a time scales version of a wirtinger type inequality and applications in this paper we derive a time scales version of a wirtinger type inequality the result is applied to derive nonoscillation of a certain second order euler type dynamic equation on time scales

analysis of variance of cross validation estimators of the generalization error this paper brings together methods from two different disciplines statistics and machine learning we address the problem of estimating the variance of cross validation cv estimators of the generalization error in particular we approach the problem of variance estimation of the cv estimators of generalization error as a problem in approximating the moments of a statistic the approximation illustrates the role of training and test sets in the performance of the algorithm it provides a unifying approach to evaluation of various methods used in obtaining training and test sets and it takes into account the variability due to different training and test sets for the simple problem of predicting the sample mean and in the case of smooth loss functions we show that the variance of the cv estimator of the generalization error is a function of the moments of the random variables y card sj sj and y card sjc sj c where sj sj are two training sets and sjc sj c are the corresponding test sets we prove that the distribution of y and y is hypergeometric and we compare our estimator with the one proposed by nadeau and bengio 2003 we extend these results in the regression case and the case of absolute error loss and indicate how the methods can be extended to the classification case we illustrate the results through simulation

reduced order controller design using coprime factor model reduction two procedures for reduced order controller design that incorporate coprime factor model reduction techniques and a related robust stabilization problem are proposed it is shown that robust stability can be achieved with these controllers the results obtained show how to reduce the controller degree systematically they clearly indicate how the maximum stability margin influences allowable model reduction error

combined 5 2 cv f test for comparing supervised classification learning algorithms dietterich 1998 reviews five statistical tests and proposes the 5 2 cv t test for determining whether there is a significant difference between the error rates of two classifiers in our experiments we noticed that the 5 2 cv t test result may vary depending on factors that should not affect the test and we propose a variant the combined 5 2 cv f test that combines multiple statistics to get a more robust test simulation results show that this combined version of the test has lower type i error and higher power than 5 2 cv proper

some integral inequalities useful in the theory of certain partial dynamic equations on time scales our aim in this paper is to investigate some integral inequalities in two independent variables on time scales which unify and extend some integral inequalities and their corresponding discrete analogues the inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

multiple positive solutions for nonlinear dynamical systems on a measure chain in this paper we consider the following dynamical system on a measure chain u 1 t f1 t u1 t u2 t 0 t a b u 2 t f2 t u1 t u2 t 0 t a b with the sturm liouville boundary value conditions ui a u i a 0 ui b ui b 0 for i 1 2 some results are obtained for the existence of three positive solutions of the above problem by using leggett williams fixed point theorem

hamiltonian systems on time scales linear and nonlinear hamiltonian systems are studied on time scales t we unify symplectic flow properties of discrete and continuous hamiltonian systems a chain rule which unifies discrete and continuous settings is presented for our so called alpha derivatives on generalized time scales this chain rule allows transformation of linear hamiltonian systems on time scales under simultaneous change of independent and dependent variables thus extending the change of dependent variables recently obtained by dosl and hilscher we also give the legendre transformation for nonlinear euler lagrange equations on time scales to hamiltonian systems on time scales

partial differential equations on time scales discrete and continuous formulations of partial differential operators are unified by a time scale formulation of partial differential operators results include an euler lagrange equation for double integral variational problems on time scales and a picone identity which implies a sturm picone comparison theorem for second order elliptic pdes on time scales

wavelet shrinkage using cross validation summary wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics this paper concentrates on the estimation of functions and images from noisy data by using wavelet shrinkage a modified form of twofold cross validation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of 2 the cross validation algorithm is then extended to data sets of any length and to multidimensional data sets the algorithms are compared with established threshold choosers by using simulation an application to a real data set arising from anaesthesia is presented

inequalities and asymptotics for riccati matrix difference operators in the first part inequalities for solutions of riccati matrix difference equations are obtained which correspond to the linear hamiltonian difference system formula whereak bk ck xk ukaren n matrices with symmetricbkandck if the matricesxkare invertible then the matricesqk ukx 1ksolve the riccati matrix difference equation formula in contrast to some recent papers dealing with these equations we do not assume that the matricesbkare invertible the second part of the paper deals with the asymptotic behaviour of solutionsqk as of the special riccati matrix difference equation which corresponds to the sturm liouville equation formula of even order 2nwith constant coefficientsr0 rn

