a graph theoretic method for determining generating sets of prime ideals in quantum matrices abstract we take a graph theoretic approach to the problem of finding generators for those prime ideals of o q m m n k which are invariant under the torus action k m n launois 2004 15 has shown that the generators consist of certain quantum minors of the matrix of canonical generators of o q m m n k and in launois 2004 14 gives an algorithm to find them in this paper we modify a classic result of lindstrom 1973 17 and gessel and viennot 1985 6 to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex disjoint directed paths in an associated directed graph

symmetric green s functions on complete manifolds on demontre d une facon constructive que l operateur de laplace admet une fonction de green symetrique g x y sur une variete de riemann complete

h reinforcement learning control of robot manipulators using fuzzy wavelet networks in this paper an h reinforcement learning controller based on a fuzzy wavelet network fwn is proposed to perform a position tracking task for a robot manipulator the proposed controller adopts the actor critic reinforcement learning control scheme the primary reinforcement is generated by a performance measurement unit the learning unit of the controller consists of an associative search network asn and an adaptive critic network acn the asn is employed to approximate unknown nonlinear functions in the robot dynamics and the acn is utilized to construct a more informative signal than the primary reinforcement alone to tune the asn since the fwn can provide accurate function approximation both the asn and acn are implemented by the fwn in addition the proposed controller requires no prior knowledge about the dynamics of the robot manipulators and no off line learning phase moreover by employing the h control theory it is possible to attenuate the effects of the approximation errors of the fwns and external disturbances to a prescribed level in contrast to the general h problem only simple equations rather than riccati equations should be solved computer simulations on a scara robot with 3 degrees of freedom confirm the effectiveness of the fwn based controller with h stabilization

double series representation of bounded signals series representations of the form f t approximately sigma sub n infinity sup infinity sigma sub k infinity sup infinity a sub n k nu t n sub e sup 2 pi kts for bounded signals f t are studied as are conditions on the unit function nu t such that coefficients a sub n k reveal the energy content of f t in the time interval n 1 2

functions analytic on the half plane as quantum mechanical states a transform between the state space of one dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half plane is constructed it gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple generalizations to higher dimensions are also discussed

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

linear growth harmonic functions on a complete manifold soit m une variete de riemann a n dimensions a courbure de ricci non negative on suppose que le volume des boules geodesiques de rayon r centrees en un point x m satisfait v b x r o r h alors la dimension de l espace des fonctions harmoniques a croissance lineaire sur m doit etre k

polynomial growth harmonic functions on groups of polynomial volume growth we consider harmonic functions of polynomial growth of some order d on cayley graphs of discrete groups of polynomial volume growth of order d w r t the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions more precisely the dimension of this space of harmonic functions is at most of order d d 1 as in the already known riemannian case this estimate is polynomial in the growth degree more generally our techniques also apply to graphs roughly isometric to cayley graphs of groups of polynomial volume growth

spectre premier de oq mn k image canonique et separation normale abstract given any commutative field k denote r oq mn k the coordinate ring of quantum n n matrices over k and assume q is a nonzero element in k which is not a root of unity recall that r is generated by n2 variables xi i 1 n 2 subject only to the following relations if x y z t is any 2 2 sub matrix of x x i then a yx q 1xy zx q 1xz tz q 1zt ty q 1yt zy yz b tx xt q q 1 yz denote r the k algebra generated by the same variables xi subject to the same relations except relations b which are replaced by c tx xt so that r is just the algebra of regular functions on some quantum affine space of dimension n2 over k the theory of derivative elimination defines a natural embedding spec r spec r and asserts that the canonical image spec r is a union of strata spec w r in the sense of goodearl letzter in cms conf proc vol 22 1998 39 58 where w describes some subset w of p 1 n 2 the sets spec w r 1 spec w r w w define the goodearl letzter h stratification of spec r in the sense of goodearl letzter trans amer math soc 352 2000 1381 1403 in this paper we give the precise description of the set w and we compute its cardinality using that description and the derivative elimination algorithm we can verify theorems 6 3 1 6 3 2 that h spec r has an h normal separation in the sense of goodearl in lecture notes in pure and appl math 210 2000 205 237 so that spec r has normal separation in the sense of brown goodearl trans amer math soc 348 1996 2465 2502 this property was conjectured by k brown and k goodearl since r is auslander regular and cohen macaulay this implies by goodearl lenagan j pure appl algebra 111 1996 123 142 that r is catenary and satisfies the tauvel s height formula

