parameters for integrating periodic functions of several variables a number theoretical method for numerical integration of periodic functions of several variables was developed some years ago this paper presents lists of numerical parameters to be used in implementing that method the parameters define quadrature formulas for functions of 2 3 8 variables error bounds for those formulas are also tabulated the derivation of the parameters and error bounds is described

best approximation of gaussian neural networks with nodes uniformly spaced this paper is aimed at exposing the reader to certain aspects in the design of the best approximants with gaussian radial basis functions rbfs the class of functions to which this approach applies consists of those compactly supported in frequency the approximative properties of uniqueness and existence are restricted to this class functions which are smooth enough can be expanded in gaussian series converging uniformly to the objective function the uniqueness of these series is demonstrated by the context of the orthonormal basis in a hilbert space furthermore the best approximation to a given band limited function from a truncated gaussian series is analyzed by an energy based argument this analysis not only gives a theoretical proof concerned with the existence of best approximations but addresses the problems of architectural selection specifically guidance for selecting the variance and the oversampling parameters is provided for practitioners

exploring regression structure using nonparametric functional estimation abstract average derivative functionals of regression are proposed for nonparametric model selection and diagnostics the functionals are of the integral type which under certain conditions allows their estimation at the usual parametric rate of n 1 2 we analyze asymptotic properties of the estimators of these functionals based on kernel regression these estimators can then be used for assessing the validity of various restrictions imposed on the form of regression in particular we show how they could be used to reduce the dimensionality of the model assess the relative importance of predictors measure the extent of nonlinearity and nonadditivity and under certain conditions help identify projection directions in projection pursuit models and decide on the number of these directions

theory of differential forms on manifolds in this paper we establish a hodge type decomposition for the lp space of differential forms on closed i e compact oriented smooth rieman nian manifolds critical to the proof of this result is establishing an lp es timate which contains as a special case the l2 result referred to by morrey as gaffney s inequality this inequality helps us show the equivalence of the usual definition of sobolev space with a more geometric formulation which we provide in the case of differential forms on manifolds we also prove the lp boundedness of green s operator which we use in developing the lp theory of the hodge decomposition for the calculus of variations we rigorously verify that the spaces of exact and coexact forms are closed in the lp norm for nonlinear analysis we demonstrate the existence and uniqueness of a solution to the a harmonic equation

inequalities for the tails of some elementary series we prove that if dn k x n k 2 n k 2 in k x in k x n 2 in x 2 with formula then we have for all x 0 dn k x k 0 and dn 1 k x 2 dn k x dn 2 k x n k 1 0

dirac harmonic equations for differential forms abstract in this paper we introduce the dirac harmonic equation for differential forms and prove some basic norm inequalities including the caccioppoli type inequality poincare type inequality and the weak reverse holder inequality for the solutions of this kind of differential equations

kolmogorov s spline network in this paper an innovative neural network architecture is proposed and elucidated this architecture based on the kolmogorov s superposition theorem 1957 and called the kolmogorov s spline network ksn utilizes more degrees of adaptation to data than currently used neural network architectures nnas by using cubic spline technique of approximation both for activation and internal functions more efficient approximation of multivariate functions can be achieved the bound on approximation error and number of adjustable parameters derived in this paper favorably compares ksn with other one hidden layer feedforward nnas the training of ksn using the ensemble approach and the ensemble multinet is described a new explicit algorithm for constructing cubic splines is presented

