is claried

asymptotic optimality of generalized cl cross validation and generalized cross validation in regression with heteroskedastic errors abstract the problem considered here is that of using a data driven procedure to select a good estimate from a class of linear estimates indexed by a discrete parameter in contrast to other papers on this subject we consider models with heteroskedastic errors the results apply to model selection problems in linear regression and to nonparametric regression estimation via series estimators nearest neighbor estimators and local regression estimators among others generalized c l gc l cross validation cv and generalized cross validation gcv procedures are analyzed the gc l and cv criteria are shown to be asymptotically optimal under general conditions a feasible version of gc l however is available only in some applications the gcv criterion is found to be asymptotically optimal only under a condition that is satisfied in some applications but not in others for example it is satisfied in the nearest neighbor estimation context but not in the series estimation local regression estimation or model selection contexts thus the cv criterion is the only feasible criterion of the three that is asymptotically optimal under general conditions the proofs rely heavily on results of li 1987

universal approximation using radial basis function networks there have been several recent studies concerning feedforward networks and the problem of approximating arbitrary functionals of a finite number of real variables some of these studies deal with cases in which the hidden layer nonlinearity is not a sigmoid this was motivated by successful applications of feedforward networks with nonsigmoidal hidden layer units this paper reports on a related study of radial basis function rbf networks and it is proved that rbf networks having one hidden layer are capable of universal approximation here the emphasis is on the case of typical rbf networks and the results show that a certain class of rbf networks with the same smoothing factor in each kernel node is broad enough for universal approximation

l2 formulation of multidimensional scalar conservation laws we show that kruzhkov s theory of entropy solutions to multidimensional scalar conservation laws kruzhkov in mat sb n s 81 123 228 255 1970 can be entirely recast in l2 and fits into the general theory of maximal monotone operators in hilbert spaces our approach is based on a combination of level set kinetic and transport collapse approximations in the spirit of previous works by brenier in c r acad sci paris ser i math 292 563 566 1981 in j diff equ 50 375 390 1983 in siam j numer anal 21 1013 1037 in methods appl anal 11 515 532 2004 giga and miyakawa in duke math j 50 505 515 1983 and tsai et al in math comp 72 159 181 2003

lipschitz metric for the novikov equation we consider the lipschitz continuous dependence of solutions for the novikov equation with respect to the initial data in particular we construct a finsler type optimal transport metric which renders the solution map lipschitz continuous on bounded sets of h 1 mathbb r cap w 1 4 mathbb r although it is not lipschitz continuous under the natural sobolev metric from an energy law due to the finite time gradient blowup by an application of thom s transversality theorem we also prove that when the initial data is in an open dense subset of h 1 mathbb r cap w 1 4 mathbb r the solution is piecewise smooth this generic regularity result helps us extend the lipschitz continuous metric to the general weak solutions our method of constructing the metric can be used to treat other kinds of quasi linear equations provided a good knowledge about the energy concentration

updating the inverse of a matrix the sherman morrison woodbury formulas relate the inverse of a matrix after a small rank perturbation to the inverse of the original matrix the history of these fomulas is presented and various applications to statistics networks structural analysis asymptotic analysis optimization and partial differential equations are discussed the sherman morrison woodbury formulas express the inverse of a matrix after a small rank perturbation in terms of the inverse of the original matrix this paper surveys the history of these formulas and we examine some applications where these formulas are helpful

weak solutions to a nonlinear variational wave equation we establish the global existence of weak solutions to the cauchy problem for a nonlinear variational wave equation where the wave speed is a given monotone function of the wave amplitude the equation arises in modeling wave motions of nematic liquid crystals long waves on a dipole chain and a few other fields we use the young measure method in the setting of lp spaces we overcome the difficulty that oscillations get amplified by the growth terms of the equation

nonparametric density estimation the l 1 view differentiation of integrals consistency lower bounds for rates of convergence rates of convergence in l1 the automatic kernel estimate l1 and pointwise convergence estimates related to the kernel estimate and the histogram estimate simulation inequalities and random variate generation the transformed kernel estimate applications in discrimination operations on density estimates estimators based on orthogonal series index

unique conservative solutions to a variational wave equation relying on the analysis of characteristics we prove the uniqueness of conservative solutions to the variational wave equation u tt c u c u u x x 0 given a solution u t x even if the wave speed c u is only hor continuous in the t x plane one can still define forward and backward characteristics in a unique way using a new set of independent variables x y constant along characteristics we prove that t x u together with other variables satisfy a semilinear system with smooth coefficients from the uniqueness of the solution to this semilinear system one obtains the uniqueness of conservative solutions to the cauchy problem for the wave equation with general initial data u 0 cdot in h 1 i r u t 0 cdot in l 2 i r

global semigroup of conservative solutions of the nonlinear variational wave equation we prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation utt c u c u ux x 0 we allow for initial data u t 0 and ut t 0 that contain measures we assume that 0 kappa 1 leqq c u leqq kappa solutions of this equation may experience concentration of the energy density u t 2 c u 2u x 2 rm d x into sets of measure zero the solution is constructed by introducing new variables related to the characteristics whereby singularities in the energy density become manageable furthermore we prove that the energy may focus only on a set of times of zero measure or at points where c u vanishes a new numerical method for constructing conservative solutions is provided and illustrated with examples

conservative solutions to a nonlinear variational wave equation we establish the existence of a conservative weak solution to the cauchy problem for the nonlinear variational wave equation utt c u c u ux x 0 for initial data of finite energy here c is any smooth function with uniformly positive bounded values

