boundedness of inhomogeneous journe s type operators on multi parameter local hardy spaces abstract in this paper we establish a boundedness criterion of a class of inhomogeneous journe s type of multi parameter singular integral operators on multi parameter local hardy spaces h p r n 1 r n k recently introduced in ding et al 2019 the consideration of such inhomogeneous journe s type of multi parameter singular integral operators is motivated by the study of the multi parameter pseudo differential operators on multi parameter local hardy spaces the lack of the vanishing moments of both the corresponding functions in the multi parameter local calderon reproducing formula and the multi parameter local littlewood paley stein functions as well as the lack of vanishing moments of atoms in the multi parameter local hardy spaces h p r n 1 r n k with large size all create substantial difficulties in our proofs

attractors of continuous difference equations abstract this paper aims at arranging the general ideas and results of the research on the nonlinear continuous difference equations and acquainting the reader with the properties of the solutions which seem unusual in the context of the theory of ordinary differential equations and have great importance in simulating sophisticated oscillatory processes such as self organization and deterministic chaos and just in those cases when it is difficult to apply ordinary differential equations

calderon zygmund operators on product hardy spaces let t be a product calderon zygmund singular integral introduced by journe using an elegant rectangle atomic decomposition of hp rn rm and journe s geometric covering lemma r fefferman proved the remarkable hp rn rm lp rn rm boundedness of t in this paper we apply vector valued singular integral calderon s identity littlewood paley theory and the almost orthogonality together with fefferman s rectangle atomic decomposition and journe s covering lemma to show that t is bounded on product hp rn rm for max nn e mm e p 1 if and only if t1 1 t2 1 0 where e is the regularity exponent of the kernel of t

non homogeneous t1 theorem for bi parameter singular integrals abstract we prove a non homogeneous t 1 theorem for certain bi parameter singular integral operators moreover we discuss the related non homogeneous journe s lemma and product bmo theory

some weighted inequalities on product domains we extend the results of r fefferman 3 on the bidisc to higher product domains via induction as an application we extend the weighted inequality for calderon zygmund operators on the bidisc to higher product domains and we also extend the result of the littlewood paley operator corresponding to the arbitrary disjoint rectangles to the weighted case 0 introduction in 3 r fefferman discovered a general technique which can be used to prove weighted results in r1 x r1 the product domain for product weights ap12 ri x r1 with 2 p oo his technique however explicitly used the special geometric properties of r2 in this paper we generalize his technique to higher dimeasion i e ri x r x r and beyond to be more specific we reduce fefferman s technique to a verification of one main estimate for a linear operatort f qqt tf x 1 dx t cf m f2 dx t q we will explain the notation in a later section having this estimate we can conclude that t is a bounded map on lp w for 2 p oo and a product weight w e ap12 verifying the main estimate for various operators t is not entirely trivial we work on two operators we prove the lp w boundedness for 2 p 00 w e api2 for the product domain generalization of calderon zygmund operators and lastly we prove the following littlewood paley result theorem let r be a set of disjoint rectangles with sides parallel to the coordinate axes let xr be their characteristic functions define si f 74 xr1 4 f we have the operator a f x si f x i bounded on lp w 2 p oo and w e ap12 r1 x r received by the editors february 9 1988 1980 mathematics subject classification 1985 revision primary 42b20 secondary 42b25 1990 american mathematical society 0002 9947 90 1 00 25 per page