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

inequalities for residuals of power expansions for the exponential function and completely monotone functions letin x be the residual after thenth term of power series for a functionf in the casef x ex x 0 inequalities forinhave been studied by many authors but there are almost no results forx 0 in this paper we present some bounds forinwithf x e xand we give some results for completely monotone functions

global weighted inequalities for operators and harmonic forms on manifolds abstract in this paper both the local and global weighted sobolev poincare imbedding inequalities and poincare inequalities for the composition t g are established where t is the homotopy operator and g is green s operator applied to a harmonic forms on manifolds

an optimal global nearest neighbor metric a quadratic metric dao x y x y t ao x y is proposed which minimizes the mean squared error between the nearest neighbor asymptotic risk and the finite sample risk under linearity assumptions a heuristic argument is given which indicates that this metric produces lower mean squared error than the euclidean metric a nonparametric estimate of ao is developed if samples appear to come from a gaussian mixture an alternative parametrically directed distance measure is suggested for nearness decisions within a limited region of space examples of some two class gaussian mixture distributions are included

chebyshev type inequalities involving permanents and their applications abstract we extend the well known chebyshev s inequality to some new cases involving permanents under the proper hypotheses our main results are per a b n per a n per b n and per a i 1 n j 1 n a i j per b i 1 n j 1 n b i j as applications we consider the problem of finding bounds of the ratios of two functions involving permanents

functional inequalities for the incomplete gamma function abstract we present several inequalities for f a x a x a 0 a 0 x 0 where a x is the incomplete gamma function one of our theorems states that the inequalities f a s p x 1 x n f a x 1 f a x n f a s q x 1 x n p q 0 hold for all nonnegative real numbers x 1 x n n 2 if and only if p m i n a 1 and q m a x a 1 here s t x 1 x n denotes the power sum of order t this extends and complements a result published by ismail and laforgia in 2006

good weights and hyperbolic kernels for neural networks projection pursuit and pattern classification fourier strategies for extracting information from high dimensional data fourier approximation and estimation of discriminant regression and density functions are considered a preference order is established for the frequency weights in multiple fourier expansions and the connection weights in single hidden layer neural networks these preferred weight vectors called good weights good lattice weights for estimation of periodic functions are generalizations for arbitrary periods of the hyperbolic lattice points of korobov 1959 and hlawka 1962 associated with classes of smooth functions of period one in each variable although previous results on approximation and quadrature are affinely invariant to the scale of the underlying periods some of our results deal with optimization over finite sets and strongly depend on the choice of scale it is shown how to count and generate good lattice weights finite sample bounds on mean integrated squared error are calculated for ridge estimates of periodic pattern class densities the bounds are combined with a table of cardinalities of good lattice weight sets to furnish classifier design with prescribed class density estimation errors applications are presented for neural networks and projection pursuit a hyperbolic kernel gradient transform is developed which automatically determines the training weights projection directions its sampling properties are discussed algorithms are presented for generating good weights for projection pursuit

inequalities involving modified bessel functions of the first kind ii the intrinsic properties including logarithmic convexity concavity of the modified bessel functions of the first kind and some other related functions are obtained several inequalities involving functions under discussion are established

the computational intractability of training sigmoidal neural networks we demonstrate that the problem of approximately interpolating a target function by a neural network is computationally intractable in particular the interpolation training problem for a neural network with two monotone lipschitzian sigmoidal internal activation functions and one linear output node is shown to be np hard and np complete if the internal nodes are in addition piecewise ratios of polynomials this partially answers a question of blum and rivest 1992 concerning the np completeness of training a logistic sigmoidal 3 node network an extension of the result is then given for networks with n monotone sigmoidal internal nodes and one convex output node this indicates that many multivariate nonlinear regression problems may be computationally infeasible

a simple lemma on greedy approximation in hilbert space and convergence rates for projection pursuit regression and neural network training a general convergence criterion for certain iterative sequences in hilbert space is presented for an important subclass of these sequences estimates of the rate of convergence are given under very mild assumptions these results establish an 0 1 f4n nonsampling convergence rate for projection pursuit regression and neural network training where n represents the number of ridge functions neurons or coefficients in a greedy basis expansion