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

primitive ideals of cq sl n abstract the primitive ideals of the quantum group c q sl n are classified in the case where q is a non zero complex number which is not a root of unity it is shown that the orbits in prim c q sl n under the action of the character group h c n 1 are parameterized naturally by w w where w is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals of c q sl n and symplectic leaves of the associated poisson algebraic group sl n c

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

gabor representation of generalized functions this paper deals with gabor representation of generalized functions by this we mean that a given function is to be expanded in a series involving gabor functions that are located in the points of a given lattice in the time frequency plane we shall mainly consider lattices in which the area of the elementary cells of the lattice equals 1 in 1946 gabor used these expansions for the simultaneous analysis of signals in time and frequency cf 181 gabor stated that the above mentioned expansions exist for every reasonable signal and that the coefficients in the expansion are uniquely determined by the signal this statement is true when interpreted carefully and the aim of this paper is to find out what kind of generalized functions are sufficiently well behaved to allow a development in a gabor series we shall also consider the question of the uniqueness of the coefficients in the expansions as well as questions concerning convergence it turns out that the uniqueness questions can be handled with the aid of the main results of io in particular 2 12 and 2 13 the questions on the existence of expansions are harder to answer and require special care depending on the particular function we consider lpfunctions 1 p 2 in detail and show existence of gabor representation for tempered distribution in general

analysis and synthesis of feedforward neural networks using discrete affine wavelet transformations a representation of a class of feedforward neural networks in terms of discrete affine wavelet transforms is developed it is shown that by appropriate grouping of terms feedforward neural networks with sigmoidal activation functions can be viewed as architectures which implement affine wavelet decompositions of mappings it is shown that the wavelet transform formalism provides a mathematical framework within which it is possible to perform both analysis and synthesis of feedforward networks for the purpose of analysis the wavelet formulation characterizes a class of mappings which can be implemented by feedforward networks as well as reveals an exact implementation of a given mapping in this class spatio spectral localization properties of wavelets can be exploited in synthesizing a feedforward network to perform a given approximation task two synthesis procedures based on spatio spectral localization that reduce the training problem to one of convex optimization are outlined

invariant prime ideals in quantizations of nilpotent lie algebras de concini kac and procesi defined a family of subalgebras u of a quantized universal enveloping algebra uq g associated to the elements of the corresponding weyl group w they are deformations of the universal enveloping algebras u n adw n where n are the nilradicals of a pair of dual borel subalgebras based on results of gorelik and joseph and an interpretation of u as quantized algebras of functions on schubert cells we construct explicitly the h invariant prime ideals of each u and show that the corresponding poset is isomorphic to w where h is the group of group like elements of uq g moreover for each h prime of u w we construct a generating set in terms of demazure modules related to fundamental representations using results of ramanathan and kempf we prove similar theorems for vanishing ideals of closures of torus orbits of symplectic leaves of related poisson structures on schubert cells in flag varieties

bayesian wavelet networks for nonparametric regression radial wavelet networks have been proposed previously as a method for nonparametric regression we analyze their performance within a bayesian framework we derive probability distributions over both the dimension of the networks and the network coefficients by placing a prior on the degrees of freedom of the model this process bypasses the need to test or select a finite number of networks during the modeling process predictions are formed by mixing over many models of varying dimension and parameterization we show that the complexity of the models adapts to the complexity of the data and produces good results on a number of benchmark test series

polynomial growth harmonic functions on connected sums of complete riemannian manifolds let m be a connected sum of complete riemannian manifolds satisfying the volume doubling condition and the poincare inequality we prove that the space of polynomial growth harmonic functions on m is finite dimensional whenever m has finitely many ends and satisfies the finite covering condition on each end this result directly generalizes that of tam and it also partially generalizes that of colding and minicozzi ii

les ideaux premiers invariants de oq mm p c abstract for q generic we give a positive answer to a conjecture of goodearl and lenagan the h invariant prime ideals of o q m m p c are generated by quantum minors