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

a discrete transform and triebel lizorkin spaces on the bidisc we use a discrete transform to study the triebel lizorkin spaces on bidisc fpaq jq and establishes the boundedness of transform sek p jq and tt jq q fp we also define the almost diagonal operator and prove its boundedness with the use of discrete transform and journe lemma we get the atomic decomposition of jq for 0 1 a e r is established too the case for p q is similar 0 introduction in this paper we use a discrete transform to study the triebel lizorkin spaces on the bidisc in 1 we define the sequence space fp and the distribution space fp 7 our fundamental formula is f eq f qq yiq where q runs over all dyadic rectangles and qq yq are translates and dilates of x y associated with q respectively fourier transforms of x y have compact support and suppqq q c l 2 2v11 1 4 in 2 we define almost diagonal operators and prove the boundedness of such operators on fp by duality we also define 3 m rectangle molecules received by the editors july 8 1993 and in revised form december 28 1993 originally communicated to the proceedings of the ams by andrew bruckner 1991 mathematics subject classification primary 42b20 supported by nnsf zpnsf and the doctoral foundation in china 1995 american mathematical society 0002 9947 95 1 00 25 per page

the surface induced spatial temporal structures in confined binary alloys abstract this paper examines surface induced ordering in confined binary alloys the hyperbolic initial boundary value problem ibvp is used to describe a scenario of spatiotemporal ordering in a disordered phase for concentration of one component of binary alloy and order parameter with non linear dynamic boundary conditions this hyperbolic model consists of two coupled second order differential equations for order parameter and concentration it also takes into account effects of the memory on the ordering of atoms and their densities in the alloy the boundary conditions characterize surface velocities of order parameter and concentration changing which is due to surface super cooling on walls confining the binary alloy it is shown that for large times there are three classes of dynamic non linear boundary conditions which lead to three different types of attractor s elements for the ibvp namely the elements of attractor are the limit periodic simple shock waves with fronts of discontinuities if is finite then the attractor contains spatiotemporal functions of relaxation type if is infinite and countable then we observe the functions of pre turbulent type if is infinite and uncountable then we obtain the functions of turbulent type

a maximal function for families of hilbert transforms along homogeneous curves let h u be the hilbert transform along the parabola t ut 2 where u in mathbb r for a set u of positive numbers consider the maximal function mathcal h u f sup h u f u in u we obtain an essentially optimal result for the l p operator norm of mathcal h u when 2 p infty the results are proved for families of hilbert transforms along more general nonflat homogeneous curves

boundedness of multi parameter fourier multiplier operators on triebel lizorkin and besov lipschitz spaces abstract the main purpose of this paper is three fold first we prove that under the limited smoothness conditions multi parameter fourier multiplier operators are bounded on multi parameter triebel lizorkin and besov lipschitz spaces by the littlewood paley decomposition and the strong maximal operator second we offer a different and more direct method to deal with the boundedness instead of transforming fourier multiplier operators into multi parameter calderon zygmund operators third we also prove the boundedness of multi parameter fourier multiplier operators on weighted multi parameter triebel lizorkin and besov lipschitz spaces when the fourier multiplier is only assumed with limited smoothness

diffuse interface model for rapid phase transformations in nonequilibrium systems a thermodynamic approach to rapid phase transformations within a diffuse interface in a binary system is developed assuming an extended set of independent thermodynamic variables formed by the union of the classic set of slow variables and the space of fast variables we introduce finiteness of the heat and solute diffusive propagation at the finite speed of the interface advancing to describe transformations within the diffuse interface we use the phase field model which allows us to follow steep but smooth changes of phase within the width of the diffuse interface governing equations of the phase field model are derived for the hyperbolic model a model with memory and a model of nonlinear evolution of transformation within the diffuse interface the consistency of the model is proved by the verification of the validity of the condition of positive entropy production and by outcomes of the fluctuation dissipation theorem a comparison with existing sharp interface and diffuse interface versions of the model is given

fuzzy linear regression with spreads unrestricted in sign abstract by the use of actual examples it is shown that when the two trends namely the fuzzy spread trend and the center line or modal trend are in conflict or inconsistent the results of the fuzzy linear regression flr model based on the approach of tanaka and coworkers frequently misinterprets the data to avoid this misinterpretation a modification of the flr procedure by allowing the spreads of the parameter to be unrestricted in sign is proposed it is shown that by the use of this unrestricted fuzzy linear regression uflr model correct prediction of the trends are obtained

