analysis and design of minimax optimal interpolators we consider a class of interpolation algorithms including the least squares optimal yen 1956 interpolator and we derive a closed form expression for the interpolation error for interpolators of this type the error depends on the eigenvalue distribution of a matrix that is specified for each set of sampling points the error expression can be used to prove that the yen interpolator is optimal the implementation of the yen algorithm suffers from numerical ill conditioning forcing the use of a regularized approximate solution we suggest a new approximate solution consisting of a sinc kernel interpolator with specially chosen weighting coefficients the newly designed sinc kernel interpolator is compared with the usual sinc interpolator using jacobian area weighting through numerical simulations we show that the sinc interpolator with jacobian weighting works well only when the sampling is nearly uniform the newly designed sinc kernel interpolator is shown to perform better than the sinc interpolator with jacobian weighting

null strings and bianchi identities the results of the theory of complexified v4 s which admit a null string are reexamined by using systematically the bianchi identities the formal reasons which permit the integration of the g 0 equations to one differential constraint in the case of the hh structures are exhibited some new results concerning complexified einstein weyl equations in the presence of a null string are given as well

an optimal recovery approach to interpolation a filter class is defined as a ball in an inner product space and some standard results on inner product spaces are applied to filter classes the filter design problem is addressed the theory of optimal recovery is reviewed and the interpolation problem is examined within an optimal recovery context it is shown that the interpolation problem can be reduced to studying a hypercircle in an inner product space the notion of the chebyshev center of a set is introduced and it is noted that the solution to the interpolation problem from the optimal recovery viewpoint is to find the chebyshev center of the hypercircle the interpolation filter is then the operator that transforms the vector of known samples into the center of the hypercircle some auxiliary results such as the linearity and time invariance of the interpolation filter are deduced it is then shown that the estimation of an unknown sample is the same as the problem of approximating the representer of the unknown sample by a linear combination of the representers of the known samples hence the interpolation problem is equivalent to minimizing the l sup 2 norm of the error frequency response from the filter design point of view

maximum likelihood reconstruction for magnetic resonance fingerprinting this paper introduces a statistical estimation framework for magnetic resonance mr fingerprinting a recently proposed quantitative imaging paradigm within this framework we present a maximum likelihood ml formalism to estimate multiple mr tissue parameter maps directly from highly undersampled noisy bf k space data a novel algorithm based on variable splitting the alternating direction method of multipliers and the variable projection method is developed to solve the resulting optimization problem representative results from both simulations and in vivo experiments demonstrate that the proposed approach yields significantly improved accuracy in parameter estimation compared to the conventional mr fingerprinting reconstruction moreover the proposed framework provides new theoretical insights into the conventional approach we show analytically that the conventional approach is an approximation to the ml reconstruction more precisely it is exactly equivalent to the first iteration of the proposed algorithm for the ml reconstruction provided that a gridding reconstruction is used as an initialization

a fast sinc function gridding algorithm for fourier inversion in computer tomography the fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the cartesian grid required for fast numerical fourier inversion although the fourier inversion method is recognized as being computationally faster than the back projection method for parallel ray projection data the artifacts resulting from inaccurate interpolation have generally limited application of the method this paper presents a computationally efficient gridding algorithm which can be used with direct fourier transformation to achieve arbitrarily small artifact levels the method has potential for application to other measurement geometries such as fan beam projections and diffraction tomography and nmr imaging

representation type of hecke algebras of type a in this paper we provide a complete classification of the representation type for the blocks for the hecke algebra of type a stated in terms of combinatorical data the computation of the complexity of young modules is a key component in the proof of this classification result

some hall polynomials for representation finite trivial extension algebras letkbe a finite field and assume that is a finite dimensional associativek algebra with 1 denote by mod the category of all finitely generated right modules and by ind the full subcategory in which every object is a representative of the isoclass of an indecomposable right module we are interested in the existance of the hall polynomial mnlfor anl m n mod for the definition see977113or section 1 below in case is directed 7has shown that has hall polynomials and in case is cyclic serial the same result has also been obtained by4 it has been conjectured in8that any representation finitek algebra has hall polynomials in this investigation we shall show that if is a representation finite trivial extension algebra then for anyl m n mod withnindecomposable has the hall polynomials mlnand mnl using these hall polynomials we can naturally structure the free abelian group with a basis ind denoted byk mod into a lie algebra and the universal enveloping algebra ofk mod z qis just h 1 z q where h 1is the degenerated hall algebra of see section 5 below

on the cohomology of specht modules we investigate the cohomology of the specht module s for the symmetric group d we show if 0 i p 2 then hi d s is isomorphic to hs i b w0 where s d d 1 2 b is the borel subgroup of the algebraic group gld k and 1d is the weight of the determinant representation we obtain similar isomorphisms of ext di s m with b cohomology which in turn yield isomorphisms of cohomology for borel subgroups of gln k for varying n d in the case i 0 and the case i 1 for certain we apply our result and known symmetric group results of james and erdmann to obtain new information about b cohomology finally we show that specht module cohomology is closely related to cohomology for the frobenius kernel b1 for small primes

