on the o spektrum of regular operators and the spectrum of measures the banach algebra m g of all bounded regular borel measures on a locally compact group g is often represented on lz g by the mapping p t 2 t zf f fel2 g this representation has the advantage of bringing the hilbert space structure into play thus the involution is preserved and the spectrum of tu 2 is easy to compute if g is abelian or compact there is nevertheless an inconvenience the spectrum of in the banach algebra m g does not coincide with the spectrum of the operator tu z in general this could be overcome by considering li g instead of lz g another possibility which conserves the hilbert space structure makes use of the fact that the operators t 2 are regular operators on the banach lattice l2 g i e linear combinations of positive operators and indeed the representation of m g in the smaller banach algebra l2 g of all regular operators on l2 g behaves nicely if g is amenable it is an algebraic and lattice isomorphism of m g onto the full subalgebra and sublattice of those regular operators which commute with the right translations consequently the spectrum of in m g is the same as the spectrum of t 2 in the banach algebra r l2 g

optimal convex error estimators for classification a cross validation error estimator is obtained by repeatedly leaving out some data points deriving classifiers on the remaining points computing errors for these classifiers on the left out points and then averaging these errors the 0 632 bootstrap estimator is obtained by averaging the errors of classifiers designed from points drawn with replacement and then taking a convex combination of this zero bootstrap error with the resubstitution error for the designed classifier this gives a convex combination of the low biased resubstitution and the high biased zero bootstrap another convex error estimator suggested in the literature is the unweighted average of resubstitution and cross validation this paper treats the following question given a feature label distribution and classification rule what is the optimal convex combination of two error estimators i e what are the optimal weights for the convex combination this problem is considered by finding the weights to minimize the mse of a convex estimator it also considers optimality under the constraint that the resulting estimator be unbiased owing to the large amount of results coming from the various feature label models and error estimators a portion of the results are presented herein and the main body of results appears on a companion website in the tabulated results each table treats the classification rules considered for the model various bayes errors and various sample sizes each table includes the optimal weights mean errors and standard deviations for the relevant error measures and the mse and mae for the optimal convex estimator many observations can be made by considering the full set of experiments some general trends are outlined in the paper the general conclusion is that optimizing the weights of a convex estimator can provide substantial improvement depending on the classification rule data model sample size and component estimators optimal convex bootstrap estimators are applied to feature set ranking to illustrate their potential advantage over non optimized convex estimators

polynomially riesz perturbations in this paper we investigate perturbations by holomorphically and polynomially riesz operators concerning the sets of upper and lower semi fredholm operators the sets of left and right fredholm operators as well as the sets of upper and lower semi browder operators and the sets of left and right browder operators we consider in particular perturbations of shifts by polynomially riesz operators 1 introduction and preliminaries let c denote the set of all complex numbers and let x be an infinite dimensional banach space we denote by b x the set of all linear bounded operators on x for a2 b x let r a and n a denote the range and the null space of a respectively and let a dimn a and a dimx r a codim r a the sets of upper and lower semi fredholm operators respectively are defined as x fa2 b x a 1 and r a is closedg and x fa 2 b x a 1g set x x x for the set of semi fredholm operators and x x x for the set of fredholm operators for a2 x the index is defined by i a a a the calkin algebra over x is the quotient algebra c x b x k x and let b x c x denote the natural homomorphism it is well known that a2 x if and only if a is invertible in c x 5 theorem 3 2 8 an operator a2 b x is left fredholm if a 1 and r a is a closed and complemented subspace of x while a is right fredholm if a 1 and n a is a complemented subspace of x set l x for the set of all left fredholm operators on x and r x for the set of all right fredholm operators on x for a 2 b x denote by asc a dsc a the ascent the descent of a 2 b x i e the smallest non negative integer n such that n a n n a n 1 r a n r a n 1 if such n does not exist then asc a 1 dsc a 1 an operator a2 b x is upper semi browder if it is upper semi fredholm with finite ascent while a is lower semi browder if it is lower semi fredholm with finite descent letb x b x denote the set of all upper lower semi browder operators we shall say that an operator a2 b x is left browder if it is left fredholm with finite ascent while a is right browder if it is right fredholm with finite descent letbl x br x denote the set of all left right browder operators an operator a2 b x is browder if it is fredholm with finite ascent and finite descent the set of all browder operators on x is denoted byb x clearly b x b x b x bl x br x