effect of rule weights in fuzzy rule based classification systems this paper examines the effect of rule weights in fuzzy rule based classification systems each fuzzy if then rule in our classification system has antecedent linguistic values and a single consequent class we use a fuzzy reasoning method based on a single winner rule in the classification phase the winner rule for a new pattern is the fuzzy if then rule that has the maximum compatibility grade with the new pattern when we use fuzzy if then rules with certainty grades the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade in this paper the effect of rule weights is illustrated by drawing classification boundaries using fuzzy if then rules with without certainty grades it is also shown that certainty grades play an important role when a fuzzy rule based classification system is a mixture of general rules and specific rules through computer simulations we show that comprehensible fuzzy rule based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy if then rules with certainty grades

extended fuzzy linear models and least squares estimates least squares regression of the fuzzy linear model is extended to overcome and interpret the occurrence of negative spreads 1 the idea is to introduce best fit difference models which exploit both the hukuhara difference and the l2 metric distance the fuzzy models use lr fuzzy numbers fitted models are compared by using the coefficient of determination in a similar way to its use in classical statistical least squares fitting the non lr fuzzy case is also considered

learning in the framework of fuzzy lattices a basis for rigorous versatile learning is introduced theoretically that is the framework of fuzzy lattices or fl framework for short which proposes a synergetic combination of fuzzy set theory and lattice theory a fuzzy lattice emanates from a conventional mathematical lattice by fuzzifying the inclusion order relation learning in the fl framework can be effected by handling families of intervals where an interval is treated as a single entity block the way explained here illustrations are provided in a lattice defined on the unit hypercube where a lattice interval corresponds to a conventional hyperbox a specific scheme for learning by clustering is presented namely spl sigma fuzzy lattice learning scheme or spl sigma fll scheme for short inspired from adaptive resonance theory art learning by the spl sigma fll is driven by an inclusion measure spl sigma of the corresponding cartesian product to be introduced here we delineate a comparison of the spl sigma fll scheme with various neural fuzzy and other models applications are shown to one medical data set and two benchmark data sets where spl sigma fll s capacity for treating efficiently real numbers as well as lattice ordered symbols separately or jointly is demonstrated due to its efficiency and wide scope of applicability the spl sigma fll scheme emerges as a promising learning scheme

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

support vector clustering we present a novel clustering method using the approach of support vector machines data points are mapped by means of a gaussian kernel to a high dimensional feature space where we search for the minimal enclosing sphere this sphere when mapped back to data space can separate into several components each enclosing a separate cluster of points we present a simple algorithm for identifying these clusters the width of the gaussian kernel controls the scale at which the data is probed while the soft margin constant helps coping with outliers and overlapping clusters the structure of a dataset is explored by varying the two parameters maintaining a minimal number of support vectors to assure smooth cluster boundaries we demonstrate the performance of our algorithm on several datasets

weighted hardy spaces in the three parameter case abstract in this paper we use the idea of the discrete littlewood paley theory developed by han and lu to carry out the three parameter weighted hardy spaces theory under a rather weak condition on the product weight w a and obtain the boundedness of singular integral operators on the weighted hardy spaces

harmonic analysis on product spaces rn x r and commute with the dilations p33 8 x2 x2 61x1 62x2 for x1 e rn x2 e rm and 61 82 0 then the corresponding operators are related to many problems in the theory of multiple fourier integrals see zygmund 21 the maximal operator which commutes with the p31382 is the jessen marcinkiewicz and zygmund 14 strong maximal operator ms defined by 1 ms f x1 x2 sup ri a y1 y2 1 dy1 dy2 xl x2 gr jr1