robust stabilization in the gap metric controller design for distributed plants the problem of robustness in the gap metric for infinite dimensional systems is considered the problem of computing the optimal controller and the optimal robustness radius for a class of systems whose normalized coprime factors have elements which are h sub infinity functions with continuous boundary values is studied the underlying hankel and related operators which are important in the gap optimization problem are studied and relations between their singular values and vectors are established a computational approach to the optimal robustness problem is developed for single input single output systems whose transfer function is an inner function in h sub infinity times a rational function the procedure is applied to a general first order delay system and a closed form formula is obtained for the optimal controller the frequency response plots of the compensated system for various values of time delay are examined

controller reduction via stable factorization and balancing abstract a controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper for a certain right coprime factorization of an lqg designed controller k s n s d 1 s we approximate using a balancing technique the pair d s n s t by a low order pair d1 s n1 s t defining a factorization of the reduced order controller k 1 s n1 s d1 1 s we show that reducing the controller order in this way is motivated in a natural way which leads to the expectation of both good stability properties and good accuracy of approximation of closed loop behaviour this is also demonstrated in some examples

dynamic picone s identity and its applications abstract we prove some picone type identities and inequalities for a class of first order nonlinear dynamic systems and derive various weighted inequalities of wirtinger type and hardy type on time scales as applications we study oscillatory and related properties of these systems including reid s roundabout theorem on disconjugacy sturm s separation and comparison theorems as well as a variational method in the oscillation theory

calculus of variations on time scales weak local piecewise crd1 solutions with variable endpoints abstract a nonlinear calculus of variations problem on time scales with variable endpoints is considered the space of functions employed is that of piecewise rd continuously differentiable functions c1prd for this problem the euler lagrange equation the transversality condition and the accessory problem are derived as necessary conditions for weak local optimality assuming the coercivity of the second variation a corresponding second order sufficiency criterion is established

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

gruss type inequalities for double integrals on time scales we prove some gruss type inequalities for double integrals on time scales and unify the corresponding continuous and discrete versions from the literature we also apply our results to the quantum calculus case

dynamic diffusion type equations on discrete space domains abstract we consider a class of partial dynamic equations on discrete space domains which includes as a special case the discrete space versions of the diffusion heat equation we focus on initial value problems and study the existence and uniqueness of forward and backward solutions moreover we discuss other topics such as sum and sign preservation maximum and minimum principles or symmetry of solutions

three positive solutions of three point boundary value problems for p laplacian dynamic equations on time scales abstract in this paper existence criteria of three positive solutions to the following boundary value problems for p laplacian dynamic equations on time scales p u t a t f u t 0 t 0 t t u 0 0 u t b 1 u 0 or u 0 b 0 u 0 u t 0 are established by using the well known leggett williams fixed point theorem an example is also given to illustrate the main results

on the computation of the gap metric abstract this paper deals with the computation of the gap metric introduced by zames and el sakkary 17 it is shown that the gap between two systems p1 p2 is precisely the maximum of the two expressions inf q h d i n i d j n j q for i j equal to 1 2 and 2 1 and ni di being normalized right coprine factorizations of pi i 1 2 in the sense of vidyasagar 12 this expression is computable using well known techniques from interpolation theory

oscillation results for second order scalar and matrix difference equations we consider the self adjoint second order scalar difference equation rn xn pnxn 1 0 1 and the matrix system rn xn pnxn 1 0 2 where rn 0 pn 0 rn 0 pn 0 are sequences of real numbers d d hermitian matrices with rn 0 rn 0 we obtain some new oscillation criteria through change of variable techniques combined with consideration of 1 and 2 as perturbations of the cases pn 0 pn 0

partial dynamic equations on time scales in this work we generalize existing ideas of the univariate case of the time scales calculus to the bivariate case formal definitions of partial derivatives and iterated integrals are offered and bivariate partial differential operators are examined in particular solutions of the homogeneous and nonhomogeneous heat and wave operators are found when initial distributions given are in terms of elementary functions by means of the generalized laplace transform for the time scale setting finally the so termed mixed time scale setting is discussed examples are given and solutions are provided in tabular form

some simplifications of the graphical nyquist criterion the nyquist criterion for feedback stability is reformulated in a very simple fashion using the motion of positive and negative crossings a number of encirclements of a point by the nyquist plot can be quickly computed with this reformulation it is a simple matter to determine the value of the feedback gain for which the system is stable