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

killing spinors and separability of rarita schwinger s equation in type 2 2 backgrounds it is shown that in a type 2 2 solution of the einstein vacuum field equations which admits a two index killing spinor a differential operator can be constructed that maps a solution of the rarita schwinger equation into a solution of its complex conjugate furthermore by considering the plebanski demianski metric which contains all the vacuum type 2 2 metrics it is shown that the separable solutions of the rarita schwinger equation are eigenfunctions of a certain differential operator with the starobinsky constant as the eigenvalue

killing vectors in algebraically special space times the form of the isometric homothetic and conformal killing vectors for algebraically special metrics which admit a shear free congruence of null geodesics is obtained by considering their complexification using the existence of a congruence of null strings the killing equations are partially integrated and the reasons which permit this reduction are exhibited in the case where the congruence of null strings has a vanishing expansion the killing equations are reduced to a single master equation

hecke algebras of classical type and their representation type the purpose of this article is to determine the representation type for all of the hecke algebras of classical type to do this we combine methods from our previous work which is used to obtain information on their gabriel quivers and recent advances in the theory of finite dimensional algebras the principal computation is for hecke algebras of type b with two parameters then we show that the representation type of hecke algebras is governed by their poincare polynomials

representation type of schur algebras we give a complete classification of the classical schur algebras and the infinitesimal schur algebras which have tame representation type in combination with earlier work of some of the authors on semisimplicity and finiteness this completes the classification of representation type of all classical and infinitesimal schur algebras in all characteristics

nonuniform interpolation of noisy signals using support vector machines the problem of signal interpolation has been intensively studied in the information theory literature in conditions such as unlimited band nonuniform sampling and presence of noise during the last decade support vector machines svm have been widely used for approximation problems including function and signal interpolation however the signal structure has not always been taken into account in svm interpolation we propose the statement of two novel svm algorithms for signal interpolation specifically the primal and the dual signal model based algorithms shift invariant mercer s kernels are used as building blocks according to the requirement of bandlimited signal the sine kernel which has received little attention in the svm literature is used for bandlimited reconstruction well known properties of general svm algorithms sparseness of the solution robustness and regularization are explored with simulation examples yielding improved results with respect to standard algorithms and revealing good characteristics in nonuniform interpolation of noisy signals

double kerr schild equivalence and hyperheavens double kerr schild equivalence of hyperheavens is studied within a spinorial formalism based on the fact of the existence of a congruence of self dual null strings in particular it is shown that a given hyperheavenly metric structure generates all members of the corresponding equivalence class in terms of a certain spinor field and the generalized key function subject to the generalized hyperheavenly equation the treatment is manifestly covariant no coordinates are employed finally a link with the classical results on hyperheavens plebanski and robinson j f plebanski and i robinson phys rev lett 37 493 1976 and finley and plebanski j d finley iii and j f plebanski j math phys 17 2207 1976 is established

the schur algebra we study a class of endomomorphism algebras of certain q permutation modules over the hecke algebra of type b whose summands involve both parabolic and quasi parabolic subgroups and prove that these algebras are integrally free and quasi hereditary and are stable under base change some consequences for decomposition numbers are discussed the notion of a q schur algebra was introduced by dipper and james dj2 who used these algebras to parametrize the irreducible representations of the finite general linear groups in non describing characteristics with hindsight these algebras had already appeared earlier in an entirely different quantum group context ji inspired by physics in ji jimbo considered the endomorphism algebras of tensor spaces as hecke algebra modules in his context a q schur algebra can be viewed as a quotient of the quantized enveloping algebra associated to gln in pw these algebras were shown to be quasi hereditary the quasi heredity property is an embodiment in classical algebra of the geometric derived category stratification exhibited by perverse sheaves ps it means more applications can be deduced from a ring theoretic point of view see e g dps3 and the possibility is raised of even deeper results in the future as suggested by cps2 certainly these algebras play a central role in the representation theories of the finite and quantum general linear groups naturally one asks are there such algebras for types other than a our paper ds showed that there were similar quasi hereditary quotients of quantized enveloping algebras for all types of root systems however no connection with hecke algebras and finite groups of lie type was found there this paper aims at the same question and constructs possible algebras directly from hecke algebras hence from finite groups of lie type we restrict attention to the type b case imitating the definition of a q schur algebra we introduce the notion of a q schur algebra these algebras are the endomorphism algebras of certain modules over the hecke algebra of type b called tensor spaces whose summands involve not only parabolic subgroups but also quasi parabolic subgroups of the weyl group of type b a main result of our paper shows that qschur algebras are quasi hereditary we speculate that similar constructions exist for other classical types only type d remains actually since type c is equivalent received by the editors march 3 1997 and in revised form october 28 1998 2000 mathematics subject classification primary 20c08 20g05 20c33 the authors would like to thank arc for support under the large grant a69530243 as well as nsf and the universities of virginia and new south wales for their cooperation the first author also thanks the newton institute at cambridge for its hospitality c 2000 american mathematical society

on the optimality of the gridding reconstruction algorithm gridding reconstruction is a method to reconstruct data onto a cartesian grid from a set of nonuniformly sampled measurements this method is appreciated for being robust and computationally fast however it lacks solid analysis and design tools to quantify or minimize the reconstruction error least squares reconstruction lsr on the other hand is another method which is optimal in the sense that it minimizes the reconstruction error this method is computationally intensive and in many cases sensitive to measurement noise hence it is rarely used in practice despite their seemingly different approaches the gridding and lsr methods are shown to be closely related the similarity between these two methods is accentuated when they are properly expressed in a common matrix form it is shown that the gridding algorithm can be considered an approximation to the least squares method the optimal gridding parameters are defined as the ones which yield the minimum approximation error these parameters are calculated by minimizing the norm of an approximation error matrix this problem is studied and solved in the general form of approximation using linearly structured matrices this method not only supports more general forms of the gridding algorithm it can also be used to accelerate the reconstruction techniques from incomplete data the application of this method to a case of two dimensional 2 d spiral magnetic resonance imaging shows a reduction of more than 4 db in the average reconstruction error