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

effacement des derivations et spectres premiers des algebres quantiques abstract given any commutative field k and any iterated ore extension r k x1 x2 2 2 xn n n satisfying some suitable assumptions we construct the so called derivative elimination algorithm it consists of a sequence of changes of variables inside the division ring f fract r starting with the indeterminates x1 xn and terminating with new variables t1 tn these new variables generate some quantum affine space r such that f fract r this algorithm induces a natural embedding spec r spec r which satisfies the following property if p is in spec r then fract r p fract r p we study both the derivative elimination algorithm and natural embedding and use them to produce for the general case a common proof of the quantum gelfand kirillov property for the prime homomorphic images of the following quantum algebras u q w g b q w g w w r q g where g denotes any complex semi simple connected simply connected lie group with associated lie algebra g and weyl group w quantum matrices algebras quantum weyl algebras and quantum euclidean respectively symplectic spaces another application will be given in g cauchon j algebra to appear in the general case the prime spectrum of any quantum matrices algebra satisfies the normal separation property

robust adaptive sliding mode control of condenser cleaning mobile manipulator using fuzzy wavelet neural network this paper presents a robust adaptive sliding mode control rasmc scheme for a class of condenser cleaning mobile manipulator ccmm in the presence of parametric uncertainties and external disturbances the development of control system is based on the fuzzy wavelet neural network fwnn first a dynamic model is obtained in view of the practical ccmm system second the fwnn is used to identify the unstructured system dynamics directly due to its ability to approximate a nonlinear continuous function to arbitrary accuracy using learning ability of neural networks rasmc can coordinately control the condenser cleaning mobile platform and the mounted manipulator with different dynamics efficiently the implementation of the control algorithm is dependent on the adaptive sliding mode control finally based on the lyapunov stability theory the stability of the whole control system the boundedness of the neural networks weight estimation errors and the uniformly ultimately boundedness of the tracking error are all strictly guaranteed moreover simulation results validate the superior control performance of the proposed adaptive control method

nonsingular terminal sliding mode control of robot manipulators using fuzzy wavelet networks this paper presents an adaptive nonsingular terminal sliding mode ntsm tracking control design for robotic systems using fuzzy wavelet networks compared with linear hyperplane based sliding control terminal sliding mode controller can provide faster convergence and higher precision control therefore a terminal sliding controller combined with the fuzzy wavelet network which can accurately approximate unknown dynamics of robotic systems by using an adaptive learning algorithm is an attractive control approach for robots in addition the proposed learning algorithm can on line tune parameters of dilation and translation of fuzzy wavelet basis functions and hidden to output weights therefore a robust control law is used to eliminate uncertainties including the inevitable approximation errors resulted from the finite number of fuzzy wavelet basis functions the proposed controller requires no prior knowledge about the dynamics of the robot and no off line learning phase moreover both tracking performance and stability of the closed loop robotic system can be guaranteed by lyapunov theory finally the effectiveness of the fuzzy wavelet network based control approach is illustrated through comparative simulations on a six link robot manipulator

torus invariant prime ideals in quantum matrices totally nonnegative cells and symplectic leaves the algebra of quantum matrices of a given size supports a rational torus action by automorphisms it follows from work of letzter and the first named author that to understand the prime and primitive spectra of this algebra the first step is to understand the prime ideals that are invariant under the torus action in this paper we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size as a corollary we obtain explicit generating sets of quantum minors for the torus invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over mathbb q

totally nonnegative cells and matrix poisson varieties we describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices that is the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by a postnikov in order to do this we relate the totally nonnegative cells to torus orbits of symplectic leaves of the poisson varieties of complex matrices in particular we describe the minors that vanish on a torus orbit of symplectic leaves we prove that such families of minors are exactly the admissible families and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices

bargmann transform zak transform and coherent states it is well known that completeness properties of sets of coherent states associated with lattices in the phase plane can be proved by using the bargmann representation or by using the kq representation which was introduced by j zak in this paper both methods are considered in particular in connection with expansions of generalized functions in what are called gabor series the setting consists of two spaces of generalized functions tempered distributions and elements of the class s which appear in a natural way in the context of the bargmann transform also a thorough mathematical investigation of the zak transform is given this paper contains many comments and complements on existing literature in particular connections with the theory of interpolation of entire functions over the gaussian integers are given