boundedness criterion of journeʼs class of singular integrals on multiparameter hardy spaces abstract this article concerns nonconvolutional type operators also known as journe s type operators associated with a multiparameter family of dilations given by x 1 x 2 x m 1 x 1 2 x 2 m x m where x 1 r n 1 x 2 r n 2 x m r n m and m 3 we are especially interested in the boundedness of such operators on the multiparameter hardy spaces this work is motivated by pipher s result on the boundedness of these operators from the multiparameter h p spaces to l p spaces for 0 p 1 and journe s counter example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter hardy spaces journe s work shows that there is a sharp difference between the situations for two and three or more parameters we establish in this paper the necessary and sufficient conditions under which the singular integral operators in journe s class are bounded on multiparameter h p spaces 0 p 1 with arbitrary number of parameters

lagrangian support vector machines an implicit lagrangian for the dual of a simple reformulation of the standard quadratic program of a linear support vector machine is proposed this leads to the minimization of an unconstrained differentiable convex function in a space of dimensionality equal to the number of classified points this problem is solvable by an extremely simple linearly convergent lagrangian support vector machine lsvm algorithm lsvm requires the inversion at the outset of a single matrix of the order of the much smaller dimensionality of the original input space plus one the full algorithm is given in this paper in 11 lines of matlab code without any special optimization tools such as linear or quadratic programming solvers this lsvm code can be used as is to solve classification problems with millions of points for example 2 million points in 10 dimensional input space were classified by a linear surface in 82 minutes on a pentium iii 500 mhz notebook with 384 megabytes of memory and additional swap space and in 7 minutes on a 250 mhz ultrasparc ii processor with 2 gigabytes of memory other standard classification test problems were also solved nonlinear kernel classification can also be solved by lsvm although it does not scale up to very large problems it can handle any positive semidefinite kernel and is guaranteed to converge a short matlab code is also given for nonlinear kernels and tested on a number of problems

finknn a fuzzy interval number k nearest neighbor classifier for prediction of sugar production from populations of samples this work introduces finknn a k nearest neighbor classifier operating over the metric lattice of conventional interval supported convex fuzzy sets we show that for problems involving populations of measurements data can be represented by fuzzy interval numbers fins and we present an algorithm for constructing fins from such populations we then present a lattice theoretic metric distance between fins with arbitrary shaped membership functions which forms the basis for finknn s similarity measurements we apply finknn to the task of predicting annual sugar production based on populations of measurements supplied by hellenic sugar industry we show that finknn improves prediction accuracy on this task and discuss the broader scope and potential utility of these techniques

h p boundedness of calderon zygmund operators on product spaces in this paper we prove the product h p boundedness of calderon zygmund operators which were considered by fefferman and stein the methods used in this paper are new even for the classical h p boundedness of calderon zygmund operators namely using some subtle estimates together with the h p l p boundedness of product vector valued calderon zygmund operators

boundedness of bi parameter pseudo differential operators on bi parameter α modulation spaces abstract though the one parameter modulation spaces have been extensively studied since it was first introduced by grobner not much is known about the bi parameter modulation spaces this paper is to introduce the bi parameter modulation spaces and establish the boundedness of bi parameter pseudo differential operators on the bi parameter modulation spaces by the special relationship between modulation spaces and modulation spaces we also obtain the boundedness of bi parameter pseudo differential operators on the bi parameter modulation spaces

estimates for multiparameter maximal operators of schrodinger type abstract multiparameter maximal estimates are considered for operators of schrodinger type sharp and almost sharp results that extend work by rogers and villarroya are obtained we provide new estimates via the integrability of the kernel which naturally appears with a t t argument and discuss the behavior at the endpoints we treat in particular the case of global integrability of the maximal operator on finite time for solutions to the linear schrodinger equation and make some comments on an open problem

boundedness of singular integrals on the flag hardy spaces on heisenberg group abstract we prove that the classical one parameter convolution singular integrals on the heisenberg group are bounded on multiparameter flag hardy spaces which satisfy the intermediate dilation between the one parameter anisotropic dilation and the product dilation on c n r implicitly