a new approach for fuzzy information retrieval based on weighted power mean averaging operators an information retrieval system can help users to retrieve documents relevant to the users queries in recent years some researchers used averaging operators i e infinite one operators waller kraft operators p norm operators and gma operators to handle and and or operations of users fuzzy queries for fuzzy information retrieval but they still have some drawbacks e g sometimes query results do not coincide with the intuition of the human being in this paper we present new averaging operators called weighted power mean averaging wpma operators based on the weighted power mean for dealing with fuzzy information retrieval to overcome the drawbacks of the existing methods furthermore we also extend the proposed wpma operators into the extended wpma operators to handle weighted fuzzy queries for fuzzy information retrieval the proposed wpma operators are more flexible and more intelligent than the existing averaging operators to handle users fuzzy queries for fuzzy information retrieval

geometrically concave univariate distributions in this paper our aim is to show that if a probability density function is geometrically concave convex then the corresponding cumulative distribution function and the survival function are geometrically concave convex too under some assumptions the proofs are based on the so called monotone form of l hospital s rule and permit us to extend our results to the case of the concavity convexity with respect to holder means to illustrate the applications of the main results we discuss in details the geometrical concavity of the probability density function cumulative distribution function and survival function of some common continuous univariate distributions moreover at the end of the paper we present a simple alternative proof to schweizer s problem related to the mulholland s generalization of minkowski s inequality

log concavity for bernstein type operators using stochastic orders abstract this paper aims to study the preservation of log concavity for bernstein type operators in particular attention is focused on positive linear operators defined on the positive semi axis admitting a probabilistic representation in terms of a process with independent increments this class includes the classical gamma szasz and szasz durrmeyer operators with respect to the first and second operators the results of this paper correct two erroneous counterexamples in 10 as a main tool in our results we use stochastic order techniques our results include as a particular case the log concavity of certain functions related to the incomplete gamma function

inequalities for the chi square distribution function abstract letpa x 1 x a0 e ttx 1 dtbe the chi square distribution function and letmt u v be the weighted power mean of ordert we prove the following ifx y 0 x y and 0 1 are fixed real numbers then the inequalities formula are valid for all real numbersa 0 if and only ifr 0 ands in particular we obtain that for alla 0 the functionx pa x is log concave on 0 this proves a conjecture of m merkle

a variant of jensen steffensen s inequality and quasi arithmetic means a variant of jensen steffensen s inequality is proved necessary and sufficient conditions for the equality in jensen steffensen s inequality are established several inequalities involving more than two monotonic functions and generalized quasi arithmetic means with not only positive weights are proved it is shown that such generalized quasi arithmetic means have the same comparability properties as those with positive weights

a contribution to the study of the metric and euclidean structures of dissimilarities this paper is concerned with the geometric properties of dissimilarity coefficients defined on finite sets and especially with their euclidean nature we present several particular transformations which preserve euclideanarity and we complete through the study of a one parameter family the current knowledge of the metric and euclidean structure of coefficients based on binary data these results are directly deduced from two theorems which prove the positive semi definite status of some quadratic forms which play a large role in some definitions of dissimilarity commonly used

weighted integral inequalities for solutions of the a harmonic equation we prove the parametric versions of ar weighted integral inequalities for differential forms satisfying the a harmonic equation these results can be considered as extensions of the classical inequalities for sobolev functions

inequalities for the generalized marcum q function abstract in this paper we consider the generalized marcum q function of order 0 real defined by q a b 1 a 1 b t e t 2 a 2 2 i 1 at d t where a b 0 i stands for the modified bessel function of the first kind and the right hand side of the above equation is replaced by its limiting value when a 0 our aim is to prove that the function q a b is strictly increasing on 0 for each a 0 b 0 and to deduce some interesting inequalities for the function q moreover we present a somewhat new viewpoint of the generalized marcum q function by showing that satisfies the new is better than used nbu property which arises in economic theory