linear hamiltonian difference systems disconjugacy and jacobi type conditions abstract we consider a linear hamiltonian difference system for the so called singular case so that discrete sturm liouville equations of higher order are included in our theory we introduce the concepts of focal points for matrix valued and generalized zeros for vector valued solutions of the system and define disconjugacy for linear hamiltonian difference systems we prove a reid roundabout theorem which gives conditions equivalent to positive definiteness of a certain discrete quadratic functional among them the strengthened jacobi s condition and a condition on a certain riccati difference equation the key to this theorem is a discrete version of picone s identity furthermore for the sake of generalization of our theorem we introduce controllability for linear hamiltonian difference systems and prove a reid roundabout theorem for a more general functional and more general boundary conditions

block regularized m 2 cross validated estimator of the generalization error a cross validation method based on replications of two fold cross validation is called an cross validation an cross validation is used in estimating the generalization error and comparing of algorithms performance in machine learning however the variance of the estimator of the generalization error in cross validation is easily affected by random partitions poor data partitioning may cause a large fluctuation in the number of overlapping samples between any two training test sets in cross validation this fluctuation results in a large variance in the cross validated estimator the influence of the random partitions on variance becomes serious as increases thus in this study the partitions with a restricted number of overlapping samples between any two training test sets are defined as a block regularized partition set the corresponding cross validation is called block regularized cross validation bcv it can effectively reduce the influence of random partitions we prove that the variance of the bcv estimator of the generalization error is smaller than the variance of cross validated estimator and reaches the minimum in a special situation an analytical expression of the variance can also be derived in this special situation this conclusion is validated through simulation experiments furthermore a practical construction method of bcv by a two level orthogonal array is provided finally a conservative estimator is proposed for the variance of estimator of the generalization error

noether theorem for birkhoffian systems on time scales birkhoff equations on time scales and noether theorem for birkhoffian system on time scales are studied first some necessary knowledge of calculus on time scales are reviewed second birkhoff equations on time scales are obtained third the conditions for invariance of pfaff action and conserved quantities are presented under the special infinitesimal transformations and general infinitesimal transformations respectively fourth some special cases are given and finally an example is given to illustrate the method and results

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

qualitative analysis of nonlinear volterra integral equations on time scales using resolvent and lyapunov functionals in this paper we use the notion of the resolvent equation and lyapunov s method to study boundedness and integrability of the solutions of the nonlinear volterra integral equation on time scales x t a t t 0 t c t s g s x s s t t 0 t in particular the existence of bounded solutions with various lp properties are studied under suitable conditions on the functions involved in the above volterra integral equation at the end of the paper we display some examples on different time scales

normalised coprime factorizations for nonstrictly proper systems some earlier results of d g meyer and g f franklin see ibid vol ac 32 pp 227 8 march 1987 are extended and a procedure is derived for obtaining directly a normalized right coprime factorization of a system that may not be strictly proper

no unbiased estimator of the variance of k fold cross validation most machine learning researchers perform quantitative experiments to estimate generalization error and compare the performance of different algorithms in particular their proposed algorithm in order to be able to draw statistically convincing conclusions it is important to estimate the uncertainty of such estimates this paper studies the very commonly used k fold cross validation estimator of generalization performance the main theorem shows that there exists no universal valid under all distributions unbiased estimator of the variance of k fold cross validation the analysis that accompanies this result is based on the eigen decomposition of the covariance matrix of errors which has only three different eigenvalues corresponding to three degrees of freedom of the matrix and three components of the total variance this analysis helps to better understand the nature of the problem and how it can make naive estimators that don t take into account the error correlations due to the overlap between training and test sets grossly underestimate variance this is confirmed by numerical experiments in which the three components of the variance are compared when the difficulty of the learning problem and the number of folds are varied

a necessary and sufficient condition for oscillation of the sturm liouville dynamic equation on time scales in this paper we investigate oscillatory properties of the second order sturm liouville dynamic equation on a time scale r t x c t x 0 using the so called trigonometric transformation we establish a necessary and sufficient condition for oscillation of this condition is then used to derive an explicit oscillation criterion for this equation