littlewood paley characterization and duality of weighted anisotropic product hardy spaces the authors study anisotropic product hardy spaces hwp a associated with a pair a a1 a2 of expansive dilations and a class of product muckenhoupt weights a a on rn rm this article is a continuation of their earlier work in bownik et al 2010 8 the authors establish the littlewood paley g function characterization and the transform characterization of hwp a 0 p 1 the authors also introduce the weighted anisotropic product campanato space lp w a and establish its transform characterization as an application the authors identify the dual space of hwp a with lp w a the results of this article improve the existing results for weighted product hardy spaces on r r and are new even in the weighted isotropic setting

interval regression analysis by quadratic programming approach when we use linear programming in possibilistic regression analysis some coefficients tend to become crisp because of the characteristic of linear programming on the other hand a quadratic programming approach gives more diverse spread coefficients than a linear programming one therefore to overcome the crisp characteristic of linear programming we propose interval regression analysis based on a quadratic programming approach another advantage of adopting a quadratic programming approach is to be able to integrate both the property of central tendency in least squares and the possibilistic property in fuzzy regression by changing the weights of the quadratic function we can analyze the given data from different viewpoints for data with crisp inputs and interval outputs the possibility and necessity models can be considered therefore the unified quadratic programming approach obtaining the possibility and necessity regression models simultaneously is proposed even though there always exist possibility estimation models the existence of necessity estimation models is not guaranteed if we fail to assume a proper function fitting to the given data as a regression model thus we consider polynomials as regression models since any curve can be represented by the polynomial approximation using polynomials we discuss how to obtain approximation models which fit well to the given data where the measure of fitness is newly defined to gauge the similarity between the possibility and the necessity models furthermore from the obtained possibility and necessity regression models a trapezoidal fuzzy output can be constructed

what is distance and why do we need the metric model for pattern learning abstract the concept of distance its role in pattern recognition and some advantages of the new model for pattern learning proposed recently by the author are discussed the universality flexibility and the ability to connect intrinsically the low level process that selects the primitives for the pattern representation with the higher level recognition process make the model clearly superior to other models proposed so far the fundamentally new analytical feature of the model which allows the learning machine to reconfigure itself efficiently is the introduction of continuity in the classical discrete computational model

and variants of multiparameter calderon zygmund theory we consider calderon zygmund operators on product domains under certain weak conditions on the kernel a singular integral operator can be proved to be bounded on h p r r r 0 p 1 if its behaviour on l 2 and on certain scalar valued and vector valued rectangle atoms is known another result concerns an extension of the authors results on l p variants of calderon zygmund theory 1 23 to the product domain setting as an application one obtains estimates for fourier multipliers and pseudo differential operators

multiparameter riesz commutators it is shown that product bmo of s y a chang and r fefferman defined on the space bbb r d 1 otimes cdots otimes bbb r d t can be characterized by the multiparameter commutators of riesz transforms this extends a classical one parameter result of r coifman r rochberg and g weiss and at the same time extends the work of m lacey and s ferguson and m lacey and e terwilleger on multiparameter commutators with hilbert transforms

product hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type the aim of this article is to develop the theory of product hardy spaces associated with operators which possess the weak assumption of davies gaffney heat kernel estimates in the setting of spaces of homogeneous type we also establish a calderon zygmund decomposition on product spaces which is of independent and use it to study the interpolation of these product hardy spaces we then show that under the assumption of generalized gaussian estimates the product hardy spaces coincide with the lebesgue spaces for an appropriate range of p