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

the semigroup generated by 2 2 conservation laws consider the cauchy problem for a strictly hyperbolic 2 2 system of conservation laws in one space dimension ie1 01 assuming that each characteristic field is either linearly degenerate or genuinely nonlinear this paper develops a new algorithm based on wave front tracking which yields a cauchy sequence of approximate solutions converging to a unique limit depending continuously on the initial data the solutions that we obtain constitute a semigroup s defined on a set ie1 02 of integrable functions with small total variation for some lipschitz constant l we have the estimate ie1 03

on the cauchy problem for a nonlinear variational wave equation with degenerate initial data abstract this paper is focused on a one dimensional nonlinear variational wave equation which is the euler lagrange equations of a variational principle arising in the theory of nematic liquid crystals and a few other physical contexts we establish the local existence and uniqueness of classical solutions to its cauchy problem with initial data given on the parabolic degenerating line

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

lipschitz metrics for a class of nonlinear wave equations the nonlinear wave equation u tt c u c u u x x 0 determines a flow of conservative solutions taking values in the space h 1 mathbb r however this flow is not continuous with respect to the natural h 1 distance the aim of this paper is to construct a new metric which renders the flow uniformly lipschitz continuous on bounded subsets of h 1 mathbb r for this purpose h 1 is given the structure of a finsler manifold where the norm of tangent vectors is defined in terms of an optimal transportation problem for paths of piecewise smooth solutions one can carefully estimate how the weighted length grows in time by the generic regularity result proved in 7 these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired lipschitz property

numerical verification of solutions for elasto plastic torsion problems abstract in this paper we consider a numerical technique which enables us to verify the existence of solutions for the elasto plastic torsion problems governed by the variational inequality based upon the finite element approximations and the explicit a priori error estimates for a simple problem we present an effective verification procedure that through numerical computation generates a set which includes the exact solution this paper is an extension of the previous paper 1 in which we mainly dealt with the obstacle problems but some special techniques are utilized to verify the solutions for nondifferentiable nonlinear equations concerned with the present problem a numerical example is illustrated

adaptive tracking of linear time variant systems by extended rls algorithms we exploit the one to one correspondences between the recursive least squares rls and kalman variables to formulate extended forms of the rls algorithm two particular forms of the extended rls algorithm are considered one pertaining to a system identification problem and the other pertaining to the tracking of a chirped sinusoid in additive noise for both of these applications experiments are presented that demonstrate the tracking superiority of the extended rls algorithms compared with the standard rls and least mean squares lms algorithms

testing for nonlinearity in time series the method of surrogate data we describe a statistical approach for identifying nonlinearity in time series the method first specifies some linear process as a null hypothesis then generates surrogate data sets which are consistent with this null hypothesis and finally computes a discriminating statistic for the original and for each of the surrogate data sets if the value computed for the original data is significantly different than the ensemble of values computed for the surrogate data then the null hypothesis is rejected and nonlinearity is detected we discuss various null hypotheses and discriminating statistics the method is demonstrated for numerical data generated by known chaotic systems and applied to a number of experimental time series which arise in the measurement of superfluids brain waves and sunspots we evaluate the statistical significance of the evidence for nonlinear structure in each case and illustrate aspects of the data which this approach identifies

reconstructions and predictions of nonlinear dynamical systems a hierarchical bayesian approach an attempt is made to reconstruct model nonlinear dynamical systems from scalar time series data via a hierarchical bayesian framework reconstruction is performed by fitting given training data with a parameterized family of functions without overfitting the reconstructed model dynamical systems are compared with respect to approximated model marginal likelihood which is a natural bayesian information criterion the best model is selected with respect to this criterion and is utilized to make predictions the results are applied to two problems i chaotic time series prediction and ii building air conditioning load prediction the former is a very good class of problems for checking the abilities of prediction algorithms for at least two reasons first since no linear dynamical systems can admit chaotic behavior an algorithm must capture the nonlinearities behind the time series second chaotic dynamical systems are sensitive to initial conditions more precisely the error grows exponentially with respect to time so that crispness of capturing nonlinearities is also important experimental results appear to indicate that the proposed scheme can capture difficult nonlinearities behind the chaotic time series data the latter class of problems air conditioning load prediction is motivated by a great amount of demand for reducing co sub 2 emissions associated with electric power generation the authors won a prediction competition using the proposed algorithm therefore it appears to be reasonably sound

sparse multivariate gaussian mixture regression fitting a multivariate gaussian mixture to data represents an attractive as well as challenging problem in especial when sparsity in the solution is demanded achieving this objective requires the concurrent update of all parameters weight centers and precisions of all multivariate gaussian functions during the learning process such is the focus of this paper which presents a novel method founded on the minimization of the error of the generalized logarithmic utility function gluf this choice which allows us to move smoothly from the mean square error mse criterion to the one based on the logarithmic error yields an optimization problem that resembles a locally convex problem and can be solved with a quasi newton method the gluf framework also facilitates the comparative study between both extremes concluding that the classical mse optimization is not the most adequate for the task the performance of the proposed novel technique is demonstrated on simulated as well as realistic scenarios