beam selection gain versus antenna selection gain we consider beam selection using a fixed beamforming network fbn at a base station with m array antennas in our setting a butler matrix is deployed at the rf stage to form m beams and then the best beam is selected for transmission we introduce some properties of the noncentral chi square distribution and prove the resulting properties of the beam selection gain verifying that beam selection is superior to antenna selection in rician channels with any k factors furthermore we find asymptotically tight stochastic bounds of the beam selection gain which yield approximate closed form expressions of the expected selection gain and the ergodic capacity beam selection has the order of growth of the ergodic capacity logm regardless of user location in contrast to log logm for antenna selection

theory of surround system and associated inequalities in this paper we define the surround system s m c l and the surround system with a center s 2 p c l under the proper hypotheses we establish several geometric inequalities for s 2 p c l as applications we give the background of the surround system with a center s 2 p c l in space science

properties of the probability density function of the non central chi squared distribution abstract in this paper we consider the probability density function pdf of a non central 2 distribution with arbitrary number of degrees of freedom for this function we prove that can be represented as a finite sum and we deduce a partial derivative formula moreover we show that the pdf is log concave when the degrees of freedom is greater or equal than 2 at the end of this paper we present some turan type inequalities for this function and an elegant application of the monotone form of l hospital s rule in probability theory is given

global poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation both local and global a r weighted poincare inequalities for green s operator applied to the solutions of the nonhomogeneous a harmonic equation in john domains are established as applications of the poincare inequalities for green s operator the a r weighted sobolev imbedding inequalities for green s operator in john domains are also proved

local greedy approximation for nonlinear regression and neural network training a criterion for local estimation and approximation in nonlinear regres sion and neural network training is introduced and motivated n th order greedy approximation for the regression or target function based on the criterion is shown to converge at rate o 1 n 1 2 in the nonsampling case

local minimax learning of functions with best finite sample estimation error bounds applications to ridge and lasso regression boosting tree learning kernel machines and inverse problems optimal local estimation is formulated in the minimax sense for inverse problems and nonlinear regression this theory provides best mean squared finite sample error bounds for some popular statistical learning algorithms and also for several optimal improvements of other existing learning algorithms such as smoothing splines and kernel regularization the bounds and improved algorithms are not based on asymptotics or bayesian assumptions and are truly local for each query not depending on cross validating estimates at other queries to optimize modeling parameters results are given for optimal local learning of approximately linear functions with side information context using real algebraic geometry in particular finite sample error bounds are given for ridge regression and for a local version of lasso regression the new regression methods require only quadratic programming with linear or quadratic inequality constraints for implementation greedy additive expansions are then combined with local minimax learning via a change in metric an optimal strategy is presented for fusing the local minimax estimators of a class of experts providing optimal finite sample prediction error bounds from random forests local minimax learning is extended to kernel machines best local prediction error bounds for finite samples are given for tikhonov regularization the geometry of reproducing kernel hilbert space is used to derive improved estimators with finite sample mean squared error mse bounds for class membership probability in two class pattern classification problems a purely local cross validation free algorithm is proposed which uses fisher information with these bounds to determine best local kernel shape in vector machine learning finally a locally quadratic solution to the finite fourier moments problem is presented after reading the first three sections the reader may proceed directly to any of the subsequent applications sections

l φ μ averaging domains and the quasi hyperbolic metric in this paper we first introduce email protected m averaging domains which are generalizations of existing domains such as john domains and l s m averaging domains then we characterize l f m averaging domains using the quasihyperbolic metric finally we give applications to quasiconformal mappings

improvement and further generalization of inequalities of ostrowski gruss type abstract we improve some inequalities of ostrowski gruss type and further generalize them we apply the obtained results to the estimation of error bounds for some numerical quadrature rules also some bounds for the differences of some special means are discussed

lagrangian mean type mappings for which the arithmetic mean is invariant we determine the class of all pairs of the lagrangian means forming mean type mappings which are invariant with respect to the arithmetic mean

higher integrability of green s operator and homotopy operator abstract we prove the higher integrability and higher order imbedding theorems of the composition of homotopy operator and green s operator we also study analogues of the homotopy operator and green s operator respectively