on boundary value problems for singular second order functional differential equations abstract positive solutions to the boundary value problem y f x y w x 0 y x y x x a x 0 y x y x x 1 x b are obtained by applying the schauder fixed point theorem where w x is a continuous function defined on 0 1 and f x y is a function defined on 0 1 0 which satisfies certain restrictions and may have singularity at y 0 the result corrects and improves an existence theorem due to erbe and kong j comput appl math 53 1994 377 388

multiple positive solutions for nonlinear eigenvalue problems with the p laplacian abstract in this paper we consider a nonlinear eigenvalue problem driven by the p laplacian differential operator and with a nonsmooth potential using degree theoretic arguments based on the degree map for certain operators of monotone type we show that the problem has at least two nontrivial positive solutions as the parameter 0 varies in a half line

on a class of degenerate and singular elliptic systems in bounded domains abstract this paper deals with the nonexistence and multiplicity of nonnegative nontrivial solutions to a class of degenerate and singular elliptic systems of the form div h 1 x u f u x u v in div h 2 x v f v x u v in where is a bounded domain with smooth boundary in r n n 2 and h i 0 h i l loc 1 h i i 1 2 are allowed to have essential zeroes at some points in f u f v f and is a positive parameter our proofs rely essentially on the critical point theory tools combined with a variant of the caffarelli kohn nirenberg inequality in p caldiroli r musina on a variational degenerate elliptic problem nodea nonlinear differential equations appl 7 2000 189 199

evolution of positive solution curves in semipositone problems with concave nonlinearities abstract we study the existence multiplicity and stability of positive solutions to u x f u x for x 1 1 u 1 0 u 1 where 0 and f 0 r is monotonically increasing and concave with f 0 0 we obtain the exact number of positive solutions as a function of f t t we follow several families of nonlinearities f for which f limt f t 0 and study how the positive solution curves to the above problem evolve also we give examples where our results apply this work extends the work of a castro and r shivaji 1988 proc roy soc edinburgh sect a108 291 302 and s h wang 1994 proc roy soc edinburgh124 no 3 507 515 by obtaining sharper results and also gives a complete study of positive solutions for concave semipositone nonlinearities

ruston riesz and perturbation classes abstract we determine the perturbation classes of fredholm and weyl elements as well the commuting perturbation classes of fredholm weyl and browder elements with respect to unbounded banach algebra homomorphism t among other things we use the ruston elements of mouton mouton and raubenheimer also we investigate the class of polynomially almost t null and the class of polynomially t riesz elements

positive solutions of second order two delay differential systems with twin parameter abstract the paper is concerned with the existence of positive solutions for a second order two delay differential systems with twin parameter firstly we present some sufficient conditions with the twin parameter belonging to an open rectangular region eigenvalue region to ensure the existence of positive solutions for the differential systems with singularity secondly we deal with the superlinear semipositone problem of the differential systems and establish the existence of positive solutions for the systems the proofs are based on a fixed point theorem in cones

multiple solutions for a class of nonlinear elliptic problems with a p laplacian type operator abstract in this paper we obtain the existence of at least two solutions for a nonlinear elliptic problem on unbounded domains involving a p laplacian type operator and nonsmooth potentials our approach is variational and it is based on the nonsmooth critical point theory for locally lipschitz functions

t fredholm analysis and application to operator theory abstract in this paper we study the general setting of fredholm theory relative to a banach algebra homomorphism t mainly we prove some perturbation results on the t browder spectrum as applications we investigate the class of polynomially fredholm perturbation operators on a banach space we obtain stability results under commuting operator perturbations that belong to this class

estimating the error rate of a prediction rule improvement on cross validation abstract we construct a prediction rule on the basis of some data and then wish to estimate the error rate of this rule in classifying future observations cross validation provides a nearly unbiased estimate using only the original data cross validation turns out to be related closely to the bootstrap estimate of the error rate this article has two purposes to understand better the theoretical basis of the prediction problem and to investigate some related estimators which seem to offer considerably improved estimation in small samples