representation type of the blocks of category os abstract in this paper the authors investigate the representation type of the blocks of the relative parabolic category o s for complex semisimple lie algebras a complete classification of the blocks corresponding to regular weights is given the main results of the paper provide a classification of the blocks in the mixed case when the simple roots corresponding to the singular set and s do not meet

cohomology and generic cohomology of specht modules for the symmetric group cohomology of specht modules for the symmetric group can be equated in low degrees with corresponding cohomology for the borel subgroup b of the general linear group gld k but this has never been exploited to prove new symmetric group results using work of doty on the submodule structure of symmetric powers of the natural gld k module together with work of andersen on cohomology for b and its frobenius kernels we prove new results about hi d s we recover work of james in the case i 0 then we prove two stability theorems one of which is a generic cohomology result for specht modules equating cohomology of sp with sp2 this is the first theorem we know relating specht modules s and sp the second result equates cohomology of s with s pa for large a

quantum affine gln via hecke algebras abstract the quantum loop algebra of gl n is the affine analogue of quantum gl n in the seminal work 1 beilinson lusztig macpherson gave a beautiful realisation for quantum gl n via a geometric setting of quantum schur algebras more precisely they used quantum schur algebras to construct a certain algebra u in 1 5 4 and proved in 1 5 7 that u is isomorphic to quantum gl n we will present in this paper a full generalisation of blm s realisation to the affine case though the realisation of the quantum loop algebra of gl n is motivated by the work 1 for quantum gl n our approach is purely algebraic and combinatorial independent of the geometric method for quantum gl n as an application we discover a presentation of the ringel hall algebra of a cyclic quiver by semisimple generators and their multiplications by the defining basis elements

tame representation type of infinitesimal q schur algebras abstract in this paper we give a classification of the infinitesimal q schur algebras which have tame representation type

first degree cohomology of specht modules and extensions of symmetric powers abstract let d denote the symmetric group of degree d and let k be a field of positive characteristic p for p 2 we give an explicit description of the first cohomology group h 1 d sp of the specht module sp over k labelled by a partition of d we also give a sufficient condition for the cohomology to be non zero for p 2 and we find a lower bound for the dimension the cohomology of specht modules has been considered in many papers including 10 12 15 and 21 our method is to proceed by comparison with the cohomology for the general linear group g gl n k and then to reduce to the calculation of ext b 1 s d e k where b is a borel subgroup of g where s d e denotes the dth symmetric power of the natural module e for g and k denotes the one dimensional b module with weight the main new input is the description of module extensions by extensions sequences coherent triples of extension sequences and coherent multi sequences of extension sequences and the detailed calculation of the possibilities for such sequences these sequences arise from the action of divided powers elements in the negative part of the hyperalgebra of g our methods are valid also in the quantised context and we aim to treat this in a separate paper

norm variance of minimax interpolation minimax optimal interpolation algorithms minimize the error resulting from the worst signal from an allowable class the result is presented that if this class lies in a hilbert space the minimax optimal algorithm is independent of or invariant to the error norm the result encompasses a broad class of inverse problems

nonuniform fast fourier transforms using min max interpolation the fast fourier transform fft is used widely in signal processing for efficient computation of the ft of finite length signals over a set of uniformly spaced frequency locations however in many applications one requires nonuniform sampling in the frequency domain i e a nonuniform ft several papers have described fast approximations for the nonuniform ft based on interpolating an oversampled fft this paper presents an interpolation method for the nonuniform ft that is optimal in the min max sense of minimizing the worst case approximation error over all signals of unit norm the proposed method easily generalizes to multidimensional signals numerical results show that the min max approach provides substantially lower approximation errors than conventional interpolation methods the min max criterion is also useful for optimizing the parameters of interpolation kernels such as the kaiser bessel function

q tensor space and q weyl modules we obtain the irreducible representations of the q schur algebra motivated by the fact that these representations give all the irreducible representations of gln q in the nondescribing characteristic the irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction the theory of polynomial representations of general linear groups is equivalent to the representation theory of schur algebras see green s book 5 and the bibliography therein in 4 we defined q analogues of schur algebras when q 1 these are the usual schur algebras and when q is a prime power representations of q schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case including all irreducible representations of these groups and important information about decomposition numbers it is natural to ask what features of the classical schur algebras have qanalogues in this paper we define q analogues of tensor space of weyl modules and of weight spaces thereby generalizing the main reslllts which appear in green s book 5 for example we classify the irreducible modules for qschur algebras we determine bases for q weyl modules compatible with weight spaces and we give results on composition multiplicities of the irreducible modules in q weyl modules the proofs are largely self contained so by specializing q to 1 we recover the corresponding results in 5 this paper therefore is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 the q schur algebra let r be a natural number let r be an integral domain and let q be a unit in r we denote the symmetric group on r letters by er the hecke algebra z is the r free r algebra with basis twlw e er where the multiplication is determined by the following rule if a i i 1 is a basic transposition in received by the editors august 15 1989 1980 mathematics subject classification 198 5 revision primary 1 6a64 1 6a6 5 secondary 20c30 this research was supported in part by nsf grant no dms 8802290 the authors gratefully acknowledge support received from nato grant no 0222 87 r 1991 american mathematical society 0002 9947 91 1 00 25 per page