differentiation of the integral hardy spaces and calderon zygmund operators in the product setting abstract this article concerns strong differentiation and operators on product hardy spaces we first show that an example created by papoulis of a function on r 2 whose integral is strongly differentiable almost everywhere but the integral of its absolute value fails to be strongly differentiable on a set of positive measure belongs to the product hardy space h 1 r r the methods that we develop enable us to present a relaxed version of chang fefferman p atoms with a lower number of required vanishing moments and no smoothness needed on the elementary particles in analogy with the proof of this result we show a generalization of a theorem of r fefferman which concludes h p l p 0 p 1 boundedness of multiparameter operators from their behavior on rectangle atoms in addition we extend a result of pipher concerning boundedness of multiparameter calderon zygmund operators from h p to l p

multiparameter hardy space theory on carnot caratheodory spaces and product spaces of homogeneous type this paper is inspired by the work of nagel and stein in which the lp 1 p theory has been developed in the setting of the product carnot caratheodory spaces m m1 mn formed by vector fields satisfying hormander s finite rank condition the main purpose of this paper is to provide a unified approach to develop the multiparameter hardy space theory on product spaces of homogeneous type this theory includes the product hardy space its dual the product bmo space the boundedness of singular integral operators and calderon zygmund decomposition and interpolation of operators as a consequence we obtain the endpoint estimates for those singular integral operators considered by nagel and stein 2004 in fact we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric all of our results are established by introducing certain banach spaces of test functions and distributions developing discrete calderon identity and discrete littlewood paley stein theory our methods do not rely on the journe type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product hardy spaces

novel fuzzy inference system fis analysis and design based on lattice theory we introduce novel set and lattice theoretic perspectives and tools for the analysis and design of fuzzy inference systems fiss we present an fis including both fuzzification and defuzzification as a device for implementing a function f rnrarr rm the family of fis functions has cardinality aleph2 2aleph1 where aleph1 is the cardinality of the set r of real numbers hence the fis family is much larger than polynomials neural networks etc furthermore a fis has a capacity for local generalization a formulation in the context of lattice theory allows us to define the set f of fuzzy interval numbers fins which includes both fuzzy numbers and intervals we present a metric dk on f which can introduce tunable nonlinearities fis design based on dk has advantages such as an alleviation of the curse of dimensionality problem and a potential for improved computer memory utilization we present a new fis classifier namely granular self organizing map grsom which we apply to an industrial fertilizer modeling application

the spatial temporal lamellar structures in the confined ideal polymer blends we consider the formation of self organized spatial temporal oscillating structures in symmetric binary polymer blends confined by two flat walls an influence of these walls on the formation of the oscillating volume structures is studied this phenomenon is simulated by an initial boundary value problem for the conserved order parameter or the concentration of one of the components in a binary mixture under a special choice the dynamical puri binder s boundary conditions these structures look like the lamellar structures the behavior of the order parameter is described by the modified cahn hilliard equation which models so called the non fickian diffusion in the symmetric binary polymer blends the nonlinear dynamical boundary conditions correspond to the process of adsorption desorption on the walls as a result these nonlinear surface processes induce into the volume the spatial temporal asymptotically periodic structures of relaxation pre turbulent or turbulent type with finite countable or non countable points of discontinuities on the period correspondingly the frequency of oscillations on the period follows a power law for the relaxation type and increases exponentially in the other cases

multi parameter local hardy spaces abstract though multi parameter hardy space theory has been well developed in the past half century not much has been studied for a local hardy space theory in the multi parameter settings such multi parameter local hardy spaces can play an important role in studying the boundedness of multi parameter pseudo differential operators multi parameter singular integrals of non convolution type and applications to partial differential equations etc by establishing a bi parameter local reproducing formula bi parameter local hardy space h p r n 1 r n 2 is introduced in this paper this space coincides with l p r n 1 n 2 when p 1 while p 1 this space is substantially different from the classical bi parameter hardy spaces h p r n 1 r n 2 we will establish its atomic decomposition of the bi parameter local hardy spaces and as an application we prove the boundedness from h p r n 1 r n 2 to l p r n 1 n 2 of the bi parameter singular integral operators in inhomogeneous journe class for p near 1 for the simplicity we have chosen to present all the results in the bi parameter setting nevertheless all of them hold for arbitrary number of parameters the multi parameter local theory developed in this paper can serve as a model case for similar theory in other multi parameter settings