the sine gordon equation on time scales abstract we formulate and discuss integrable analogue of the sine gordon equation on arbitrary time scales this unification contains the sine gordon equation discrete sine gordon equation and the hirota equation doubly discrete sine gordon equation as special cases we present the lax pair check compatibility conditions and construct the darboux backlund transformation finally we obtain a soliton solution on arbitrary time scale the solution is expressed by the so called cayley exponential function

controller approximation approaches for preserving h sub spl infin performance investigates the design of reduced order controllers using an h sub spl infin framework given a stabilizing controller which satisfies a prespecified level of closed loop h sub spl infin performance sufficient conditions are derived for another controller to be stabilizing and satisfy the same level of h sub spl infin performance such controllers are said to be p spl gamma admissible where p is the model of the plant under consideration and spl gamma is the required level of prespecified h sub spl infin performance the conditions are expressed as norm bounds on particular frequency weighted errors where the weights are selected to make a specific transfer function a contraction the design of reduced order p spl gamma admissible controllers is then formulated as a frequency weighted model reduction problem it is advantageous for the required weights to be large in some sense solutions which minimize either the trace or the determinant of the inverse weights are characterized we show that the procedure for minimizing the determinant of the inverse weights always gives a direction where the weights are the best possible to conclude we demonstrate by way of a numerical example that when used in conjunction with a combined model reduction convex optimization scheme the proposed design procedures are effective in substantially reducing controller complexity

double integral calculus of variations on time scales we consider a version of the double integral calculus of variations on time scales which includes as special cases the classical two variable calculus of variations and the discrete two variable calculus of variations necessary and sufficient conditions for a local extremum are established among them an analogue of the euler lagrange equation

feedback stability under simultaneous gap metric uncertainties in plant and controller abstract the stability robustness of a feedback system is studied in this paper by assuming that the plant and the controller are subject to independent uncertainties and that the uncertainties are measured by the gap metric a fairly complete solution is obtained by exploring the trigonometric structure of the graphs of the plant and the controller

averaging techniques for self adjoint matrix equations on a measure chain abstract we apply averaging techniques to establish oscillation and nonoscillation criteria for the self adjoint matrix dynamic equation lx t p t x t q t x t 0 on a measure chain t these results extend and unify earlier results for the differential and difference equation case

unstable rational function approximation the problem is considered of approximating a transfer function with stable and unstable poles by a lower order transfer function with the same number of unstable poles a method is suggested and compared with alternative known methods based on hankel norm approximation examples suggest that no one method always outperforms other methods in terms of minimizing the approximation error

upper and lower bounds for approximation in the gap metric upper and lower bounds for the closest approximant of degree k n in the gap metric to a plant of degree n are obtained the bounds are expressed in terms of the singular values of two hankel operators defined in terms of the symbol of the graph of the plant the question of robust stability and performance of feedback systems is examined in the context of approximation of plant and controller in the gap metric

block regularized repeated learning testing for estimating generalization error abstract repeated learning testing rlt is a popular cross validation method that is usually adopted in estimation and comparison of algorithm performance in machine learning however the variance of the estimator of the generalization error in the rlt method is easily affected by random partitions poor data partitioning may cause a large overlap between any two training sets of rlt and enlarge the variance in the rlt estimator of the generalization error thus in this study we constrain numbers of overlapping samples between any two training sets and construct a novel data partitioning schema in which rlt is called the block regularized rlt brlt we theoretically prove that the brlt estimator has a smaller variance than does the rlt estimator specifically the variance of the rlt estimator reaches its minimum when samples in a data set equally occur in all training sets and all the numbers of overlapping samples are equal furthermore we provide easy to use construction algorithms of brlt s partition set for several training set size and partition count settings by adopting two level orthogonal arrays we also illustrate brlt s optimal properties with several simulated and real world examples

multiple positive solutions for functional dynamic equations on time scales in this paper we study the following functional dynamic equation on time scales f u d t a t f u t u m t 0 email protected 0 t t u t email protected t email protected r 0 t u 0 b 0 u d 0 0 u d t 0 where f r r is an increasing homeomorphism and a positive homomorphism and f 0 0 by using the well known leggett williams fixed point theorem existence criteria for multiple positive solutions are established an example is also given to illustrate the main results

sturm liouville eigenvalue problems on time scales for sturm liouville eigenvalue problems on time scales with separated boundary conditions we give an oscillation theorem and establish rayleigh s principle our results not only unifly the corresponding theories for differential and difference equations but are also new in the discrete case