schur weyl duality for orthogonal groups we prove schur weyl duality between the brauer algebra bn m and the orthogonal group om k over an arbitrary infinite field k of odd characteristic if m is even we show that each connected component of the orthogonal monoid is a normal variety this implies that the orthogonal schur algebra associated to the identity component is a generalized schur algebra as an application of the main result an explicit and characteristic free description of the annihilator of n tensor space v n in the brauer algebra bn m is also given

on simple polynomial g r t modules abstract using the general framework of polynomial representations defined by doty and generalizing the definition given by doty nakano and peters for g gl n we consider polynomial representations of g r t for an arbitrary closed reductive subgroup scheme g gl n and a maximal torus t of g in positive characteristic we give sufficient conditions on g making a classification of simple polynomial g r t modules similar to the case g gl n possible and apply this to recover the corresponding result for gl n with a different proof extending it to symplectic similitude groups levi subgroups of gl n and in a weaker form to odd orthogonal similitude groups we also consider orbits of the affine weyl group and give a condition for equivalence of blocks of polynomial representations for g r t in the case g gl n

schur weyl duality for infinitesimal q schur algebras sq 2 r 1 abstract using the result of s r doty d k nakano k m peters polynomial representations of frobenius kernels of gl 2 in contemp math vol 194 1996 pp 57 67 s konig c xi when is a cellular algebra quasi hereditary math ann 315 1999 281 293 we prove that a non semisimple infinitesimal schur algebra s 2 r 1 is not cellular furthermore we determine the structure of the endomorphism ring of tensor space as a module for the infinitesimal schur algebra s 2 r 1 up to morita equivalence both results generalize to the quantum case

a linear time varying system framework for noniterative discrete time band limited signal extrapolation a technique for the analysis and design of noniterative algorithms for discrete time band limited signal extrapolation is described the approach involves modeling the extrapolation process as a linear time varying ltv system or filter together with a previously developed fourier theory for ltv systems this model provides a frequency domain transfer function representation for the extrapolation system this representation serves as a powerful tool for characterizing and comparing the reconstruction properties of several well known least squares optimal algorithms for band limited extrapolation moreover the frequency domain setting provides a conceptually attractive means for understanding the process of extrapolation itself additionally a least squares approximation methodology for designing ltv filters for band limited extrapolation is developed the design technique is shown to unify a broad class of algorithms for extrapolating discrete time data and further to provide a means for designing new and improved extrapolation algorithms

all algebraically degenerate h spaces via hh spaces starting from a canonical hh space which by virtue of the self dual part of its conformal curvature tensor being algebraically degenerate has a single congruence of totally null extremal two surfaces along which the coordinates are built we specialize to the case where the anti self dual conformal curvature vanishes giving us an algebraically degenerate h space with coordinates especially adapted to its degeneration we then solve the hh equation in this case and obtain at one stroke all algebraically degenerate h spaces in a simple compact form useful for later applications the generic solution depends on four arbitrary holomorphic functions of two variables and any special petrov type desired is easily distinguished also a special comparison is made for all such spaces of type d showing the relation of their parameters to the usual real type d parameters of mass newman unti tamburino nut parameter rotation and acceleration lastly a contraction to the special cases in which the leav

support varieties for modules over symmetric groups 1 1 in the late 1970s alperin a defined an invariant called the complexity of a module as a way to relate the modules with the complexes and resolutions that they admit several years later carlson ca1 ca2 defined affine algebraic varieties corresponding to modules over group algebras these varieties are subvarieties of the spectrum of the cohomology ring which was earlier described by quillen q they are known in present day language as support varieties it was discovered early on that the complexity of a module is equal to the dimension of the support variety of the module geometric methods involving support varieties have played a fundamental role in understanding the interplay between the modular representation theory and cohomology for finite groups despite substantial progress in this direction there have been few explicit computations of support varieties for important classes of modules over certain groups the goal of this paper is to introduce methods and techniques for computing support varieties for modules over the symmetric group d in the process we will provide explicit computations of support varieties for certain classes

selection of a convolution function for fourier inversion using gridding computerised tomography application in the technique known as gridding the data samples are weighted for sampling density and convolved with a finite kernel then resampled on a grid preparatory to a fast fourier transform the authors compare the artifact introduced into the image for various convolving functions of different sizes including the kaiser bessel window and the zero order prolate spheroidal wave function pswf they also show a convolving function that improves upon the pswf in some circumstances

approximate factorizations of fourier matrices with nonequispaced knots abstract we propose a new algorithm for the multiplication of vectors with fourier matrices of the form a tf e 2 i x k v j n n 2 1 j k n 2 where both x k and v j are arbitrary knots in n 2 n 2 the algorithm is based on an approximate factorization of the transform matrix a tf into sparse matrices which entries are chosen to minimize the frobenius norm of certain error matrices numerical experiments demonstrate that the approximation error introduced by our new algorithm is about 10 2 times smaller than the approximation error of previously reported algorithms with the gaussian or b splines as window functions and as good as a previous algorithm with the kaiser bessel function as window function