density function approximation using reduced sufficient statistics for joint estimation of linear and nonlinear parameters a new algorithm is presented for the joint estimation of linear and nonlinear parameters of a deterministic signal embedded in additive gaussian noise the algorithm is an approximation to the reduced sufficient statistics rss method introduced by kulhavy 1990 which estimates the posterior parameter density via minimization of the cross entropy kullback leibler distance in the modified rss algorithm presented the components of the posterior density representing the nonlinear parameter are modeled using haar basis scale functions and the components corresponding to the linear parameters are represented by gaussian densities in the additive gaussian noise measurement model the rss algorithm employs a parallel bank of modified least squares estimators for the linear parameters coupled with a nonlinear estimator for the nonlinear parameters simulation results are presented for the problem of estimating parameters of a chirp signal embedded in multipath and the averaged squared error ase of the parameter estimates is compared with the cramer rao bound finally an application of the algorithm is presented in which the delay multipath coefficients and doppler shift of a digitally modulated waveform received over a fading channel are jointly estimated

winding invariant prime ideals in quantum 3 3 matrices abstract a complete determination of the prime ideals invariant under winding automorphisms in the generic 3 3 quantum matrix algebra o q m 3 k is obtained explicit generating sets consisting of quantum minors are given for all of these primes thus verifying a general conjecture in the 3 3 case the result relies heavily on certain tensor product decompositions for winding invariant prime ideals developed in an accompanying paper in addition new methods are developed here which show that certain sets of quantum minors not previously manageable generate prime ideals in o q m n k

combinatorics of h primes in quantum matrices abstract for q c transcendental over q we give an algorithmic construction of an order isomorphism between the set of h primes of o q m n c and the sub poset s of the reverse bruhat order of the symmetric group s 2 n consisting of those permutations that move any integer by no more than n positions further we describe the permutations that correspond via this bijection to rank t h primes more precisely we establish the following result imagine that there is a barrier between positions n and n 1 then a 2n permutation s corresponds to a rank t h invariant prime ideal of o q m n c if and only if the number of integers that are moved by from the right to the left of this barrier is exactly n t the existence of such an order isomorphism was conjectured by goodearl and lenagan

quotients premiers deoq mn k resume consider a fieldk some nonzero elementqofkwhich is not a root of unity and some nonnegative integern k r goodearl and e s letzter have proved 1991 prime factor algebras of the coordinate ring of quantum matrices preprint that any prime factor algebra of the coordinate ringoq m n k of quantumn nmatrices overkis a domain in that paper it is proved that the division ring of fractions of such a domain is always isomorphic to the division ring of fractions of the coordinate ring of some quantum space of dimension at mostn2over some fieldkwhich is an extension ofk a similar result is also proved for quantum weyl algebras

predictive uncertainty of chaotic daily streamflow using ensemble wavelet networks approach 1 perfect or even mediocre weather predictions over a long period are almost impossible because of the ultimate growth of a small initial error into a significant one even though the sensitivity of initial conditions limits the predictability in chaotic systems an ensemble of prediction from different possible initial conditions and also a prediction algorithm capable of resolving the fine structure of the chaotic attractor can reduce the prediction uncertainty to some extent all of the traditional chaotic prediction methods in hydrology are based on single optimum initial condition local models which can model the sudden divergence of the trajectories with different local functions conceptually global models are ineffective in modeling the highly unstable structure of the chaotic attractor this paper focuses on an ensemble prediction approach by reconstructing the phase space using different combinations of chaotic parameters i e embedding dimension and delay time to quantify the uncertainty in initial conditions the ensemble approach is implemented through a local learning wavelet network model with a global feed forward neural network structure for the phase space prediction of chaotic streamflow series quantification of uncertainties in future predictions are done by creating an ensemble of predictions with wavelet network using a range of plausible embedding dimensions and delay times the ensemble approach is proved to be 50 more efficient than the single prediction for both local approximation and wavelet network approaches the wavelet network approach has proved to be 30 50 more superior to the local approximation approach compared to the traditional local approximation approach with single initial condition the total predictive uncertainty in the streamflow is reduced when modeled with ensemble wavelet networks for different lead times localization property of wavelets utilizing different dilation and translation parameters helps in capturing most of the statistical properties of the observed data the need for taking into account all plausible initial conditions and also bringing together the characteristics of both local and global approaches to model the unstable yet ordered chaotic attractor of a hydrologic series is clearly demonstrated

