quantum function algebra at roots of 1 abstract we introduce a form of the quantum function algebra on a drinfeld jimbo quantum group over the ring z q q 1 specializing q to a root of 1 we show that over the cyclotomic field this algebra is a projective module over its central sub algebra which is the usual coordinate algebra of the group we study the induced poisson lie structure of the group a bundle of algebras on a complex simply connected lie group with hamiltonian flows in the bundle is constructed some representations of the quantum function algebra in a root of 1 are constructed as an application an estimate of the dimension of an arbitrary representation is given

double partition quantum cluster algebras abstract a family of quantum cluster algebras is introduced and studied in general these algebras are new but sub classes have been studied previously by other authors the algebras are indexed by double partitions or double flag varieties equivalently they are indexed by broken lines l by grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of one broken line to another compatible pairs can be written down the algebras are equal to their upper cluster algebras the variables of the quantum seeds are given by elements of the dual canonical basis

wavelet analysis on the circle the construction of a wavelet analysis over the circle is presented the spaces of infinitely times differentiable functions tempered distributions and square integrable functions over the circle are analyzed by means of the wavelet transform

characterizations of function spaces on the sphere using frames in this paper we introduce a polynomial frame on the unit sphere s d 1 of r d for which every distribution has a wavelet type decomposition more importantly we prove that many function spaces on the sphere s d 1 such as l p h p and besov spaces can be characterized in terms of the coefficients in the wavelet decompositions as in the usual euclidean case r d we also study a related nonlinear m term approximation problem on s d 1 in particular we prove both a jackson type inequality and a bernstein type inequality associated to wavelet decompositions which extend the corresponding results obtained by r a devore b jawerth and v popov compression of wavelet decompositions amer j math 114 1992 no 4 737 785

exact wavelets on the ball we develop an exact wavelet transform on the three dimensional ball i e on the solid sphere which we name the flaglet transform for this purpose we first construct an exact transform on the radial half line using damped laguerre polynomials and develop a corresponding quadrature rule combined with the spherical harmonic transform this approach leads to a sampling theorem on the ball and a novel three dimensional decomposition which we call the fourier laguerre transform we relate this new transform to the well known fourier bessel decomposition and show that band limitedness in the fourier laguerre basis is a sufficient condition to compute the fourier bessel decomposition exactly we then construct the flaglet transform on the ball through a harmonic tiling which is exact thanks to the exactness of the fourier laguerre transform from which the name flaglets is coined the corresponding wavelet kernels are well localised in real and fourier laguerre spaces and their angular aperture is invariant under radial translation we introduce a multiresolution algorithm to perform the flaglet transform rapidly while capturing all information at each wavelet scale in the minimal number of samples on the ball our implementation of these new tools achieves floating point precision and is made publicly available we perform numerical experiments demonstrating the speed and accuracy of these libraries and illustrate their capabilities on a simple denoising example

wavelet based segmentation on the sphere abstract segmentation a useful powerful technique in pattern recognition is the process of identifying object outlines within images there are a number of efficient algorithms for segmentation in euclidean space that depend on the variational approach and partial differential equation modelling wavelets have been used successfully in various problems in image processing including segmentation inpainting noise removal super resolution image restoration and many others wavelets on the sphere have been developed to solve such problems for data defined on the sphere which arise in numerous fields such as cosmology and geophysics in this work we propose a wavelet based method to segment images on the sphere accounting for the underlying geometry of spherical data our method is a direct extension of the tight frame based segmentation method used to automatically identify tube like structures such as blood vessels in medical imaging it is compatible with any arbitrary type of wavelet frame defined on the sphere such as axisymmetric wavelets directional wavelets curvelets and hybrid wavelet constructions such an approach allows the desirable properties of wavelets to be naturally inherited in the segmentation process in particular directional wavelets and curvelets which were designed to efficiently capture directional signal content provide additional advantages in segmenting images containing prominent directional and curvilinear features we present several numerical experiments applying our wavelet based segmentation method as well as the common k means method on real world spherical images including an earth topographic map a light probe image solar data sets and spherical retina images these experiments demonstrate the superiority of our method and show that it is capable of segmenting different kinds of spherical images including those with prominent directional features moreover our algorithm is efficient with convergence usually within a few iterations