positive solutions of quasilinear boundary value problems abstract this paper deals with the existence of positive solutions for the boundary value problem formula formula where f is either superlinear or sublinear at and f t 0 may be negative and p is a positive continuous function the results extend several known results for semilinear equations our approach is based on fixed point theory for completely continuous operators which leave invariant a suitable cone in a banach space of continuous functions

impact of error estimation on feature selection given a large set of potential features it is usually necessary to find a small subset with which to classify the task of finding an optimal feature set is inherently combinatoric and therefore suboptimal algorithms are typically used to find feature sets if feature selection is based directly on classification error then a feature selection algorithm must base its decision on error estimates this paper addresses the impact of error estimation on feature selection using two performance measures comparison of the true error of the optimal feature set with the true error of the feature set found by a feature selection algorithm and the number of features among the truly optimal feature set that appear in the feature set found by the algorithm the study considers seven error estimators applied to three standard suboptimal feature selection algorithms and exhaustive search and it considers three different feature label model distributions it draws two conclusions for the cases considered 1 depending on the sample size and the classification rule feature selection algorithms can produce feature sets whose corresponding classifiers possess errors far in excess of the classifier corresponding to the optimal feature set and 2 for small samples differences in performances among the feature selection algorithms are less significant than performance differences among the error estimators used to implement the algorithms moreover keeping in mind that results depend on the particular classifier distribution pair for the error estimators considered in this study bootstrap and bolstered resubstitution usually outperform cross validation and bolstered resubstitution usually performs as well as or better than bootstrap

boundary value problems for singular second order functional differential equations abstract we consider general linear boundary value problems for equations of the form y f x y x 0 0 x x is continuous and f x y has a singularity at y 0 the results improve and extend earlier results for the case x x due to taliaferro 1979 and gatica 1989

exact multiplicity results for a p laplacian problem with concave convex concave nonlinearities we study the exact number of positive solutions of a two point dirichlet boundary value problem involving the p laplacian operator we consider the case p 2 as well as the case p 1 when the nonlinearity f satisfies f 0 0 and has two distinct simple positive zeros and such that f changes sign exactly twice on 0 note that we may allow that f changes sign more than twice on 0 some interesting examples of quartic polynomials are given in particular for f u u2 u 1 u 2 we study the evolution of the bifurcation curves of the p laplacian problem as p increases from 1 to infinity and hence are able to determine the exact multiplicity of positive solutions for each p 1

necessary and sufficient conditions of existence for a system involving the p laplacian 1 p n in this work we study the existence of positive solutions for the system p pu f x u v inrn pv g x u v inrn lim x u x 0 lim x v x where pu div u p 2 u is the p laplacian 1 in some case we can have because these parameters depend on p n and the growth of f and g the existence result is related to a sub super solution methods and the nonexistence result is a consequence of picone s identity

positive solutions for a class of nonlinear boundary value problems with neumann robin boundary conditions abstract we consider the two point boundary value problem u x f u x 0 x 1 u 0 0 u 1 u 1 0 where 0 and 0 are parameters and f c2 0 1 we discuss the existence of nonnegative solutions for superlinear nonlinearities by developing a quadrature method we study the positone f 0 0 case as well as the semipositone f 0

multiple positive solutions of nonlinear two point boundary value problems we improve the results obtained by erbe hu and wang in a recent paper we show that there exist at least two positive solutions of two point boundary value problems under conditions weaker than those used by erbe hu and wang

existence of positive solutions for boundary value problem of second order fde abstract we use a fixed point theorem in cones to investigate the existence of positive solutions for boundary value problem of a second order functional differential equation fde with the form y t r t y t 0 0 y t y t t r t 0 y t y t n t 1 t 1 a where y t y t for a we allow that r t has the singularity at the endpoints t 0 and t 1 of 0 1 our results include the situations that f is either superlinear or sublinear

an existence result on positive solutions for a class of p laplacian systems abstract consider the system p u f v in p v g u in u v 0 on where pz div z p 2 z p 1 is a positive parameter and is a bounded domain in rn with smooth boundary we prove the existence of a large positive solution for large when lim x f m g x 1 p 1 x p 1 0 for every m 0 in particular we do not assume any sign conditions on f 0 or g 0