a modified blm approach to quantum affine mathfrak gl _n we introduce a spanning set of beilinson lusztig macpherson type a j r a j for affine quantum schur algebras boldsymbol mathcal s vartriangle n r and construct a linearly independent set a j a j for an associated algebra boldsymbol widehat mathcal k vartriangle n we then establish explicitly some multiplication formulas of simple generators e vartriangle h h 1 mathbf 0 by an arbitrary element a j in boldsymbol widehat mathcal k vartriangle n via the corresponding formulas in boldsymbol mathcal s vartriangle n r and compare these formulas with the multiplication formulas between a simple module and an arbitrary module in the ringel hall algebras boldsymbol mathfrak h vartriangle n associated with cyclic quivers this allows us to use the triangular relation between monomial and pbw type bases for boldsymbol mathfrak h vartriangle n established in deng and du adv math 191 276 304 2005 to derive similar triangular relations for boldsymbol mathcal s vartriangle n r and boldsymbol widehat mathcal k vartriangle n using these relations we then show that the subspace boldsymbol mathfrak a vartriangle n of boldsymbol widehat mathcal k vartriangle n spanned by a j a j contains the quantum enveloping algebra mathbf u vartriangle n of affine type a as a subalgebra as an application we prove that when this construction is applied to quantum schur algebras boldsymbol mathcal s n r the resulting subspace boldsymbol mathfrak a vartriangle n is in fact a subalgebra which is isomorphic to the quantum enveloping algebra of mathfrak gl n we conjecture that boldsymbol mathfrak a vartriangle n is a subalgebra of boldsymbol widehat mathcal k vartriangle n

integral affine schur weyl reciprocity abstract let d n be the double ringel hall algebra of the cyclic quiver n and let d n be the modified quantum affine algebra of d n we will construct an integral form d n z for d n such that the natural algebra homomorphism from d n z to the integral affine quantum schur algebra is surjective furthermore we will use hall algebras to construct the integral form u z g l n of the universal enveloping algebra u g l n of the loop algebra g l n g l n q q t t 1 and prove that the natural algebra homomorphism from u z g l n to the affine schur algebra over z is surjective in a subsequent paper fu 10 we will use affine schur algebras to give blm realization of u z g l n and this enables us to give a new proof of the statements about u z g l n given in this paper

block based methods for the reconstruction of finite length signals from nonuniform samples two novel block based algorithms are presented for the reconstruction of uniform samples given the nonuniform samples the first algorithm uses a sinc interpolator whereas the second one uses a dft based interpolator it is shown that the proposed algorithms are stable and the error due to noise and sampling jitter is bounded by the corresponding error norms of noise and jitter respectively we show that both of the block based algorithms provide nearly perfect reconstruction for a class of practically time and bandlimited signals boundary effects are considered and single and multiblock processing is discussed a modified block based algorithm is developed by using the windowing technique in order to improve the mean squared error mse performance for nonbandlimited signals it is shown that this algorithm performs better than a group of alternative algorithms including yen s third algorithm for a variety of signal noise and sampling grids

on schur algebras and related algebras ii this paper is a continuation of 7 the purpose of 7 was to set up in the context of reductive algebraic groups a theory of generalized schur algebras a special case of our construction gives the schur algebras s n r studied extensively in lo we concentrate here on the schur algebras themselves and show that our construction yields new results in their representation theory by means of the schur functor see 10 chap 6 we obtain new results in the representation theory of the symmetric groups the main results are as follows the schur algebras s n r defined over a principal ideal domain r have finite global dimension see also 1 i over a field or complete valuation ring r every finitely generated projective s n r module has a weyl module filtration we prove the irreducibility of the socle of certain weyl modules see also 15 we describe a connection between weights of projective indecomposable modules for the schur algebras in terms of the involution on irreducible representations of symmetric groups given by tensoring with the sign representation we show that over a complete valuation ring or a field every young module for a symmetric group has a specht series in particular the projective indecomposables have specht series finally we show that for r n and a complete valuation ring or field r the blocks of the schur algebra s n r are given by the rule embodied in nakayama s conjecture the notations and conventions are as in the first part of this work 7

on the good filtration dimension of weyl modules for a linear algebraic group let g be a linear algebraic group over an algebraically closed field of characteristic p whose corresponding root system is irreducible in this paper we calculate the weyl filtration di mension of the induced g modules and the simple g modules l fora regular weight we use this to calculate some ext groups of the form ext extl l and ext where are regular and is the weyl module of highest weight we then deduce the projective dimensions and injective dimensions for l and for a regular weight in associated generalised schur algebras we also deduce the global dimension of the schur algebras for gln s n r when p n and for s mp p with m an integer

a practical guide to the recovery of wavelet coefficients from fourier measurements in a series of recent papers b adcock a c hansen and c poon appl comput harmon anal 36 2014 pp 387 415 b adcock m gataric and a c hansen siam j imaging sci 7 2014 pp 1690 1723 adcock et al siam j math anal 47 2015 pp 1196 1233 it was shown that one can optimally recover the wavelet coefficients of an unknown compactly supported function from pointwise evaluations of its fourier transform via the method of generalized sampling while these papers focused on the optimality of generalized sampling in terms of its stability and error bounds the current paper explains how this optimal method can be implemented to yield a computationally efficient algorithm in particular we show that generalized sampling has a computational complexity of mathcal o left m n log n right when recovering the first n boundary corrected wavelet coefficients of an unknown compactly supported function from m n fourier samples therefore due to the linear correspondences bet

extrapolating a band limited function from its samples taken in a finite interval a band limited signal of finite energy can be reconstructed from its samples taken at the nyquist rate moreover the reconstruction is stable a feature crucial for implementation a small error in the sample values generates only a correspondingly small error in the resulting signal the nyquist sample values are mutually independent so that knowledge of them in a given interval t leq t leq 0 provides hardly any information about the behavior of the signal outside the interval however when the samples are taken at a greater rate a case referred to as over sampling they are interrelated and this redundancy can be exploited in various ways to improve the behavior of the reconstruction procedure a natural question is whether it can also be used to form accurate estimates of the signal outside the interval of observation this problem is relevant as well to prediction theory with oversampling when t infty so that the samples are known on the entire half line t they determine the signal everywhere although the reconstruction is now no longer stable here we examine the case of finite t of course a finite amount of data can yield only limited accuracy we prove that the samples can be used to form an approximation to the signal outside the sampling interval with an error which asymptotically as t rightarrow infty decreases exponentially in t over a range which grows linearly with t however as in the limiting case these approximations are not useful in practice since they require the sample values to be known exactly in the presence of measurement error the nature of the results changes good approximations are now available for only a bounded distance outside the interval of observation regardless of its length and their accuracy and range of validity can be increased only by improving the precision of sample reading since physical measurements are never perfect it is this conclusion which counts for applications the same results hold for the extrapolation of bounded band limited signals

reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters this paper considers the problem of reconstructing a class of nonuniformly sampled bandlimited signals of which a special case occurs in e g time interleaved analog to digital converter adc systems due to time skew errors to this end we propose a synthesis system composed or digital fractional delay filters the overall system i e nonuniform sampling and the proposed synthesis system can be viewed as a generalization of time interleaved adc systems to which the former reduces as a special case compared with existing reconstruction techniques our method has major advantages from an implementation point of view to be precise 1 we can perform the reconstruction as well as desired in a certain sense by properly designing the digital fractional delay filters and 2 if properly implemented the fractional delay filters need not be redesigned in case the time skews are changed the price to pay for these attractive features is that we need to use a slight oversampling it should be stressed however that the oversampling factor is less than two as compared with the nyquist rate the paper includes error and quantization noise analysis the former is useful in the analysis of the quantization noise and when designing practical fractional delay filters approximating the ideal filters

the weighted fusion category algebra and the q schur algebra for gl 2 q abstract we show that the weighted fusion category algebra of the principal 2 block b 0 of gl 2 q is the quotient of the q schur algebra s 2 q by its socle for q an odd prime power as a consequence we get a canonical bijection between the set of isomorphism classes of simple k gl 2 q b 0 modules and the set of conjugacy classes of b 0 weights in this case

representation theory of reductive normal algebraic monoids new results in the representation theory of semisimple algebraic monoids are obtained based on renner s monoid version of chevalley s big cell the semisimple algebraic monoids have been classified by renner the rational representations of such a monoid are the same thing as polynomial representations of the associated reductive group of units in the monoid and this representation category splits into a direct sum of subcategories by homogeneous degree we show that each of these homogeneous subcategories is a highest weight category in the sense of cline parshall and scott and so equivalent with the module category of a certain finite dimensional quasihereditary algebra which we show is a generalized schur algebra in s donkin s sense

accelerating the nonuniform fast fourier transform the nonequispaced fourier transform arises in a variety of application areas from medical imaging to radio astronomy to the numerical solution of partial differential equations in a typical problem one is given an irregular sampling of n data in the frequency domain and one is interested in reconstructing the corresponding function in the physical domain when the sampling is uniform the fast fourier transform fft allows this calculation to be computed in o n log n operations rather than o n 2 operations unfortunately when the sampling is nonuniform the fft does not apply over the last few years a number of algorithms have been developed to overcome this limitation and are often referred to as nonuniform ffts nuffts these rely on a mixture of interpolation and the judicious use of the fft on an oversampled grid a dutt and v rokhlin siam j sci comput 14 1993 pp 1368 1383 in this paper we observe that one of the standard interpolation or gridding schemes based on gaussians can be accelerated by a significant factor without precomputation and storage of the interpolation weights this is of particular value in two and three dimensional settings saving either 10 d n in storage in d dimensions or a factor of about 5 10 in cputime independent of dimension

globally irreducible weyl modules abstract in the representation theory of split reductive algebraic groups it is well known that every weyl module with minuscule highest weight is irreducible over every field also the adjoint representation of e 8 is irreducible over every field in this paper we prove a converse to these statements as conjectured by gross if a weyl module is irreducible over every field it must be either one of these or trivially constructed from one of these we also prove a related result on non degeneracy of the reduced killing form

nonlinear approximation by sums of exponentials and translates in this paper we discuss the numerical solution of two nonlinear approximation problems many applications in electrical engineering signal processing and mathematical physics lead to the following problem let h be a linear combination of exponentials with real frequencies determine all frequencies all coefficients and the number of summands if finitely many perturbed uniformly sampled data of h are given we solve this problem by an approximate prony method apm and prove the stability of the solution in the square and uniform norm further an apm for nonuniformly sampled data is proposed too the second approximation problem is related to the first one and reads as follows let varphi be a given 1 periodic window function as defined in section 4 further let f be a linear combination of translates of varphi determine all shift parameters all coefficients and the number of translates if finitely many perturbed uniformly sampled data of f are given using fourier technique this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by apm the stability of the solution is discussed in the square and uniform norm too numerical experiments show the performance of our approximation methods

the representation type of ariki koike algebras and cyclotomic q schur algebras abstract we give a necessary and sufficient condition on parameters for ariki koike algebras resp cyclotomic q schur algebras to be of finite representation type

reconstruction algorithms in irregular sampling a constructive solution of the irregular sampling problem for band limited functions is given we show how a band limited function can be com pletely reconstructed from any random sampling set whose density is higher than the nyquist rate and give precise estimates for the speed of convergence of this iteration method variations of this algorithm allow for irregular sampling with derivatives reconstruction of band limited functions from local averages and irregular sampling of multivariate band limited functions in the irregular sampling problem one is asked whether and how a band limited function can be completely reconstructed from its irregularly sam pled values f xi this has many applications in signal and image processing seismology meteorology medical imaging etc finding constructive solutions of this problem has received considerable attention among mathematicians and engineers the mathematical literature provides several uniqueness results 1 2 17 18 19 it is now part of the folklore that for stable sampling the sampling rate must be at least the nyquist rate 18 these results as deep as they are have had little impact for the applied sciences because they were not constructive if the sampling set is just a perturbation of the regular oversampling then a reconstruction method has been obtained in a seminal paper by duffin and schaeffer 6 see also 29 if for some l 0 a 0 and o 0 the sampling points xk k e z satisfy a xk ok a k i then the norm equivalence a ir f x 2dx with w n o this norm equivalence implies that it is possible to reconstruct through an iterative procedure the so called frame method most of the later work on constructive methods consists of variations of this method 3 21 22 26 the above conditions on the sampling set exclude random irregular sampling sets e g sets with regions of higher sampling density a partial but undesirable remedy to handle highly irregular sampling sets would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration

on the hyperalgebra of the loop algebra glˆn abstract let u z gl n be the garland integral form of u gl n introduced by garland 19 where u gl n is the universal enveloping algebra of gl n using ringel hall algebras a certain integral form u z gl n of u gl n was constructed in 15 we prove that the garland integral form u z gl n coincides with u z gl n let k be a commutative ring with unity and let u k gl n u z gl n k for h 1 we use ringel hall algebras to construct a certain subalgebra denoted by u n h of u k gl n we refer to u n h as the hth frobenius kernel associated with gl n the first frobenius kernel associated with quantum gl n was realized by beilinson lusztig macpherson 1 6 we will give a realization of the hth frobenius kernel u n h associated with gl n for each h 1 the infinitesimal and little schur algebras are certain subalgebras of schur algebras which are closely related to the frobenius kernel associated with gl n we will investigate the affine version of infinitesimal and little schur algebras

the representation type of hecke algebras of type b abstract this paper determines the representation type of the iwahori hecke algebras of type b when q 1 in particular we show that a single parameter non semisimple iwahori hecke algebra of type b has finite representation type if and only if q is a simple root of the poincare polynomial confirming a conjecture of uno s j algebra 149 1992 287

a computational study of reconstruction algorithms for diffraction tomography interpolation versus filtered backpropagation from the standpoint of reporting a new contribution this paper shows that by using bilinear interpolation followed by direct two dimensional fourier inversion one can obtain reconstructions of quality which is comparable to that produced by the filtered backpropagation algorithm proposed recently by devaney for an n n image reconstructed from n diffracted projections the former approach requires approximately 4n fft s whereas the backpropagation technique requires approximately n2fft s we have also taken this opportunity to present the reader with a tutorial introduction to diffraction tomography an area that is becoming increasingly important not only in medical imaging but also in underwater and seismic mapping with microwaves and sound the main feature of the tutorial part is the statement of the fourier diffraction projection theorem which is an extension of the traditional fourier slice theorem to the case of image formation with diffracting illumination

gauge fields in algebraically special space times it is shown that in an algebraically special space time which admits a congruence of null strings a source free gauge field aligned with the congruence is determined by a matrix potential which has to satisfy a second order differential equation with quadratic nonlinearities the einstein yang mills equations are then reduced to a scalar and two matrix equations in the case of self dual gauge fields in a self dual space time the existence of an infinite set of conservation laws of an associated linear system and of infinitesimal backlund transformations is demonstrated all the results apply for an arbitrary gauge group

analysis of error in reconstruction of two dimensional signals from irregularly spaced samples we consider the problem of estimating a band limited two dimensional 2 d signal based on a finite set of irregularly spaced samples we derive the minimum mean squared error estimator of the form of a sum of weighted interpolating functions that are identical in shape but are centered at the irregularly spaced sample points and show that this estimator is identical to the minimum energy band limited interpolator that has been previously obtained by others the relation between the mean squared error and the set of irregularly spaced sampling points is studied using the minimax principle the maximum of the mean squared error over a class of signals is minimized by proper choice of the set of sampling points three criteria for evaluating the performance of the sampling point set are derived from a candidate group of sets of irregularly spaced sampling points these criteria may be used to select that point set which is optimal for recovery over a class of signals

resampling of data between arbitrary grids using convolution interpolation for certain medical applications resampling of data is required in magnetic resonance tomography mrt or computer tomography ct e g data may be sampled on nonrectilinear grids in the fourier domain for the image reconstruction a convolution interpolation algorithm often called gridding can be applied for resampling of the data onto a rectilinear grid resampling of data from a rectilinear onto a nonrectilinear grid are needed e g if projections of a given rectilinear data set are to be obtained in this paper the authors introduce the application of the convolution interpolation for resampling of data from one arbitrary grid onto another the basic algorithm can be split into two steps first the data are resampled from the arbitrary input grid onto a rectilinear grid and second the rectilinear data is resampled onto the arbitrary output grid furthermore the authors like to introduce a new technique to derive the sampling density function needed for the first step of their algorithm for fast sampling pattern independent determination of the sampling density function the voronoi diagram of the sample distribution is calculated the volume of the voronoi cell around each sample is used as a measure for the sampling density it is shown that the introduced resampling technique allows fast resampling of data between arbitrary grids furthermore it is shown that the suggested approach to derive the sampling density function is suitable even for arbitrary sampling patterns examples are given in which the proposed technique has been applied for the reconstruction of data acquired along spiral radial and arbitrary trajectories and for the fast calculation of projections of a given rectilinearly sampled image

chebyshev polynomials on symmetric matrices in this paper we evaluate chebyshev polynomials of the second kind on a class of symmetric integer matrices namely on adjacency matrices of simply laced dynkin and extended dynkin diagrams as an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type

direct fourier reconstruction in computer tomography direct fourier transform ft reconstruction of images in computerized tomography ct is not widely used because of the difficulty of precisely interpolating from polar to cartesian samples in this paper an exact interpolation scheme is proposed which in practice can be approached with arbitrary accuracy using well conditioned algorithms several features of the direct ft method are discussed a method that allows angular band limiting of the data before processing to avoid angular aliasing artifacts in the reconstructed image is discussed and experimentally verified the experimental results demonstrate the feasibility of direct ft reconstruction of ct data

kaiser bessel basis for particle mesh interpolation in this work we introduce the kaiser bessel interpolation basis for the particle mesh interpolation in the fast ewald method a reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated charge systems and is shown to be effective in optimizing the shape parameter of the kaiser bessel basis in terms of accuracy by comparing the optimized kaiser bessel basis with the traditional b spline basis we demonstrate that the former is more accurate than the latter in part of the working parameter space say a relatively small real space cutoff a relatively small reciprocal space mesh and a relatively large truncation of basis in some cases the kaiser bessel basis is found to be more than one order of magnitude more accurate

fourier based forward and back projectors in iterative fan beam tomographic image reconstruction fourier based forward and back projection methods can reduce computation in iterative tomographic image reconstruction recently an optimized nonuniform fast fourier transform nufft approach was shown to yield accurate parallel beam projections in this paper we extend the nufft approach to describe an o n sup 2 log n projector backprojector pair for fan beam transmission tomography simulations and experiments with real ct data show that fan beam fourier based forward and back projection methods can reduce computation for iterative reconstruction while still providing accuracy comparable to their o n sup 3 space based counterparts

multiple overlapping k space junctions for investigating translating objects mojito it is a well known property in fourier transform magnetic resonance imaging mri that rigid body translational motion in image space results in linear phase accumulation in k space this work describes multiple overlapping k space junctions for investigating translating objects mojito a correction scheme based on phase differences at trajectory intersections caused by 2 d alterations in the position of an object during mr imaging the algorithm allows both detection and correction of motion artifacts caused by 2 d rigid body translational motion although similar in concept to navigator echoes mojito does not require a repeating path through k space uses k space data from a broader region of k space and uses the repeated data in image reconstruction this provides the potential for a highly efficient self navigating motion correction method here the concept and theoretical basis of mojito is demonstrated using the continuous sampling bowtie trajectory in simulation and mr experiments this particular trajectory is selected since it is well suited for such an algorithm due to numerous trajectory intersections specifically the validity of the technique in the presence of noise and off resonance effects is demonstrated through simulation

on the structure of end uk 2 ωk r let uk 2 be the infinitesimal quantum gl2 over k where k is a field containing an lth primitive root e of 1 with l 3 odd we will determine the basic algebra for end uk 2 k r where k is the natural module for uk 2

fast higher order mr image reconstruction using singular vector separation medical resonance imaging mri conventionally relies on spatially linear gradient fields for image encoding however in practice various sources of nonlinear fields can perturb the encoding process and give rise to artifacts unless they are suitably addressed at the reconstruction level accounting for field perturbations that are neither linear in space nor constant over time i e dynamic higher order fields is particularly challenging it was previously shown to be feasible with conjugate gradient iteration however so far this approach has been relatively slow due to the need to carry out explicit matrix vector multiplications in each cycle in this work it is proposed to accelerate higher order reconstruction by expanding the encoding matrix such that fast fourier transform can be employed for more efficient matrix vector computation the underlying principle is to represent the perturbing terms as sums of separable functions of space and time compact representations with this property are found by singular vector analysis of the perturbing matrix guidelines for balancing the accuracy and speed of the resulting algorithm are derived by error propagation analysis the proposed technique is demonstrated for the case of higher order field perturbations due to eddy currents caused by diffusion weighting in this example image reconstruction was accelerated by two orders of magnitude

the affineq schur algebra abstract we introduce an analogue of the q schur algebra associated to coxeter systems of type a n 1 we give two constructions of this algebra the first construction realizes the algebra as a certain endomorphism algebra arising from an affine hecke algebra of type a r 1 where n r this generalizes the original q schur algebra as defined by dipper and james and the new algebra contains the ordinary q schur algebra and the affine hecke algebra as subalgebras using this approach we can prove a double centralizer property the second construction realizes the affine q schur algebra as the faithful quotient of the action of a quantum group on the tensor power of a certain module analogous to the construction of the ordinary q schur algebra as a quotient of u g l n