separable projection integrals for higher order correlators of the cosmic microwave sky we present a case study describing efforts to optimise and modernise modal the simulation and analysis pipeline used by the planck satellite experiment for constraining general non gaussian models of the early universe via the bispectrum or three point correlator of the cosmic microwave background radiation we focus on one particular element of the code the projection of bispectra from the end of inflation to the spherical shell at decoupling which defines the cmb we observe today this code involves a three dimensional inner product between two functions one of which requires an integral on a non rectangular domain containing a sparse grid we show that by employing separable methods this calculation can be reduced to a one dimensional summation plus two integrations reducing the overall dimensionality from four to three the introduction of separable functions also solves the issue of the non rectangular sparse grid this separable method can become unstable in certain scenarios and so the slower non separable integral must be calculated instead we present a discussion of the optimisation of both approaches we demonstrate significant speed ups of 100 arising from a combination of algorithmic improvements and architecture aware optimisations targeted at improving thread and vectorisation behaviour the resulting mpi openmp hybrid code is capable of executing on clusters containing processors and or coprocessors with strong scaling efficiency of 98 6 on up to 16 nodes we find that a single coprocessor outperforms two processor sockets by a factor of 1 3 and that running the same code across a combination of both microarchitectures improves performance per node by a factor of 3 38 by making bispectrum calculations competitive with those for the power spectrum or two point correlator we are now able to consider joint analysis for cosmological science exploitation of new data

spin weighted spherical wavelets derived from approximate identities abstract spin weighted functions introduced in 1966 by newman and penrose are used primarily for description of the cosmic microwave background radiation in the present paper an analysis tool for spin weighted functions is proposed namely spin weighted spherical wavelets based on approximate identities in the spin 0 case there exist two different definitions of continuous wavelet transforms the group theoretical one introduced by antoine et al and that based on approximate identities pioneered by freeden et al the first one has been recently extended to the case of spin weighted functions by mcewen et al and the present research is the approximate identities counterpart of their work wavelets presented here can be constructed for and the wavelet transform can be applied to functions of each integer spin weight further the set of parameters scale and position is continuous a big advantage of the construction is that the wavelet transform is invertible directly by an integral last but not least wavelets can reveal directional features such that the directional intensity of a signal can be probed

stable adaptive regulation of arbitrary nth order plants this paper presents an algorithm for adaptively stabilizing and asymptotically regulating an arbitrary single input single output linear time invariant plant which is controllable and observable of known order n and has unknown parameters no further assumptions are made no external probing signal is required

admissible wavelets associated with the affine automorphism group of the siegel upper half plane abstract letp nambe the minimal parabolic subgroup ofsu n 1 1 which can be regarded as the affine automorphism group of the siegel upper half planeun 1 palso acts on the heisenberg grouphn the boundary ofun 1 thereforephas a natural representationuonl2 hn we decomposel2 hn into the direct sum of the irreducible invariant closed subspaces underu the restrictions ofuon these subspaces are square integrable we give the characterization of the admissible condition in terms of the fourier transform and define the wavelet transform with respect to admissible wavelets the wavelet transform gives isometric operators from the irreducible invariant closed subspaces ofl2 hn tol2 p

identifier tracking model reference adaptive control model reference adaptive control is a major design method for controlling plants with uncertain parameters the primary objective of this paper is to develop a new design approach for a differentiator free model reference adaptive control of a single input single output linear time invariant plant the proposed method called the identifier tracking model reference adaptive control uses a stacked identifier structure that is new to the field of adaptive control the goal is to make the output of the plant asymptotically track the output of the first identifier and then driving the output of the first identifier to track that of the second identifier and so forth up to the qth identifier where q is the relative degree of the plant lastly the output of the qth identifier is forced to converge to that of the reference model simulation results show the superiority of the proposed method over the traditional model reference adaptive control with augmented error in terms of the transient response sinc

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

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

quantized rank r matrices abstract first some old as well as new results about p i algebras ore extensions and degrees are presented then quantized n r matrices as well as certain quantized factor algebras m r 1 q n of m q n are analyzed for r 1 n 1 m r 1 q n is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all r 1 r 1 quantum subdeterminants and a certain localization of this algebra is proved to be isomorphic to a more manageable one in almost all cases the quantum parameter is a primitive m th root of unity the degrees and centers of the algebras are determined when m is a prime and the general structure is obtained for arbitrary m

a phase sensitive approach to filtering on the sphere this paper examines filtering on a sphere by first examining the roles of spherical harmonic magnitude and phase we show that phase is more important than magnitude in determining the structure of a spherical function we examine the properties of linear phase shifts in the spherical harmonic domain which suggest a mechanism for constructing finite impulse response fir filters we show that those filters have desirable properties such as being associative mapping spherical functions to spherical functions allowing directional filtering and being defined by relatively simple equations we provide examples of the filters for both spherical and manifold data