wave propagation and sampling theory part ii sampling theory and complex waves morlet et al 1982 this issue showed the advantages of using complex values for both waves and characteristics of the media we simulated the theoretical tools we present here using the goupillaud kunetz algorithm now we present sampling methods for complex signals or traces corresponding to received waves and sampling methods for complex characterization of multilayered or heterogeneous media regarding the complex signals we present a twodimecsional 2 d method of sampling in the time frequency domain using a special or extended gabor expansion on a set of basic wavelets adapted to phase preservation such a 2 d expansion permits us to handle in a proper manner instantaneous frequency spectra we show the differences between wavelet resolution and sampling grid resolution we also show the importance of phase preservation in high resolution seismic regarding the media we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution analytical studies of wave propagation in periodic structures are well known in solid state physics and lead to the so called bloch waves the introduction of complex waves leads to replacing the classical wave equation by a schriidinger equation finally we show that complex wave equations gabor expansion and bloch waves are three different ways of introducing the tools of quantum mechanics in highresolution seismic gabor 1946 kittel 1976 morlet 1975 and conversely the goupillaud kunetz algorithm and an extended gabor expansion may be of some use in solid state physics

wavelet transform of multifractals we present the wavelet transform as a mathematical microscope which is well suited for studying the local scaling properties of fractal objects we apply this technique to probability measures on self similar cantor sets and to the golden mean trajectories on two tori at the onset of chaos

a tree structured adaptive network for function approximation in high dimensional spaces nonlinear function approximation is often solved by finding a set of coefficients for a finite number of fixed nonlinear basis functions however if the input data are drawn from a high dimensional space the number of required basis functions grows exponentially with dimension leading many to suggest the use of adaptive nonlinear basis functions whose parameters can be determined by iterative methods the author proposes a technique based on the idea that for most of the data only a few dimensions of the input may be necessary to compute the desired output function additional input dimensions are incorporated only where needed the learning procedure grows a tree whose structure depends upon the input data and the function to be approximated this technique has a fast learning algorithm with no local minima once the network shape is fixed and it can be used to reduce the number of required measurements in situations where there is a cost associated with sensing three examples are given controlling the dynamics of a simulated planar two joint robot arm predicting the dynamics of the chaotic mackey glass equation and predicting pixel values in real images from pixel values above and to the left

adaptive fuzzy wavelet network control design for nonlinear systems this paper presents a new adaptive fuzzy wavelet network controller a fwnc for control of nonlinear affine systems inspired by the theory of multiresolution analysis mra of wavelet transforms and fuzzy concepts the proposed adaptive gain controller which results from the direct adaptive approach has the ability to tune the adaptation parameter in the then part of each fuzzy rule during real time operation each fuzzy rule corresponds to a sub wavelet neural network sub wnn and one adaptation parameter each sub wnn consists of wavelets with a specified dilation value the degree of contribution of each sub wnn can be controlled flexibly orthogonal least square ols method is used to determine the number of fuzzy rules and to purify the wavelets for each sub wnn since the efficient procedure of selecting wavelets used in the ols method is not very sensitive to the input dimension the dimension of the approximated function does not cause the bottleneck for constructing fwn fwn is constructed based on the training data set of the nominal system and the constructed fuzzy rules can be adjusted by learning the translation parameters of the selected wavelets and also determining the shape of membership functions then the constructed adaptive fwn controller is employed such that the feedback linearization control input can be best approximated and the closed loop stability is guaranteed the performance of the proposed a fwnc is illustrated by applying a second order nonlinear inverted pendulum system and compared with previously published methods simulation results indicate the remarkable capabilities of the proposed control algorithm it is worth noting that the proposed controller significantly improves the transient response characteristics and the number of fuzzy rules and on line adjustable parameters are reduced