higher order journe commutators and characterizations of multi parameter bmo abstract we characterize l p boundedness of iterated commutators of multiplication by a symbol function and tensor products of riesz and hilbert transforms we obtain a two sided norm estimate that shows that such operators are bounded on l p if and only if the symbol belongs to the appropriate multi parameter bmo class we extend our results to a much more intricate situation commutators of multiplication by a symbol function and paraproduct free journe operators we show that the boundedness of these commutators is also determined by the inclusion of their symbol function in the same multi parameter bmo class in this sense the tensor products of riesz transforms are a representative testing class for journe operators previous results in this direction do not apply to tensor products and only to journe operators which can be reduced to calderon zygmund operators upper norm estimates of journe commutators are new even in the case of no iterations lower norm estimates for iterated commutators only existed when no tensor products were present in the case of one dimension lower estimates were known for products of two hilbert transforms and without iterations new methods using journe operators are developed to obtain these lower norm estimates in the multi parameter real variable setting

fuzzy least squares abstract several models for simple least squares fitting of fuzzy valued data are developed criteria are given for when fuzzy data sets can be fitted to the models and analogues of the normal equations are derived

representation of bi parameter singular integrals by dyadic operators abstract we prove a dyadic representation theorem for bi parameter singular integrals that is we represent certain bi parameter operators as averages of rapidly decaying sums of what we call bi parameter shifts a new version of the product space t 1 theorem is established as a consequence

some obstacles in characterising the boundedness of bi parameter singular integrals the famous t1 theorem for classical calderon zygmund operators is a characterisation for their boundedness in l 2 in the bi parameter case on the other hand the current t1 theorem is merely a collection of sufficient conditions this difference in mind we study a particular dyadic bi parameter singular integral operator namely the full mixed bi parameter paraproductp which is precisely the operator responsible for the outstanding problems in the bi parameter theory we make several remarks about p the common theme of which is to demonstrate the delicacy of the problem of finding a completely satisfactory product t1 theorem for example p need not be unconditionally bounded if it is conditionally bounded a major difference compared to the corresponding one parameter model operators moreover currently the theory even lacks a characterisation for the potentially easier unconditional boundedness the product bmo condition is sufficient but far from necessary we show by example that unconditional boundedness does not even imply the weaker rectangular bmo condition

function approximation fast convergence neural approach based on spectral analysis we propose a constructive approach to building single hidden layer neural networks for nonlinear function approximation using frequency domain analysis we introduce a spectrum based learning procedure that minimizes the difference between the spectrum of the training data and the spectrum of the network s estimates the network is built up incrementally during training and automatically determines the appropriate number of hidden units this technique achieves similar or better approximation with faster convergence times than traditional techniques such as backpropagation

variable selection algorithm for the construction of mimo operating point dependent neurofuzzy networks an input variable selection procedure is introduced for the identification and construction of multi input multi output mimo neurofuzzy operating point dependent models the algorithm is an extension of a forward modified gram schmidt orthogonal least squares procedure for a linear model structure which is modified to accommodate nonlinear system modeling by incorporating piecewise locally linear model fitting the proposed input nodes selection procedure effectively tackles the problem of the curse of dimensionality associated with lattice based modeling algorithms such as radial basis function neurofuzzy networks enabling the resulting neurofuzzy operating point dependent model to be widely applied in control and estimation some numerical examples are given to demonstrate the effectiveness of the proposed construction algorithm

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

sharp lorentz space estimates for rough operators we demonstrate the h 1 l 1 2 or l p l p 2 mapping properties of several rough operators in all cases these estimates are sharp in the sense that the lorentz exponent 2 cannot be replaced by any lower number