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

continuous wavelet transforms on the sphere in this very short paper we shall construct a continuous wavelet analysis based on dilations translations and rotations on the sphere it is the analog of the construction proposed by murenzi in his thesis 1990 on r2 at small scale we shall recover the euclidian structure of the sphere at large scale we obtain that the wavelet transform decays rapidly because the sphere is compact

the center of the quantized matrix algebra abstract we compute the center in the case where q is a root of unity the main steps are to compute the degree of an associated quasipolynomial algebra and to compute the dimensions of some interesting irreducible modules

biorthogonal radial multiresolution in dimension three in this paper we present the definition and the relative theorems of the biorthogonal radial multiresolution in dimension three unlike the orthogonal case there exist real valued dual radial scaling functions with compact support in the biorthogonal case the associated mallat algorithm can be simply performed in terms of classical biorthogonal filters

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

wavelets on the n sphere and related manifolds we present a purely group theoretical derivation of the continuous wavelet transform cwt on the n 1 sphere sn 1 based on the construction of general coherent states associated to square integrable group representations the parameter space of the cwt x similar to so n xr is embedded into the generalized lorentz group so0 n 1 via the iwasawa decomposition so that x similar or equal to so0 n 1 in where n similar or equal to rn 1 then the cwt on sn 1 is derived from a suitable unitary representation of so0 n 1 acting in the space l 2 sn 1 d mu of finite energy signals on sn 1 which turns out to be square integrable over x we find a necessary condition for the admissibility of a wavelet in the form of a zero mean condition which entails all the usual filtering properties of the cwt next the euclidean limit of this cwt on sn 1 is obtained by redoing the construction on a sphere of radius r and performing a group contraction for r infinity from which one recovers the usual cwt on flat euclidean space finally we discuss the extension of this construction to the two sheeted hyperboloid hn 1so0 n 1 1 so n 1 and some other riemannian symmetric spaces c 1998 american institute of physics s0022 2488 98 00308 9

pi degree parity in q skew polynomial rings abstract for k a field of arbitrary characteristic and r a k algebra we show that the pi degree of an iterated skew polynomial ring r x 1 1 1 x n n n agrees with the pi degree of r x 1 1 x n n when each i i satisfies a q i skew relation for q i k and extends to a higher q i skew i derivation we confirm the quantum gel fand kirillov conjecture for various quantized coordinate rings and calculate their pi degrees we extend these results to completely prime factor algebras

eigenfunctions of non integrable systems in generalised phase spaces the authors use the generalised coherent state distribution to study in phase space the eigenfunctions of non integrable hamiltonians the methods available in the weyl group the husimi distribution are extended to the more complicated phase spaces that arise from other lie groups they demonstrate the feasibility of numerical calculations in two models with w2 and su 3 symmetries emphasising the similarities and showing the emergence of classical invariant structures in the semiclassical limit

on contractions of semisimple lie groups a limiting formula is given for the representation theory of the cartan motion group associated to a riemannian symmetric pair g k in terms of the representation theory of g introduction let g be a connected lie group with lie algebra g and h a closed subgroup with subalgebra b the coset space g h is called reductive 9 p 389 if h admits an adg h invariant complement m in g i e a subspace m c g such that i g b m direct sum ii adg h m c m in this case we can form the semidirect product m a h with respect to the adjoint action of h on m in this paper we shall restrict ourselves to the case where g is semisimple with finite centre and g h is a riemannian symmetric pair 10 p 209 hence h is contained in the fixed point set ha of an analytic involution a of g it contains the identity component ha e and adc h is compact following custom we write k and f rather than h and b in this instance it is well known that k is compact 10 p 252 and connected if g is noncompact furthermore f is the 1 eigenspace of doe we make the natural choice for m namely the 1 eigenspace v of dae when g is noncompact dae is a cartan involution then v is usually denoted p and g f is called a cartan decomposition 10 p 182 when g is compact one can choose a real form g0 of the complexification gc of g and a cartan

fast directional continuous spherical wavelet transform algorithms we describe the construction of a spherical wavelet analysis through the inverse stereographic projection of the euclidean planar wavelet framework introduced originally by antoine and vandergheynst and developed further by wiaux fast algorithms for performing the directional continuous wavelet analysis on the unit sphere are presented the fast directional algorithm based on the fast spherical convolution algorithm developed by wandelt and goacuterski provides a savings of o radicnpix over a direct quadrature implementation for npix pixels on the sphere and allows one to perform a directional spherical wavelet analysis of a 106 pixel map on a personal computer

an optimal dimensionality sampling scheme on the sphere with fast spherical harmonic transforms we develop a sampling scheme on the sphere that permits accurate computation of the spherical harmonic transform and its inverse for signals band limited at l using only l 2 samples we obtain the optimal number of samples given by the degrees of freedom of the signal in harmonic space the number of samples required in our scheme is a factor of two or four fewer than existing techniques which require either 2l 2 or 4l 2 samples we note however that we do not recover a sampling theorem on the sphere where spherical harmonic transforms are theoretically exact nevertheless we achieve high accuracy even for very large band limits for our optimal dimensionality sampling scheme we develop a fast and accurate algorithm to compute the spherical harmonic transform and inverse with computational complexity comparable with existing schemes in practice we conduct numerical experiments to study in detail the stability accuracy and computational complexity of the proposed transforms we also highlight the advantages of the proposed sampling scheme and associated transforms in the context of potential applications

phase space localization theorem for ondelettes it is proven that an orthonormal basis of ondelettes can never have exponential localization in both position space and momentum space

second generation curvelets on the sphere curvelets are efficient to represent highly anisotropic signal content such as a local linear and curvilinear structure first generation curvelets on the sphere however suffered from blocking artefacts we present a new second generation curvelet transform where scale discretized curvelets are constructed directly on the sphere scale discretized curvelets exhibit a parabolic scaling relation are well localized in both spatial and harmonic domains support the exact analysis and synthesis of both scalar and spin signals and are free of blocking artefacts we present fast algorithms to compute the exact curvelet transform reducing computational complexity from mathcal o l 5 to mathcal o l 3 log 2 l for signals band limited at l the implementation of these algorithms is made publicly available finally we present an illustrative application demonstrating the effectiveness of curvelets for representing directional curve like features in natural spherical images

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

continuous wavelets on compact manifolds let m be a smooth compact oriented riemannian manifold and let m be the laplace beltrami operator on m say 0 neq f in mathcal s mathbb r and that f 0 0 for t 0 let kt x y denote the kernel of f t2 m we show that kt is well localized near the diagonal in the sense that it satisfies estimates akin to those satisfied by the kernel of the convolution operator f t2 on mathbb r n we define continuous mathcal s wavelets on m in such a manner that kt x y satisfies this definition because of its localization near the diagonal continuous mathcal s wavelets on m are analogous to continuous wavelets on mathbb r n in mathcal s mathbb r n in particular we are able to characterize the holder continuous functions on m by the size of their continuous mathcal s wavelet transforms for holder exponents strictly between 0 and 1 if m is the torus mathbb t 2 or the sphere s2 and f s se s the mexican hat situation we obtain two explicit approximate formulas for kt one to be used when t is large and one to be used when t is small

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

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

a combined direct indirect and variable structure method for robust adaptive control direct methods and indirect methods for the adaptive control of linear plants with unknown parameters are currently well known methods for combining direct and indirect methods have also been suggested in the paper the latter method is combined with the variable structure method new adaptive laws are suggested which overcome the principal drawbacks of the variable structure method while the direct and indirect components of the new method assure stability the variable structure component improves the transient response of the system simulation results are presented to show that the overall system is asymptotically stable in the ideal case and exhibits robustness and improved performance in the presence of perturbations

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

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

optimal filters on the sphere we derive optimal filters on the sphere in the context of detecting compact objects embedded in a stochastic background process the matched filter and the scale adaptive filter are derived on the sphere in the most general setting allowing for directional template profiles and filters the performance and relative merits of the two optimal filters are discussed the application of optimal filter theory on the sphere to the detection of compact objects is demonstrated on simulated data a naive detection strategy is adopted with an initial aim of illustrating the application of the new filter theory derived on the sphere nevertheless this simple object detection strategy is demonstrated to perform well even at low signal to noise ratio

nearly tight frames and space frequency analysis on compact manifolds let m be a smooth compact oriented riemannian manifold of dimension n without boundary and let be the laplace beltrami operator on m say 0 neq f in mathcal s mathbb r and that f 0 0 for t 0 let kt x y denote the kernel of f t2 suppose f satisfies daubechies criterion and b 0 for each j write m as a disjoint union of measurable sets ej k with diameter at most baj and measure comparable to ba j n if baj is sufficiently small take xj k ej k we then show that the functions phi j k x mu e j k 1 2 overline k a j x j k x form a frame for i p l2 m for b sufficiently small here p is the projection onto the constant functions moreover we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1 so that the frame quickly becomes nearly tight for b sufficiently small moreover based upon how well localized a function f i p l2 is in space and in frequency we can describe which terms in the summation f sim sf sum j sum k langle f phi j k rangle phi j k are so small that they can be neglected if n 2 and m is the torus or the sphere and f s se s the mexican hat situation we obtain two explicit approximate formulas for the j k one to be used when t is large and one to be used when t is small

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