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

multidimensional wavelet frames pati and krishnaprasad 1990 first studied the connection between neural networks and wavelet transforms zhang and benveniste 1992 gave a different treatment of this connection however the problem of constructing multidimensional wavelet frames for use in neural networks has not been satisfactorily studied in this paper one dimensional wavelet frame is generalized to the multidimensional case by using single scaling and multiscaling parameters the construction of multidimensional wavelet frames is also discussed these results provide more insight on the use of wavelets in neural networks

ondelettes and phase cell cluster expansions a vindication y meyer has recently developed a particularly useful o n basis forl2 r d expansions using this basis i e expansions into ondelettes or wavelets have yielded important new results in soft and hard analysis the expansion into ondelettes of a boson scalar field naturally leads to phase cell cluster expansions a formalism already developed by the authors using other related bases adoption of ondelettes expansions into the phase cell program gives improvements of some extant results and excises an early error

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

frames in the bargmann space of entire functions we look at the decomposition of arbitrary f in l2 r in terms of the family of functions cp x w exp iimnab iman i x nb 2 with a b 0 we derive bounds and explicit formulas for the minimal expansion coefficients in the case where ab 2w n n an integer 2 2 transported to the hilbert space f of entire functions introduced by v bargmann these results are expressed as inequalities of the form

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

generalized haar dwt and transformations between decision trees and neural networks the core contribution of this paper is a three fold improvement of the haar discrete wavelet transform dwt it is modified to efficiently transform a multiclass rather than numerical valued function over a multidimensional rather than low dimensional domain or transform a multiclass valued decision tree into another useful representation we prove that this multidimensional multiclass dwt uses dynamic programming to minimize within its framework the number of nontrivial wavelet coefficients needed to summarize a training set or decision tree it is a spatially localized algorithm that takes linear time in the number of training samples after a sort convergence of the dwt to benchmark training sets seems to degrade with rising dimension in this test of high dimensional wavelets which have been seen as difficult to implement this multiclass multidimensional dwt has tightly coupled applications from learning dyadic decision trees directly from training data rebalancing or converting preexisting decision trees to fixed depth boolean or threshold neural networks in effect parallelizing the evaluation of the trees or learning rule exception sets represented as a new form of tree called an e tree which could greatly help interpretation visualization of a dataset

generalized gabor expansions of discrete time signals in l sup 2 z via biorthogonal like sequences in this paper a biorthogonal like sequences bls theory and its application to the generalized gabor expansions equivalently the generalized short time fourier transform filterbank summation are presented a pair of bls s are defined to be two sequences satisfying a biorthogonal like condition blc which is a moment equation and equivalent to a linear difference equation we show that two collections in a hilbert space generated by a pair of bls s in the joint time frequency domain are complete either can be used as an analysis filter and the other can be used as a synthesis filter for a generalized gabor expansion of discrete time signals a sufficient and necessary condition on the existence of bls s based on the moment equation is presented which is simpler to use than frame theory given a filter generating a frame its bls s also generate frames the dual frame is one of them given a fir analysis synthesis filter there is a fir synthesis analysis filter if bls s exist the algorithm to compute fir analysis and synthesis filters based on the linear difference equation is presented in this paper which is simpler than frame operator

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

lower bounds for multivariate approximation by affine invariant dictionaries the problem of approximating locally smooth multivariate functions by linear combinations of elements from an affine invariant redundant dictionary is considered augmenting previous upper bound results for approximation we establish lower bounds on the performance of such schemes the lower bounds are tight to within a logarithmic factor in the number of elements used in the approximation using a previously introduced notion of nonlinear approximation we show that the approximation ability may be completely characterized by the pseudodimension of the approximation space with respect to a finite set of points this result establishes a useful link between the problems of approximation and estimation or learning the latter often being conveniently characterized at least in terms of upper bounds by the pseudodimension

painless nonorthogonal expansions in a hilbert space h discrete families of vectors hj with the property that f j hj f hj for every f in h are considered this expansion formula is obviously true if the family is an orthonormal basis of h but also can hold in situations where the hj are not mutually orthogonal and are overcomplete the two classes of examples studied here are i appropriate sets of weyl heisenberg coherent states based on certain non gaussian fiducial vectors and ii analogous families of affine coherent states it is believed that such quasiorthogonal expansions will be a useful tool in many areas of theoretical physics and applied mathematics

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed